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+arXiv:2301.01910v1  [math.DS]  5 Jan 2023
+Differentiability of the largest Lyapunov exponent for planar open
+billiards
+Amal Al Dowais a,b
+Abstract
+In this paper, we estimate the largest Lyapunov exponent for open billiards in the plane.
+We show that the largest Lyapunov exponent is differentiable with respect to a billiard defor-
+mation.
+keywords. Open billiards; Lyapunov exponents; Non-wandering set; Billiard deformation
+Mathematics Subject Classification (2010). 37D50, 37B10, 37D20, 34D08
+1
+Introduction
+The stability and instability of a dynamical system can be studied by means of Lyapunov expo-
+nents. A dynamical system is considered chaotic if it has a positive Lyapunov exponent. Examples
+of chaotic systems are the dispersing billiards or so-called Sinai billiards (see [15], [16]). Billiards
+are dynamical systems in which a particle moves with constant speed and hits the billiard’s wall
+(boundary of the billiard’s domain) according to the law of geometrical optics,“the angle of inci-
+dence equals the angle of reflection”. Open billiards are a particular case of billiards in unbounded
+domains. The domain is the exterior of finitely many strictly convex compact obstacles satisfying
+the no-eclipse condition (H) of Ikawa [6]: the convex hull of any two obstacles does not intersect
+with another obstacle; in other words, there does not exist a straight line that intersects more
+than two obstacles. It follows from Sinai [15], [16] (see also [14]) that the non-wandering set of
+the open billiard map is hyperbolic (i.e. there exist positive and negative Lyapunov exponents).
+Many studies have investigated Lyapunov exponents for billiards (see [22], [1], [4], [9], [10]). In
+this paper, we estimate the largest Lyapunov exponent for open billiard in R2. We demonstrate
+that the Lyapunov exponent depends continuously on a parameter α related to a deformation of
+the billiard as defined in [21]. Moreover, we prove that the Lyapunov exponent is differentiable
+with respect to the deformation parameter α.
+Here we state the main results:
+In the following theorems, we denote the billiard deformation by K(α) where α ∈ [0, b].
+See
+Section 4 for the precise definition.
+aDepartment of Mathematics and Statistics, School of Physics, Mathematics and Computing, University of
+Western Australia, Perth, WA 6009, Australia
+Email address: [email protected]
+bDepartment of Mathematics, College of Science and Arts, Najran University, Najran, Saudi Arabia
+Email address: [email protected]
+1
+
+Theorem 1.1. (Continuity) Let K(α) be a C4,2 billiard deformation in R2. Let λ1(α) be the
+largest Lyapunov exponent for K(α). Then the largest Lyapunov exponent is continuous as a
+function of α.
+Theorem 1.2. (Differentiability) Let K(α) be a C5,3 billiard deformation in R2. Let λ1(α) be
+the largest Lyapunov exponent for K(α). Then λ1(α) is C1 with respect to α.
+There are many works studying continuity properties of Lyapunov exponents (see e.g. [20], [3]).
+However, to our knowledge, in all these continuity is established generically, i.e. with respect to
+“most” (typical) values of the parameters/perturbations involved. In the case of the open billiard
+considered in the present paper we establish continuity and even differentiability for all values
+of the parameter that appears in the perturbation, which is a truly remarkable property of this
+physical system.
+2
+Preliminaries
+This section provides some preliminary concepts for open billiards, billiard flow, symbolic coding
+and stable/unstable manifolds. We also describe some notations related to curvatures, distances,
+and collision angles. In the last part of this section, we state the Oseledets multiplicative ergodic
+theorem and its consequence for open billiards.
+2.1
+Open billiard
+Let Ki, where i = 1, 2, 3, ..., z0, be strictly convex compact domains with smooth boundaries ∂Ki
+in R2. In this paper, we assume that K = �
+i Ki satisfies the following condition(H) of Ikawa [6]:
+for any i ̸= j ̸= k the convex hull of Ki ∪Kk does not have any common points with Kj. Let Ω be
+the exterior of K (i.e., Ω = R2\K). Let Φt, t ∈ R, be the billiard flow such that for any particle
+x = (q, v), where q ∈ Ω represents the position of x and v is the unit velocity of the particle x,
+then Φt(x) = (qt, vt) = (q + tv, v). When the particle hits the boundary, then the velocity follows
+the collision law vnew = vold − 2 < vold, n > n, where n is the outwards unit normal vector to ∂K
+at q, and φ the angle between n = n(q) and v.
+We denote the time of the j-th reflection of x by tj(x) ∈ (−∞, ∞) for j ∈ Z. We say tj(q) = ∞
+(tj(q) = −∞) if the forwards (backwards) trajectory of x has less than j reflections. We denote
+the non-wandering set of the flow Φt by Λ = {x ∈ �Ω, |tj(x)| < ∞, for all j ∈ Z}, where
+�Ω = {(q, v) ∈ int Ω × S1 or (q, v) ∈ ∂Ω × S1 : ⟨n(q), v⟩ ≥ 0}.
+Now let M = {x = (q, v) ∈ ∂K × S1 : ⟨n(q), v⟩ ≥ 0} and let π : M → ∂K be the canonical
+projection map defined by π(q, v) = q. Let t1(x) be the time of the first reflection of x and let
+M1 = {x ∈ M : t1(x) < ∞}. Define the billiard ball map B : M1 → M by B(x) = Φt1(x)(x), (e.g.
+if y = (p0, w0), where p0 lies on ∂Ki then B(y) = B(p0, w0) = (p1, w1) where p1 = p0+t1w0 ∈ ∂Kj
+and w1 = w0 − 2 < w0, n > n). The non-wandering set of the open billiard map is M0 = {x ∈
+M : |tj(x)| < ∞} which is a subset of Λ. Finally, let B : M0 → M0 be the restriction of the
+open billiard map on the non-wandering set M0. It is obvious that the non-wandering set is an
+invariant set. See [15], [16], [4], [5], [14], for general information about billiard dynamical systems.
+2
+
+2.2
+Symbolic coding for open billiards
+Each particular x ∈ M0 can be coded by a bi-infinite sequence
+ξ(x) = (..., ξ−1, ξ0, ξ1, ...) ∈ {1, 2, ..., z0}Z,
+in which ξi ̸= ξi+1, for all i ∈ Z, and ξj indicates the obstacle Kξj such that πBj(x) ∈ ∂Kξj. For
+example, if there are three obstacles K1, K2 and K3 as above and a particular q repeatedly hits
+K1, K3, K2, K1, K3, K2, then the bi-infinite sequence is (..., 1, 3, 2, 1, 3, 2, ...). Let Σ be the symbol
+space which is defined as:
+Σ = {ξ = (..., ξ−1, ξ0, ξ1, ...) ∈ {1, 2, ..., z0}Z : ξi ̸= ξi+1, ∀i ∈ Z}.
+Define the representation map R : M0 → Σ by R(x) = ξ(x). Let σ : Σ → Σ be the two-sided
+subshift map defined by σ(ξi) = ξi+1. Given θ ∈ (0, 1) define the metric dθ on Σ by:
+dθ(ξ, η) =
+� 0
+if
+ξi = ηi for all i ∈ Z
+θn
+if
+n = max{j ≥ 0 : ξi = ηi for all |i| < j}
+Then σ is continuous with respect to dθ ([2]).
+It is also known that the representation map
+R : M0 → Σ is a homeomorphism (see e.g. [14]). See [6], [8], [11], [14], [17], for topics related to
+symbolic dynamics for open billiards.
+2.3
+Lyapunov exponents
+Here we state a consequence of Oseledets Multiplicative Ergodic Theorem for billiards (see e.g.
+Ch. 2 in [13], also see [12], [20], [7]).
+For the open billiard map B : M0 −→ M0 in R2 we will use the coding R : M0 −→ Σ from Section
+2.2, which conjugates B with the shift map σ : Σ −→ Σ, to define Lyapunov exponents. It is
+well known that there are ergodic σ-invariant measures µ on Σ. Let µ be an ergodic σ-invariant
+probability measure on Σ. The following is a consequence of Oseledets Multiplicative Ergodic
+Theorem:
+Theorem 2.1 (A Consequence of Oseledets Multiplicative Ergodic Theorem). There exist real
+numbers λ1 > 0 > −λ1 and one-dimensional vector subspaces Eu(x) and Es(x) of Tx(∂K),
+x ∈ M0, depending measurably on R(x) ∈ Σ such that:
+(i) Eu(x) and Es(x) for almost all x ∈ M0;
+(ii) DxB(Eu(x)) = Eu(B(x)) and DxB(Es(x)) = Es(B(x)) for almost all x ∈ M0, and
+(iii) For almost all x ∈ M0 there exists
+lim
+n→∞
+1
+n log ∥DxBn(w)∥ = λ1
+whenever 0 ̸= w ∈ Eu(x).
+Here ”for almost all x” means ”for almost all R(x)” with respect to µ. The numbers λ1 > 0 > −λ1
+are called Lyapunov exponents, while the invariant subspaces Eu(x) and Es(x) are called Oseledets
+subspaces.
+3
+
+2.4
+Propagation of unstable manifolds for open billiards
+We describe a formula which is useful in getting estimates for
+lim
+m→∞
+1
+m log ∥DxBmw∥, (0 ̸= w ∈ Eu(x), x ∈ M0).
+Let M0 be the non-wandering set of the billiard ball map B of an open billiard. Then
+Λ = {Φt(x) : x ∈ M0 , t ∈ R},
+is the non-wandering set for the billiard flow Φt. For x ∈ Λ and a sufficiently small ǫ > 0 let
+�
+W s
+ǫ (x) = {y ∈ Λ : d(Φt(x), Φt(y)) ≤ ǫ for all t ≥ 0 , d(Φt(x), Φt(y)) →t→∞ 0 },
+�
+W u
+ǫ (x) = {y ∈ Λ : d(Φt(x), Φt(y)) ≤ ǫ for all t ≤ 0 , d(Φt(x), Φt(y)) →t→−∞ 0 }
+be the (strong) stable and unstable manifolds of size ǫ for the billiard flow. Then �Eu(x) = Tx�
+W u
+ǫ (x)
+and �Es(x) = Tx�
+W s
+ǫ (x). In a similar way one defines stable/unstable manifolds for the billiard
+ball map B. For any x = (q, v) ∈ M0 define
+W s
+ǫ (x) = {y ∈ M0 : d(Bn(x), Bn(y)) ≤ ǫ for all n ∈ N , d(Bn(x), Bn(y)) →n→∞ 0 },
+W u
+ǫ (x) = {y ∈ M0 : d(B−n(x), B−n(y)) ≤ ǫ for all n ∈ N , d(B−n(x), B−n(y)) →n→∞ 0 }.
+In what follows we will just write W u(x) and W s(x) for W u
+ǫ (x) and W s
+ǫ (x), assuming some
+appropriately chosen sufficiently small ǫ > 0 is involved. Similarly for �
+W u and �
+W s.
+It is well-known that there is an one-to-one correspondence between the stable/unstable man-
+ifolds for the billiard ball map and these for the flow.
+Geometrically the easiest (and most
+convenient way) to describe this is as follows.
+Given x = (q, v) ∈ M0 (so q ∈ ∂K and v ∈ S1), and a small 0 < r < t1(x), set y = (q + rv, v).
+Then there is a 1-1 correspondence
+ϕ : W u(x) −→ �
+W u(y)
+such that ϕ(z, w) = (z + t w, w) for all (z, w) ∈ W u(x), where t = t(z, w) > 0. Similarly, there is
+a correspondence between W s(x) and �
+W s(y). Moreover
+Dϕ(x) : TxM0 −→ TyΛ
+is so that D��(x)(Eu(x)) = �Eu(y) and Dϕ(x)(Es(x)) = �Es(y).
+It is known that �
+W u(y) has the form �
+W u(y) = �Y , where
+�Y = {(p, νY (p)) : p ∈ Y }
+for some smooth curve Y in R2 containing the point y such that Y is strictly convex with respect
+to the unit normal field νY , i.e. the curvature of Y is strictly positive.
+4
+
+Next, let x and y be as above and let x1 = (q1, v1) = B(x). Then q1 = q + t1 v. Define
+y1 = (q1 + r′v1, v1) for some small 0 < r′ < t2(x) − t1(x), where 0 = t0(x) < t1(x) < t2(x). Then
+there is a 1-1 correspondence
+ϕ1 : W u(x1) −→ ˜W u(y1)
+defined as above. Again, we can write �
+W u(y1) = �Y1, where
+�Y1 = {(p1, νY (p1)) : p1 ∈ Y1}
+for some smooth curve Y1 in R2 containing the point y1 such that Y1 is strictly convex with respect
+to the unit normal field νY1. Moreover the following diagram is commutative, where t = t1 + r′:
+W u(x)
+B
+−→
+W u(x1)
+�ϕ
+�ϕ1
+�
+W u(y) = �Y
+Φt
+−→
+�
+W u(y1) = �Y1
+Similarly, the following diagram is commutative:
+Eu(x)
+DB(x)
+−→
+Eu(x1)
+�Dϕ
+�Dϕ1
+�Eu(y)
+DΦt(y)
+−→
+�Eu(y1)
+Since the derivatives Dϕ and Dϕ1 are uniformly bounded, the above conjugacy can be used later
+to calculate the Lyapunov exponents of the billiard ball map using propagation of appropriate
+convex curves Y which we describe as follows.
+Let x0 = (q0, v0) ∈ M0 and let W u
+ǫ (x0) be the local unstable manifold for x0 for sufficiently
+small ǫ > 0. Let t1(x0) be the time of the first reflection of x0. Then �
+X = W u
+ǫ (x0) = {(q, nX(q)) :
+q ∈ X} for some C3 curve X in Ω such that q0 ∈ X and X is strictly convex curve with respect to
+the outer unit normal field nX(q). Let X be parametrized by q(s), s ∈ [0, a], such that q(0) = q0,
+and has unit normal field nX(q(s)).
+Set q0(s) = q(s).
+Let qj(s), j ≥ 1 be the jth-reflection
+points of the forward billiard trajectory γ(s) generated by x(s) = (q(s), nX(q(s)). We assume
+that a > 0 is sufficiently small so that the jth-reflection points qj(s) belong to the same boundary
+component ∂Kξj for every s ∈ [0, a]. Let 0 = t0(x(s)) < t1(x(s)) < ... < tm+1(x(s)) be the times
+of the reflections of the ray γ(s) at ∂K. Let κj(s) be the curvature of ∂Kξj at qj(s) and φj(s)
+be the collision angle between the outward unit normal to ∂K and the reflection ray of γ(s) at
+qj(s). Also, let dj(s) be the distance between two reflection points i.e. dj(s) = ∥qj+1(s) − qj(s)∥,
+j = 0, 1, . . . , m.
+Given a large m ≥ 1, let tm(x(s)) < t < tm+1(x(s)). Set Φt( �
+X) = �
+Xt. Let π(Φt(x(s))) = p(s).
+Then p(s), s ∈ [0, a], is a parametrization of the C3 curve Xt = π(Φt( �
+X).
+Next, let k0(s) > 0 be the curvature of X at q(s).
+Let tj(x(s)) < τ < tj+1(x(s)), j =
+1, 2, . . . , m. Denote by uτ(s) be the shift of (q(s), n(q(s))) along the forward billiard trajectory
+γ(s) after time τ > 0. Then Xτ = {uτ(s) : s ∈ [0, a]} is a C3 convex curve with respect to the
+outward normal field n(uτ(s)). Let kj(s) > 0 be the curvature of Xtj = limτցtj(s) Xτ at qj(s). It
+follows from Sinai [15] that
+kj+1(s) =
+kj(s)
+1 + dj(s)kj(s) + 2
+κj+1(s)
+cos φj+1(s)
+,
+0 ≤ j ≤ m − 1 .
+(2.1)
+5
+
+Moreover, the curvature of Xτ at uτ(s) is
+kτ(s) =
+kj(s)
+1 + (τ − tj(s))kj(s).
+(2.2)
+Set
+δj(s) =
+1
+1 + dj(s)kj(s)
+,
+1 ≤ j ≤ m .
+(2.3)
+Theorem 2.2. [18] For all s ∈ [0, a] we have
+∥ ˙q(s)∥ = ∥ ˙p(s)∥δ1(s)δ2(s) . . . δm(s) .
+(2.4)
+This was proved in [18] in the 2D case and in [19] in the general case.
+Finally, we want to introduce some notation related to the maximum and minimum of previous
+billiard characteristies dj(s),κj(s), φj(s) and kj(s).
+For all j, we have dmin ≤ dj(s) ≤ dmax,
+where dmax and dmin are constants independent of j such that dmax = max{d(Ki, Kk)} and
+dmin = min{d(Ki, Kk)} for i ̸= k. Also, since ∂K is strictly convex, we have constants κmin > 0
+and κmax > 0 independent of j such that κmin ≤ κj(s) ≤ κmax. And it follows from the condition
+(H) that there exists a constant φmax ∈ (0, π
+2 ) such that 0 ≤ φj(s) ≤ φmax < π
+2 , (see e.g. [17]).
+Let kj(s) be as in equation (2.1). It follows easily that kmin ≤ kj(s) ≤ kmax, where kmin = 2κmin
+and kmax =
+1
+dmin +
+2κmax
+cos φmax .
+3
+Estimation of the largest Lyapunov exponent for open billiards
+A formula for the largest Lyapunov exponents for a rather general class of billiards can be found
+in [5], see Theorem 3.41 there. In our case we derive this formula again (see (3.1) below) and then
+we use Theorem 2.2 to derive important regularity properties of the largest Lyapunov exponent.
+Assume that µ is an ergodic σ-invariant measure on Σ, and let x0 = (q0, v0) ∈ M0 correspond
+to a typical point in Σ with respect to µ via the representation map R. That is as in Theorem
+2.1, we have
+λ1 = lim
+m→∞
+1
+m log ∥Dx0Bm(w)∥,
+with 0 ̸= w ∈ Eu(x0). As in Sect. 2.4, let X be a (small) C3 strictly convex curve containing q0 and
+having a unit normal field nX so that nX(q0) = v0. As in Sect. 2.4 again, let X be parametrised
+by arc length via q(s), s ∈ [0, a], such that q(0) = q0. Let again qj(s), j = 1, 2, . . . , m + 1, be the
+consecutive reflection points of the billiard trajectory γ(s) determined by x(s) = (q(s), nX(q(s)).
+Given an integer m > 0 and assuming the interval [0, a] is sufficiently small, the jth reflection
+points qj(s) belong to the same boundary component ∂Kξj for all s ∈ [0, a]. Next, define dj(s),
+tj(x(s)), etc. as in Sect. 2.4, let tm(x(0)) < t < tm+1(x(0)), and let p(s) be the parametrisation
+of �
+Xt corresponding to q(s). Then the formula (2.4) in Theorem 2.2 (holds with ∥ ˙q(s)∥ = 1 from
+our assumptions). Now the discussion in Sect. 2.4 implies that there exist some global constants
+c1 > c2 > 0, independent of x0, X, m, etc. such that
+c2∥ ˙p(s)∥ ≤ ∥Dx0Bm(w)∥ ≤ c1∥ ˙p(s)∥
+6
+
+for all s ∈ [0, a]. So, by (2.4),
+c2
+δ1(0)δ2(0) . . . δm(0) ≤ ∥Dx0Bm(w)∥ ≤
+c1
+δ1(0)δ2(0) . . . δm(0)
+for all s ∈ [0, a]. Using this for s = 0, taking logarithms and limits as m → ∞, we obtain
+− lim
+m→∞
+1
+m log (δ1(0)δ2(0) . . . δm(0)) ≤ lim
+m→∞
+1
+m log ∥Dx0Bm(w)∥
+≤ − lim
+m→∞
+1
+m log (δ1(0)δ2(0) . . . δm(0)) .
+Hence,
+λ1 = lim
+m→∞ − 1
+m
+m
+�
+i=1
+log δi(0).
+This implies that the largest Lyapunov exponent at the initial point x0, so at almost every point
+wilt respect to the given measure µ, is given by
+λ1 = lim
+m→∞
+1
+m
+m
+�
+i=1
+log
+�
+1 + di(0)ki(0)
+�
+.
+(3.1)
+From equation (3.1), we can estimate the largest Lyapunov exponent from below and above as
+log (1 + dminkmin) ≤ λ1 ≤ log (1 + dmaxkmax).
+4
+Billiard deformations
+In this section, we consider some changes to the billiards in the plane, such as moving, rotating,
+and changing the shape of one or multiple obstacles. This kind of billiard transformation is called a
+billiard deformation as defined in [21]. We describe this deformation by adding an extra parameter
+α ∈ [0, b] for some b ∈ R+, which is called the deformation parameter, to the parametrization of
+the boundary of obstacles i.e., if the boundary of an obstacle parametrized by ϕ(u), it will become
+ϕ(u, α). In this section, we provide the definition a billiard deformation as defined in [21]. In
+addition, we describe the propagation of unstable manifolds for billiard deformations. We also
+estimate the higher derivatives of some of the billiard characteristics such as distance, collision
+angle and curvature, with respect to deformation parameter α.
+Let α ∈ I = [0, b], for some b ∈ R+, be a deformation parameter and let ∂Ki(α) be
+parametrized counterclockwise by ϕi(ui, α) and parametrized by arc-length ui. Let qi = ϕi(ui, α)
+be a point that lies on ∂Ki(α). Denote the perimeter of ∂Ki(α) by Li(α), and let Pi = {(ui, α) :
+α ∈ I, ui ∈ [0, Li(α)]}.
+Definition 4.1. [21] For any α ∈ I = [0, b], let K(α) be a subset of R2. For integers r ≥ 4, r′ ≥ 2,
+we call K(α) a Cr,r′-billiard deformation (i.e. Cr with respect to u and Cr′ with respect to α) if
+the following conditions hold for all α ∈ I:
+1. K(α) = �z0
+i=1 Ki(α) satisfies the no-eclipse condition (H).
+7
+
+2. Each Ki(α) is a compact, strictly convex set with Cr boundary and total arc length Li(α).
+3. Each Ki is parametrized counterclockwise by arc-length with Cr,r′ functions ϕi : Pi → R2.
+4. For all integers 0 ≤ l ≤ r, 0 ≤ l′ ≤ r′ (apart from l = l′ = 0), there exist constants C(l,l′)
+ϕ
+depending only on the choice of the billiard deformation and the parametrizations ϕi, such
+that for all integers i = 1, 2, 3, ..., z0,
+��� ∂l+l′ϕi
+∂ul
+i∂αl′
+��� ≤ C(l,l′)
+ϕ
+.
+Let Bα be the open billiard map on a non-wandering set Mα for K(α). Let Σ defined in Sec.
+2.2, we defined Rα : Mα → Σ by Rα(x(α)) = ξ(x(α)). We can write the points that correspond
+to the billiard trajectories according to the parameterization in previous definition as follows,
+π(Bj(x(α))) = qξj(α) = ϕξj(uξj(α), α) ∈ ∂Kξj(α), where uξj(α) ∈ [0, Lξj(α)]. For brevity, we will
+write qj(α) = ϕj(uj(α), α).
+The next corollary shows that uj(α) = uξj(α) for a fixed ξ ∈ Σ, is differentiable with respect
+to α. This corollary is proved in [21].
+Theorem 4.2. [21] Let K(α) be a Cr,r′ billiard deformation with r, r′ ≥ 2.
+Then uj(α) is
+Cmin{r−1,r′−1} with respect to α, and there exist constants C(n)
+u
+> 0 such that
+���dnuj(α)
+dαn
+��� ≤ C(n)
+u .
+The next corollary follows from Definition 4.1 and Theorem 4.2.
+Corollary 4.3. Let K(α) be a Cr,r′ billiard deformation with r, r′ ≥ 2. Let qj(α) belongs to ∂Kξj.
+Then qj(α) is Cn, where n = min{r − 1, r′ − 1}, with respect to α, and there exist constants
+C(n)
+q
+> 0 such that
+���dnqj(α)
+dαn
+��� ≤ C(n)
+q
+.
+4.1
+Propagation of unstable manifolds for billiard deformations
+We described the unstable manifolds propagation in Section 2.4 for open billiards. Here in this
+section, we describe it for billiard deformations.
+Let K(α), α ∈ [0, b] be a Cr,r′ billiard deformation as in Definition 4.1 with r ≥ 3, r′ ≥ 1.
+x0(α) = (q0(α), v0(α)) ∈ Mα and let W u
+ǫ (x0(α)) be the local unstable manifold for x0(α) for
+sufficiently small ǫ > 0. Take a curve Xα containing q0(α) such that Xα = {q0(s, α) : s ∈ [0, a]}
+is a convex curve with outer unit normal field nX(q0(s, α)) = v0(α) and C3 with respect to s.
+It follows from Sinai [15], [16] that W u
+ǫ (x0(α)) = {(q0(s, α), nX(q0)) : s ∈ [0, a]}.
+Set �
+Xα =
+W u
+ǫ (x0(α)). Let a ∈ R+ be small enough such that all reflection points qj(s, α), j = 1, 2, ..., m,
+that are generated by x0(s, α) = (q0(s, α), nX(q0(s, α))) belong to the same boundary ∂Kξj(α).
+Let dj(s, α) = ∥qj+1(s, α) − qj(s, α)∥ be the distance between two reflection points qj+1(s, α) and
+qj(s, α). Denote the curvature of ∂K(α) at qj(s, α) by κj(s, α), the collision angle between the
+unit normal to ∂K(α) and the reflection vector at qj(s, α) by φj(s, α), and the curvature of X at
+q0(s, α) by k0(s, α) .
+8
+
+Let tj(x(s, α)) = tj(s, α) be the time of the j-th reflection. Given t with tj < t < tj+1 for
+some j = 1, 2, ..., m, set π(Φt( �
+Xα)) = Xαt. Then Xαt = {uαt(s, α) : s ∈ [0, a]} is C3 with respect
+to s and a convex curve with outer unit normal field nXαt(uαt(s, α)). Denote the curvature of
+Xαtj(s,α) at qj(s, α) by kj(s, α), where Xαtj(s,α) = limtցtj(s,α) Xαt. As in equation (2.1), we can
+define kj(s, α) as follows:
+kj+1(s, α) =
+kj(s, α)
+1 + dj(s, α)kj(s, α) + 2
+κj+1(s, α)
+cos φj+1(s, α)
+,
+0 ≤ j ≤ m − 1 .
+(4.1)
+From now on, we will need to use previous characteristics in the case s = 0, so for brevity,
+we will write dj(α) = dj(0, α), etc. Also, we denote the billiard deformation by K(α), so all of
+its characteristics will be denoted dj(α), kj(α), etc. The initial open billiard is K(0) so all of its
+characteristics will be denoted dj(0), etc.
+4.2
+The higher derivatives of billiard characteristics
+Let K(α) be a Cr,r′ billiard deformation as in
+Definition 4.1 with r ≥ 4, r′ ≥ 2. Recall that
+∂Kξj(α) is parametrized by arc-length ujand qj(α) = ϕj(uj(α), α) ∈ ∂Kξj(α). Here, we state
+some corollaries related to bounds of the higher derivatives of curvature, distance and collision
+angle of a billiard deformation. These corollaries are forthright consequences of condition 4 in
+Definition 4.1.
+Corollary 4.4. Let K(α) be a Cr,r′ billiard deformation with r ≥ 4, r′ ≥ 2. Then the curvature
+κj(α) at qj(α) is Cn, where n = min{r − 3, r′ − 1} with respect to α and there exist constants
+C(n)
+κ
+> 0 depending only on n such that
+���dnκ
+dαn
+��� ≤ C(n)
+κ .
+Proof. Suppose K(α) is a a Cr,r′ billiard deformation with r ≥ 3, r′ ≥ 1.
+Since ∂Kj(α) is
+paramitrized by arc-length uj, then the curvature of ∂Kj(α) at qj(α) = ϕj(uj(α), α) is κj = ∂2ϕj
+∂u2
+j ,
+for j = 0, 1, ..., m. Then κj(α) is Cmin{r−3,r′−1} with respect to α.
+For the first derivative, we have
+���dκj
+dα
+��� =
+���∂3ϕj
+∂u3
+j
+∂uj
+∂α + ∂3ϕj
+∂u2
+j∂α
+��� ≤ C(1)
+κ ,
+this estimate was obtained in [21]. Next, we continue to estimate the second derivative, so we
+have
+���d2κj
+dα2
+��� =
+���∂4ϕj
+∂u4
+j
+�∂uj
+∂α
+�2 + ∂3ϕj
+∂u3
+j
+∂u2
+j
+∂α2 + 2 ∂4ϕj
+∂u3
+j∂α
+∂uj
+∂α +
+∂4ϕj
+∂u2
+j∂α2
+���.
+By using condition 4 in Definition 4.1 and Theorem 4.2, there exists a constant C(2)
+κ
+> 0 such
+that
+9
+
+���d2κj
+dα2
+��� ≤ C(4,0)
+ϕ
+(C(1)
+u )2 + C(3,0)
+ϕ
+C(2)
+u
++ 2Cϕ(3,1)C(1)
+u
++ C(2,2)
+ϕ
+= C(2)
+κ .
+Continuing by induction we see that the n-th derivative, where n = min{r −3, r′ −1}, is bounded
+by a constant C(n)
+κ
+> 0 which depends only on n such that
+���dnκ
+dαn
+��� ≤ C(n)
+κ .
+Corollary 4.5. Let K(α) be a Cr,r′ billiard deformation with r ≥ 3, r′ ≥ 1. Then the distance
+dj(α) between two points qj+1(α) and qj(α) is Cn, where n = min{r − 1, r′ − 1} with respect to α
+and there exist constants C(n)
+d
+> 0 depending only on n such that
+���dndj
+dαn
+��� ≤ C(n)
+d .
+Proof. Since dj = ∥qj+1(α) − qj(α)∥ = ∥ϕj+1(uj+1(α), α) − ϕj(uj(α), α)∥ for j = 0, 1, ..., m, then
+dj is Cmin{r−1,r′−1}. The first derivative is
+ddj
+dα =
+� ϕj+1(uj+1(α), α) − ϕj(uj(α), α)
+∥ϕj+1(uj+1(α), α) − ϕj(uj(α), α)∥, ∂ϕj+1
+∂uj+1
+∂uj+1
+∂α
++ ∂ϕj+1
+∂α
++ ∂ϕj
+∂uj
+∂uj
+∂α + ∂ϕj
+∂α
+�
+.
+And then
+���ddj
+dα
+��� =
+���∂ϕj+1
+∂uj+1
+∂uj+1
+∂α
++ ∂ϕj+1
+∂α
++ ∂ϕj
+∂uj
+∂uj
+∂α + ∂ϕj
+∂α
+��� ≤ C(1)
+d ,
+which was estimated in [21]. For the second derivative, using condition 4 in Definition 4.1 and
+Theorem 4.2 it follows that
+���d2dj
+dα2
+��� =
+���∂2ϕj+1
+∂u2
+j+1
+�∂uj+1
+∂α
+�2
++ ∂ϕj+1
+∂uj+1
+∂2uj+1
+∂α2
++ 2 ∂2ϕj+1
+∂uj+1∂α
+∂uj+1
+∂α
++ ∂2ϕj+1
+∂α2
++ ∂2ϕj
+∂u2
+j
+�∂uj
+∂α
+�2
++ ∂ϕj
+∂uj
+∂2uj
+∂α2 + 2 ∂2ϕj
+∂uj∂α
+∂uj
+∂α + ∂2ϕj
+∂α2
+���.
+By using condition 4 in Definition 4.1 and Theorem 4.2, there exists a constant C(2)
+κ
+> 0 such
+that
+���d2dj
+dα2
+��� ≤ 2C(2,0)
+ϕ
+(C(1)
+u )2 + 2C(2)
+u
++ 4C(1,1)
+ϕ
+(C(1)
+u )2 + 2C(0,2)
+ϕ
+= C(2)
+d .
+Continuing by induction, we can see that there exists a constant C(n)
+d
+> 0 depends only on n such
+that
+���dndj
+dαn
+��� ≤ C(n)
+d
+.
+Corollary 4.6. Let K(α) be a Cr,r′ billiard deformation with r ≥ 4, r′ ≥ 2. Then cos φj(α) is
+Cmin{r−1,r′−1} and there exists a constant C(n)
+φ
+> 0 depending only on n such that
+���dn cos φj
+dαn
+��� ≤ C(n)
+φ .
+10
+
+Proof. We can write
+cos 2φj =
+�
+qj+1(α) − qj(α)
+�
+·
+�
+qj(α) − qj−1(α)
+�
+|qj+1(α) − qj(α)||qj(α) − qj−1(α)|
+=
+�
+ϕj+1(, uj+1, α) − ϕj(uj, α)
+�
+·
+�
+ϕj(uj, α) − ϕj−1(uj−1α)
+�
+|ϕj+1(uj+1, α) − ϕj(uj, α)||ϕj(uj, α) − ϕj−1(uj−1, α)|
+.
+And then, cos φj(α) =
+�
+cos 2φj(α)+1
+2
+. Therefore, the statement follows from condition 4 in Defi-
+nition 4.1 and Corollary 4.2.
+The next corollary follows from Corollaries 4.4, 4.6.
+Corollary 4.7. Let K(α) be a Cr,r′ billiard deformation with r ≥ 4, r′ ≥ 2. Then the expression
+gj(α) =
+2κj
+cos φj is Cmin{r−3,r′−1} and there exist constants C(n)
+g
+> 0 depending only on n such that
+���dngj
+dαn
+��� ≤ C(n)
+g
+.
+The next corollary concerning the curvature kj, defined in (4.1), follows from Corollaries 4.5 and
+4.7.
+Corollary 4.8. Let K(α) be a Cr,r′ billiard deformation with r ≥ 4, r′ ≥ 2. Then the curvature
+kj(α) is Cn, where n = min{r − 3, r′ − 1} and here exist constants C(n)
+k
+depending only on n such
+that
+���dnkj
+dαn
+��� ≤ C(n)
+k .
+Proof. First, we recall
+kj+1(α) =
+kj(α)
+1 + dj(α)kj(α) + 2
+κj+1(α)
+cosφj+1(α)
+,
+0 ≤ j ≤ m − 1 .
+We will write kj+1(α) simply as follows
+kj+1(α) =
+kj(α)
+1 + dj(α)kj(α) + gj+1(α),
+where gj+1(α) =
+2κj+1
+cos φj+1 . [21] contains an estimate that the first derivative of kj(α) with respect
+to α is bounded by a constant C(1)
+k . Here, we use the same argument in [21] and show that the
+second derivative of kj(α) with respect to α is also bounded. These estimates are useful and will
+be used later in Section 5.
+Next, we start with the first derivative of kj+1 with respect to α and we will use the notation
+˙k, ¨k,...etc. to simplify equations. So, we have
+˙kj+1 =
+˙kj
+(1 + djkj)2 −
+˙djk2
+j
+(1 + djkj)2 + ˙gj+1.
+And for the second derivative, we have
+¨kj+1 =
+¨kj
+(1 + djkj)2 −
+k2
+j ( ¨dj + ¨djdjkj − 2 ˙d2
+jkj) + 2˙kj(˙kjdj + 2 ˙djkj)
+(1 + djkj)3
++ ¨gj+1.
+11
+
+Let
+βj =
+1
+(1 + djkj)2 ,
+ηj = −
+k2
+j( ¨dj + ¨djdjkj − 2 ˙d2
+jkj) + 2˙kj(˙kjdj + 2 ˙djkj)
+(1 + djkj)3
++ ¨gj+1
+,
+0 ≤ j ≤ m − 1 .
+From Corollaries 4.5 and 4.7, and the estimate of ˙kj, we have
+|βj| ≤ βmax =
+1
+(1 + dminkmin)2 ,
+|ηj| ≤ ηmax = k2
+max(C(2)
+d
++ C(2)
+d dmaxkmax + 2(C(1)
+d )2kmax)
+(1 + dminkmin)3
++ 2C(1)
+k (C(1)
+k dmax + 2C(1)
+d kmax)
+(1 + dminkmin)3
++ C(2)
+g .
+Then, we have
+¨km(α) = ηm−1 + βm−1¨km−1(α)
+= ηm−1 + βm−1 ηm−2 + .... + βm−1....β1 η0 + βm−1....β0 ¨k0(α).
+To solve this equation, we assume that (q(α), v(α)) is periodic such that Bm
+α (q(α), v(α)) =
+(q(α), v(α)). Then km(α) = k0(α). From this, we can solve the previous equation as follows
+¨km(α) − βm−1....β0 ¨k(α) = ηm−1 + βm−1 ηm−2 + .... + βm−1....β1 η0
+¨km(α) =
+1
+1 − βm−1....β0
+�
+ηm−1 + βm−1 ηm−2 + .... + βm−1....β1
+�
+By the maximum value of ηj and βi, we have
+|¨km(α)| ≤
+ηmax
+1 − βm
+max
+�
+1 + βmax + .... + βm−1
+max
+�
+=
+ηmax
+1 − βm
+max
+�1 − βm
+max
+1 − βmax
+�
+=
+ηmax
+1 − βmax
+.
+This means there exists a constant C(2)
+k
+> 0 does not depend on m or α such that |¨kj(α)| ≤ C(2)
+k ,
+for every j = 0, 1, ..., m. Continuing by induction we can see that the n-th derivative of kj(α)
+with respect to α is bounded by constant C(n)
+k
+> 0 that depending only on n.
+5
+Continuity of the largest Lyapunov exponent
+In this section, we show that the largest Lyapunov exponent λ1 depends continuously on a planar
+billiard deformation. Let K(α) be a billiard deformation as defined in Definition 4.1 and let K(0)
+be the initial open billiard. Let kj(α), kj(0) and dj(α), di(0) be the curvatures and the distances
+that are described in section 4.1.
+12
+
+For every α ∈ [0, b], let Mα be the non-wandering set for the billiard map and let Rα :
+Mα −→ Σ be the analogue of the conjugacy map R : M0 −→ Σ, so that the following diagram is
+commutative:
+Mα
+Bα
+−→
+Mα
+�Rα
+�Rα
+Σ
+σ
+−→
+Σ
+where Bα is the billiard ball map on Mα. By Theorem 2.1 there exists a subset Aα of Σ with
+µ(Aα) = 1 so that
+λ1(α) = lim
+m→∞
+1
+m log ∥Dx0Bm
+α (w)∥
+(5.1)
+for all x ∈ Mα with Rα(x) ∈ Aα. Similarly, let A0 be the set with µ(A0) = 1 which we get from
+Theorem 2.1 for α = 0.
+Lemma 5.1. Given an arbitrary sequence
+α1, α2, . . . , αp, . . .
+of elements of [0, b], for µ-almost all ξ ∈ Σ the formula (5.1) is valid for α = αp and x = R−1
+α (ξ)
+for all p = 1, 2, . . . and also for α = 0 and x = R−1(ξ).
+Proof. The set A = A0 ∩ ∩∞
+p=1Aαp has µ(A) = 1 since
+Σ \ A = (Σ \ A0) ∪ ∪∞
+p (Σ \ Aαp)
+has measure zero as a countable union of sets of measure zero. If α = αp for some p and Rα(x) ∈ A,
+then Rα(x) ∈ Aαp so formula (5.1) holds. Similarly (5.1) holds for α = 0 as well.
+Thus, using the notation x(0, α) ∈ Mα, we can choose ξ ∈ Σ so that formula (5.1) applies for
+α = αp and x = x(0, αp) for all p = 1, 2, . . ., and also for α = 0 and x = (0, 0).
+From the formula for the largest Lyapunov exponent (3.1), we can write the Lyapunov expo-
+nents for K(α) and K(0) as follows:
+λ1(α) = lim
+m→∞
+1
+m
+m
+�
+j=1
+log
+�
+1 + dj(α)kj(α)
+�
+= lim
+m→∞ λ(m)
+1
+(α),
+λ1(0) = lim
+m→∞
+1
+m
+m
+�
+j=1
+log
+�
+1 + dj(0)kj(0)
+�
+= lim
+m→∞ λ(m)
+1
+(0),
+where
+λ(m)
+1
+(α) = 1
+m
+m
+�
+j=1
+log
+�
+1 + dj(α)kj(α)
+�
+and
+λ(m)
+1
+(0) = 1
+m
+m
+�
+j=1
+log
+�
+1 + dj(0)kj(0)
+�
+.
+(5.2)
+Now, we prove Theorem 1.1
+13
+
+Proof of Theorem 1.1: Let K(α) be a C4,2 billiard deformation in R2, and let
+α ∈ [0, b]. Assume that λ1(α) is not continuous at α = 0. Then there exists ε > 0 and a sequence
+α1 > α2 > ... > αp > ... → 0 in [0, b] with αp → 0 such that |λm
+1 (αk) − λm
+1 (0)| ≥ ε for all p ≥ 1.
+By using Lemma 5.1 and the previous expressions of λm
+1 (α) for α = αp and λm
+1 (0) in (5.2), we
+have
+�����λm
+1 (αp) − λm
+1 (0)
+����� =
+�����
+1
+m
+m
+�
+j=1
+(log δj(αp) − log δj(0))
+�����
+=
+�����
+−1
+m
+m
+�
+j=1
+(log(1 + dj(αp)kj(αp)) − log(1 + dj(0)kj(0)))
+�����
+≤ 1
+m
+m
+�
+j=1
+����� log(1 + dj(αp)kj(αp)) − log(1 + dj(0)kj(0))
+�����
+≤ 1
+m
+m
+�
+j=1
+�����
+1 + dj(αp)kj(αp) − (1 + dj(0)kj(0))
+1 + min{dj(αp)kj(αp), dj(0)kj(0)}
+�����
+= 1
+m
+m
+�
+j=1
+�����
+dj(αp)kj(αp) − dj(0)kj(0)
+1 + dminkmin
+�����
+= 1
+m C0
+m
+�
+j=1
+�����dj(αp)kj(αp) − dj(0)kj(0)
+�����
+= 1
+m C0
+m
+�
+j=1
+�����(dj(αp) − dj(0))kj(αp) + dj(0)(kj(αp) − kj(0))
+�����,
+where C0 =
+1
+1+dminkmin > 0 is a global constant independent of αp.
+Fix a small δ > 0; we will state later how small δ > 0 should be. Next consider p sufficiently large so
+that αp < δ. For all p, we have |kj(αp)−kj(0)| = αp|˙kj(s(αp))| and |dj(αp)−dj(0)| = αp| ˙dj(r(αp))|,
+for some s(αp), r(αp) ∈ [0, αp]. From Corollaries 4.5 and 4.8 , there exist constants Ck and Cd
+such that |˙kj(s(αp))| ≤ Ck and | ˙dj(s(αp))| ≤ Cd. Therefore for all j,
+|kj(αp) − kj(0)| ≤ αpCk < δCk, and |dj(αp) − dj(0)| ≤ αpCd < δCd. Then
+���λm
+1 (αp) − λm
+1 (0)
+��� ≤ 1
+m C0
+m
+�
+j=1
+����dj(αp) − dj(0)
+���kj(αp) + dj(0)
+���kj(αp) − kj(0)
+���
+�
+< 1
+m C0
+m
+�
+j=1
+δ(Cdkmax + Ckdmax)
+= C0δ(Cdkmax + Ckdmax) < ε,
+if we take δ <
+ε
+Cdkmax+Ckdmax . We now have a contradiction because with the choice of the sequence
+α1 > α2 > ... > αp > ... → 0 in [0, b]. Therefore the statement is proved.
+14
+
+6
+Differentiability of the largest Lyapunov exponent
+Here we prove Theorem 1.2
+Proof of Theorem 1.2: We will prove differentiability at α = 0. From this differentiability at any
+α ∈ [0, b] follows. To prove the differentiability at α = 0, we have to show that there exists
+lim
+α→0
+λ1(α) − λ1(0)
+α
+.
+Equivalently, there exists a number F such that
+lim
+p→∞
+λ1(αp) − λ1(0)
+αp
+= F,
+for any sequence α1 > α2 > ... > αp > ... → 0 as p → ∞ in [0, b].
+Let K(α) ⊂ R2 be a C5,3 billiard deformation and α ∈ [0, b] for a positive number b. Let λ1(α) be
+the largest Lyapunov exponent for K(α) and λ1(0) be the largest Lyapunov exponent for K(0).
+By using Lemma 5.1 and the expressions of λm
+1 (α) for α = αp and λm
+1 (0) in (5.2), we have
+λ(m)
+1
+(αp) → λ1(αp) and λ(m)
+1
+(0) → λ1(0) when m → ∞. Also,
+λ(m)
+1
+(αp) − λ(m)
+1
+(0)
+αp
+= − 1
+m
+m
+�
+j=1
+log δj(αp) − log δj(0)
+αp
+= − 1
+m
+m
+�
+j=1
+log
+�
+1 + dj(αp)kj(αp)
+�
+− log
+�
+1 + dj(0)kj(0)
+�
+αp
+.
+Set fj(αp) = log
+�
+1 + dj(αp)kj(αp)
+�
+and fj(0) = log
+�
+1 + dj(0)kj(0)
+�
+. Then
+λ(m)
+1
+(αp) − λ(m)
+1
+(0)
+αp
+= − 1
+m
+m
+�
+j=1
+fj(αp) − fj(0)
+αp
+.
+Taylor’s formula gives
+fj(αp) = fj(0) + αp ˙fj(0) + α2
+p
+2
+¨fj(rj(αp))
+for some rj(αp) ∈ [0, αp]. Then
+fj(αp) − fj(0)
+αp
+− ˙fj(0) = αp
+2
+¨fj(rj(αp)).
+Let
+Fm = 1
+m
+m
+�
+j=1
+˙fj(0).
+Summing up the above for j = 1, 2, ..., m, we get
+λ(m)
+1
+(αp) − λ(m)
+1
+(0)
+αp
+− Fm = − 1
+m
+m
+�
+j=1
+�fj(αp) − fj(0)
+αp
+− ˙fj(0)
+�
+.
+15
+
+From the definition of fj(αp),
+˙fj(αp) =
+˙dj(αp)kj(αp) + dj(αp)˙kj(αp)
+1 + dj(αp)kj(αp)
+,
+and therefore,
+¨fj(αp) =
+� ¨dj(αp)kj(αp) + 2 ˙dj(αp)˙kj(αp) + dj(αp)¨kj(αp)
+��
+1 + dj(αp)kj(αp)
+�
+�
+1 + dj(αp)kj(αp)
+�2
+−
+� ˙dj(αp)k(αp) + dj(αp)˙kj(αp)
+�2
+�
+1 + dj(αp)kj(αp)
+�2
+.
+Then from Corollaries 4.5 and 4.8, we get
+��� ˙fj(αp)
+��� ≤ C(1)
+d kmax + dmaxC(1)
+k
+1 + dminkmin
+= C1,
+��� ¨fj(αp)
+��� ≤
+�
+C(2)
+d kmax + 2C(1)
+d C(1)
+k
++ dmaxC(2)
+k
+��
+1 + dmaxkmax
+�
+�
+1 + dminkmin
+�2
++
+�
+C(1)
+d kmax + dmaxC(1)
+k
+�2
+�
+1 + dminkmin
+�2
+= C2.
+Therefore
+| ¨fj(rj(αp))| ≤ C2,
+for some constant C2 > 0 independent of rj(αp) and j. This implies
+���λ(m)
+1
+(αp) − λ(m)
+1
+(0)
+αp
+− Fm
+��� ≤ 1
+m
+m
+�
+j=1
+αp
+2
+��� ¨fj(tj(αp))
+���
+≤ C2
+2 αp.
+Since | ˙fj(αp)| ≤ C1, we have |Fm| ≤
+1
+m
+�m
+j=1 | ˙fj(0)| ≤ C1, for all m. Therefore, the sequence
+{Fm} has convergent subsequences. Let for example Fmh → F, for some sub-sequence {mh}.
+Then
+���λ(mh)
+1
+(αp) − λ(mh)
+1
+(0)
+αp
+− Fmh
+��� ≤ C2
+2 αp,
+for all h ≥ 1. So, letting h → ∞, we get
+���λ1(αp) − λ1(0)
+αp
+− F
+��� ≤ C2
+2 αp,
+and letting αp → 0 as p → ∞ we get that there exists
+lim
+p→∞
+λ1(αp) − λ1(0)
+αp
+= F.
+for every sequence α1 > α2 > ... > αp > ... → 0 as p → ∞ in [0, b].
+Thus, there exists
+F = limm→∞ 1
+m
+�m
+j=1 ˙fj(0). This is true for every subsequence {mh}, so for any subsequence we
+have Fmh → F. Hence, Fm converges to F as well. This implies that there exists
+16
+
+lim
+α→0
+λ1(α) − λ1(0)
+α
+= F,
+so λ1 is differentiable at α = 0 and ˙λ1(0) = F.
+Corollary 6.1. Let K(α) be a C5,3 billiard deformation. Then there exists a constant Cλ1 > 0
+such that
+���dλ1(α)
+dα
+��� ≤ Cλ1,
+for all α ∈ [0, b].
+Proof. We have
+λ1(α) = lim
+m→∞
+1
+m
+m
+�
+j=1
+log(1 + dj(α)kj(α)).
+By Theorem 1.2, λ1(α) is C1. So, from the formula in the previous proof that
+˙λ1(0) = limm→∞ 1
+m
+�m
+j=1 ˙fj(0), we have
+dλ1
+dα = lim
+m→∞
+1
+m
+m
+�
+j=1
+ddj
+dα kj(α) + dj(α)dkj
+dα
+1 + dj(α)kj(α)
+.
+From Corollaries 4.5 and 4.8, there exist constants C(1)
+d , C(1)
+k
+> 0 such that
+���ddj
+dα
+��� ≤ C(1)
+d
+and
+���dkj
+dα
+��� ≤ C(1)
+k . Then, we have
+���dλ1
+dα
+��� ≤ lim
+m→∞
+1
+m
+m
+�
+j=1
+C(1)
+d kmax + dmaxC(1)
+k
+1 + dminkmin
+= C(1)
+d kmax + dmaxC(1)
+k
+1 + dminkmin
+= Cλ1.
+This proves the statement.
+Acknowledgment
+The author would like to thank Prof. Luchezar Stoyanov for his suggestions, comments, and help.
+This work was supported by a scholarship from Najran University, Saudi Arabia.
+References
+[1] L. Barreira and Ya. Pesin, Lyapunov exponents and smooth ergodic theory. Univ. Lect. Series 23,
+American Mathematical Society, Providence, RI, 2001.
+[2] R. Bowen, Symbolic dynamics for hyperbolic flows. Amer. J. Math. 95 (1973), 429-460.
+[3] P. Duarte, S. Klein and M. Poletti, H¨older continuity of the Lyapunov exponents of linear cocycles over
+hyperbolic maps. Math. Z. 302 (2022), 2285–2325.
+17
+
+[4] N. Chernov, Entropy, Lyapunov exponents, and mean free path for billiards. Journal of Statistical
+Physics, 88 (1997), 1-29.
+[5] N. Chernov and R. Markarian, Chaotic Billiards. Math. Surveys and Monographs Vol. 127, Amer.
+Math. Soc. 2006.
+[6] M. Ikawa, Decay of solutions of the wave equation in the exterior of several strictly convex bodies. Ann.
+Inst. Fourier 38 (1988), 113-146.
+[7] A. Katok and J. M. Strelcyn, Invariant Manifolds, Entropy and Billiards; Smooth Maps with Singular-
+ities. Lecture Notes in Mathematics 1222, Springer, 1986.
+[8] A. Lopes and R. Markarian, Open billiards: invariant and conditionally invariant probabilities on
+Cantor sets. SIAM J. Appl. Math. 56 (1996), 651-680.
+[9] R. Markarian, Billiards with Pesin Region of Measure one. Comm. in Math Phys. 118 (1988), 87-97.
+[10] R. Markarian, New ergodic Billiards: exact results. Nonlinearity 6. (1993), 819-841
+[11] T. Morita, The symbolic representation of billiards without boundary condition. Trans. Amer. Math.
+Soc. 325 (1991), 819-828.
+[12] V. I. Oseledets, A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical
+systems. Trans. Moscow Math. Soc. 19 (1968), 197-221.
+[13] M. Pollicott, Lectures on ergodic theory and Pesin theory on compact manifolds. Cambridge Univ.
+Press, Cambridge 1993.
+[14] V. Petkov and L. Stoyanov, Geometry of Reflecting Rays and Inverse Spectral Problems. Wiley, Chich-
+ester, (1992).
+[15] Ya. Sinai, Dynamical systems with elastic reflections. Russian Math. Surveys 25 (1970), 137-190.
+[16] Ya. Sinai, Development of Krylov’s ideas, An addendum to: N.S.Krylov ”Works on the foundations
+of statistical physics”. Princeton Univ. Press, Princeton 1979, 239-281.
+[17] L. Stoyanov, Exponential instability and entropy for a class of dispersing billiards. Ergod. Th. &
+Dynam. Sys. 19 (1999), 201-226.
+[18] L. Stoyanov, Spectrum of the Ruelle operator and exponential decay of correlation for open billiard
+flows. Amer. J. Math. 123 (2001), 715-759.
+[19] L. Stoyanov, Non-integrability of open billiard flows and Dolgopyat-type estimates. Ergodic Th. & Dyn.
+Systems 32 (2012), 295-313.
+[20] M. Viana, Lectures on Lyapunov exponents, Cambridge Studies in Adv. Math. vol.145, Cambridge
+Univ. Press 2014.
+[21] P. Wright, Differentiability of the Hausdorff dimension of the non-wandering set in a planar open
+billiard, Discrete & Continuous Dynamical Systems 36(7) (2016), 3993-4014.
+[22] M, P. Wojtkowski, Principles for the design of billiards with nonvanishing Lyapunov exponents. Com-
+mun. Math. Phys. 105 (1986), 391-414.
+18
+

4tFKT4oBgHgl3EQfRy39/content/tmp_files/2301.11773v1.pdf.txt

@@ -0,0 +1,3963 @@
+1
+Automatic Modulation Classification with Deep
+Neural Networks
+Clayton A. Harper, Mitchell A. Thornton, and Eric C. Larson
+Darwin Deason Institute for Cyber Security
+{caharper, mitch, eclarson}@smu.edu
+Abstract—Automatic modulation classification is a desired
+feature in many modern software-defined radios. In recent years,
+a number of convolutional deep learning architectures have
+been proposed for automatically classifying the modulation used
+on observed signal bursts. However, a comprehensive analysis
+of these differing architectures and importance of each design
+element has not been carried out. Thus it is unclear what
+tradeoffs the differing designs of these convolutional neural
+networks might have. In this research, we investigate numerous
+architectures for automatic modulation classification and perform
+a comprehensive ablation study to investigate the impacts of
+varying hyperparameters and design elements on automatic
+modulation classification performance. We show that a new
+state of the art in performance can be achieved using a subset
+of the studied design elements. In particular, we show that
+a combination of dilated convolutions, statistics pooling, and
+squeeze-and-excitation units results in the strongest performing
+classifier. We further investigate this best performer according
+to various other criteria, including short signal bursts, common
+misclassifications, and performance across differing modulation
+categories and modes.
+Index Terms—Automatic modulation classification, deep learn-
+ing, convolutional neural network.
+I. INTRODUCTION
+A
+UTOMATIC modulation classification (AMC) is of par-
+ticular interest for radio frequency (RF) analysis and in
+modern software-defined radios to perform numerous tasks
+including “spectrum interference monitoring, radio fault detec-
+tion, dynamic spectrum access, opportunistic mesh network-
+ing, and numerous regulatory and defense applications” [1].
+Upon detection of an RF signal with unknown characteristics,
+AMC is a crucial initial procedure in order to demodulate the
+signal. Efficient AMC allows for maximal usage of transmis-
+sion mediums and can provide resilience in modern cognitive
+radios. Systems capable of adaptive modulation schemes can
+monitor current channel conditions with AMC and adjust
+exercised modulation schemes to maximize usage across the
+transmission medium.
+Moreover, for receivers that have a versatile demodulation
+capability, AMC is a requisite task. The correct demodulation
+scheme must be applied to recover the modulated message
+within a detected signal. In systems where the modulation
+scheme is not known a priori, AMC allows for efficient predic-
+tion of the employed modulation scheme. Higher performing
+AMC can increase the throughput and accuracy of these
+systems; therefore, AMC is currently an important research
+topic in the fields of machine learning and communication
+systems, specifically for software-defined radios.
+Typical benchmarks are constructed on the premise that the
+AMC model must classify not only the mode of modulation
+(e.g., QAM), but the exact variant of that mode of modulation
+(e.g., 32QAM). While many architectures have proven to be
+effective at high signal to noise ratios (SNRs), performance
+degrades significantly at lower SNRs that often occur in real-
+world applications. Other works have investigated increasing
+classification performance at lower SNR levels through the
+use of SNR-specific modulation classifiers [2] and clustering
+based on SNR ranges [3]. To perform classification, a variety
+of signal features have been investigated. Historically, AMC
+has relied upon statistical moments and higher order cumulants
+[4]–[6] derived from the received signal. Recent approaches
+[1], [7]–[9] use raw time-domain in-phase (I) and quadrature
+(Q) components as features to predict the modulation variant
+of a signal. Further works have investigated additional features
+including I/Q constellation plots [10]–[12].
+After selecting the signal input features, machine learning
+models are used to determine statistical patterns in the data
+for the classification task. Support vector machines, decision
+trees, and neural networks are commonly used classifiers for
+this application [1], [3], [7]–[10], [13], [14]. Residual neural
+networks (ResNets), along with convolutional neural networks
+(CNNs), have been shown to achieve high classification perfor-
+mance for AMC [1], [3], [7]–[10]. Thus, deep learning based
+methods in AMC have become more prevalent due to their
+promising performance and their ability to generalize to large,
+complex datasets.
+While other works have contributed to increased AMC
+performance, the importance of many design elements for
+AMC remains unclear and a number of architectural elements
+have yet to be investigated. Therefore, in this work, we aim
+to formalize the impact of a variety of architectural changes
+and model design decisions on AMC performance. Numerous
+modifications to architectures from previous works, including
+our own [7], and novel combinations of elements applied to
+AMC are considered. After an initial investigation, we provide
+a comprehensive ablation study in this work to investigate
+the performance impact of various architectural modifications.
+Additionally, we achieve new state-of-the-art classification
+performance on the RadioML 2018.01A dataset [15]. Using
+the best performing model, we provide additional analyses
+that characterize its performance across modulation modes and
+arXiv:2301.11773v1  [cs.LG]  27 Jan 2023
+
+2
+Fig. 1. ResNet architecture used in [1]. Each block represents a unit in the network, which may be comprised of several layers and connections as shown
+on the right of the figure. Dimensions of the tensors on the output of each block are also shown where appropriate.
+signal burst duration.
+II. RELATED WORK
+The area of AMC has been investigated by several research
+groups. We provide a summary of results in AMC to provide
+context and motivation for our contributions to AMC and the
+corresponding ablation study described in this paper.
+Corgan et al. [8] illustrate that deep convolutional neural
+networks are able to achieve high classification performance
+particularly at low SNRs on a dataset comprising 11 different
+types of modulation. It was found that CNNs exceeded perfor-
+mance over expertly crafted features. Comparing results with
+architectures in [8] and [1], [16] improved AMC performance
+utilizing self-supervised contrastive learning. First, an encoder
+is pre-trained in a self-supervised manner through creating
+contrastive pairs with data augmentation. By creating different
+views of the input data through augmentation, contrastive loss
+is used to maximize the cosine similarity between positive
+pairs (augmented views of the same input). Once converged,
+the encoder is frozen (i.e., the weights are set to fixed
+values) and two fully-connected layers are added following the
+encoder to form the classifier. The classifier is trained using
+supervised learning to predict the 11 different modulation
+schemes. Chen et al. applied a novel architecture to the
+same dataset where the input signal is sliced and transformed
+into a square matrix and apply a residual network to predict
+the modulation schemes [17]. Other work has investigated
+empirical and variational mode decomposition to improve few-
+shot learning for AMC [18]. In our work, we utilize a larger,
+more complex dataset consisting of 24 modulation schemes,
+as well as modeling improvements.
+Spectrograms and I/Q constellation plots in [19] were found
+to be effective input features to a traditional CNN achieving
+nearly equivalent performance as the baseline CNN network
+in [1] which used raw I/Q signals.
+Further, [10]–[12] also used I/Q constellations as an input
+feature in their machine learning models on a smaller scale
+of four or eight modulation types. Other features have been
+used in AMC— [20], [21] utilized statistical features and
+support vector machines while [22], [23] used fusion methods
+in CNN classifiers. Mao et al. utilized various constellation
+diagrams at varying symbol timings alleviating symbol timing
+synchronization concerns [24]. A squeeze-and-excitation [25]
+inspired architecture was used as an attention mechanism to
+focus on the most important diagrams.
+Although spectrograms and constellation plots have shown
+promise, they require additional processing overhead and have
+had comparable performance to raw I/Q signals. In addition,
+models that use raw I/Q signals could be more adept at
+handling varying-length signals than constellation plots be-
+cause they are not limited by periodicity constraints for short
+duration signals (i.e., burst transmissions). Consequently, we
+utilize raw I/Q signals in our work.
+Tridgell, in his dissertation [26], builds upon these works by
+investigating these architectures when deployed on resource-
+limited Field Programmable Gate Arrays (FGPAs). His work
+stresses the importance of reducing the number of parameters
+for modulation classifiers because they are typically deployed
+in resource-constrained embedded systems.
+Fig. 2. X-Vector architecture overview. The convolutional activations imme-
+diately before pooling are shown. These activations are fed into two statistical
+pooling layers that collapse the activations over time, creating a fixed-length
+tensor that can be further processed by fully connected dense layers.
+
+ResNet Architecture
+Residual Stack
+Residual Unit
+Input ↓ Batch size ×1024 ×2
+Batch size x 128 x 32
+Batch size x 512
+Residual Stack
+Residual Stack
+Dense + SeLU (128)
+Input
+Conv1D + Linear (32, 1)
+Conv1D + ReLU (32, 3)
+ Batch size × 512 × 32
+↓ Batch size x× 64 × 32
+Batch size x 128
+Residual Stack
+Residual Stack
+Dense + SeLU (128)
+Residual Unit
+Conv1D + Linear (32, 3)
+↓ Batch size × 256 × 32
+Batch size x 32 x 32
+Batch size x 128
+Dense + Softmax (24)
+Residual Unit
+Residual Stack
+Residual Stack
+Batch size x 16 × 32
+Batch size x 24
+Max Pooling (stride=2)
+ndno
+Flatten
+Prediction
+↑ andno
+Conv1D
+(number of filters, filter size)Time
+μ
+Mean
+Statistics Pooling
+Dense
+Channels
+Across Channels
+Layers
+0
+Variance
+Fixed-length
+Convolutional Activations
+X-Vector
+Pooled
+Statistics3
+Fig. 3. Proposed CNN Architecture in [7]. This is the first work to employ an X-Vector inspired architecture for AMC showing strong performance. This
+architecture is used as a baseline for the modifications investigated in this paper. The f and k variables shown designate the number of kernels and size of
+each kernel, respectively, in each layer. These parameters are investigated for optimal sizing in our initial investigation.
+In [1], Oshea et al. created a dataset with 24 different
+types of modulation, known as RadioML 2018.01A, and
+achieved high classification performance using convolutional
+neural networks—specifically using residual connections (see
+Figure 1) within the network (ResNet). A total of 6 residual
+stacks were used in the architecture. A residual stack is defined
+as a series of a convolutional layers, residual units, and a max
+pooling operation as shown in Figure 1. The ResNet employed
+by [1] attained approximately 95% classification accuracy at
+high SNR values.
+Harper et al. proposed the use of X-Vectors [27] to increase
+classification performance using CNNs [7]. X-Vectors are tra-
+ditionally used in speaker recognition and verification systems
+making use of aggregate statistics. X-Vectors employ statistical
+moments, specifically mean and variance, across convolutional
+filter outputs. It can be theorized that taking the mean and
+variance of the embedding layer helps to eliminate signal-
+specific information, leaving global, modulation-specific char-
+acteristics. Figure 2 illustrates the X-Vector architecture where
+statistics are computed over the activations from a convolu-
+tional layer producing a fixed-length vector.
+Additionally,
+this
+architecture
+maintains
+a
+fully-
+convolutional structure enabling variable size inputs into
+the network. Using statistical aggregations allows for this
+property to be exploited. When using statistical aggregations,
+the input to the first dense layer is dependent upon the
+number of filters in the final convolutional layer. The number
+of filters is a hyperparameter, independent of the length in
+time of the input signal into the neural network.
+Without the statistical aggregations, the input signals into
+a traditional CNN or ResNet would need to be resampled,
+cropped or padded to a fixed-length in time such that there is
+not a size mismatch with the final convolutional output and
+the first dense layer. While the dataset used in this work has
+uniformly sized signals in terms of duration, (1024 × 2), this
+is an architectural advantage in our deployment as received
+signals may vary in duration. Instead of modifying the inputs
+to the network via sampling, cropping, padding, etc., the X-
+Vector architecture can directly operate with variable-length
+inputs without modifications to the network or input signal.
+Figure 3 outlines the employed X-Vector architecture in [7]
+where F = [f1, f2, ..., f7] = 64 and K = [k1, k2, ..., k7] = 3.
+Mean and variance pooling are performed on the final con-
+volutional outputs, concatenated, and fed through a series of
+dense layers creating the fixed-length X-Vector. A maximum
+of 98% accuracy was achieved at high SNR levels.
+Fig. 4.
+Accuracy comparison of the reproduced ResNet in [1] and the X-
+Vector inspired model from [7] over varying SNRs. This accuracy comparison
+shows the superior performance of the X-Vector architecture, especially at
+higher SNRs, and supports using this architecture as a baseline for the
+improvements investigated in this paper.
+The work of [7] replicated the ResNet architecture from
+[1] and compared the results with the X-Vector architectures
+as seen in Figure 4. Harper et al. [7] were able to reproduce
+this architecture achieving a maximum of 93.7% accuracy. The
+authors attribute the difference in performance to differences in
+the train and test set separation they used since these parame-
+ters were unavailable. As expected, the classifiers perform with
+a higher accuracy as the SNR value increases. In signals with a
+low SNR value, noise becomes more dominant and the signal
+is harder to distinguish. In modern software-defined radio
+
+Input
+Batch size x 1024 x 2
+ Batch size x 1024 × f4
+Batch size x 1024 x fz
+Conv1D + ReLU (f5, k5, 1)
+Statistics Pooling
+Conv1D + ReLU (f1, k1, 1)
+Batch size x 1024 x f1
+I Batch size x 1024 x fs 
+Batch size x (f*2)
+Conv1D + ReLU (f2, k2, 1)
+Conv1D (f6, k6, 1)
+Dense + SeLU (128)
+Conv1D (number of filters,
+filter size, dilation rate)
+Batch size x 1024 x f2
++ Batch size x 1024 x fe 
+Batch size x 128
+Conv1D + ReLU (f3, k3, 1)
+Conv1D + ReLU (f7, k7, 1)
+Dense + SeLU (128)
+Batch size x 1024 x f3
+Batch size x 128
+Conv1D + ReLU (f4, k4, 1)
+Dense + Softmax (24)
+Batch size x 24
+Prediction1
+0.8
+Accuracy
+0.6
+0.4
+0.2
+0
+-20
+-10
+10
+20
+30
+0
+SNR (dB)4
+applications, a high SNR value is not always a given. However,
+there is still significant improvement compared to random
+chance, even at low SNR values. Moreover, in systems where
+the modulation type must be classified quickly, this could
+become crucially important as fewer demodulation schemes
+would need to be applied in a trial and error manner to discover
+the correct scheme.
+One challenge of AMC is that performance is desired to
+work well across a large range of SNRs. For instance, Figure 4
+illustrates modulation classification performance plateaued in
+peak performance beyond +8dB SNR and approached chance
+classification performance below −8dB SNR on the RadioML
+2018.01A dataset. This range is denoted by the shaded region.
+Harper et al. investigated methods to improve classification
+performance in this range by employing an SNR regression
+model to aid separate modulation classifiers (MCs). While
+other works have trained models to be as resilient as possible
+under varying SNR conditions, Harper et al. employed SNR-
+specific MCs [2].
+TABLE I
+SNR GROUPINGS FOR TRAINING SNR-SPECIFIC CLASSIFIERS AND
+DEMULTIPLEXED CLASSIFICATION RANGES FOR EACH PREDICTED SNR.
+Training Range (dB)
+Demultiplexed Classification Range (dB)
+[-20, -8]
+(−∞, -8)
+[-8, -4]
+[-8, -4)
+[-4, 0]
+[-4, 0)
+[0, 4]
+[0, 4)
+[4, 8]
+[4, 8)
+[8, 30]
+[8, ∞)
+Six MCs were created by discretizing the SNR range to
+ameliorate performance between −8dB to +8dB SNR (see
+Figure 5). These groupings were chosen in order to provide
+sufficient training data to avoid overfitting the MCs and
+provide enough resolution so that combining MCs provided
+more value than a single classifier.
+By first predicting the SNR of the received signal with
+a regression model, an SNR-specific MC that was trained
+on signals with the predicted SNR is applied to make the
+final prediction. Although the SNR values in the dataset
+are discrete, SNR is measured on a continuous scale in a
+deployment scenario and can vary over time. As a result,
+regression is used over classification to model SNR. Using this
+approach, different classifiers can tune their feature processing
+for differing SNR ranges. Each MC in this approach uses the
+same architecture as that proposed in [7]; however, each MC
+is trained with signals within each MC’s SNR training range
+(see Table I).
+Highlighting improvements across varying SNR values, Fig-
+ure 6 shows the overall performance improvement (in percent-
+age accuracy) using the SNR-assisted architecture compared to
+the baseline classification architecture described in [7]. While
+a slight decrease in performance was observed for −8dB and
+a larger decrease for −2dB, improvement is shown under most
+SNR conditions—particularly in the target range of −8dB to
++8dB. A possible explanation for the decrease in performance
+at particular SNRs is that the optimization for a particular
+MC helped overall performance for a grouping at the expense
+of a single value in the group. That is, the MC for [−4, 0)
+Fig. 5. The architecture using SNR regression and SNR-specific classifiers
+from [2]. Each MC block shown employs the same architecture as the baseline
+from [7], but specifically trained to perform AMC within a more narrow range
+of SNRs (denoted as dB ranges in each block).
+boosted the overall performance by performing well at −4
+and 0dB at the expense of −2dB. Due to the large size of
+the testing set, these small percentage gains are impactful
+because thousands more classifications are correct. All results
+are statistically significant based on a McNemar’s test [28],
+therefore achieving new state-of-the-art performance at the
+time.
+Soltani et al. [3] found SNR regions of [−10, −2]dB,
+[0, 8]dB, and [10, 30]dB having similar classification patterns.
+Instead of predicting exact modulation variants, the authors
+group commonly confused variants into a more generic,
+coarse-grained label. This grouping increases performance of
+AMC by combining modulation variants that are commonly
+confused. However, it also decreases the sensitivity of the
+model to the numerous possible variants.
+Cai et al. utilized a transformer based architecture to aid
+performance at low SNR levels with relatively few training pa-
+rameters (approximately 265,0000 parameters) [29]. A multi-
+scale network along with center loss [30] was used in [31].
+It was found that larger kernel sizes improved AMC perfor-
+mance. We further explore kernel size performance impacts
+in this work. Zhang et al. proposed a high-order attention
+mechanism using the covariance matrix achieving a maximum
+accuracy of 95.49% [32].
+Although many discussed works use the same RadioML
+2018.01A dataset, there is a lack of a uniform dataset split
+to establish a benchmark for papers to report performance.
+In an effort to make AMC work more reproducible and
+comparable across publications, we have made our dataset split
+and accompanying code available on GitHub.1
+While numerous works have investigated architectural im-
+provements, we aim to improve upon these works by intro-
+ducing additional modifications as well as a comprehensive
+ablation study that illustrates the improvement of each mod-
+ification. With the new modifications, we achieve new state-
+of-the-art AMC performance.
+III. DATASET
+To evaluate different machine learning architectures, we
+use the RadioML 2018.01A dataset that is comprised of 24
+1https://github.com/caharper/Automatic-Modulation-Classification-with-
+Deep-Neural-Networks
+
+SNR Regression
+Model
+DEMUX
+MC
+MC
+MC
+MC
+MC
+MC
+(-8, -8)
+[-8, -4)
+[-4, 0]
+[0, 4]
+[4, 8]
+(8, 8)5
+Fig. 6. Summary of residual improvement in accuracy over [7] that was first
+published in [2]. This work showed how the baseline architecture could be
+tuned to specific SNR ranges. Positive improvement is observed for most SNR
+ranges.
+different modulation types [1], [15]. Due to the complexity
+and variety of modulation schemes in the dataset, it is fairly
+representative of typically encountered modulation schemes.
+Moreover, this variety increases the likelihood that AMC
+models will generalize to more exotic or non-existing modu-
+lation schemes in the training data that are derived from these
+traditional variants.
+There are a total of 2.56 million labeled signals, S(T),
+each consisting of 1024 time domain digitized intermediate
+frequency (IF) samples of in-phase (I) and quadrature (Q)
+signal components where S(T) = I(T) + jQ(T). The data
+was collected at a 900MHz IF with an assumed sampling
+rate of 1MS/sec such that each 1024 time domain digitized
+I/Q sample is 1.024 ms [33]. The 24 modulation types and
+the representative groups that we chose for each are listed as
+follows:
+• Amplitude: OOK, 4ASK, 8ASK, AM-SSB-SC, AM-
+SSB-WC, AM-DSB-WC, and AM-DSB-SC
+• Phase: BPSK, QPSK, 8PSK, 16PSK, 32PSK, and
+OQPSK
+• Amplitude and Phase: 16APSK, 32APSK, 64APSK,
+128APSK, 16QAM, 32QAM, 64QAM, 128QAM, and
+256QAM
+• Frequency: FM and GMSK
+Each modulation type includes a total of 106, 496 obser-
+vations ranging from −20dB to +30dB SNR in 2dB steps
+for a total of 26 different SNR values. SNR is assumed
+to be consistent over the same window length as the I/Q
+sample window. For evaluation, we divided the dataset into 1
+million different training observations and 1.5 million testing
+observations under a random shuffle split, stratified across
+modulation type and SNR. Because of this balance, the
+expected performance for a random chance classifier is 1/24
+or 4.2%. With varying SNR levels across the dataset, it is
+expected that the classifier would perform with a higher degree
+of accuracy as the SNR value is increased. For consistency,
+each model investigated in this work was trained and evaluated
+on the same train and test set splits.
+IV. INITIAL INVESTIGATION
+In this work, we use the architecture described in [7] as
+the baseline architecture. We note that [2] improved upon the
+baseline; however, each individual MC used the baseline archi-
+tecture except trained on specific SNR ranges. Therefore, the
+base architectural elements were similar to [7], but separated
+for different SNRs. In this work, our focus is to improve upon
+the employed CNN architecture for an individual MC rather
+than the use of several MCs. Therefore, we use the architecture
+from [7] as our baseline.
+Before exploring an ablation study, we make a few notable
+changes from the baseline architecture in an effort to increase
+AMC performance. This initial exploration is for clarity as
+it reserves the ablation study that follows from requiring an
+inordinate number of models. It also introduces the general
+training procedures that assist and orient the reader in fol-
+lowing the ablation study—the ablation study mirrors these
+procedures. We first provide an initial investigation exploring
+these notable changes.
+We train each model using the Adam optimizer [34] with
+an initial learning rate lr = 0.0001, a decay factor of 0.1 if
+the validation loss does not decrease for 12 epochs, and a
+minimum learning rate of 1e-7. If the validation loss does not
+decrease after 20 epochs, training is terminated and the models
+are deemed converged. For all experiments, mini-batches of
+size 32 are used. As has been established in most programming
+packages for neural networks, we refer to fully connected
+neural network layers as dense layers, which are typically
+followed by an activation function.
+A. Architectural Changes
+A common property of neural networks is using fewer but
+larger kernels in the early layers of the network, and an
+increase of smaller kernels are used in the later layers than
+the baseline architecture. This is commonly referred to as the
+information distillation pipeline [35]. By utilizing a smaller
+number of large kernels in early layers, we are able to increase
+the temporal context of the convolutional features without
+dramatically increasing the number of trainable parameters.
+Numerous, but smaller kernels are used in later convolu-
+tional layers to create more abstract features. Configuring
+the network in this manner is especially popular in image
+classification where later layers represent more abstract, class-
+specific features.
+We investigate this modification in three stages, using the
+baseline architecture described in Figure 3 [7]. We denote
+number of filters in the network and the filter sizes as
+F = [f1, f2, ..., f7] and K = [k1, k2, ...k7] in Figure 3. The
+baseline architecture used f = 64 (for all layers) and k = 3
+(consistent kernel size for all layers). Our first modification to
+the baseline architecture is F = [32, 48, 64, 72, 84, 96, 108],
+but keeping k = 3 for all layers. Second, we use the baseline
+architecture, but change the size of filters in the network where
+f = 64 (same as baseline) and K = [7, 5, 7, 5, 3, 3, 3]. Third,
+we make both modifications and compare the result to the
+baseline model where F = [32, 48, 64, 72, 84, 96, 108] and
+K = [7, 5, 7, 5, 3, 3, 3]. These modifications are not exhaustive
+
+0.318
+0.306
+0.3 -
+0.2600.259
+Residual Improvement (0-100%)
+0.235
+0.229
+0.222
+0.216
+0.2
+0.192
+0.172
+0.170
+0.165
+0.1540.157
+0.142
+0.149
+0.134
+0.124
+0.1
+0.020
+0.008
+0.0
+-0.012 -0.008
+-0.008
+-0.060
+-0.1
+0.094
+-0.111
+20-18-16-14-12-10
+-8
+-6
+-4
+0
+2
+4
+6
+8
+10
+12
+16
+20
+22
+24
+26
+2830
+SNR (dB)6
+searches; rather, these modifications are meant to guide future
+changes to the network by understanding the influence of filter
+quantity and filter size in a limited context.
+TABLE II
+INITIAL INVESTIGATION PERFORMANCE OVERVIEW. ALL
+ARCHITECTURES EMPLOY THE BASELINE WITH VARYING NUMBERS OF
+KERNELS AND KERNEL SIZES.
+Notes
+# Params
+Avg.
+Accuracy
+Max
+Accuracy
+Reproduced ResNet [1]
+165,144
+59.2%
+93.7%
+X-Vector in [7]
+110,680
+61.3%
+98.0%
+More Filters
+(Same Filter Sizes)
+149,168
+61.0%
+96.1%
+Larger Filter Sizes
+(Same # Filters)
+143,960
+62.6%
+98.2%
+Combined
+174,000
+62.9%
+98.6%
+B. Initial Investigation Results
+As shown in Table II, increasing the size of the filters
+in earlier layers increases both average and maximum test
+accuracy over [7]; but, at the cost of additional parameters.
+A possible explanation for the increase in performance is the
+increase in temporal context due to the larger kernel sizes.
+Increasing the number of filters without increasing temporal
+context decreases performance. This is possibly because it in-
+creases the complexity of the model without adding additional
+signal context.
+Fig. 7.
+SNR vs. accuracy comparison of the initial investigation using the
+baseline architecture. Noticeable improvements can be observed across all
+SNRs.
+Figure 7 illustrates the change in accuracy with varying
+SNR. The combined model, utilizing various kernel sizes
+and numbers of filters, consistently outperforms the other
+architectures across changing SNR conditions.
+Although increasing the number of filters decreases per-
+formance alone, combining the approach with larger kernel
+sizes yields the best performance in our initial investigation.
+Increasing the temporal context may have allowed additional
+filters to better characterize the input signal.
+Because increased temporal context improves AMC perfor-
+mance, we are inspired to investigate additional methods such
+as squeeze-and-excitation blocks and dilated convolutions that
+can increase global and local context [25], [36].
+V. ABLATION STUDY ARCHITECTURE BACKGROUND
+Building upon our findings from our initial investigation,
+we make additional modifications to the baseline architecture.
+For the MCs, we introduce dilated convolutions, squeeze-
+and-excitation blocks, self-attention, and other architectural
+changes. We also investigate various kernel sizes and the
+quantity of kernels employed from the initial investigation.
+Our goal is to improve upon existing architectures while
+investigating the impact of each modification on classification
+accuracy through an ablation study. In this section, we describe
+each modification performed.
+A. Squeeze-and-Excitation Networks
+Fig. 8.
+Squeeze-and-Excitation block proposed in [25]. One SE block is
+shown applied to a single layer convolutional output activation. Two paths
+are shown, a scaling path and an identity path. The scaling vector is applied
+across channels to the identity path of the activations.
+Squeeze-and-Excitation (SE) blocks introduce a channel-
+wise attention mechanism first proposed in [25]. Due to the
+limited receptive field of each convolutional filter, SE blocks
+propose a recalibration step based on global statistics across
+channels (average pooling) to provide global context. Although
+initially utilized for image classification tasks [25], [37], [38],
+we argue the use of SE blocks can provide meaningful global
+context to the convolutional network used for AMC over the
+time domain.
+Figure 8 depicts an SE block. The squeeze operation is de-
+fined as temporal global average pooling across convolutional
+filters. For an individual channel, c, the squeeze operation is
+defined as:
+zc = Fsq(xc) = 1
+T
+T
+�
+i=1
+xi,c
+(1)
+where X
+∈
+RT ×C
+=
+[x1, x2, ..., xC], Z
+∈
+R1×C
+=
+[z1, z2, ..., zC], T is the number of samples in time, and C is
+the total number of channels. To model nonlinear interactions
+between channel-wise statistics, Z is fed into a series of dense
+layers followed by nonlinear activation functions:
+s = Fex(z, W) = σ(g(z, W)) = σ(W2δ(W1z))
+(2)
+where δ is the rectified linear (ReLU) activation function,
+W1 ∈ R
+C
+r ×C, W2 ∈ RC× C
+r , r is a dimensionality reduction
+ratio, and σ is the sigmoid activation function. The sigmoid
+function is chosen as opposed to the softmax function so that
+multiple channels can be accentuated and are not mutually-
+exclusive. That is, the normalization term in the softmax
+can cause dependencies among channels, so the sigmoid
+activation is preferred. W1 imposes a bottleneck to improve
+generalization performance and reduce parameter counts while
+W2 increases the dimensionality back to the original number
+of channels for the recalibration operation. In our work, we
+
+0.8
+2
+0.6
+Accurac
+0.4
+0.2
+0
+-20
+-10
+0
+10
+20
+30
+SNR (dB)
+Model - X-Vector Model from [7] - More Filters (Same Filter Sizes)
+- Larger Filter Sizes (Same # Filters) → Combined - - Random ChanceX
+Fex(·, W)
+Time (T)
+Fsq(·)
+1 ×C
+1 × C
+Channels (C)
+Fscale(·, ·)
+T×C
+T×C7
+Fig. 9. Proposed architecture with modifications including SENets, dilated convolutions, optional ReLU activation before statistics pooling, and self-attention.
+The output tensor sizes are also shown for each unit in the diagram. An * denotes where the sizes differ from the baseline architecture.
+use r = 2 for all SE blocks to ensure a reasonable number
+of trainable parameters without over-squashing the embedding
+size.
+The final operation in the SE block, scaling or recalibration,
+is obtained by scaling the the input X by s:
+ˆxc = Fscale(xc, sc) = scxc
+(3)
+where ˆX ∈ RT ×C = [ ˆx1, ˆx2, ..., ˆ
+xC].
+B. Dilated Convolutions
+Fig. 10.
+Dilated convolutions diagram. The top shows a traditional kernel
+applied to sequential time series points. The middle and bottom diagram
+illustrate dilation rates of two and three, respectively. These dilations serve
+to increase the receptive field of the filter without increasing the number of
+trainable variables in the kernel.
+Proposed in [36], Figure 10 depicts dilated convolutions
+where the convolutional kernels are denoted by the colored
+components. In a traditional convolution, the dilation rate
+is equal to 1. Dilated convolutions build temporal context
+by increasing the receptive field of the convolutional kernels
+without increasing parameter counts as the number of entries
+in the kernel remains the same.
+Dilated convolutions also do not downsample the signals
+like strided convolutions. Instead, the output of a dilated
+convolution can be the exact size of the input after properly
+handling edge effects at the beginning and end of the signal.
+C. Final Convolutional Activation
+We also investigate the impact of using an activation func-
+tion (ReLU) after the last convolutional layer, just before
+statistics pooling. Because ReLU transforms the input se-
+quence to be non-negative, the distribution characterized by
+the pooling statistics may become skewed. In [7] and [2],
+no activation was applied after the final convolutional layer
+as shown in Figure 3. We investigate if this transformation
+impacts classification performance.
+D. Self-Attention
+Self-attention allows the convolutional outputs to interact
+with one another enabling the network to learn to focus on
+important outputs. Self-attention before statistics pooling es-
+sentially creates a weighted summation over the convolutional
+outputs weighting their importance similarly to [39]–[41].
+We use the attention mechanism described by Vaswani et
+al. in [42] where each output element is a weighted sum of
+the linearly transformed input where the dimensionality of K
+is dk as seen in Equation (4).
+Attention(Q, K, V ) = softmax
+� QKT
+|√dk|
+�
+V
+(4)
+In the case of self-attention, Q, K, and V are equal. A scaling
+factor of
+1
+|√dk| is applied to counteract vanishing gradients in
+the softmax output when dk is large.
+VI. ABLATION STUDY ARCHITECTURE
+Applying the specified modifications to the architecture in
+[7], Figure 9 illustrates the proposed architecture with every
+modification included in the graphic. Each colored block
+represents an optional change to the architecture that will be
+investigated in the ablation study. That is, each combination
+of network modifications are analyzed to aid understanding of
+each modification’s impact on the network.
+Each convolutional layer has the following parameters:
+number of filters, kernel size, and dilation rate. The asterisk
+next to each dilation rate represents the changing of dilation
+rates in the ablation study. If dilated convolutions are used,
+
+Time
+Dilation rate = 1
+Dilation rate = 2
+Dilation rate = 3Input +
+Batch size × 1024 x 2
+1 Batch size × 1024 × 64
+1 Batch size x 1024 x 84
+IBatch size x 1024 x 108
+Conv1D + ReLU (32, 7, 1*)
+SE Block
+Conv1D + ReLU (96, 3, 2*)
+Statistics Pooling
++ Batch size × 1024 x 32
+I Batch size x 1024 × 64
+↓ Batch size × 1024 x 96
+Batch size x 216
+Conv1D (number of filters,
+SE Block
+Conv1D + ReLU (72, 5, 2*)
+SE Block
+Dense + SeLU (128)
+filter size, dilation rate)
+ Batch size × 1024 × 32
+I Batch size x 1024 ×72
+I Batch size x 1024 × 96
+Batch size × 128
+Equals 1 for the
+initial investigation
+Conv1D + ReLU (48, 5, 2*)
+SE Block
+Conv1D (108, 3, 1*)
+Dense + SeLU (128)
+Not included in the
+initial investigation
+Batch size × 1024 × 48
+I Batch size x 1024 x 72
+ Batch size x 1024 × 108
+Batch size x 128
+SE Block
+Conv1D + ReLU (84, 3, 2*)
+ReLU
+Dense + Softmax (24)
++ Batch size x 1024 × 48
+I Batch size x 1024 × 84
+I Batch size × 1024 × 108
+Batch size x 24
+Conv1D + ReLU (64, 7, 3*)
+SE Block
+Self-Attention
+Prediction8
+TABLE III
+ABLATION STUDY PERFORMANCE OVERVIEW.
+Model Name
+Notes
+SENet
+Dilated
+Convolutions
+Final
+Activation
+Attention
+# Params
+Avg.
+Accuracy
+Max
+Accuracy
+—
+Reproduced ResNet [1]
+—
+—
+—
+—
+165,144
+59.2%
+93.7%
+—
+X-Vector in [7]
+—
+—
+—
+—
+110,680
+61.3%
+98.0%
+0000
+Best performing model from
+the initial investigation
+—
+—
+—
+—
+174,000
+62.9%
+98.6%
+0001
+—
+—
+—
+
+221,088
+62.3%
+97.6%
+0010
+—
+—
+
+—
+174,000
+62.8%
+98.6%
+0011
+—
+—
+
+
+221,088
+62.3%
+97.5%
+0100
+—
+
+—
+—
+174,000
+63.2%
+98.9%
+0101
+—
+
+—
+
+221,088
+63.1%
+97.9%
+0110
+—
+
+
+—
+174,000
+63.2%
+98.9%
+0111
+—
+
+
+
+221,088
+63.0%
+98.0%
+1000
+
+—
+—
+—
+202,880
+62.9%
+98.5%
+1001
+
+—
+—
+
+249,968
+62.6%
+98.2%
+1010
+
+—
+
+—
+202,880
+62.6%
+98.3%
+1011
+
+—
+
+
+249,968
+62.8%
+98.1%
+1100
+
+
+—
+—
+202,880
+62.8%
+98.2%
+1101
+
+
+—
+
+249,968
+63.0%
+97.7%
+1110
+Overall best performing model
+
+
+
+—
+202,880
+63.7%
+98.9%
+1111
+
+
+
+
+249,968
+63.0%
+97.8%
+then the dilation rate value in the graphic is used. If dilated
+convolutions are not used, each dilation rate is set to 1. That
+is, a traditional convolution is applied. All convolutions use a
+stride of 1, and the same training procedure from the initial
+investigation is used.
+VII. EVALUATION METRICS
+We present several evaluation metrics to compare the dif-
+ferent architectures considered in the ablation study. In this
+section, we will discuss each evaluation technique used in the
+results section.
+Due to the varying levels of SNRs in the employed dataset,
+we plot classification accuracy over each true SNR value. This
+allows for a visualization of the tradeoff in performance as
+noise becomes more or less dominant in the received signals.
+Additionally, we report average accuracy and maximum ac-
+curacy across the entire test set for each model. While we
+note that average accuracy is not indicative of the model’s
+performance, as accuracy is highly correlated to the SNR of
+the input signal, we share this result to give other researchers
+the ability to reproduce and compare works.
+As discussed in [26], AMC is often implemented on
+resource-constrained devices. In these systems, using larger
+models in terms of parameter counts may not be feasible. We
+report the number of parameters for each model in the ablation
+study to examine the tradeoff in AMC performance and model
+size.
+Additional analyses are also carried out. However, due to
+the large number of models investigated in this study, we
+will select the best performing model from the ablation study
+for brevity and analyze the performance of this model in
+greater detail. For example, confusion matrices for the best
+performing model from the ablation study are provided to
+show common misclassifications for each modulation type.
+Additionally, there exist several use-cases where relatively
+short signal bursts are received. For example, a wide-band
+scanning receiver may only detect a short signal burst. There-
+fore, signal duration in the time domain versus AMC perfor-
+mance is investigated to determine the robustness of the best
+performing model when short signal bursts are received.
+VIII. ABLATION RESULTS
+A. Overall Performance
+Table III lists the maximum and average accuracy perfor-
+mance for each model in the ablation study. A binary naming
+convention is used to indicate the various methods used for
+each architecture. Similarly to the result found in Section IV,
+increasing the temporal context typically results in increased
+performance. Models that incorporate dilated convolutions
+tended to have higher average accuracies than models without
+dilated convolutions.
+The best performing model, in terms of average accuracy
+across all SNR conditions included SE blocks, dilated convolu-
+tions, and a ReLU activation prior to statistics pooling (model
+1110) with an average accuracy of approximately 63.7%. This
+model also achieved the highest maximum accuracy of about
+98.9% at a 22dB level.
+SE blocks did not increase performance compared to model
+0000 with the exception of models 1110 and 1111. However,
+SE blocks were incorporated in the best performing model,
+1110. Self-attention was not found to aid classification perfor-
+mance in general with the proposed architecture. Self-attention
+introduces a large number of trainable parameters possibly
+forming a complex loss space.
+Table IV lists the performances of single modification (from
+baseline) architectures. Each component of the ablation study,
+with the exception of dilated convolutions, decreased perfor-
+mance when applied individually. When combined, however,
+the best performing model was found. Therefore, we conclude
+that each component could possibly aid the optimization of
+
+9
+Fig. 11. Ablation study results in terms of classification accuracy across SNR ranges. The best performing model is in the second to last row and displays
+strong performance across SNR values.
+TABLE IV
+INDIVIDUAL NETWORK MODIFICATION PERFORMANCE OVERVIEW.
+ENTRIES ARE REPEATED FROM TABLE III FOR CLARITY.
+Model Name
+Notes
+SENet
+Dilated
+Convolutions
+Final
+Activation
+Attention
+# Params
+Avg.
+Accuracy
+Max
+Accuracy
+—
+X-Vector in [7]
+—
+—
+—
+—
+110,680
+61.3%
+98.0%
+0000
+—
+—
+—
+—
+174,000
+62.9%
+98.6%
+0001
+—
+—
+—
+
+221,088
+62.3%
+97.6%
+0010
+—
+—
+
+—
+174,000
+62.8%
+98.6%
+0100
+—
+
+—
+—
+174,000
+63.2%
+98.9%
+1000
+
+—
+—
+—
+202,880
+62.9%
+98.5%
+1110
+Best performer
+
+
+
+—
+202,880
+63.7%
+98.9%
+each other—and, in general, dilated convolutions tend to have
+the most dramatic performance increases.
+B. Accuracy Over Varying SNR
+Figure 11 summarizes the ablation study in terms of classi-
+fication accuracy over varying SNR levels. We add this figure
+for completeness and reproducibility for other researchers.
+The accuracy within each SNR band is shown along with the
+modifications used, similar to Table III. The coloring in the
+figure denotes the accuracy in each SNR band. Performance
+follows a trend similar to that of a sigmoid function, where
+the rate at which peak classification accuracy is achieved is
+the most distinguishing feature between the different models.
+With the improved architectures, a maximum of 99% accuracy
+is achieved at high SNR levels (starting around 12dB SNR).
+While the proposed changes to the architectures gener-
+ally improve performance at higher SNR levels, the largest
+improvements occur between −12dB and 12dB compared
+to the baseline model in [7]. For example, at 4dB, the
+performance increases from 75% up to 82%. Incorporating
+these modifications to the network may prove to be critical
+in real-world situations where noisy signals are likely to be
+obtained. Improving AMC performance at lower SNR ranges
+(< −12dB) is still an open research topic, with accuracies
+near chance level.
+One observation is the best performing model can vary with
+SNR. In systems that have available memory and processing
+power, an approach similar to [2] may be used to utilize several
+models and intelligently chose predictions based on estimated
+SNR conditions. That is, if the SNR of the signal of interest is
+known, a model can be tuned to increase performance slightly,
+as shown in [2]. Using the results presented here, researchers
+could also choose the architecture differences that perform best
+for a given SNR range (although performance differences are
+subtle).
+C. Parameter Count Tradeoff
+Fig. 12.
+Ablation study parameter count tradeoff. The x-axis shows the
+number of trainable variables in each model and the y-axis shows max or
+average accuracy. The callout for each point denotes the model name as shown
+in Table III.
+An overview of each model’s complexity and overall per-
+formance across the entire testing set is shown in Table III.
+This information is also shown graphically in Figure 12 for
+the maximum accuracy over SNR and the average accuracy
+across all SNRs. Whether looking at the maximum or the
+average measures of performance, the conclusions are similar.
+The previously described binary model name also appears in
+the figure. We found a slight correlation between the number
+of model parameters and overall model performance; however,
+with the architectures explored, there was a general parameter
+count where performance peaked. Models with parameter
+counts between approximately 170k to 205k generally per-
+formed better than smaller and larger models. We note that the
+
+Dilated
+Final
+ Model Name
+Notes
+SENet
+Activation
+Attention
+ Modulation Classification Results
+Convolutions
+一
+Reproduced ResNet [1]
+-
+-
+一
+0.04
+15
+0.93
+0.94
+0.94
+0.94
+0.93
+-
+X-Vector in [7]
+-
+-
+-
+-
+0.04
+0.91
+0.94
+0.97
+0.98
+0.98
+0.98
+0.98
+0.98
+0.98
+0.98
+0.98
+000
+-
+-
+-
+-
+一
+0.04
+0.95
+0.97
+0.98
+0.98
+86°0
+66°0
+66°0
+66°0
+66°0
+0.99
+0.99
+0001
+-
+一
+-
+-
+0.0
+0.94
+0.97
+0.97
+0.97
+0.97
+0.98
+0.98
+0.98
+0.98 
+0010
+一
+-
+V
+0.98
+0.98
+0.99
+0.99
+0011
+-
+V
+0.89
+0.94
+0.96
+0.97
+0.97
+0.97
+0.97
+0.97
+0.97
+0.97
+0100
+一
+V
+-
+一
+0.0
+0.92
+0.97
+0.98
+0.99
+0.99
+0.99
+0.99
+0.99
+0.99
+0.99
+0.99
+0.99
+0.99
+0101
+一
+-
+一
+0.04
+0.92
+0.97
+0.98
+0.98
+0.98
+0.98
+0.98
+0110
+-
+-
+0.04
+0.97
+0.99
+66:0
+0.99
+0.99
+0.99
+0.99
+0111
+-
+一
+V
+0.04
+0.97
+0.98
+0.98
+0.98
+0.98
+0.98
+0.98
+0.98
+1000
+一
+v
+-
+-
+一
+0.04
+0.98
+0.98
+0.98
+0.98
+0.98
+1001
+-
+v
+-
+-
+0.04
+0.98
+1010
+-
+-
+v
+-
+0.98
+0.98
+0.98
+0.98
+0.98
+0.98
+1011
+-
+v
+-
+v
+v
+0.98
+0.98
+0.96
+0.98
+0.98
+0.98
+0.98
+0.98
+0.98
+0.98
+1100
+-
+v
+-
+一
+0.04
+0.91
+0.96
+0.97
+0.98
+0.98
+0.98
+86°0
+0.98
+0.98
+0.98
+0.98
+0.98
+0.98
+1101
+-
+-
+0.04
+0.81
+0.91
+0.95
+0.97
+0.97
+0.98
+86°0
+0.98
+86°0
+0.98
+0.98
+0.98
+0.98
+0.98
+1110
+Overall best performing model
+V
+0.28
+0.36
+0.46
+0.58
+0.69
+0.82
+0.92
+0.97
+0.98
+0.99
+66°0
+66°0
+66°0
+66°0
+66°0
+66°0
+0.99
+0.99
+66:0
+1111
+-
+0.81
+0.91
+0.96
+0.97
+0.97
+0.98
+0.98
+0.98
+86°0
+0.98
+0.98
+0.98
+0.98
+20
+-18
+-16
+-14
+-12
+-10
+-6
+-2
+10
+12
+14
+16
+18
+20
+22
+24
+4
+6
+8
+28
+30
+SNR (dB)1.00
+Overall Accuracy
+_0110
+0100-
+←1110
+1000→
+Max Accuracy
+0.98
+LLO
+0101
+X-Vector Model from [7]
+0001
+0011
+0.96
+0.94
+%)
+Reproduced ResNet from [1]
+acy
+0.92
+ra
+0.64
+iccur
+^—1110
+0100-
+→-0110
+0101
+1101.
+1000+1100
+toiii
+0.63
+1111
+A
+0000-
+—0010
+←1011
+-i010
+1001→
+0.62
+X-Vector Model from [7]
+0.61
+0.60
+Reproduced ResNet from [1]
+0.59
+# Params10
+(a)
+(b)
+(c)
+Fig. 13. Accuracy over varying SNR conditions for model 1110 with (a), (b), and (c) showing the top-1, top-2, and top-5 accuracy respectively. Random
+chance for each is defined as 1/24, 2/24, and 5/24.
+models with more than 205k parameters included self-attention
+which was found to decrease model performance with the
+proposed architectures. This implies that one possible reason
+self-attention did not perform as well as other modifications
+is because of the increase in parameters, resulting in a more
+difficult loss space from which to optimize.
+IX. BEST PERFORMING MODEL INVESTIGATION
+Due to the large volume of models, we focus upon the
+best performing model, (model 1110), for the remainder of
+this work. As previously mentioned, this model employs all
+modifications except self-attention.
+A. Top-K Accuracy
+As discussed, in systems where the modulation schemes
+must be classified quickly, it is advantageous to apply fewer
+demodulation schemes in a trial and error fashion. This is
+particularly significant at lower SNR values where accuracy is
+mediocre. Top-k accuracy allows an in-depth view on the ex-
+pected number of trials before finding the correct modulation
+scheme. Although traditional accuracy (top-1 accuracy) char-
+acterizes the performance of the model in terms of classifying
+the exact variant, top-k accuracy characterizes the percentage
+of the classifier predicting the correct variant among the top-
+k predictions (sorted by descending class probabilities). We
+plot the top-1, top-2, and top-5 classification accuracy over
+varying SNR conditions for each modulation grouping defined
+in Section III in Figure 13.
+Although performance decays to approximately random
+chance for the overall (all modulation schemes) performance
+curves for each top-k accuracy, it is notable that some modu-
+lation group performances drop below random chance. The
+models are trained to maximize the overall model perfor-
+mance. This could explain why certain modulation groups dip
+below random chance but the overall performance and other
+modulation groups remain at or above random chance.
+Using the proposed method greatly reduces the correct
+modulation scheme search space. While high performance in
+top-1 accuracy is increasingly difficult to achieve with low
+SNR signals, top-2 and top-5 accuracy converge to higher
+values at a much faster rate. This indicates our proposed
+method greatly reduces the search space from 24 modulation
+candidates to fewer candidate types when employing trial and
+error methods to determine the correct modulation scheme.
+Further, if the group of modulation is known (e.g., FM), one
+can view a more specific tradeoff curve in terms of SNR and
+top-k accuracy given in Figure 13.
+B. Short Duration Signal Bursts
+Due to the rapid scanning characteristic of some modern
+software-defined radios, we investigate the performance trade-
+off of varying signal duration and AMC performance. This
+analysis is meant to emulate the situation wherein a receiver
+only detects a short RF signal burst. We investigate signal
+burst durations of 1.024 ms (full length signal from original
+dataset), 512 µs, 256 µs, 128 µs, 64 µs, 32 µs, and 16 µs.
+We assume the same 1MS/sec sampling rate as in the previous
+analyses such that 16 µs burst is captured in 16 I/Q samples.
+Fig. 14. Tradeoff in accuracy for various signal lengths across SNR, grouped
+by modulation category for the best performing model 1110. The top plot
+shows the baseline performance using the full sequence. Subsequent plots
+show the same information using increasingly smaller signal lengths for
+classification.
+In this section, we use the same test set as our other
+investigations; however, a uniformly random starting point is
+
+0.8
+0.6
+0.4
+Overall
+一
+ Amplitude
+←Phase
+- Amplitude and Phase
+0.2
+ Frequency
+- - Random Chance
+0
+-20
+-10
+0
+10
+20
+30
+SNR (dB)0.8
+0.6
+0.4
+0.2
+0
+-20
+-10
+0
+10
+20
+30
+SNR (dB)0.8
+0.6
+0.4
+0.2
+0
+-20
+-10
+0
+10
+20
+30
+SNR (dB)1.024 ms (n=1024)
+1
+0.8
+Overall
+0.6
+Amplitude
+Phase
+0.4
+Amplitude and Phase
+Frequency
+0.2
+Random Chance
+0
+-20
+-10
+0
+10
+20
+30
+512 μs (n=512)
+256 μs (n=256)
+1
+1
+Accuracy
+0.8
+0.8
+0.6
+0.6
+0.4
+0.4
+0.2 
+0.2
+0
+0
+-20
+-10
+0
+10
+20
+30
+-20
+-10
+0
+10
+20
+30
+128 μs (n=128)
+64 μs (n=64)
+1
+1
+0.8
+0.8
+0.6
+0.6
+0.4
+0.4
+0.2 
+0.2
+0
+0
+-20
+-10
+0
+10
+20
+30
+-20
+-10
+0
+10
+20
+30
+32 μs (n=32)
+16 μs (n=16)
+1
+1
+0.8
+0.8
+0.6
+0.6
+0.4
+0.4
+0.2
+0.2
+0
+0
+20
+-10
+0
+10
+20
+30
+-20
+0
+10
+20
+-10
+30
+SNR11
+(a)
+(b)
+(c)
+Fig. 15. Confusion matrices for (a) model 1110 (best performing model from this work), (b) the reproduced ResNet model from [1], and (c) the X-Vector
+inspired model from [19] with SNR ≥ 0dB.
+determined for each signal such that a contiguous sample of
+the desired duration, starting at the random point, is chosen.
+Thus, the chosen segment from a test set sample is randomly
+assigned.
+We also note that, although the sample length for the evalu-
+ation is changed, the best performing model is the same archi-
+tecture with the exact same trained weights because this model
+uses statistics pooling from the X-Vector inspired modification.
+A significant benefit to the X-Vector inspired architecture is
+its ability to handle variable-length inputs without the need
+of padding, retraining, or other network modifications. This
+is achieved by taking global statistics across convolutional
+channels producing a fixed-length vector, regardless of signal
+duration. Due to this flexibility, the same model (model 1110)
+weights are used for each duration experiment. This fact also
+emphasizes the desirability of using X-vector inspired AMC
+architectures for receivers that are deployed in an environment
+where short-burst and variable duration signals are anticipated
+to be present.
+For each signal duration in the time domain, we plot the
+overall classification accuracy over varying SNR conditions
+as well as the accuracy for each modulation grouping de-
+fined in Section III. Figure 14 demonstrates the tradeoff for
+various signal durations where n is the number of samples
+from the time domain I/Q signal. The first observation is,
+as we would expect, that classification performance degrades
+with decreased signal duration. For example, the maximum
+accuracy begins to degrade at 256 µs and is more noticeable
+at 128 µs. This is likely a result of using sample statistics
+that result in unstable or biased estimates for short signal
+lengths since the number of received signal data points are
+insufficient to characterize the sample statistics used during
+training. Random classification accuracy is approximately 4%
+and is shown in the black dotted line in Figure 14. Although
+classification performance decreases with decreased duration,
+we are still able to achieve significantly higher classification
+accuracy than random chance down to 16 µs of signal capture.
+FM (frequency modulation) signals were typically more
+resilient to noise interference than AM (amplitude modulation)
+and AM-PM (amplitude and phase modulation) signals in our
+AMC. This was observed across all signal burst durations and
+our top-k accuracy analysis. This behavior indicates that the
+performance of our AMC for short bursts, in the presence
+of increasing amounts of noise, is more robust for signals
+modulated by changes in the carrier frequency and is more
+sensitive to signals modulated by varying the carrier amplitude.
+We attribute this behavior to our AMC architecture, the
+architecture of the receiver, or a combination of both of the
+AMC and receiver.
+C. Confusion Matrices
+While classification accuracy provides a holistic view of
+model performance, it lacks the granularity to investigate
+where misclassifications are occurring. Confusion matrices are
+used to analyze the distribution of classifications for each given
+class. For each true label, the proportion of correctly classified
+samples is calculated along with the proportion of incorrect
+predictions for each opposing class. In this way, we can see
+which classes the model is struggling to distinguish from
+one another. A perfect classifier would be the identity matrix
+where the diagonal values represent the true class matches the
+predicted class. Each matrix value represents the percentage
+of classifications for the true label and each row sums to 1
+(100%).
+Figure 15 illustrates the class confusion matrices for SNR
+levels greater than or equal to 0dB for models 1110, the
+reproduced ResNet architecture from [1], and the baseline X-
+Vector architecture from [7] respectively. Shown in [7], the
+X-Vector architecture was able to distinguish PSK and AM-
+SSB variants to a higher degree and performed better overall
+than [1]. Both architectures struggled to differentiate QAM
+variants.
+Model 1110 improved upon these prior results for QAM
+signals and in general has higher diagonal components than
+the other architectures. This again supports a conclusion that
+model 1110 achieves a new state-of-the-art in AMC perfor-
+mance.
+X. CONCLUSION
+A comprehensive ablation study was carried out with regard
+to AMC architectural features using the extensive RadioML
+2018.01A dataset. This ablation study built upon a strong
+performance of a new baseline model that was also intro-
+duced in the initial investigation of this study. This initial
+investigation informed the design of a number of AMC ar-
+chitecture modifications—specifically, the use of X-Vectors,
+dilated convolutions, and SE blocks. With the combined
+
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+modifications, we achieved a new state-of-the-art in AMC
+performance. Among these modifications, dilated convolutions
+were found to be the most critical architectural feature for
+model performance. Self-attention was also investigated but
+was not found to increase performance—although increased
+temporal context improved upon prior works.
+REFERENCES
+[1] T. O’Shea, T. Roy, and T. C. Clancy, “Over-the-air deep learning based
+radio signal classification,” in IEEE Journal of Selected Topics in Signal
+Processing, vol. 12:1.
+IEEE, 2018, pp. 168–179.
+[2] C. A. Harper, A. Sinha, M. A. Thornton, and E. C. Larson, “SNR-
+boosted automatic modulation classification,” in 2021 55th Asilomar
+Conference on Signals, Systems, and Computers.
+IEEE, 2021, pp.
+372–375.
+[3] N. Soltani, K. Sankhe, S. Ioannidis, D. Jaisinghani, and K. Chowdhury,
+“Spectrum awareness at the edge: Modulation classification using smart-
+phones,” in 2019 IEEE International Symposium on Dynamic Spectrum
+Access Networks (DySPAN).
+IEEE, 2019, pp. 1–10.
+[4] S. Soliman and S.-Z. Hsue, “Signal classification using statistical mo-
+ments,” IEEE Transactions on Communications, vol. 40, no. 5, pp. 908–
+916, 1992.
+[5] A. Swami and B. Sadler, “Hierarchical digital modulation classification
+using cumulants,” IEEE Transactions on Communications, vol. 48, no. 3,
+pp. 416–429, 2000.
+[6] M. Abdelbar, W. H. Tranter, and T. Bose, “Cooperative cumulants-based
+modulation classification in distributed networks,” IEEE Transactions on
+Cognitive Communications and Networking, vol. 4, no. 3, pp. 446–461,
+2018.
+[7] C. A. Harper, L. Lyons, M. A. Thornton, and E. C. Larson, “Enhanced
+automatic modulation classification using deep convolutional latent
+space pooling,” in 2020 54th Asilomar Conference on Signals, Systems,
+and Computers, 2020.
+[8] T. O’Shea, J. Corgan, and T. Clancy, “Convolutional radio modulation
+recognition networks,” in International Conference on Engineering
+Applications of Neural Networks., 2016, p. 213–226.
+[9] T. Huynh-The, C.-H. Hua, J.-W. Kim, S.-H. Kim, and D.-S. Kim,
+“Exploiting a low-cost cnn with skip connection for robust automatic
+modulation classification,” in 2020 IEEE Wireless Communications and
+Networking Conference (WCNC), 2020, pp. 1–6.
+[10] S. Peng, H. Jiang, H. Wang, H. Alwageed, and Y. Yao, “Modulation
+classification using convolutional neural network based deep learning
+model,” in 2017 26th Wireless and Optical Communication Conference
+(WOCC), 2017, pp. 1–5.
+[11] S. Peng, H. Jiang, H. Wang, H. Alwageed, Y. Zhou, M. M. Sebdani,
+and Y. Yao, “Modulation classification based on signal constellation
+diagrams and deep learning,” IEEE Transactions on Neural Networks
+and Learning Systems, vol. 30, no. 3, pp. 718–727, 2019.
+[12] F. Wang, Y. Wang, and X. Chen, “Graphic constellations and DBN
+based automatic modulation classification,” in 2017 IEEE 85th Vehicular
+Technology Conference (VTC Spring), 2017, pp. 1–5.
+[13] K. Hassan, I. Dayoub, W. Hamouda, C. N. Nzeza, and M. Berbineau,
+“Blind digital modulation identification for spatially-correlated mimo
+systems,” IEEE Transactions on Wireless Communications, vol. 11,
+no. 2, pp. 683–693, 2012.
+[14] M. Abdelbar, B. Tranter, and T. Bose, “Cooperative modulation classi-
+fication of multiple signals in cognitive radio networks,” in 2014 IEEE
+International Conference on Communications (ICC), 2014, pp. 1483–
+1488.
+[15] DeepSig
+Incorporated,
+“RF
+datasets
+for
+machine
+learning,”
+https://www.deepsig.ai/datasets, 2018, accessed: 2021-
+04-29.
+[16] D. Liu, P. Wang, T. Wang, and T. Abdelzaher, “Self-contrastive learning
+based semi-supervised radio modulation classification,” in MILCOM
+2021-2021 IEEE Military Communications Conference (MILCOM).
+IEEE, 2021, pp. 777–782.
+[17] Z. Chen, H. Cui, J. Xiang, K. Qiu, L. Huang, S. Zheng, S. Chen,
+Q. Xuan, and X. Yang, “Signet: A novel deep learning framework for
+radio signal classification,” IEEE Transactions on Cognitive Communi-
+cations and Networking, vol. 8, no. 2, pp. 529–541, 2022.
+[18] T. Chen, S. Gao, S. Zheng, S. Yu, Q. Xuan, C. Lou, and X. Yang, “Emd
+and vmd empowered deep learning for radio modulation recognition,”
+IEEE Transactions on Cognitive Communications and Networking, pp.
+1–1, 2022.
+[19] A. J. Uppal, M. Hegarty, W. Haftel, P. A. Sallee, H. B. Cribbs, and
+H. H. Huang, “High-performance deep learning classification for radio
+signals,” in 2019 53rd Asilomar Conference on Signals, Systems, and
+Computers, 2019.
+[20] C. Park, J. Choi, S. Nah, W. Jang, and D. Y. Kim, “Automatic modulation
+recognition of digital signals using wavelet features and SVM,” in 2008
+10th International Conference on Advanced Communication Technology,
+vol. 1, 2008, pp. 387–390.
+[21] X. Teng, P. Tian, and H. Yu, “Modulation classification based on spectral
+correlation and SVM,” in 2008 4th International Conference on Wireless
+Communications, Networking and Mobile Computing, 2008, pp. 1–4.
+[22] Z. Zhang, C. Wang, C. Gan, S. Sun, and M. Wang, “Automatic
+modulation classification using convolutional neural network with fea-
+tures fusion of SPWVD and BJD,” IEEE Transactions on Signal and
+Information Processing over Networks, vol. 5, no. 3, pp. 469–478, 2019.
+[23] S. Zheng, P. Qi, S. Chen, and X. Yang, “Fusion methods for CNN-based
+automatic modulation classification,” IEEE Access, vol. 7, pp. 66 496–
+66 504, 2019.
+[24] Y. Mao, Y.-Y. Dong, T. Sun, X. Rao, and C.-X. Dong, “Attentive siamese
+networks for automatic modulation classification based on multitiming
+constellation diagrams,” IEEE Transactions on Neural Networks and
+Learning Systems, 2021.
+[25] J. Hu, L. Shen, and G. Sun, “Squeeze-and-excitation networks,” in
+Proceedings of the IEEE conference on computer vision and pattern
+recognition, 2018, pp. 7132–7141.
+[26] S. Tridgell, “Low latency machine learning on fpgas,” Ph.D. dissertation,
+The University of Sydney Australia, 2019.
+[27] D. Snyder, D. Garcia-Romero, G. Sell, D. Povey, and S. Khudanpur,
+“X-vectors: Robust DNN embeddings for speaker recognition,” in 2018
+IEEE International Conference on Acoustics, Speech and Signal Pro-
+cessing (ICASSP).
+IEEE, 2018, pp. 5329–5333.
+[28] Q. McNemar, “Note on the sampling error of the difference between
+correlated proportions or percentages,” Psychometrika, vol. 12, no. 2,
+pp. 153–157, 1947.
+[29] J. Cai, F. Gan, X. Cao, and W. Liu, “Signal modulation classification
+based on the transformer network,” IEEE Transactions on Cognitive
+Communications and Networking, vol. 8, no. 3, pp. 1348–1357, 2022.
+[30] Y. Wen, K. Zhang, Z. Li, and Y. Qiao, “A discriminative feature
+learning approach for deep face recognition,” in European conference
+on computer vision.
+Springer, 2016, pp. 499–515.
+[31] H. Zhang, F. Zhou, Q. Wu, W. Wu, and R. Q. Hu, “A novel automatic
+modulation classification scheme based on multi-scale networks,” IEEE
+Transactions on Cognitive Communications and Networking, vol. 8,
+no. 1, pp. 97–110, 2022.
+[32] D. Zhang, Y. Lu, Y. Li, W. Ding, and B. Zhang, “High-order convo-
+lutional attention networks for automatic modulation classification in
+communication,” IEEE Transactions on Wireless Communications, pp.
+1–1, 2022.
+[33] T. J. O’Shea, J. Corgan, and T. C. Clancy, “Convolutional radio modula-
+tion recognition networks,” in International conference on engineering
+applications of neural networks.
+Springer, 2016, pp. 213–226.
+[34] D. P. Kingma and J. Ba, “Adam: A method for stochastic optimization,”
+CoRR, vol. abs/1412.6980, 2015.
+[35] F. Chollet, Deep learning with Python.
+Simon and Schuster, 2021.
+[36] F. Yu and V. Koltun, “Multi-scale context aggregation by dilated
+convolutions,” in International Conference on Learning Representations
+(ICLR), May 2016.
+[37] X. Zhang, X. Zhou, M. Lin, and J. Sun, “Shufflenet: An extremely effi-
+cient convolutional neural network for mobile devices,” in Proceedings
+of the IEEE conference on computer vision and pattern recognition,
+2018, pp. 6848–6856.
+[38] M. Tan and Q. Le, “Efficientnet: Rethinking model scaling for con-
+volutional neural networks,” in International conference on machine
+learning.
+PMLR, 2019, pp. 6105–6114.
+[39] K. Okabe, T. Koshinaka, and K. Shinoda, “Attentive Statistics Pooling
+for Deep Speaker Embedding,” in Proc. Interspeech 2018, 2018, pp.
+2252–2256.
+[40] P. Safari, M. India, and J. Hernando, “Self-attention encoding and
+pooling for speaker recognition,” in Proc. Interspeech 2020, 2020, pp.
+941–945.
+[41] G. Sammit, Z. Wu, Y. Wang, Z. Wu, A. Kamata, J. Nese, and E. C.
+Larson, “Automated prosody classification for oral reading fluency
+with quadratic kappa loss and attentive x-vectors,” in ICASSP 2022-
+2022 IEEE International Conference on Acoustics, Speech and Signal
+Processing (ICASSP).
+IEEE, 2022.
+
+13
+[42] A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez,
+Ł. Kaiser, and I. Polosukhin, “Attention is all you need,” Advances in
+neural information processing systems, vol. 30, 2017.
+Clayton A. Harper received his B.S. in mathematics
+and computer engineering in 2019 and M.S. in
+data engineering in 2021 from Southern Methodist
+University in Dallas, TX, where he specialized in
+machine learning and audio signal processing. His
+main research area is the analysis of time series
+signal processing in computer systems, especially
+pertaining to security and privacy. He is a student
+member of IEEE and is currently pursuing his Ph.D.
+in computer science at Southern Methodist Univer-
+sity with the co-advisors Dr. Eric C. Larson and Dr.
+Mitchell A. Thornton.
+Mitchell (Mitch) A. Thornton is currently the Cecil
+H. Green Chair of Engineering and Professor in
+the Department of Electrical and Computer Engi-
+neering at Southern Methodist University in Dallas,
+Texas. He also serves as the Executive Director of
+the Darwin Deason Institute for Cyber Security, a
+research-only unit, and as Program Director for the
+interdisciplinary M.S. in Data Engineering degree
+program within the Lyle School of Engineering at
+SMU. His main research interests are in the areas
+of cyber security and quantum informatics. His past
+industrial experience includes full-time employment at the Amoco Research
+Center, E-Systems, Inc. (now L3Harris Technologies Inc.), and the Cyrix
+Corporation. Dr. Thornton is a member of several professional and honor
+societies including the IEEE and the ACM where he is a senior member in
+each organization. He was elected as Chair of the IEEE Technical Community
+on Multiple-Valued Logic (TCMVL, 2010-11) and has served in various roles
+for other IEEE/ACM committees. He is an author or co-author of five books
+and more than 300 technical articles. He is a named inventor on over 20
+US/PCT/WIPO patents and patents pending. He holds P.E. licenses in the
+states of Texas, Mississippi and Arkansas. He received the Ph.D. in computer
+engineering from SMU in 1995, M.S. in computer science from SMU in 1993,
+M.S. in electrical engineering from the University of Texas at Arlington in
+1990, and B.S. in electrical engineering from Oklahoma State University in
+1985.
+Eric C. Larson is an Associate Professor in the de-
+partment of Computer Science in the Bobby B. Lyle
+School of Engineering, Southern Methodist Univer-
+sity. His main research interests are in machine
+learning, sensing, and signal / image processing
+for various applications, in particular, for healthcare
+and security applications. His work in both areas
+has been commercialized and he holds a variety of
+patents for sustainability sensing and mobile phone-
+based health sensing. Dr. Larson has authored one
+textbook and over 70 technical articles. He is active
+in signal processing education for computer scientists and is an active member
+of IEEE and the ACM. He received his Ph.D. in 2013 from the University
+of Washington, where he was co-advised by Shwetak N. Patel and Les Atlas.
+He received his B.S. and M.S. in Electrical Engineering in 2006 and 2008,
+respectively, at Oklahoma State University, where he was advised by Damon
+Chandler.
+

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+Mapi-Pro: An Energy Efficient Memory Mapping Technique
+for Intermittent Computing
+SATYAJASWANTH BADRI, MUKESH SAINI, and NEERAJ GOEL, Indian Institute of Technology,
+Ropar
+Battery-less technology evolved to replace battery usage in space, deep mines, and other environments to
+reduce cost and pollution. Non-volatile memory (NVM) based processors were explored for saving the system
+state during a power failure. Such devices have a small SRAM and large non-volatile memory. To make the
+system energy efficient, we need to use SRAM efficiently. So we must select some portions of the application
+and map them to either SRAM or FRAM. This paper proposes an ILP-based memory mapping technique for
+Intermittently powered IoT devices. Our proposed technique gives an optimal mapping choice that reduces
+the system’s Energy-Delay Product (EDP). We validated our system using a TI-based MSP430FR6989 and
+MSP430F5529 development boards. Our proposed memory configuration consumes 38.10% less EDP than
+the baseline configuration and 9.30% less EDP than the existing work under stable power. Our proposed
+configuration achieves 15.97% less EDP than the baseline configuration and 21.99% less EDP than the existing
+work under unstable power. This work supports intermittent computing and works efficiently during frequent
+power failures.
+Additional Key Words and Phrases: NVM, MSP430FR6989, ILP, Intermittent power, Memory-Mapping
+1
+INTRODUCTION
+The Internet of Things (IoT) is a network of sensors and nodes that allows nearby objects to
+communicate and collaborate easily. Batteries are the most common source of power for IoT devices.
+Because of the battery’s limited capacity and short lifespan [15], replacement is costly. IoT may
+consist of billions of sensors and systems by the end of 2050 [9]. Replacing and disposing billions
+of battery-operated devices is expensive and hazardous to the environment. As a result, we need
+battery-free IoT devices.
+Energy harvesters are a promising alternative to battery-powered devices. The energy harvester
+collects energy from the environment and stores energy in capacitors. Energy harvesting is un-
+reliable, power failures are unavoidable, and the application’s execution is irregular. This type of
+computing is known as intermittent computing [14, 27, 34].
+For intermittently powered IoT devices, energy harvesting is the primary energy source. Energy
+harvesting sources like piezo-electric materials and radio-frequency devices extract a small amount
+of energy from their surroundings. We must use energy efficiently in both stable and unstable
+power supply scenarios.
+In order to utilize energy efficiently and to make the system energy efficient, we primarily have
+two choices. The first choice is to reduce energy consumption by proposing new techniques that
+use energy efficiently. The second choice is to increase the number of different energy harvesters,
+which will accumulate more energy while increasing maintenance costs. We need to maintain
+these many energy harvesters, which is not a feasible solution. Thus, our main concern is to reduce
+energy consumption by proposing new techniques which help to design an energy-efficient system.
+Gonzalez et al. [10] mentioned energy as not an ideal metric for evaluating system efficiency. By
+simply reducing supply voltage or load capacitance, energy can be reduced. Instead of using energy
+as a metric, they suggested using the Energy-Delay Product (EDP) as the energy-efficient design
+Authors’ address: SatyaJaswanth Badri, [email protected]; Mukesh Saini, [email protected]; Neeraj Goel, neeraj@
+iitrpr.ac.in, Indian Institute of Technology, Ropar, S.Ramanujan Block, IIT Ropar Main Campus, Ropar, Punjab, India, 140001.
+arXiv:2301.11967v1  [cs.AR]  27 Jan 2023
+
+2
+S.J Badri, et al.
+metric. The EDP considers both performance and energy simultaneously in a design. If a design
+minimizes the EDP, we can call such a design energy-efficient. We define EDP in the equation 1.
+𝐸𝐷𝑃 = 𝐸𝑠𝑦𝑠𝑡𝑒𝑚 × 𝑁𝑢𝑚_𝑐𝑦𝑐𝑙𝑒𝑠
+(1)
+Where 𝐸𝑠𝑦𝑠𝑡𝑒𝑚 is the system’s energy consumption, 𝑁𝑢𝑚_𝑐𝑦𝑐𝑙𝑒𝑠 is the number of CPU cycles.
+During these frequent power failures, executing IoT applications becomes more difficult because
+all computed data may be lost, and the application’s execution must restart from the beginning.
+During power failures, we need an additional procedure to backup/checkpoint the volatile memory
+contents to non-volatile memory (NVM).
+Flash memory was the prior NVM technology used by modern microcontrollers at the main
+memory level, such as MSP430F5529 [24]. Flash is ineffective for frequent backups and checkpointing
+because its erase/write operations require a lot of energy. Emerging NVMs outperform flash,
+including spin-transfer-torque RAM (STT-RAM) [4, 28], phase-change memory (PCM) [25], resistive
+RAM (ReRAM), and ferroelectric RAM (FRAM) [16]. Previous works have been demonstrated by
+incorporating these emerging NVMs into low-power-based microcontrollers (MCUs) [16, 18, 24].
+Recent non-volatile processors (NVPs), such as the flash-based MSP430F5529 and the FRAM-based
+MSP430FR6989, encourage the use of hybrid main memory. The flash-based NVP, MSP430F5529, is
+made up of SRAM and flash, while the FRAM-based NVP, MSP430FR6989, is made up of SRAM
+and FRAM at the main memory level. The challenges associated with hybrid main memory-based
+NVPs, such as MSP430FR6989, are as follows.
+(1) FRAM consumes 2x times more energy and latency than SRAM. This design degrades system
+performance and consumes extra energy even during normal operations.
+(2) SRAM loses contents during a power failure and needs to execute the application from the
+beginning, which consumes extra energy and time. For large-size applications, this design
+will not be helpful. Anyway, using only SRAM performs better during regular operations.
+(3) We can design a hybrid main memory to get the benefits from both SRAM and FRAM. The
+following questions need to be answered and analyzed to use the hybrid main memory design.
+(a) How do we choose the appropriate sections of a program and map them to either SRAM
+or FRAM regions? A significant challenge is mapping a program’s stack, code, and data
+sections to either SRAM or FRAM.
+(b) How and where should volatile contents be backed up to the NVM region during frequent
+power failures?
+The main question is which section of an application should be placed in which memory region;
+this is essentially a memory mapping problem. Concerning all of the challenges mentioned earlier,
+this article makes the following contributions:
+• To the best of our knowledge, this is the first work on the Integer-Linear Programming (ILP)
+based memory mapping technique for intermittently powered IoT devices.
+• We formulated the memory mapping problem to cover all the possible design choices. We
+also formulated our problem in such a way that it supports large-size applications.
+• We proposed a framework that efficiently consumes low energy during regular operation
+and frequent power failures. Our proposed framework supports intermittent computing.
+• We evaluated the proposed techniques and frameworks in actual hardware boards.
+Our proposed ILP model recommends placing each section in either SRAM or FRAM. We com-
+pared the proposed memory configuration and techniques with the baseline memory configurations
+under both stable and unstable power scenarios. Our proposed memory configuration consumes
+38.10% less EDP than baseline-1 and 9.30% less EDP than the existing work under stable power.
+
+Mapi-Pro: An Energy Efficient Memory Mapping Technique for Intermittent Computing
+3
+Our proposed configuration achieves 15.97% less EDP than baseline-1 and 21.99% less EDP than the
+existing work under unstable power.
+Paper organization: Section 2 discusses the background and related works. Section 3 explains
+the motivation behind the proposed framework. Section 4 explains the system model and gives an
+overview of the problem definition. Section 5 explains about proposed ILP-based memory mapping
+technique and framework that supports during frequent power failures. The experimental setup
+and results are described in section 6. We conclude this work in section 7.
+2
+BACKGROUND AND RELATED WORKS
+SRAM and DRAM are used to design registers, caches, and main memory in traditional processors.
+For an intermittently aware design, we replace a regular processor’s volatile memory model
+with an NVM. STT-RAM, PCM, flash, and FRAM are all relatively new NVM technologies [4–
+6, 11, 17, 25, 28, 29, 31, 37]. FRAM consumes less energy than other NVM technologies, such as flash.
+FRAM can be helpful for IoT devices that are operating at low power. These NVM technologies
+motivated researchers because of their appealing characteristics, such as non-volatility, low cost,
+and high density [2, 3].
+Researchers started using real-time NVPs for intermittent computing [16, 24, 30, 32]. Researchers
+observed that using only NVMs at the cache or main memory level degrades the system’s perfor-
+mance and consumes more energy, which gives an idea to explore hybrid memories. Recent NVPs
+such as MSP430FR6989 [16] consists of both SRAM and FRAM. We need to utilize the SRAM and
+FRAM efficiently and correctly; otherwise, we may degrade system performance and consume
+extra energy. To make the system more efficient, we need to map the application contents to either
+SRAM or FRAM. This is actually a memory mapping problem, similar to scratch-pad memories.
+Researchers explored a similar mapping problem in scratch-pad memories (SPMs) [12, 26, 33].
+Chakraborty et al. [1] documented the existing and standard memory mapping techniques on SPMs.
+In earlier works, memory mapping was done mainly between SPMs and main memory. Memory
+mapping can be done statically and dynamically [21, 22]. In static memory mapping, either ILP
+or the compiler can assist in determining the best placement [12, 26, 33]. ILP-solver takes inputs
+obtained from profilers and memory sizes as constraints in ILP-based memory mapping works. The
+ILP-solver provides the best placement option based on the objective function. In dynamic memory
+allocation [7, 8, 35, 36], either the user-defined program or the compiler will decide on an optimal
+placement choice at run time.
+However, our problem differs from the memory mapping techniques in SPMs because intermittent
+computing brings new constraints. During intermittent computation, the challenges were the
+forward progress of an application, data consistency, environmental consistency, and concurrency
+between the tasks. Due to these challenges, the execution model and development environment
+differ from the SPM-based memory mapping techniques. As a result, we require a memory mapping
+technique that supports intermittent computation.
+Researchers have explored memory mapping techniques and analysis for the MSP430FR6989
+MCU. In FRAM-based MCUs, Jayakumar et al. [18] implement a checkpointing policy. They save the
+system state to FRAM during a power failure. Jayakumar et al. [19, 20] propose an energy-efficient
+memory mapping technique for TI-based applications in FRAM-based MCUs. Kim et al. [23] present
+a detailed analysis of energy consumption for all memory sections in FRAM-based MCUs under
+different memory mappings.
+Earlier works investigated this problem by analyzing the possibilities to make the system efficient.
+The authors [19, 20, 23] have not covered all the design choices and possibilities. In addition, there
+is significantly less contribution towards memory mappings in FRAM-based MCUs that supports
+
+4
+S.J Badri, et al.
+intermittent computation. Our work proposes an energy-efficient memory mapping technique for
+intermittently powered IoT devices that experience frequent power failures.
+3
+MOTIVATION
+This section discusses the advantages of using hybrid SRAM and FRAM for these MSP430-based
+MCUs over unified SRAM or unified FRAM designs, as well as the importance of an efficient
+memory allocation.
+SRAM is 2KB, and FRAM is 128KB in a FRAM-based MCU, MSP430FR6989. The first naive
+approach is to use the entire 128KB of FRAM in both stable and unstable power scenarios, resulting
+in longer execution cycles and higher energy consumption. Similarly, we have a second naive
+approach to use the entire 2KB SRAM for small applications (whichever fits within the SRAM size),
+which has advantages during regular operation. Unfortunately, it loses all 2KB SRAM data during
+a power failure and takes more time to backup 2KB contents to FRAM during a power failure.
+These two approaches are treated as baselines 1 and 2 for this work. As shown in figure 1, for
+the baseline-1 design, we map all three sections to FRAM and all three sections to SRAM for the
+baseline-2 design.
+int glob1, glob2,..., globn;
+func_1(){
+local_variables 
+}
+func_2(){
+local_variables 
+}
+func_n(){
+local_variables 
+}
+Text
+Data
+Stack
+For func_1 ()
+Text
+Data
+Stack
+For func_2 ()
+For func_n ()
+Program
+Global_Variables
+Functions
+Consists of
+Local Variables
+For global_vars
+Data
+.bss
+Text
+Data
+Stack
+SRAM
+SRAM (2 KB)
+FRAM (128 KB)
+Memory
+Stack(func_1)
+Stack(func_n)
+Text(func_1)
+Data(func_1)
+Data(func_n)
+Text(func_n)
+   Map to SRAM
+FRAM
+SRAM (2 KB)
+FRAM (128 KB)
+Memory
+Stack(func_1)
+Stack(func_n)
+Text(func_1)
+Data(func_1)
+Data(func_n)
+Text(func_n)
+   Map to FRAM
+Baseline-1 Design
+Baseline-2 Design
+global_vars
+global_vars
+Fig. 1. Overview of the Baseline-1 and Baseline-2 memory mappings in MSP430FR6989
+
+Mapi-Pro: An Energy Efficient Memory Mapping Technique for Intermittent Computing
+5
+We compared baseline-1 and baseline-2 in both stable and unstable power scenarios. Baseline-1
+performs better during frequent power failures, while baseline-2 performs better during regular
+operations (without any power failures), as shown in figure 2. On average, baseline-1 consumes
+47.9% more energy than baseline-2 during a stable power, as shown in figure 2 (a). On average,
+baseline-2 consumes 32.7% more energy than baseline-1 during an unstable power, as shown in
+figure 2 (b). We also observed that MCU would pitch an error to either increase the SRAM space or
+use FRAM space for any computations. For large-size applications will not run using only SRAM, it
+requires FRAM as well. Thus, large applications consume more energy in baseline-2 during a stable
+power scenario.
+These two designs motivate us to propose a hybrid memory design that effectively uses both
+SRAM and FRAM. We also encountered that baseline-2 is ineffective for larger applications. As a
+result, we had to use a hybrid memory and figure out how and where to place the sections. To the
+best of our knowledge, only one work explored the memory mapping issue for these MCUs [20].
+We analyzed the mapping decisions using their empirical model. Jayakumar et al. [20] calculated
+the energy consumption values for each configuration. The authors suggested that allocate the
+sections to either SRAM or FRAM based on the energy values.
+0
+0.2
+0.4
+0.6
+0.8
+1
+16bit_2dim
+aes
+basicmath_small
+basicmath_large
+bf
+crc
+dhrystone
+dijkstra
+fft
+fir
+matrix_mult
+patricia
+qsort_small
+qsort_large
+sha
+susan
+Normalized Energy Consumption 
+(Normalized with Baseline-1)
+Benchmarks
+Baseline-1
+Baseline-2
+(a) Under Stable Power
+0
+0.2
+0.4
+0.6
+0.8
+1
+16bit_2dim
+aes
+basicmath_small
+basicmath_large
+bf
+crc
+dhrystone
+dijkstra
+fft
+fir
+matrix_mult
+patricia
+qsort_small
+qsort_large
+sha
+susan
+Normalized Energy Consumption 
+(Normalized with Baseline-1)
+Benchmarks
+Baseline-1
+Baseline-2
+(b) Under Unstable Power
+Fig. 2. Comparison between Baseline-1 and 2 configurations under Stable and Unstable Power Scenarios
+Table 1. Analysis of the Empirical Methods Used by Jayakumar et al. [20] for qsort_small under stable and
+unstable power supply scenarios
+Configuration
+Text
+Data
+Stack
+𝐸𝑛𝑒𝑟𝑔𝑦𝑠𝑡𝑎𝑏𝑙𝑒 (𝑚𝐽)
+𝐸𝑛𝑒𝑟𝑔𝑦𝑢𝑛𝑠𝑡𝑎𝑏𝑙𝑒 (𝑚𝐽)
+1 {SSS}
+SRAM
+SRAM
+SRAM
+16.70
+79.56
+2 {SSF}
+SRAM
+SRAM
+FRAM
+21.08
+66.34
+3 {SFS}
+SRAM
+FRAM
+SRAM
+28.75
+33.79
+4 {SFF}
+SRAM
+FRAM
+FRAM
+35.97
+52.10
+5 {FSS}
+FRAM
+SRAM
+SRAM
+39.48
+68.24
+6 {FSF}
+FRAM
+SRAM
+FRAM
+57.64
+54.75
+7 {FFS}
+FRAM
+FRAM
+SRAM
+64.14
+45.83
+8 {FFF}
+FRAM
+FRAM
+FRAM
+92.09
+36.07
+The empirical method used by the authors is as follows. The authors considered functions as the
+basic unit. They explored all configurations and calculated the energy values, as shown in table 1.
+The authors have eight configurations because they have two memory regions (SRAM or FRAM)
+and need to map three sections (stack, data, text). Using the author’s model, we calculated the
+
+6
+S.J Badri, et al.
+energy values for the qsort_small application. For instance, the SSS configuration performs better
+during a stable power supply, and during a power failure, SFS consumes less energy than all other
+configurations. As a result, authors allocate text and stack sections to SRAM and data sections to
+FRAM.
+We observed that this empirical method becomes ineffective as the number of configurations
+increases. The authors considered all global variables, arrays, and constants as data sections. Instead,
+why can’t we map each global variable or array to either SRAM or FRAM? This increases the
+number of configurations, and calculating/tracking energy values is challenging. Our design space
+grows enormously and makes our mapping problem challenging.
+This new set of challenges motivated us to propose an energy-efficient memory mapping tech-
+nique. Our proposed memory mapping framework supports large-size applications and covers all
+possible configurations.
+4
+SYSTEM MODEL AND PROBLEM DEFINITION
+This section discusses the system model for embedded MCUs and defines the mapping problem for
+these MCUs.
+4.1
+System Model
+We consider a simple, customized RISC instruction set with a Von-Neumann architecture, where
+the instructions and data share the same address space that supports at least 16-bit addressing. Base
+architecture doesn’t have a cache to avoid uncertainty. To make things simple, we assume single
+cycle execution of the processor. Base architecture has a small SRAM memory and a larger NVM.
+The MSP430 is an example of such a processor. Non-volatile memory sizes range from 1 kilobyte
+(KB) to 256 KB, while volatile RAM sizes range from 256 bytes to 2KB. Both SRAM and NVM can
+be accessed by instructions using a compiler/linker script. We can modify the linker script to map
+memory according to the memory ranges specified by the user. MSP430 doesn’t have any operating
+system.
+4.2
+Problem Definition
+Definition 4.1: Optimal Memory Mapping Problem: Given a program that consists of various
+functions and global variables, sizes of SRAM and FRAM, the number of reads and writes for each
+function/variable, and the energy required per read/write to the SRAM/FRAM. What is the optimal
+memory mapping for these functions/variables in order to reduce the system’s EDP?
+The inputs are : Number of functions; number of global variables; energy per write to SRAM
+and FRAM; energy per read to SRAM and FRAM; SRAM and FRAM sizes; Number of CPU cycles
+per each function; the number of reads; the number of writes.
+The output is: Mapping information for all functions and global variables, under which the
+system’s EDP is minimized.
+Definition 4.2: Support for Intermittent Computing: During power failures, we must safely
+backup the volatile contents to NVM. As previously stated, we must use SRAM efficiently for
+energy savings; but again, how can we save the contents of SRAM? There are two significant issues
+with intermittent computation. First, during a power failure, all SRAM’s mapping information
+and register contents are lost, causing the system to become inconsistent. Second, how do we
+backup/restore the mapping information and register contents to ensure system consistency?
+
+Mapi-Pro: An Energy Efficient Memory Mapping Technique for Intermittent Computing
+7
+5
+MAPI-PRO: AN ENERGY EFFICIENT MEMORY MAPPING FOR INTERMITTENT
+COMPUTING
+In this section, we discuss the details of the proposed mapping technique. Our main objective is to
+pick the optimal mapping choice among all the design choices, which reduces the system’s EDP. To
+achieve this, we proposed an ILP-based mapping technique. The overview of the proposed mapping
+technique is shown in figure 3. We also discuss how we support intermittent computing for these
+MCUs.
+int glob_1, glob_2,..., glob_n;
+func_1(){
+local_variables 
+}
+func_2(){
+local_variables 
+}
+func_n(){
+local_variables 
+}
+Text
+Data
+Stack
+For func_1 ()
+Text
+Data
+Stack
+For func_2 ()
+For func_n ()
+Program
+Global_Variables
+Functions
+Consists of
+Local Variables
+Proposed ILP based
+Mapping Technique
+For global_vars
+Data
+.bss
+Text
+Data
+Stack
+Placement
+Decision
+for SRAM
+Placement
+Decision
+for FRAM
+Stack(func_2)
+Stack(func_n)
+Text(func_1)
+Data(func_2)
+Text(func_n)
+Data(func_n)
+Data(func_1)
+Data(func_2)
+Stack(func_1)
+glob_3,....,glob_n
+Text(func_1)
+Text(func_2)
+Data(func_n)
+SRAM (2 KB)
+FRAMn (125 KB)
+Memory
+Backup Region
+FRAMb (3 KB)
+glob_1, glob_2
+Fig. 3. Overview of the proposed memory mappings in MSP430FR6989
+5.1
+ILP Formulation for Data Mapping
+We present the ILP formulation for the memory mapping problem mentioned in definition 4.1. We
+divide this ILP formulation into two parts, one is for global variables, and the second is for the
+functions. We have shown the overview block diagram of the proposed ILP framework in figure 4.
+Application
+Profiling and one-time
+characterization
+Assembly
+Code
+Number of reads & writes to
+each variable
+Number of reads & writes to
+each function
+Energy per read/write to
+SRAM
+Energy per read/write to
+FRAM
+ILP Solver
+Number of CPU cycles
+required for eachfunctions
+and variable
+Number of Functions
+Number of Global variables
+SRAM and FRAM sizes
+Mapping Information for each
+Variable and Function
+MSP430FR6989
+Fig. 4. Overview of the Proposed ILP Framework
+
+8
+S.J Badri, et al.
+For Global Variables: Let the number of global variables in a program be ‘G’. Let the number
+of reads and writes to variable ‘i’ are 𝑟𝑖 and 𝑤𝑖. We divided FRAM’s 128 KB into two regions, i.e.,
+𝐹𝑅𝐴𝑀𝑛 and 𝐹𝑅𝐴𝑀𝑏, 𝐹𝑅𝐴𝑀𝑛 memory region has 125 KB, and the 𝐹𝑅𝐴𝑀𝑏 memory region has 3 KB.
+We have two memory regions represented as 𝑀𝑒𝑚𝑗 as shown in the equation 2; when j=1, we
+select the memory region as SRAM, and we use 𝐹𝑅𝐴𝑀𝑛 for j=2.
+𝑀𝑒𝑚𝑗 =
+�
+𝑗 = 1
+; SRAM
+𝑗 = 2
+; 𝐹𝑅𝐴𝑀𝑛
+(2)
+Let the sizes of SRAM/FRAM as 𝑆𝑖𝑧𝑒(𝑀𝑒𝑚𝑗) as shown in equation 3, when j=1, we refer as
+SRAM memory size in bytes, and when j=2, we refer as 𝐹𝑅𝐴𝑀𝑛 memory size in bytes.
+𝑆𝑖𝑧𝑒(𝑀𝑒𝑚𝑗) =
+�
+𝑗 = 1
+; SRAM
+𝑗 = 2
+; 𝐹𝑅𝐴𝑀𝑛
+(3)
+Let the energy required for each read/write to 𝑀𝑒𝑚𝑗 is 𝐸𝑟_𝑗 and 𝐸𝑤_𝑗. Let the number of CPU
+cycles required to execute a global variable 𝑣𝑖 be 𝑁𝐶𝑣𝑖, where ∀𝑖 ∈ [1,𝐺]). Using one-time charac-
+terization and static profiling, we gathered data such as per read/write energy to SRAM/FRAM and
+the number of cycles.
+We define a binary variable (BV); 𝐼𝑗 (𝑣𝑖), which refers to a variable 𝑣𝑖 is allocated to memory
+region 𝑗. If 𝐼𝑗 (𝑣𝑖)=1 then the variable 𝑣𝑖 is allocated and 𝐼𝑗 (𝑣𝑖)=0 indicates that the variable 𝑣𝑖 is
+not allocated. 𝐼𝑗 (𝑣𝑖), where (∀𝑗 ∈ [1, 𝑀𝑒𝑚𝑗], ∀𝑖 ∈ [1,𝐺]) is defined as shown in the equation 4.
+𝐼𝑗 (𝑣𝑖) =
+�
+1
+𝑣𝑖 is allocated to memory region 𝑗
+0
+otherwise
+(4)
+Constraints: There are two constraints, one is for BV; 𝐼𝑗 (𝑣𝑖) and one is a memory size constraint.
+In any case, a variable 𝑣𝑖 is allocated to only one memory region, which means 𝑣𝑖 is allocated to
+either SRAM or FRAM but not both. This constraint is defined in the equation 5.
+𝑀𝑒𝑚𝑗
+∑︁
+𝑗=1
+𝐼𝑗 (𝑣𝑖) = 1
+(∀𝑖 ∈ [1,𝐺])
+(5)
+The other constraint is related to memory sizes. The allocated variables 𝑣𝑖 and its 𝑆𝑖𝑧𝑒(𝑣𝑖);
+∀𝑖 ∈ [1,𝐺]) should not be greater than the 𝑆𝑖𝑧𝑒(𝑀𝑒𝑚𝑗). This constraint is defined in the equation
+6.
+𝐺
+∑︁
+𝑖=1
+𝐼𝑗 (𝑣𝑖) ∗ 𝑆𝑖𝑧𝑒(𝑣𝑖) ≤ 𝑆𝑖𝑧𝑒(𝑀𝑒𝑚𝑗)
+(∀𝑗 ∈ [1, 𝑀𝑒𝑚𝑗])
+(6)
+Objective 4.1: The challenge of mapping global variables in a program to either SRAM or FRAM
+is to reduce EDP and improve system performance. 𝐸𝑔𝑙𝑜𝑏𝑎𝑙 is defined in the equation 7. Where
+𝐸𝑔𝑙𝑜𝑏𝑎𝑙 is the energy required to allocate global variables to either SRAM or FRAM.
+𝐸𝑔𝑙𝑜𝑏𝑎𝑙 =
+𝑀𝑒𝑚𝑗
+∑︁
+𝑗=1
+𝐺
+∑︁
+𝑖=1
+[𝐸𝑟_𝑗 × 𝑟𝑖 + 𝐸𝑤_𝑗 × 𝑤𝑖]
+(7)
+𝐸𝐷𝑃𝑔𝑙𝑜𝑏𝑎𝑙 is defined in the equation 8. Where 𝐸𝐷𝑃𝑔𝑙𝑜𝑏𝑎𝑙 is the energy-delay product required to
+allocate global variables to either SRAM or FRAM.
+
+Mapi-Pro: An Energy Efficient Memory Mapping Technique for Intermittent Computing
+9
+𝐸𝐷𝑃𝑔𝑙𝑜𝑏𝑎𝑙 =
+𝑀𝑒𝑚𝑗
+∑︁
+𝑗=1
+𝐺
+∑︁
+𝑖=1
+𝐼𝑗 (𝑣𝑖) [𝐸𝑔𝑙𝑜𝑏𝑎𝑙 × 𝑁𝐶𝑣𝑖]
+(8)
+For Functions: Let the number of functions in a program be ‘𝑁 ′
+𝑓 . Let the number of reads and
+writes to 𝑖𝑡ℎ function are 𝑟 (𝐹𝑖) and 𝑤(𝐹𝑖), where ∀𝑖 ∈ [1, 𝑁𝑓 ]. Functions consist of procedural
+parameters, local variables, and return variables. Internally the code/data of functions are divided
+into the text, data, and stack sections. We map at least one section among these three sections to
+either SRAM or FRAM regions, i.e., 𝑀𝑒𝑚𝑗 and 𝑆𝑒𝑐𝑘 (𝑖) defines section ‘k’ of 𝑖𝑡ℎ function as shown
+in the equation 9, when k=1, we refer to the text section of 𝑖𝑡ℎ function, when k=2, we refer to the
+data section of 𝑖𝑡ℎ function, and when k=3, we refer to the stack section of 𝑖𝑡ℎ function.
+𝑆𝑒𝑐𝑘 (𝑖) =
+
+
+𝑘 = 1
+; Text
+𝑘 = 2
+; Data
+𝑘 = 3
+; Stack
+;∀𝑖 ∈ [1, 𝑁𝑓 ]
+(9)
+We define a BV; 𝐼𝑗 (𝑆𝑒𝑐𝑘 (𝑖)), which refers to a section 𝑆𝑒𝑐𝑘 of 𝑖𝑡ℎ function is allocated to only
+one memory region 𝑗. If 𝐼𝑗 (𝑆𝑒𝑐𝑘 (𝑖))=1 then the section 𝑆𝑒𝑐𝑖 is allocated and 𝐼𝑗 (𝑆𝑒𝑐𝑘 (𝑖))=0 that
+indicates the section 𝑆𝑒𝑐𝑖 is not allocated. 𝐼𝑗 (𝑆𝑒𝑐𝑘 (𝑖)), where (∀𝑗 ∈ [1, 𝑀𝑒𝑚𝑗], ∀𝑖 ∈ [1, 𝑁𝑓 ]),
+∀𝑘 ∈ [1,𝑆𝑒𝑐𝑘 (𝑖)]) is defined as shown in the equation 10.
+𝐼𝑗 (𝑆𝑒𝑐𝑘 (𝑖)) =
+�
+1
+𝑆𝑒𝑐𝑘 of 𝑖𝑡ℎ function is allocated to 𝑗
+0
+otherwise
+(10)
+Constraints: There are two constraints, one is for BV; 𝐼𝑗 (𝑆𝑒𝑐𝑘 (𝑖)) and one is a memory size
+constraint. In any case, a 𝑆𝑒𝑐𝑘 of 𝑖𝑡ℎ function is allocated to only one memory region, which means
+𝑆𝑒𝑐𝑘 of 𝑖𝑡ℎ function is either allocated to either SRAM or FRAM but not both. This constraint is
+defined in the equation 11.
+3
+∑︁
+𝑘=1
+𝑀𝑒𝑚𝑗
+∑︁
+𝑗=1
+𝐼𝑗 (𝑆𝑒𝑐𝑘 (𝑖))) = 1
+(∀𝑖 ∈ [1, 𝑁𝑓 ])
+(11)
+The other constraint is related to memory sizes. The allocated sections 𝑆𝑒𝑐𝑘 (𝑖) and its 𝑆𝑖𝑧𝑒(𝐹𝑖);
+∀𝑘 ∈ [1,𝑆𝑒𝑐𝑘 (𝑖)]), ∀𝑗 ∈ [1, 𝑀𝑒𝑚𝑗], ∀𝑖 ∈ [1, 𝑁𝑓 ] should not be greater than the 𝑆𝑖𝑧𝑒(𝑀𝑒𝑚𝑗). This
+constraint is defined in the equation 12.
+𝐺
+∑︁
+𝑖=1
+𝐼𝑗 (𝑣𝑖) ∗ 𝑆𝑖𝑧𝑒(𝑣𝑖) +
+3
+∑︁
+𝑘=1
+𝑁𝑓
+∑︁
+𝑖=1
+𝐼𝑗 (𝑆𝑒𝑐𝑘 (𝑖)) ∗ 𝑆𝑖𝑧𝑒(𝐹𝑖) ≤ 𝑆𝑖𝑧𝑒(𝑀𝑒𝑚𝑗)
+(12)
+Objective 4.2: The challenge of mapping sections of these functions in a program to either
+SRAM or FRAM is to minimize EDP and improve system performance. 𝐸𝑓 𝑢𝑛𝑐 is defined in the
+equation 13, where 𝑀𝑐𝑖 is the number of the times 𝑖𝑡ℎ functions called.
+𝐸𝑓 𝑢𝑛𝑐 =
+𝑀𝑒𝑚𝑗
+∑︁
+𝑗=1
+𝑁𝑓
+∑︁
+𝑖=1
+[𝐸𝑟_𝑗 × 𝑟 (𝐹𝑖) + 𝐸𝑤_𝑗 × 𝑤(𝐹𝑖)] × 𝑀𝑐𝑖
+(13)
+𝐸𝐷𝑃𝑓 𝑢𝑛𝑐 is defined in the equation 14. Where 𝐸𝐷𝑃𝑓 𝑢𝑛𝑐 is the energy-delay product required to
+allocate all functions to either SRAM or FRAM. Where 𝐸𝑓 𝑢𝑛𝑐 is the energy required to allocate
+
+10
+S.J Badri, et al.
+functions to either SRAM or FRAM. Where 𝑁𝐶𝐹𝑖 is the number of CPU cycles required to execute
+a function 𝐹𝑖.
+𝐸𝐷𝑃𝑓 𝑢𝑛𝑐 =
+3
+∑︁
+𝑘=1
+𝑀𝑒𝑚𝑗
+∑︁
+𝑗=1
+𝑁𝑓
+∑︁
+𝑖=1
+𝐼𝑗 (𝑆𝑒𝑐𝑘 (𝑖)) [𝐸𝑓 𝑢𝑛𝑐 × 𝑁𝐶𝐹𝑖]
+(14)
+The overall system EDP, 𝐸𝐷𝑃𝑠𝑦𝑠𝑡𝑒𝑚, is the sum of both 𝐸𝐷𝑃𝑔𝑙𝑜𝑏𝑎𝑙 and 𝐸𝐷𝑃𝑓 𝑢𝑛𝑐 as shown in the
+equation 15.
+𝐸𝐷𝑃𝑠𝑦𝑠𝑡𝑒𝑚 = 𝐸𝐷𝑃𝑔𝑙𝑜𝑏𝑎𝑙 + 𝐸𝐷𝑃𝑓 𝑢𝑛𝑐
+(15)
+Our objective function is shown in the equation 16. Our main objective is to minimize the
+system’s EDP by choosing the optimal placement choice.
+Objective Function: Minimize 𝐸𝐷𝑃𝑠𝑦𝑠𝑡𝑒𝑚
+(16)
+5.2
+Implementing Mapping Technique in MSP430FR6989
+Once we obtain the placement information from the 𝐼𝐿𝑃_𝑠𝑜𝑙𝑣𝑒𝑟, we map the respective variables
+and the sections of a function to either SRAM or FRAM. We modify the linker script accordingly
+for mapping the sections or variables to either SRAM or FRAM. In our proposed mapping policy,
+placing global variables is straightforward, i.e., mapping the respective variable to either SRAM or
+FRAM based on the ILP decision.
+We observed that from the linker script, we can map the whole stack section of each function
+to either SRAM or FRAM. We analyzed the mappings of the stack section for each function by
+modifying the linker script. We used the inbuilt attributes to differentiate mappings between SRAM
+and FRAM; for instance, we used the inbuilt attribute (__𝑎𝑡𝑡𝑟𝑖𝑏𝑢𝑡𝑒__((𝑟𝑎𝑚𝑓𝑢𝑛𝑐)) that maps that
+function to SRAM. If we want to place the stack section to SRAM, we modify the linker script by
+replacing the default setting with " .stack: {} > RAM (HIGH) ". If we want to place the stack section
+to FRAM, we modify the linker script by replacing the default setting with " .stack: {} > FRAM".
+Similarly, for the text section, we observed that placing the text section in either SRAM or FRAM
+shows an impact on EDP. This effect is because the majority of access in the text section are read
+accesses, as we observed that the energy consumption for each read access to SRAM/FRAM differs.
+Table 3 shows that approximately FRAM consumes 2x more read energy than SRAM. Thus, we
+analyzed each application where to map the text section based on the free space available. If we
+have enough space available in SRAM, we place the text section in SRAM itself; otherwise, we
+place the text section in FRAM. We included the following four lines in our linker script to check
+the above condition and map the text section.
+(1) #𝑖𝑓 𝑛𝑑𝑒𝑓 __𝐿𝐴𝑅𝐺𝐸_𝐶𝑂𝐷𝐸_𝑀𝑂𝐷𝐸𝐿__
+(2) .text : {} > FRAM
+(3) #else
+(4) .text : {} » SRAM
+We modified the linker script for mapping the data section by using the inbuilt compiler directives.
+We followed the below three steps.
+(1) Allocate a new memory block, for instance, 𝑁𝐸𝑊 _𝐷𝐴𝑇𝐴𝑆𝐸𝐶𝑇𝐼𝑂𝑁. We can declare the start
+address and size of the data section in the linker script.
+(2) Define a segment (.Localvars) which stores in this memory block (𝑁𝐸𝑊 _𝐷𝐴𝑇𝐴𝑆𝐸𝐶𝑇𝐼𝑂𝑁).
+
+Mapi-Pro: An Energy Efficient Memory Mapping Technique for Intermittent Computing
+11
+(3) Use #pragma 𝐷𝐴𝑇𝐴_𝑆𝐸𝐶𝑇𝐼𝑂𝑁 (𝑓𝑢𝑛𝑐𝑡_𝑛𝑎𝑚𝑒,𝑠𝑒𝑔_𝑛𝑎𝑚𝑒) in the program to define functions
+in this segment. Where 𝑓𝑢𝑛𝑐𝑡_𝑛𝑎𝑚𝑒 is the function name and𝑠𝑒𝑔_𝑛𝑎𝑚𝑒 is the created segment
+name. For instance, #pragma 𝐷𝐴𝑇𝐴_𝑆𝐸𝐶𝑇𝐼𝑂𝑁 (𝑓𝑢𝑛𝑐_1, .𝐿𝑜𝑐𝑎𝑙𝑣𝑎𝑟𝑠)
+Once we are done with creating the different sections, we can allocate these sections to either
+SRAM or FRAM based on ILP decisions. For instance, placing " 𝑁𝐸𝑊 _𝐷𝐴𝑇𝐴𝑆𝐸𝐶𝑇𝐼𝑂𝑁: {} > FRAM"
+in the linker script, which maps the 𝑁𝐸𝑊 _𝐷𝐴𝑇𝐴𝑆𝐸𝐶𝑇𝐼𝑂𝑁 to FRAM.
+5.3
+Support for Intermittent Computing
+When the power is stable, everything works properly. Because of the static allocation scheme, we
+map all functions/variables to SRAM/FRAM for the first time. During a power failure, SRAM and
+registers lose all of their contents, including mapping information. When power is restored, we
+don’t know what functions/variables were allocated to SRAM before the failure. As a result, we
+must either restart the execution from the beginning or end up with incorrect results. Restarting
+the application consumes extra energy and time, making our system inefficient in terms of energy
+consumption and performance.
+We propose a backup strategy during frequent power failures. FRAM was divided into 𝐹𝑅𝐴𝑀𝑛
+and 𝐹𝑅𝐴𝑀𝑏 as shown in the figure 3. 𝐹𝑅𝐴𝑀𝑛 has a size of 125 KB and is used for regular mappings.
+𝐹𝑅𝐴𝑀𝑏 has a size of 3 KB that serves as a backup region (BR) during power failures. So, during a
+power failure, we back up all register and SRAM contents to FRAM. Whenever power is restored,
+we restore the register and SRAM contents from 𝐹𝑅𝐴𝑀𝑏 to SRAM and resume the application
+execution. The proposed backup strategy reduces extra energy consumption and makes the system
+more energy efficient.
+6
+EXPERIMENTAL SETUP AND RESULTS
+6.1
+Experimental Setup
+We used TI’s MSP430FR6989 for all experiments. We experimented on mixed benchmarks, which
+have both Mi-Bench [13] and TI-based benchmarks. We have shown the experimental setup in the
+table 2. The development platform and experimental setup are shown in figure 5. We performed
+experiments to determine the energy required for a single read/write to SRAM/FRAM, as shown in
+the table 3. We collected the number of reads/writes for each global variable and functions as part
+of a one-time characterization. We also used TI’s MSP430F5529 for comparing flash with FRAM.
+We performed experiments to determine the energy required for a single read/write to flash, as
+shown in the table 3.
+Table 2. Experimental Setup
+Component
+Description
+Target Board
+TI MSP430FR6989 Launchpad
+Core
+MSP430 (1.8-3.6 V; 16 MHz)
+Memory
+2KB SRAM and 128KB FRAM
+IDE
+Code Composer Studio
+Energy Profiling
+Energy Trace++
+ILP Solver
+LPSolve_IDE
+Benchmarks
+Mixed benchmarks (MiBench and TI-based)
+MCU, which we experimented has MSP430 architecture, which is more suitable for IoT devices.
+The majority of MSP430 software is written in C and compiled with one of TI’s recommended
+
+12
+S.J Badri, et al.
+compilers ( IAR Embedded Code Bench, Code-Composer Studio (CCS), or msp430-gcc). The IAR
+Embedded Code Bench and CCS compilers are part of integrated development environments (IDEs).
+We used the widely used, freely available, and easily extended tool, i.e., CCS, for all experiments
+in this article. EnergyTrace++ technology allows us to calculate energy and power consumption
+directly. According to the datasheet for the MSP430FR6989, the number of cycles required to
+read/write in FRAM is twice that of SRAM.
+Table 3. Energy Values for each read/write to SRAM and FRAM
+Memory
+Per Read Energy (nJ)
+Per Write Energy (nJ)
+SRAM
+5500
+5600
+FRAM
+10325
+13125
+Flash
+23876
+31198
+Fig. 5. (a) TI-based MSP430 Launchpad Development Boards (b) Working with EnergyTrace++ on CCS
+6.2
+Evaluation Benchmarks
+We chose benchmarks from both the MiBench suite and TI benchmarks. One of the primary
+motivations for using the MiBench suite is that most of the TI-based benchmarks were small in size
+and easily fit into either SRAM or FRAM. In these cases, we don’t require any hybrid memory design.
+Most of the TI-based benchmarks have only one or two functions and 3-4 global variables, which is
+not useful for the hybrid main-memory design. Thus we used mixed benchmarks consisting of 4
+TI-based benchmarks and 12 from the MiBench suite.
+For the MiBench suite, we first make MCU-compatible benchmarks by adding MCU-related
+header files and watchdog timers. All benchmarks may not be compatible with the MCU. Thus, we
+need to choose the benchmarks from the MiBench suite, which are compatible with the MSP430
+boards. Once we have benchmarks, we execute them on board for the machine code. Using the
+.asm file, we calculate the inputs that are required by the ILP solver, as shown in figure 4.
+6.3
+Baseline Configurations
+We chose five different memory configurations to compare with the proposed memory configuration.
+We directly map all the functions/variables to FRAM in the baseline configuration 1, as shown in
+figure 1. We use configuration-1 to compare our proposed memory configuration during stable and
+unstable power scenarios.
+
+Code
+Composer
+Studio
+MSP430F5529MSP430FR6989
+EnergyTrace++Mapi-Pro: An Energy Efficient Memory Mapping Technique for Intermittent Computing
+13
+We directly map all the functions/variables to SRAM in the baseline configuration 2, as shown in
+figure 1. We use configuration-2 to compare our proposed memory configuration during stable and
+unstable power scenarios.
+In baseline configuration 3, we used the empirical method of Jayakumar et al. [20]. We compare
+this configuration-3 with our proposed configuration during stable and unstable power scenarios
+to observe the importance of the proposed than the existing work.
+In baseline configuration 4, we used the proposed ILP technique for the flash-based msp430
+board [24]. We compare this configuration-4 with our proposed configuration during stable and
+unstable power scenarios to observe the difference between FRAM and Flash technologies.
+In baseline configuration 5, we only have a proposed memory mapping technique and no BR.
+We compare this configuration-5 with our proposed configuration during frequent power failures
+to observe the importance of BR. The overview of all baseline configurations is shown in table 4.
+The experimental setup for all baseline configurations is the same as the one proposed.
+Table 4. Overview of the Baseline Configurations
+Configuration
+FRAM
+SRAM
+Flash
+Backup Region (BR)
+ILP
+Baseline-1
+✓
+✗
+✗
+✗
+✗
+Baseline-2
+✗
+✓
+✗
+✗
+✗
+Baseline-3 ( Jayakumar et al. [20])
+✓
+✓
+✗
+✗
+✗
+Baseline-4
+✗
+✓
+✓
+✗
+✓
+Baseline-5
+✓
+✓
+✗
+✗
+✓
+Proposed
+✓
+✓
+✓
+✓
+✓
+✓- Supported , ✗- Not Supported
+6.4
+Results
+The proposed memory configuration is evaluated in this section under stable and unstable power.
+The proposed memory configuration is compared with five baseline memory configurations as
+discussed in the section 6.3.
+6.4.1
+Under Stable Power: Our main objective of the proposed memory configuration is to
+minimize the system’s EDP. All values shown in figure 6 are normalized with baseline-1. Compared
+to baseline-1, the proposed gets 38.10% lesser EDP, as shown in figure 6. Because there are no
+power interruptions in this scenario, this improvement is totally from the proposed ILP model. In
+configuration-1, we place everything to FRAM, where FRAM consumes more energy and the number
+of cycles than SRAM, as shown in the table 3. Our proposed memory configuration incorporates
+the placement recommendation from the proposed ILP model and suggests utilizing both SRAM
+and FRAM.
+Under a stable power scenario, the proposed gets 9.30% less EDP than baseline-3, as shown in
+figure 6. We discussed the author’s empirical model and assumptions in the previous section 3. The
+authors assumed that the data section included all global variables, constants, and arrays. As a
+result, our proposed ILP-based mapping differs from the author’s mapping in that our proposed
+mapping outperforms the existing work. Under stable power, baseline-3 receives 24.57% less EDP
+than baseline-1, as shown in figure 6. This advantage is primarily due to baseline-3’s hybrid memory.
+In comparison to baseline-4, the proposed reduces EDP by 18.55%, as shown in figure 6. We
+used flash+SRAM with our proposed ILP framework in baseline-4. As shown in table 3, the above
+benefit is primarily due to FRAM because flash consumes more energy. Baseline-3 outperforms
+
+14
+S.J Badri, et al.
+0
+0.2
+0.4
+0.6
+0.8
+1
+16bit_2dim
+aes
+basicmath_small
+basicmath_large
+bf
+crc
+dhrystone
+dijkstra
+fft
+fir
+matrix_mult
+patricia
+qsort_small
+qsort_large
+sha
+susan
+Normalized EDP (Normalized with 
+Baseline-1)
+Benchmarks
+Baseline-2
+Jayakumar et al. [20]
+Baseline-4
+Proposed
+Fig. 6. Comparison between Baseline configurations and the Proposed under Stable Power
+baseline-4 during stable power. Because of FRAM in baseline-3, even our proposed ILP model is
+ineffective in this case. We encountered that baseline-3 achieves 9.19% less EDP than baseline-4, and
+this benefit is because of smaller applications. From figure 6, baseline-4 performs better for large
+applications than baseline-4. Jayakumar et al. [20] empirical method suggests placing more content
+on SRAM because SRAM is sufficient for placing the entire small-size application. As a result, the
+performance of baseline 3 is dependent on the application size, as for large-size applications, even
+FRAM does not outperform flash.
+Baseline 2 outperforms the proposed and all other baselines under stable power conditions.
+We noticed that this benefit is primarily due to SRAM, but it only applies to smaller applications.
+Baseline 2 achieves 36.19% less EDP than the proposed for smaller applications, as shown in figure
+6. We also looked at large applications where the proposed outperforms the baseline-2 by a small
+margin. When the SRAM is full, the MCU must wait for the space to be released, which consumes
+extra energy and cycles. For more extensive applications, baseline-2 achieves 2.94% more EDP than
+proposed.
+We also evaluated our proposed framework with another MSP430F5529 MCU with flash and
+SRAM for completeness. This comparison assists the user in selecting the most appropriate NVM
+technology, such as FRAM or flash, as needed. To be fair, we used the same sizes of SRAM (2 KB)
+and Flash (128 KB) in this comparison. We compared FRAM-based and flash-based MCUs under
+stable power conditions. We used the proposed frameworks and techniques in both MCUs. We
+discovered that the proposed FRAM-based configuration outperforms the flash-based configuration.
+Flash-based configurations consume 26.03% more EDP than FRAM-based configurations, as shown
+in figure 7. Flash consumes more energy, as shown in table 3.
+6.4.2
+Under Unstable power: We used the default TI-based compute through power loss (ctpl)
+tool for migration. During a power failure, we need to migrate the SRAM contents to a FRAM-based
+backup region (𝐹𝑅𝐴𝑀𝑏), i.e., the backup process. Whenever power comes back, we need to migrate
+the (𝐹𝑅𝐴𝑀𝑏) contents to SRAM, i.e., the restoration process. So, all these migrations are done using
+ctpl() functions. We introduce a power failure by changing the low power modes mentioned in
+the MSP430FR6989 design document. We used ctpl() for creating power failures. We assume that
+the number of power failures is spread equally within the execution period. For instance, if the
+
+Mapi-Pro: An Energy Efficient Memory Mapping Technique for Intermittent Computing
+15
+0
+0.2
+0.4
+0.6
+0.8
+1
+16bit_2dim
+aes
+basicmath_small
+basicmath_large
+bf
+crc
+dhrystone
+dijkstra
+fft
+fir
+matrix_mult
+patricia
+qsort_small
+qsort_large
+sha
+susan
+Normalized EDP (Normalized with 
+MSP430F5529)
+Benchmarks
+MSP430F5529
+MSP430FR6989
+Fig. 7. Comparison between MSP430FR6989 (FRAM-based MCU) and MSP430F5529 (Flash-based MCU)
+under Stable Power
+total execution period for an application is 20 milliseconds (ms), and let’s say the number of power
+failures is four, then for every 5 ms, we experience a power failure.
+We performed experiments under unstable power to compare the proposed memory configuration
+with baseline configurations. All values shown in figure 8 are normalized with baseline-2. Compared
+to baseline-2, the proposed gets 15.97% lesser EDP, as shown in figure 8. We observed that migration
+overhead is less than the energy consumed to execute the application from FRAM, and this migration
+overhead depends on the number of power failures. For instance, one backup migration consumes
+approximately 16.88 mJ of energy, and one restore migration consumes approximately 11.606 mJ of
+energy in a qsort application. The above benefit to our proposed configuration is using a hybrid
+memory.
+Under an unstable power scenario, the proposed gets 21.99% less EDP than baseline-3, as shown
+in figure 8. We discussed the author’s empirical model and assumptions in the previous section 3.
+As already stated, the Jayakumar et al. empirical method is more beneficial for small applications.
+In contrast, the author’s empirical method suggests placing more content on SRAM because SRAM
+is sufficient for placing the entire small-size application. Thus, for [20] work, backup/restore
+operations take more energy during a power failure. Our proposed mapping outperforms the
+existing work. During frequent power failures, baseline-3 receives 6.91% less EDP than baseline-1,
+as shown in figure 8. This advantage is primarily due to baseline-3’s hybrid memory.
+Compared to baseline-4, the proposed reduces EDP by 23.05%, as shown in figure 8. We used
+flash+SRAM with our proposed ILP framework in baseline-4. As shown in table 3, the above benefit
+is primarily due to FRAM because flash consumes more energy. Baseline-3 outperforms baseline-4
+during stable power. Because of FRAM in baseline-3, even our proposed ILP model is ineffective
+for this comparison. We encountered that baseline-3 achieves 6.28% less EDP than baseline-4 for
+smaller applications. The above benefit for baseline-3 is minimal because the size of backup/restores
+increases, which even neutralizes the flash for some applications, as shown in figure 8. Baseline-4
+achieves 2.69% less EDP than baseline-3 for large applications, as shown in figure 8. As a result, the
+
+16
+S.J Badri, et al.
+0
+0.2
+0.4
+0.6
+0.8
+1
+16bit_2dim
+aes
+basicmath_small
+basicmath_large
+bf
+crc
+dhrystone
+dijkstra
+fft
+fir
+matrix_mult
+patricia
+qsort_small
+qsort_large
+sha
+susan
+Normalized EDP (Normalized with 
+Baseline-2)
+Benchmarks
+Baseline-1
+Jayakumar et al. [20]
+Baseline-4
+Baseline-5
+Proposed
+Fig. 8. Comparison between Baseline configurations and the Proposed under Unstable Power
+performance of baseline 3 is dependent on the application size, as for large-size applications, even
+FRAM does not outperform flash.
+The proposed outperforms all baselines under unstable power conditions. This benefit is primarily
+due to a hybrid memory and the proposed mapping technique. Baseline 2 achieves 42.98% less EDP
+than the proposed, as shown in figure 8.
+When we remove BR, all the mapping information of SRAM is lost because our model is static.
+We introduce a BR in the FRAM memory region to save this mapping information. During a power
+failure, we migrate the SRAM contents to 𝐹𝑅𝐴𝑀𝑏, and whenever power comes back, we restore
+the 𝐹𝑅𝐴𝑀𝑏 contents to the SRAM.
+We experimented to know the importance of BR, where we compared the proposed memory
+configuration with baseline-5. Compared to baseline-5, the proposed gets 23.94% lesser EDP, as
+shown in figure 8. This benefit is because we need to re-execute the application four times from
+the beginning, which consumes extra time and energy. The number of times re-executing the
+application is equal to the number of power failures.
+We also evaluated our proposed framework with another MSP430F5529 MCU, which consists of
+flash and SRAM for completeness. This comparison assists the user in selecting the most appropriate
+NVM technology, such as FRAM or flash, as needed. To be fair, we used the same sizes of SRAM (2
+KB) and Flash (128 KB) in this comparison. We also used BR for both baselines; the only difference
+is that we replaced FRAM with the flash in the proposed configurations, and everything is the
+same. We compared FRAM-based and flash-based MCUs under unstable power conditions. We
+used the proposed frameworks and techniques in both MCUs. We discovered that the proposed
+FRAM-based configuration outperforms the flash-based configuration. Flash-based configurations
+consume 16.50% more EDP than FRAM-based configurations, as shown in figure 9. Flash consumes
+more energy, as shown in table 3.
+6.5
+Summary of the Proposed Mapping Technique
+We outline the proposed ILP-based memory mapping technique in this section. Following all
+of these analyses, we observed that the mappings shown below consume less EDP than other
+design choices, as shown in the table. To keep things simple, we only showed the final mapping
+
+Mapi-Pro: An Energy Efficient Memory Mapping Technique for Intermittent Computing
+17
+0
+0.2
+0.4
+0.6
+0.8
+1
+16bit_2dim
+aes
+basicmath_small
+basicmath_large
+bf
+crc
+dhrystone
+dijkstra
+fft
+fir
+matrix_mult
+patricia
+qsort_small
+qsort_large
+sha
+susan
+Normalized EDP (Normalized with 
+MSP430F5529)
+Benchmarks
+MSP430F5529
+MSP430FR6989
+Fig. 9. Comparison between MSP430FR6989 (FRAM-based MCU) and MSP430F5529 (Flash-based MCU)
+under Unstable Power
+configurations for each application’s stack, data, and text sections, keeping out the final mappings
+for global variables.
+Table 5. Optimal Placement for different Applications in MSP430FR6989
+Benchmarks
+Stack
+Text
+Data
+16bit_2dim
+SRAM
+SRAM
+SRAM
+aes
+SRAM
+FRAM
+FRAM
+basicmath_small
+SRAM
+SRAM
+FRAM
+basicmath_large
+SRAM
+FRAM
+FRAM
+bf
+SRAM
+SRAM
+FRAM
+crc
+SRAM
+FRAM
+SRAM
+dhrystone
+FRAM
+SRAM
+FRAM
+dijkstra
+SRAM
+FRAM
+SRAM
+fft
+SRAM
+SRAM
+FRAM
+fir
+SRAM
+SRAM
+FRAM
+matrix_mult
+SRAM
+SRAM
+SRAM
+patricia
+SRAM
+FRAM
+SRAM
+qsort_small
+SRAM
+SRAM
+FRAM
+qsort_large
+SRAM
+FRAM
+FRAM
+sha
+SRAM
+FRAM
+FRAM
+susan
+SRAM
+FRAM
+FRAM
+Table 5 shows that, with the exception of the dhrystone application, the remaining three TI
+benchmark applications (fir, matrix, and 16bit_2dim) are very small and can easily be placed in SRAM.
+We don’t need FRAM for these types of smaller applications, but there is a disadvantage during
+frequent power failures. Backup and restore sizes to FRAM are larger for these applications during
+frequent power failures. As a result, our proposed backup/restore strategy should be intelligent
+
+18
+S.J Badri, et al.
+enough to reduce EDP. The dhrystone application, on the other hand, has a larger stack section
+that requires FRAM to accommodate the entire stack section.
+As we can see from the table 5, many applications used both SRAM and FRAM for the Mi-
+Bench applications. As a result, we can conclude that a hybrid main memory design is required
+for many applications. Using a hybrid main memory design helps to reduce EDP during stable
+power scenarios. Even so, determining how and where to backup the volatile contents can be
+difficult during frequent power outages. However, our proposed memory mapping technique and
+the framework suggest using a hybrid main memory design that supports intermittent computing.
+7
+CONCLUSIONS
+This paper proposed an ILP-based memory mapping technique that reduces the system’s energy-
+delay product. For both global variables and functions, we formulated an ILP model. Functions
+consist of data, stack, and code sections. Our ILP model suggests placing each section on either SRAM
+or FRAM. Under both stable and unstable power scenarios, we compared the proposed memory
+configuration to the baseline memory configurations. We evaluated our proposed frameworks and
+techniques on actual boards. We added a backup region in FRAM to support intermittent computing.
+We compared the proposed framework with the recent related work.
+Under stable power, our proposed memory configuration consumes 38.10% less EDP than baseline-
+1 and 9.30% less EDP than the existing work. Under unstable power, our proposed configuration
+achieves 15.97% less EDP than baseline-1 and 21.99% less EDP than the existing work. Under stable
+power, our proposed memory configuration consumes 18.55% less EDP than baseline-4. We also
+compared FRAM-based MSP430FR6989 with flash-based MSP430F5529. Compared to the flash,
+the FRAM-based hybrid main memory design consumes less EDP. FRAM-based design consumes
+26.03% less EDP than flash-based design during stable power and 16.50% less EDP than flash based
+during frequent power failures.
+REFERENCES
+[1] Prasenjit Chakraborty, Preeti Ranjan Panda, and Sandeep Sen. 2016. Partitioning and Data Mapping in Reconfigurable
+Cache and Scratchpad Memory–Based Architectures. ACM Transactions on Design Automation of Electronic Systems
+(TODAES) 22, 1 (2016), 1–25.
+[2] Yiran Chen, Weng-Fai Wong, Hai Li, and Cheng-Kok Koh. 2011. Processor caches built using multi-level spin-transfer
+torque ram cells. In IEEE/ACM International Symposium on Low Power Electronics and Design. IEEE, 73–78.
+[3] Yu-Ting Chen, Jason Cong, Hui Huang, Bin Liu, Chunyue Liu, Miodrag Potkonjak, and Glenn Reinman. 2012. Dynami-
+cally reconfigurable hybrid cache: An energy-efficient last-level cache design. In 2012 Design, Automation & Test in
+Europe Conference & Exhibition (DATE). IEEE, 45–50.
+[4] Yu-Der Chih, Yi-Chun Shih, Chia-Fu Lee, Yen-An Chang, Po-Hao Lee, Hon-Jarn Lin, Yu-Lin Chen, Chieh-Pu Lo,
+Meng-Chun Shih, Kuei-Hung Shen, et al. 2020. 13.3 a 22nm 32Mb embedded STT-MRAM with 10ns read speed, 1M
+cycle write endurance, 10 years retention at 150 c and high immunity to magnetic field interference. In 2020 IEEE
+International Solid-State Circuits Conference-(ISSCC). IEEE, 222–224.
+[5] Ju-Hee Choi and Gi-Ho Park. 2017. NVM way allocation scheme to reduce NVM writes for hybrid cache architecture
+in chip-multiprocessors. IEEE Transactions on Parallel and Distributed Systems 28, 10 (2017), 2896–2910.
+[6] Tim Daulby, Anand Savanth, Alex S Weddell, and Geoff V Merrett. 2020. Comparing NVM technologies through
+the lens of Intermittent computation. In Proceedings of the 8th International Workshop on Energy Harvesting and
+Energy-Neutral Sensing Systems. 77–78.
+[7] Jean-Francois Deverge and Isabelle Puaut. 2007. WCET-directed dynamic scratchpad memory allocation of data. In
+19th Euromicro Conference on Real-Time Systems (ECRTS’07). IEEE, 179–190.
+[8] Bernhard Egger, Jaejin Lee, and Heonshik Shin. 2008. Dynamic scratchpad memory management for code in portable
+systems with an MMU. ACM Transactions on Embedded Computing Systems (TECS) 7, 2 (2008), 1–38.
+[9] Hêriş Golpîra, Syed Abdul Rehman Khan, and Sina Safaeipour. 2021. A review of logistics internet-of-things: Current
+trends and scope for future research. Journal of Industrial Information Integration (2021), 100194.
+[10] Ricardo Gonzalez and Mark Horowitz. 1996. Energy dissipation in general purpose microprocessors. IEEE Journal of
+solid-state circuits 31, 9 (1996), 1277–1284.
+
+Mapi-Pro: An Energy Efficient Memory Mapping Technique for Intermittent Computing
+19
+[11] Laura M Grupp, Adrian M Caulfield, Joel Coburn, Steven Swanson, Eitan Yaakobi, Paul H Siegel, and Jack K Wolf. 2009.
+Characterizing flash memory: Anomalies, observations, and applications. In Proceedings of the 42nd Annual IEEE/ACM
+International Symposium on Microarchitecture. 24–33.
+[12] Yibo Guo, Qingfeng Zhuge, Jingtong Hu, Juan Yi, Meikang Qiu, and Edwin H-M Sha. 2013. Data placement and
+duplication for embedded multicore systems with scratch pad memory. IEEE Transactions on Computer-Aided Design of
+Integrated Circuits and Systems 32, 6 (2013), 809–817.
+[13] Matthew R Guthaus et al. 2001. MiBench: A free, commercially representative embedded benchmark suite. In Proceedings
+of the fourth annual IEEE international workshop on workload characterization. IEEE, IEEE, 3–14.
+[14] Josiah Hester and Jacob Sorber. 2017. The future of sensing is batteryless, intermittent, and awesome. In Proceedings of
+the 15th ACM conference on embedded network sensor systems. 1–6.
+[15] Xiaosong Hu, Le Xu, Xianke Lin, and Michael Pecht. 2020. Battery lifetime prognostics. Joule 4, 2 (2020), 310–346.
+[16] Texas Instruments. 2018. MSP430FR5969 launchpad development kit.
+[17] Pulkit Jain, Umut Arslan, Meenakshi Sekhar, Blake C Lin, Liqiong Wei, Tanaya Sahu, Juan Alzate-Vinasco, Ajay
+Vangapaty, Mesut Meterelliyoz, Nathan Strutt, et al. 2019. 13.2 A 3.6 Mb 10.1 Mb/mm 2 embedded non-volatile ReRAM
+macro in 22nm FinFET technology with adaptive forming/set/reset schemes yielding down to 0.5 V with sensing time
+of 5ns at 0.7 V. In 2019 IEEE International Solid-State Circuits Conference-(ISSCC). IEEE, 212–214.
+[18] Hrishikesh Jayakumar, Arnab Raha, and Vijay Raghunathan. 2014. QuickRecall: A low overhead HW/SW approach for
+enabling computations across power cycles in transiently powered computers. In 2014 27th International Conference on
+VLSI Design and 2014 13th International Conference on Embedded Systems. IEEE, 330–335.
+[19] Hrishikesh Jayakumar, Arnab Raha, and Vijay Raghunathan. 2016. Energy-aware memory mapping for hybrid FRAM-
+SRAM MCUs in IoT edge devices. In 2016 29th International Conference on VLSI Design and 2016 15th International
+Conference on Embedded Systems (VLSID). IEEE, 264–269.
+[20] Hrishikesh Jayakumar, Arnab Raha, Jacob R Stevens, and Vijay Raghunathan. 2017. Energy-aware memory mapping
+for hybrid FRAM-SRAM MCUs in intermittently-powered IoT devices. ACM Transactions on Embedded Computing
+Systems (TECS) 16, 3 (2017), 1–23.
+[21] Morteza Mohajjel Kafshdooz and Alireza Ejlali. 2015. Dynamic shared SPM reuse for real-time multicore embedded
+systems. ACM Transactions on Architecture and Code Optimization (TACO) 12, 2 (2015), 1–25.
+[22] Mahmut Kandemir, J Ramanujam, Mary Jane Irwin, Narayanan Vijaykrishnan, Ismail Kadayif, and Amisha Parikh.
+2001. Dynamic management of scratch-pad memory space. In Proceedings of the 38th Design Automation Conference
+(IEEE Cat. No. 01CH37232). IEEE, 690–695.
+[23] Mirae Kim, Jungkeol Lee, Youngil Kim, and Yong Ho Song. 2018. An analysis of energy consumption under various
+memory mappings for FRAM-based IoT devices. In 2018 IEEE 4th World Forum on Internet of Things (WF-IoT). IEEE,
+574–579.
+[24] Markus Koesler and Franz Graf. 2002. Programming a flash-based MSP430 using the JTAG Interface. SLAA149, TEXAS
+INSTRUMENTS (2002), 4–13.
+[25] Benjamin C Lee, Engin Ipek, Onur Mutlu, and Doug Burger. 2009. Architecting phase change memory as a scalable
+dram alternative. In Proceedings of the 36th annual international symposium on Computer architecture. 2–13.
+[26] Lian Li, Hui Feng, and Jingling Xue. 2009. Compiler-directed scratchpad memory management via graph coloring.
+ACM Transactions on Architecture and Code Optimization (TACO) 6, 3 (2009), 1–17.
+[27] Brandon Lucia, Vignesh Balaji, Alexei Colin, Kiwan Maeng, and Emily Ruppel. 2017. Intermittent computing: Challenges
+and opportunities. 2nd Summit on Advances in Programming Languages (SNAPL 2017) (2017).
+[28] Sheel Sindhu Manohar and Hemangee K Kapoor. 2022. CAPMIG: Coherence Aware block Placement and MIGration in
+Multi-retention STT-RAM Caches. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems
+(2022).
+[29] Sparsh Mittal and Jeffrey S Vetter. 2015. A survey of software techniques for using non-volatile memories for storage
+and main memory systems. IEEE Transactions on Parallel and Distributed Systems 27, 5 (2015), 1537–1550.
+[30] Keni Qiu, Mengying Zhao, Zhenge Jia, Jingtong Hu, Chun Jason Xue, Kaisheng Ma, Xueqing Li, Yongpan Liu, and
+Vijaykrishnan Narayanan. 2020. Design insights of non-volatile processors and accelerators in energy harvesting
+systems. In Proceedings of the 2020 on Great Lakes Symposium on VLSI. 369–374.
+[31] Arindam Sarkar, Newton Singh, Varun Venkitaraman, and Virendra Singh. 2021. DAM: Deadblock Aware Migration
+Techniques for STT-RAM-Based Hybrid Caches. IEEE Computer Architecture Letters 20, 1 (2021), 62–4.
+[32] Sophiane Senni, Lionel Torres, Gilles Sassatelli, and Abdoulaye Gamatie. 2016. Non-volatile processor based on MRAM
+for ultra-low-power IoT devices. ACM Journal on Emerging Technologies in Computing Systems (JETC) 13, 2 (2016),
+1–23.
+[33] Vivy Suhendra, Chandrashekar Raghavan, and Tulika Mitra. 2006. Integrated scratchpad memory optimization and
+task scheduling for MPSoC architectures. In Proceedings of the 2006 international conference on Compilers, architecture
+and synthesis for embedded systems. 401–410.
+
+20
+S.J Badri, et al.
+[34] Milijana Surbatovich, Brandon Lucia, and Limin Jia. 2020. Towards a formal foundation of intermittent computing.
+Proceedings of the ACM on Programming Languages 4, OOPSLA (2020), 1–31.
+[35] Sumesh Udayakumaran, Angel Dominguez, and Rajeev Barua. 2006. Dynamic allocation for scratch-pad memory
+using compile-time decisions. ACM Transactions on Embedded Computing Systems (TECS) 5, 2 (2006), 472–511.
+[36] Manish Verma, Lars Wehmeyer, and Peter Marwedel. 2004. Dynamic overlay of scratchpad memory for energy
+minimization. In Proceedings of the 2nd IEEE/ACM/IFIP international conference on Hardware/software codesign and
+system synthesis. 104–109.
+[37] Mimi Xie, Chen Pan, and Chun Jason Xue. 2018. A Novel STT-RAM-Based Hybrid Cache for Intermittently Powered
+Processors in IoT Devices. IEEE Micro 39, 1 (2018), 24–32.
+

9NE4T4oBgHgl3EQfdgy1/content/tmp_files/2301.05092v1.pdf.txt

@@ -0,0 +1,3087 @@
+arXiv:2301.05092v1  [gr-qc]  12 Jan 2023
+,
+Slowly rotating Kerr metric derived from the Einstein equations in affine-null
+coordinates
+Thomas M¨adler
+∗
+Escuela de Obras Civiles and Instituto de Estudios Astrof´ısicos,
+Facultad de Ingenier´ıa y Ciencias, Universidad Diego Portales,
+Avenida Ej´ercito Libertador 441, Casilla 298-V, Santiago, Chile.
+Emanuel Gallo
+†
+FaMAF, UNC; Instituto de F´ısica Enrique Gaviola (IFEG), CONICET,
+Ciudad Universitaria, (5000) C´ordoba, Argentina.
+Using a quasi-spherical approximation of an affine-null metric adapted to an asymptotic Bondi
+inertial frame, we present high order approximations of the metric functions in terms of the specific
+angular momentum for a slowly rotating stationary and axi-symmetric vacuum spacetime.
+The
+metric is obtained by following the procedure of integrating the hierarchy of Einstein equations in a
+characteristic formulation utilizing master functions for the perturbations. It is further verified its
+equivalence with the Kerr metric in the slowly rotation approximation by carrying out an explicit
+transformation between the Boyer-Lindquist coordinates to the employed affine-null coordinates.
+PACS numbers:
+I.
+INTRODUCTION
+At the dawn of the ’Golden Era of General Relativity’
+in the 60ties of the last century, two important space-
+time metrics were found, the Bondi-Sachs metric [1–3]
+and the Kerr metric [4, 5]. The first settled the question
+that an isolated system looses mass via gravitational radi-
+ation and that this effect is a non-linear effect of General
+Relativity; while the second describes a stationary and
+rotating isolated black hole that is expected to be the
+end product of a gravitational collapse of a massive star
+or a merger of two compact objects.
+One of the defining features of the Bondi-Sachs metric
+is that one coordinate is constant along a family of null
+hypersurfaces while a radial coordinate along these null
+hypersurfaces is an areal distance that can be related to a
+luminosity distance [6]. As such, the first long term sta-
+ble evolution of black hole space times were made using a
+Bondi-Sachs metric in a null cone-world tube formalism
+[7], also see [8, 9] for review. Apart from usage in numer-
+ical relativity simulations, the Bondi-Sachs metric is now
+frequently used in high energy physics addressing ques-
+tions of the AdS/CFT correspondence [10] (and citations
+thereof). It also became popular to discuss gravitational
+wave memory effects [11–15]. A pleasant property of the
+Bondi-Sachs formalism is that the Einstein equation can
+be solved in a hierarchical manner when initial data on
+a null hypersurface and boundary conditions at a world
+tube or vertex are given. However, the radial coordinate
+of the Bondi-Sachs metric has the unpleasant property
+∗Electronic address: thomas.maedler˙.at.˙mail.udp.cl
+†Electronic address: egallo˙.at.˙unc.edu.ar
+that it breaks down when an apparent horizon forms due
+to the focusing of the surface-forming null rays and their
+vanishing expansion.
+This can be overcome in choos-
+ing an affine parameter as radial coordinate, because an
+affine parameter only becomes singular at a caustic. But,
+the Einstein equations resulting from an affine-null met-
+ric do not provide the hierarchical structure as the Bondi
+Sachs metric [9] and the hierarchical structure needs to
+be reestablished by various new definitions of variables
+[16–18]. Moreover, it turns out that also the hierarchy
+of equations in the affine-null metric formulation breaks
+down in the events of apparent horizon formation, but
+fortunately the equations can be regularized so that it is
+possible to follow up the formation of black holes up to
+singularity [19, 20].
+Despite the success and popularity of the Bondi-Sachs
+metric in the various areas, an explicit closed analytical
+representation of the Kerr metric in Bondi-Sachs form
+without bad behaviour in the exterior region or related
+metrics with one or two null coordinates is missing. Var-
+ious attempts have been made to derive a null metric
+representation, numerically [21, 22] as well as analyti-
+cally [23–25]. In all of the approaches, the authors start
+out with the Kerr metric and then calculate the respec-
+tive null metric via a coordinate transformation. After
+these transformations the resulting metric can still posses
+a conical singularity at the axis of symmetry (see [22]
+for a complete discussion). In addition, the final met-
+ric is determined by integrals of non-elementary func-
+tions. Arga˜naraz and Moreschi’s approach [22] differs to
+the aforementioned ones that the authors aim to find a
+double–null representation of the Kerr metric by geomet-
+rically adopting the coordinates to in- and outgoing null
+geodesics adapted to the center of mass [27]. In this way,
+the authors were successful in finding null coordinates
+
+2
+that are not only regular at every point of the external
+communication region (unlike the previous formulations)
+but also that they are regular at the event horizon, thus
+allowing a way to study the evolution of different matter
+fields (as scalar fields) in such background even when they
+cross the event horizon[26]. Unfortunately, even in their
+construction arises a differential equation that needs to
+be solved numerically and an explicit closed form repre-
+sentation of the double null version of the Kerr metric is
+not possible. The work of Bai and collaborators [28, 29]
+also starts with the Kerr metric (in Boyer-Lindquist co-
+ordinates) and then makes coordinate transformation to
+a Bondi-Sachs metric valid near future null infinity (in a
+compactified version of the metric). The authors are able
+to calculate the Newman-Penrose quantities and multi-
+poles at large distances and show the peeling property of
+the Weyl tensor at large radii and the vanishing of the
+so-called Newman-Penrose constants.
+In this article, in contrast to all the previous works
+which start with the Kerr metric expressed in Boyer-
+Lindquist coordinates and attempt to find a null coordi-
+nate version of it, we will directly solve the Einstein equa-
+tions in a characteristic formulation based on an affine-
+null metric formulation of the Einstein equations. In ad-
+dition, inspired by the Hartle-Thorne methods for obtain-
+ing solutions for slowly rotating compact stars [30], we
+will employ a quasi-spherical approximation of the field
+equations to find a high order approximation of the Kerr
+metric in out-going polar null coordinates.
+To obtain
+our solution, we assume stationarity and axial symme-
+try. We further require an asymptotic inertial observer
+as well as that that Weyl scalar Ψ0 is regular everywhere
+where the background solution is regular. A study of vac-
+uum stationary metrics with a smooth future null infinity
+in affine-null coordinates has recently be carried out by
+Tafel in [31] by considering power series of the metric
+components in terms of the inverse affine distance.
+Throughout the article, we will use signature +2, units
+G = c = 1 and the Einstein sum convention for indices
+as well as products of associated Legendre polynomials.
+The article is organised as follows:
+Sec. II recalls
+the affine-null metric formulation, makes the necessary
+symmetry assumptions for archiving our goal and de-
+fines the perturbative variables; in Sec. III, we determine
+the background model (Sec. III A), define useful recur-
+sively re-appearing functions in the perturbation analysis
+(Sec. III B), solve the perturbation equations (Sec. III C-
+III F) and in Sec. III G the affine-null metric functions
+for the null are expressed in terms of the mass and spe-
+cific angular momentum, in Sec. IV, to verify our re-
+sults, we calculate the affine-null version of Kerr metric
+in a Bondi frame via a coordinate transformation with a
+method adopted from [28], in Sec. V position of the outer
+ergosphere and event (past) horizon of the black hole we
+discuss the and Sec. VI contains the final discussion of our
+work. The article finishes with two appendices: App. A
+lists relations between associated Legendre polynomials
+and App. B presents a derivation of the expression of the
+Komar charges relevant for this work.
+II.
+AFFINE-NULL METRIC FORMULATION
+FOR STATIONARY AND AXIAL SYMMETRIC
+SPACETIMES
+Here we review the necessary properties of character-
+istic initial value formulation of the Einstein equations
+in affine-null coordinates, discuss the implications of the
+imposed symmetry assumptions and present the notation
+used in our analysis.
+Taking coordinates xa = (u, λ, xA), where u is an out–
+going null coordinate, λ an affine parameter, and xA are
+angular coordinates, a generic line element for an affine-
+null metric defined with respect to a family of outgoing
+null hypersurfaces u = const is [16–18, 32]
+gabdxadxb = −Wdu2 − 2dudλ
++R2hAB(dxA − W Adu)(dxB − W Bdu). (2.1)
+The determinant det(hAB) = det(qAB) = sin2 θ is the
+determinant of a round unit sphere metric qAB.
+Con-
+sequently hAB is transverse-traceless and has only two
+degrees of freedom. Thus, the function R relates to the
+area of cuts du = dλ = 0. The inverse metric is given by
+guλ = −1 , gλλ = W , gλA = −W A , gAB = hAB
+R2 ,
+(2.2)
+where W A = (W θ, W φ) and hABhBC = δC
+A and in par-
+ticular [33]
+hABdxAdxB =
+�
+e2γdθ2 + sin2 θ
+e2γ dφ2�
+cosh(2δ)
++2 sinθ sinh(2δ)dθdφ .
+(2.3)
+A complex null dyad to represent the 2-metric hAB like
+hAB = m(A ¯mB) with mAmBhAB = mA ¯mBhAB − 1 = 0
+is
+mA∂A =
+1
+√
+2eγ
+�
+cosh δ − i sinh δ
+�
+∂y
++
+ieγ
+√
+2 sin θ
+�
+cosh δ + i sinh δ
+�
+∂φ,
+(2.4)
+Like in any Bondi-Sachs type metric [9], the vacuum field
+equations Rab = 0 with Rab being the Ricci tensor can
+be grouped into supplementary equations Si = 0 with
+Si = (Ruu, Ruθ, Ruφ),
+(2.5)
+one trivial equation Ruλ = 0 and the six main equations
+H(γ)
+K
+= 0, K ∈ (1, 2, 3, 4)) and H(δ)
+k
+= 0, k ∈ (1, 2)) with
+H(γ)
+K
+=
+�
+Rλλ, Rλθ, hABRAB, ℜe(mAmBRAB)
+�
+,
+H(δ)
+k
+=
+�
+Rλφ, ℑm(mAmBRAB)
+�
+,
+(2.6)
+with ℜe(x) and ℑm(x) the real an imaginary part of x re-
+spectively. We assume that the spacetime is axisymmet-
+ric and stationary with associated Killing vectors fields
+
+3
+∂u and ∂φ. Therefore the metric functions do not depend
+on u and φ. The Killing symmetries imply two conserved
+quantities, the Komar mass, Km, and the Komar angu-
+lar momentum, KL, which can be calculated from their
+respective integrals (also see App. B)
+Km := K(∂u) = 1
+8π lim
+λ→∞
+� �
+W,λ−R2hABW AW B
+,λ
+�
+R2d2q
+(2.7)
+while for the axial Killing vector we have
+KL := K(∂φ) = − 1
+16π lim
+λ→∞
+� �
+R4hφBW B
+,λ
+�
+d2q
+(2.8)
+where dq = sin θdθdφ is the surface area element of the
+unit sphere.
+Let us assume there is a smooth one parameter family
+of stationary and axially symmetric metrics gab(ε), where
+ε is a small dimensionless parameter such that ε = 0 is a
+corresponds to a (static) spherically symmetric spacetime
+solution of the vacuum Einstein equations. Then there is
+an expansion of the metric fields like
+R(λ, θ) = r(λ) + R[1](λ, θ)ε + R[2](λ, θ)ε2 + R[3](λ, θ)ε3 + O(ε4),
+(2.9a)
+W(λ, θ) = V (λ) + W[1](λ, θ)ε + W[2](λ, θ)ε2 + W[3](λ, θ)ε3 + O(ε4),
+(2.9b)
+W A(λ, θ) = W A
+[1](λ, θ)ε + W A
+[2](λ, θ)ε2 + W A
+[3](λ, θ)ε3 + O(ε4),
+(2.9c)
+γ(λ, θ) = γ[1](λ, θ)ε + γ[2](λ, θ)ε2 + γ[3](λ, θ)ε3 + O(ε4),
+(2.9d)
+δ(λ, θ) = δ[1](λ, θ)ε + δ[2](λ, θ)ε2 + δ[3](λ, θ)ε3 + O(ε4).
+(2.9e)
+Inserting (2.9) in (2.7) and (2.8) implies Km = O(ε0) and
+KL = O(ε). We make the requirements
+Km(ε) = Km(−ε) , KL(ε) = −KL(−ε).
+(2.10)
+These
+conditions
+imply
+that
+under
+the
+change
+ε → −ε the sense of rotation is reversed (recall that
+K(∂φ) = −K(∂(−φ))).
+From the metric (2.1), we see
+that the 2-surfaces with u = u0 and λ = λ0, defined
+such that R(u0, λ0, θ) =const have the induced metric
+R2hABdxAdxB with area 4πR2(u0, λ0).
+We assume
+that the area of these 2-surfaces is invariant under
+the change ε → −ε, which implies that R2 is an even
+function of ε. Therefore R is either an even or an odd
+function of ε.
+However, if R were an odd function,
+we had R(ε = 0) = 0, which is a non admissible solu-
+tion.
+In addition, ds2(∂φ, ∂φ) and ds2(∂θ, ∂θ) must be
+independent of the sense of rotation implying that hφφ
+and hθθ are even. However, due to the frame dragging
+effect ds2(∂θ, ∂φ) must depend on the sense of rotation.
+Therefore hθφ is an odd function of ε.
+Using similar
+arguments, because the Komar angular momentum KL
+is an odd function of ε and taking into account (2.8)
+and the parity behaviour of hAB and R2, we have that
+W θ is even and W φ odd. Similarly, since Km must be a
+even function of ε, W must be even in ε. Therefore,
+R[2n+1] = W[2n+1] = 0,
+(2.11a)
+W θ
+[2n+1] = 0,
+(2.11b)
+W φ
+[2n] = 0,
+(2.11c)
+γ[2n+1] = δ[2n] = 0.
+(2.11d)
+To arrive at the last conditions (2.11d) we have taken
+into account the odd parity of hθφ, which gives us
+sinh(δ(ε)) = − sinh(δ(−ε)).
+Hence, δ must be odd in
+ε. Similarly, for hθθ and hφφ be even, γ(ε) must satisfies
+e2γ(ε) = e2γ(−ε), which implies that γ is a even function
+of ε.
+We conclude
+R = r + R[2]ε2 + +R[4]ε4 + O(ε6),
+(2.12a)
+W = V + W[2]ε2 + W[4]ε4 + O(ε6),
+(2.12b)
+W θ = W θ
+[2]ε2 + W θ
+[4]ε4 + O(ε4),
+(2.12c)
+W φ = W φ
+[1]ε + W φ
+[3]ε3 + O(ε5),
+(2.12d)
+γ = γ[2]ε2 + γ[4]ε4 + O(ε6),
+(2.12e)
+δ = δ[1]ε + δ[3]ε3 + O(ε5).
+(2.12f)
+A similar expansion was made by Hartle [30] in the
+derivation of a metric for slowly rotating stars using a
+a 3+1 decomposition of the metric. From (2.9) follows
+that the Ricci tensor has the expansions
+Rab = R[0]ab + R[1]abε + R[2]abε2 + R[3]abε3 + ... (2.13)
+In fact, with the notation f[i] ∈ {γ[i], δ[i], R[i], W A
+[i], W[i]},
+it turns out for a perturbation at order n > 1 that
+S[n]i = ˆSi(f[n]) + s[i](f[m<n])
+(2.14)
+H(γ)
+K
+= ˆH(γ)
+K (f[n]) + h(γ)
+K (f[m<n])
+(2.15)
+H(δ)
+k
+= ˆH(δ)
+k (f[n]) + h(δ)
+k (f[m<n])
+(2.16)
+where ˆSi, ˆH(γ)
+K
+and ˆH(δ)
+k
+are linear differential operators
+of the indicated arguments. The functions s[i], h(γ)
+K
+and
+
+4
+h(δ)
+k
+are nonlinear functions of the lower order perturba-
+tions f[m] for m < n.
+For the computations, it is useful to change the an-
+gular coordinate according to y = − cosθ, introduce
+s(y) =
+�
+1 − y2 and transform W θ = s−1W y.
+In ad-
+dition, for a perturbation at order n it will shown useful
+to make the following decomposition of the perturbation
+f[n] in terms of associated Legendre polynomials, P m
+ℓ (y),
+R[n](λ, y) =R[n.ℓ](λ)P 0
+ℓ (y)
+(2.17)
+W y
+[n](λ, y) =W φ
+[n.ℓ](λ)
+�
+s(y)P 1
+ℓ (y)
+�
+(2.18)
+W φ
+[n](λ, y) =W φ
+[n.ℓ](λ)
+�P 1
+ℓ (y)
+s(y)
+�
+(2.19)
+W[n](λ, y) =W[n,ℓ](λ)P 0
+ℓ (y)
+(2.20)
+γ[n](λ, y) =γ[n.ℓ](λ)P 2
+ℓ (y)
+(2.21)
+δ[n](λ, y) =δ[n.ℓ](λ)P 2
+ℓ (y) .
+(2.22)
+We remark that this decomposition with respect to the
+associated Legendre polynomials is in fact a decomposi-
+tion in terms of axi-symmetric spin-weighed harmonics
+(up to normalisation) obtained by setting m = 0 in the
+standard sYℓm(y, φ).
+III.
+SOLUTION OF THE BACKGROUND AND
+PERTURBATION EQUATIONS
+A.
+Solution background equations
+The main equations for the background model are
+0 = r,λλ
+r
+(3.1a)
+0 = [(r2),λV − 2λ],λ ,
+(3.1b)
+From which we deduce the simple relations
+r(λ) = r1(λ − λ0) + r0,
+(3.2)
+V (λ) = 2(λ − λ0) + 2r0r1V0 + A
+2rr1
+,
+(3.3)
+where A is a constant of integration. Since we have the
+freedom of rescaling the affine parameter λ → αλ+β, we
+can take without loss of generality
+r(λ) = λ , V (λ) = 1 − A
+2λ.
+(3.4)
+The resulting spacetime is the Schwarzschild metric in
+Eddington-Finkelstein coordinates, with a total Bondi
+mass m0 related to the integration constant A by A =
+4m0.
+Moreover, λ = A/2 corresponds to location the
+horizon of the Schwarzschild horizon.
+B.
+Recurrent operators in the equations of the
+perturbations
+Assuming a formal expansion like
+R(λ, y) ≈ λ + R[n](λ, y)εn
+(3.5a)
+W(λ, y) ≈ 1 − A
+2λ + W[n](λ, y)εn,
+(3.5b)
+W A(λ, y) ≈ W A
+[n](λ, y)εn,
+(3.5c)
+γ(λ, y) ≈ γ[n](λ, y)εn,
+(3.5d)
+δ(λ, y) ≈ δ[n](λ, y)εn.
+(3.5e)
+Then
+the
+Ricci
+tensor
+also
+has
+the
+expansion
+Rab ≈ Rab[n]εn after dropping terms of higher order.
+The O(εn) coefficients of the supplementary equations
+gives
+ˆS1(R[n], W[n], W y
+[n]) =
+1
+2λ2
+�
+1 − A
+2λ
+� �
+λ2W[n],λ + AR[n]
+λ
+�
+,λ
++ (s2W[n],y),y
+2λ2
+− A
+4λ2 W y
+[n],y
+(3.6a)
+ˆS2(W[n], W y
+[n]) =
+1
+2λ2
+�
+1 − A
+2λ
+� (λ4W y
+[n]),λ
+s
++ s
+2W[n],λy +
+W y
+[n]
+s
+(3.6b)
+ˆS3(W φ
+[n]) =
+s2
+2λ2
+�
+1 − A
+2λ
+�
+(λ4W φ
+[n],λ),λ + 1
+2
+�
+s4W φ
+[n],y
+�
+,y
+(3.6c)
+
+5
+Those for ˆH(γ)
+K
+are
+ˆH(γ)
+1
+(R[n]) = − 2
+λR[n],λλ
+(3.7a)
+ˆH(γ)
+2
+(R[n], γ[n], W y
+[n]) =
+1
+2λ2
+(λ4W y
+[n],λ),λ
+s
+− s
+�R[n],y
+λ
+�
+,λ
++ (γ[n],λs2),y
+s
+(3.7b)
+ˆH(γ)
+3
+(R[n], γ[n], W y
+[n], W[n]) = −
+�
+λW[n]
+�
+,λ −
+��
+1 − A
+2λ
+�
+(λR[n]),λ
+�
+,λ
+−
+�
+λ[n],ys2�
+,y
+λ
++
+�
+λ4W y
+[n]
+�
+,λy
+2λ2
++(γ[n],ys4),y
+s2
+− 2γ[n]
+(3.7c)
+ˆH(γ)
+4
+(γ[n], W y
+[n]) = −
+�
+λ
+�
+λ − A
+2
+�
+γ[n],λ
+�
+,λ
++ s2
+2
+�
+λ2W y
+[n]
+s2
+�
+,λy
+(3.7d)
+and those for ˆH(δ)
+k
+ˆH(δ)
+1 (δ[n], W φ
+[n]) =
+s2
+2λ2 (λ4W φ
+[n],λ),λ + (δ[n],λs2),y
+(3.8a)
+ˆH(δ)
+2 (δ[n], W φ
+[n]) = −
+�
+λ
+�
+λ − A
+2
+�
+δ[n],λ
+�
+,λ
+− s2
+2
+�
+λ2W φ
+[n]
+�
+,λy
+(3.8b)
+We observe that (3.7b) and (3.7d) as well as (3.8a) and
+(3.8b) can be combined (see e.g. in [34]) to two fourth
+order (master) equations
+0 =M(γ[n]) − s2R[n],λλyy
+(3.9a)
+0 =M(δ[n])
+(3.9b)
+where
+M(F) := 1
+λ2 [λ4(λF),λλλ],λ + [(λF),λλys2],y
++
+� A
+2λ + 2 − 4
+s2
+�
+(λF),λλ − A
+2 [λ(λF),λλλ],λ
+(3.10)
+We emphasize that Eqs. (3.9a) and (3.9b) (similarly to
+the Teukolsky master equations in 3+1 perturbation the-
+ory) are the key equations to solve the system, because
+they provide the initial data γ[n] or δ[n] needed to in-
+tegrate the hypersurface equations of the characteristic
+initial value problem.
+C.
+First order perturbations
+Since γ[1], R[1], W y
+[1] and W[1] are zero, we only have
+to consider the equations
+0 = ˆS3(δ[1], W φ
+[1])
+(3.11)
+0 = ˆH(δ)
+1 (δ[1], W φ
+[1])
+(3.12)
+0 = ˆH(δ)
+2 (δ[1], W φ
+[1])
+(3.13)
+whose explicit form can be read off from (3.6c), (3.8a)
+and (3.8b). The corresponding master equation is
+0 = 1
+λ2 [λ4(λδ[1]),λλλ],λ + [(λδ[1]),λλys2],y
++
+� A
+2λ + 2 − 4
+s2
+�
+(λδ[1]),λλ − A
+2
+�
+λ(λδ[1]),λλλ
+�
+,λ
+(3.14)
+which is in fact a second order equation for the variable
+ψ[1] := (λδ[1]),λλ ,
+(3.15)
+namely
+0 =λ(2λ − A)ψ[1],λλ + (8λ − A)ψ[1],λ + Aψ[1]
+λ
++ 2
+�
+(s2ψ[1],y),y +
+��
+2 − 4
+s2
+��
+ψ[1]
+�
+,
+(3.16)
+which admit a solution by separation of variables by set-
+ting ψ[1](λ, y) = p[1](λ)S(y),
+0 =λ(2λ − A)p[1],λλ + (8λ − A)p[1],λ +
+�A
+λ + 2k
+�
+p[1],
+(3.17)
+0 = d
+dy
+�
+s2 dS
+dy
+�
++
+�
+2 − k − 4
+s2
+�
+S,
+(3.18)
+with k a constant. Identifying 2 − k = ℓ(ℓ + 1), we see
+that (3.18) is an associated Legendre differential equation
+
+6
+(A3), whose general solution is
+Sℓ(y) = B0kP(ℓ, 2, y) + B1kQ(ℓ, 2, y),
+(3.19)
+where P(·) and Q(·) are the Legendre functions of first
+kind and of second kind, respectively.
+Requiring a regular solution at the poles y = ±1, im-
+poses that ℓ must be a nonnegative integer and B1k = 0,
+because P(ℓ, 2, −1) blows up at the pole y = −1 and
+Q(ℓ, 2, ±1) blows up at the poles y = ±1.
+Then the
+remaining Legendre function P(·) is the associated Leg-
+endre polynomial P 2
+ℓ (y).
+To find a solution for (3.15) and (3.16), we set
+ψ[1](λ, y) = ψ[1.ℓ](λ)P 2
+ℓ (y) , δ[1](λ, y) = δ[1.ℓ](λ)P 2
+ℓ (y),
+(3.20)
+where a sum in ℓ is understood.
+Note that P 2
+0 (y) =
+P 2
+1 (y) = 0 , consequently, δ[1.0] = δ[1.1] = 0 without loss
+of generality. Subsequent insertion into (3.16) while using
+eq.(A3) gives us
+0 = − 1
+2λ(A − 2λ)d2ψ[1.ℓ]
+dλ2
++
+�
+4λ − A
+2
+� dψ[1.ℓ]
+dλ
++
+�
+2 − ℓ(ℓ + 1) + A
+2λ
+�
+ψ[1.ℓ] .
+(3.21)
+Using the parameter transformation x = 4λ
+A − 1, similar
+to [30], we find
+0 =(1 − x)d2ψ[1.ℓ]
+dx2
+− 4x + 2
+x + 1
+dψ[1.ℓ]
+dx
++ ℓ(ℓ + 1)(x + 1) − 2x − 4
+(x + 1)2
+ψ[1.ℓ],
+(3.22)
+which can also be written as
+0 = d
+dy
+�
+(1 − x2) d
+dx(1 − x)ψ[1.ℓ]
+�
++
+�
+ℓ(ℓ + 1) −
+4
+1 − x2
+�
+(1 − x)ψ[1.ℓ].
+(3.23)
+Eq. (3.23) is an associated Legendre differential equation,
+like (A3), with the general solution
+ψ[1.ℓ](x) = B[1.ℓ]P 2
+ℓ (x) + B[2.ℓ]Q2
+ℓ(x)
+1 − x
+(3.24)
+Inverting the parameter transformation from x to λ yields
+the general solution of (3.21) so that using (3.20)
+ψ[1](λ, y) =
+� AB[1.ℓ]
+2A − 4λ
+�
+P 2
+ℓ
+�4λ
+A − 1
+�
+P 2
+ℓ (y)
++
+� AB[2.ℓ]
+2A − 4λ
+�
+Q2
+ℓ
+�4λ
+A − 1
+�
+P 2
+ℓ (y)
+(3.25)
+The field ψ[1] is related to the Weyl scalar Ψ0,
+Ψ0 = −iψ[1]
+λ ε + O(ε2) .
+(3.26)
+Inspection of (3.24) shows that Ψ becomes infinite for
+λ → A/2 and for λ → ∞ if ℓ ≥ 2.
+The first case
+corresponds to the unperturbed location of the horizon
+while the second one corresponds to the asymptotic re-
+gion.
+Consequently, also Ψ0 becomes infinite in these
+cases. We require regularity of the scalar curvature Ψ0
+at these locations, which implies B[1.ℓ] = B[2.ℓ] = 0. This
+leaves us with the trivial solution ψ[1] = 0.
+Integration of (3.15) with this trivial solution while
+using (3.20) yields
+δ[1.ℓ](λ) = Bδ
+[0.ℓ] +
+Bδ
+[1.ℓ]
+λ
+,
+(3.27)
+where as aforementioned, since δ[1.0] = δ[1.1] = 0 we get
+that Bδ
+[0.0] = Bδ
+[0.1] = Bδ
+[1.0] = Bδ
+[1.1] = 0. These modes
+are physically irrelevant because δ[1] is expressed by the
+angular base of P 2
+ℓ −associated Legendre polynomials and
+P 2
+0 = P 2
+1 = 0. Since δ[1] is now known, we are now in
+position to integrate the hypersurface equation (3.12).
+We insert (3.27) into (3.12), while using (A5), to find
+(λ4W φ
+[1],λ),λ = 2λ2
+�dδ[1.ℓ]
+dλ
+� KℓP 1
+ℓ (y)
+s
+.
+(3.28)
+where
+Kℓ = 2 − ℓ(ℓ + 1) = (1 − ℓ)(2 + ℓ).
+(3.29)
+Then setting
+W φ
+[1](λ, y) =W φ
+[1.ℓ](λ)P 1
+ℓ (y)
+s
+,
+(3.30)
+gives us
+d
+dλ
+�
+λ4 W φ
+[1.ℓ]
+dλ
+�
+= 2Bδ
+[1.ℓ]Kℓ;
+(3.31)
+or after integration
+W φ
+[1.ℓ] =Bφ
+[0.ℓ] −
+KℓBδ
+[1.ℓ]
+λ2
+−
+Bφ
+[3.ℓ]
+3λ3 .
+(3.32)
+This give us for the first order axisymmetric perturba-
+tions
+δ[1](λ, y) =
+�
+Bδ
+[0.ℓ] +
+Bδ
+[1.ℓ]
+λ
+�
+P 2
+ℓ (y)
+(3.33)
+W φ
+[1](λ, y) =
+�
+Bφ
+[0.ℓ] −
+KℓBδ
+[1.ℓ]
+λ2
+−
+Bφ
+[3.ℓ]
+3λ3
+�
+P 1
+ℓ (y)
+s
+(3.34)
+Again we set the unphysical modes Bφ
+[0.0] = Bφ
+[3.0] = 0,
+because of behavior of the angular base functions of W φ
+[1].
+Inserting the obtained solutions into (3.13) yields while
+using (A8)
+0 = A
+2
+�
+λδ[1.ℓ],λ
+�
+,λ−
+�
+λ2δ[1.ℓ],λ
+�
+,λ− 1
+2
+�
+λ2W φ
+[1.ℓ]
+�
+,λ (3.35)
+
+7
+and together with (3.27) and (3.32) this gives us for any
+ℓ ≥ 2
+0 = −Bφ
+[0.ℓ]λ + 1
+λ2
+�
+A
+2 Bδ
+[0.ℓ] −
+Bφ
+[3.ℓ]
+6
+�
+.
+(3.36)
+Hence for any ℓ ≥ 2,
+Bφ
+[0.ℓ] = 0 , Bδ
+[0.ℓ] =
+Bφ
+[3.ℓ]
+3A .
+(3.37)
+Moreover, inserting the obtained solution into the sup-
+plementary equation (3.11) while using (A9), we find
+ˆS3 =
+�
+Bφ
+[0,ℓ]
+2
+−
+ℓ(ℓ + 1)Bδ
+[1.ℓ]
+2λ2
++
+3ABδ
+[1.ℓ] − Bφ
+[3.ℓ]
+6λ3
+�
+× Kℓ × s(y)P 1
+ℓ (y) .
+(3.38)
+Considering (3.38) for the various modes of ℓ gives us:
+ℓ = 0 is trivial because P 1
+0 = 0; the ℓ = 1 coefficient van-
+ishes since K1 = 0. Therefore the coefficients Bφ
+[0.1] and
+Bφ
+[3.1] are unconstrained by the supplementary equation
+ˆS3. Finally considering ˆS3 = 0 for the ℓ > 1 coefficients
+while using (3.37) gives us
+0 =ℓ(ℓ + 1)Bδ
+[1.ℓ]
+(3.39)
+which implies
+Bδ
+[1.ℓ] = 0
+:
+∀ℓ > 1 .
+(3.40)
+Furthermore, requiring an asymptotic Bondi frame (a
+non-rotating inertial observer at large distances), i.e.
+gabdxadxb → −du2 − dλdu + λ2qABdxAdxB
+(3.41)
+annuls the integration constants,
+W φ
+[0.1] = Bδ
+[0.ℓ] = 0.
+(3.42)
+From the above requirements, the final solution of the
+linear perturbations are
+δ[1](y, λ) = 0 ,
+W φ
+[1](y, λ) = − B
+3λ3
+P 1
+ℓ (y)
+s
+= − B
+3λ3
+y
+s,
+(3.43)
+where we redefined B := Bφ
+[3,1] for notational convenience
+because it is the only remaining integration constant.
+D.
+Quadratic perturbations
+Using the notation of Sec. III B, the relevant main
+equations (i.e. only those containing γ[2], R[2], W y
+[2] and
+W[2]) for the quadratic perturbations are found to be
+0 = ˆS1(W[2], W y
+[2]) + B2s2
+2λ6
+�
+1 − A
+2λ
+�
+(3.44a)
+0 = ˆS2(W[2], W y
+[2])
+(3.44b)
+0 = ˆH(γ)
+1
+(R[2])
+(3.44c)
+0 = ˆH(γ)
+2
+(R[2], γ[2], W y
+[2])
+(3.44d)
+0 = ˆH(γ)
+3
+(W[2], R[2], γ[2], W y
+[2]) − B2s2
+4λ4
+(3.44e)
+0 = ˆH(γ)
+4
+(γ[2], W y
+[2]) + B2s2
+4λ4
+(3.44f)
+The first hypersurface equation (3.44c) is readily inte-
+grated
+R[2] = CR20(y) + CR11(y)r.
+(3.45)
+Similarily to (3.9b), we can deduce a master equation for
+γ[2]
+0 =M(γ[2]) − s2R2,λλyy − 5B2
+2λ5 s2.
+(3.46)
+For finding a solution of the remaining fields γ[2], W y
+[2]
+and W[2], we need to solve the master equation (3.46).
+Defining
+ψ[2] = (λγ[2]),λλ
+(3.47)
+with Legendre decomposition
+ψ[2] = ψ[2.ℓ](λ)P 2
+ℓ (y)
+(3.48)
+while using (3.44c) gives us after insertion of (3.45),
+(3.47) and (3.48) into (3.46)
+0 =
+�
+−1
+2r(A − 2λ)d2ψ[2.ℓ]
+dλ2
++
+�
+4r − A
+2
+� dψ[2.ℓ]
+dy
++
+�
+2 − ℓ(ℓ + 1) + A
+2λ
+�
+ψ[2.ℓ]
+�
+P 2
+ℓ − 5B2
+2λ5 s2
+(3.49)
+To fully factor out the Legendre polynomials P 2
+ℓ , we re-
+call that P 2
+2 (y) = 3s2. This allows us to write
+0 =
+��
+−1
+2r(A − 2λ)d2ψ[2.ℓ]
+dλ2
++
+�
+4r − A
+2
+� dψ[2.ℓ]
+dy
++
+�
+2 − ℓ(ℓ + 1) + A
+2λ
+�
+ψ[2.ℓ]
+�
+δℓ′
+ℓ − 5B2
+6λ5 δℓ′
+2
+�
+P 2
+ℓ′(y)
+(3.50)
+We can see that (3.50) resembles (3.21) if B = 0. It
+is in fact a inhomogeneous version of (3.21). We seek
+solutions of (3.50) as a superposition of a homogeneous
+solution, ψ(hom)
+[2.ℓ]
+for B = 0, and a particular solution
+ψ(part)
+[2.ℓ]
+for B ̸= 0, i.e.
+ψ[2.ℓ] = ψ(hom)
+[2.ℓ]
++ ψ(part)
+[2.ℓ]
+.
+(3.51)
+
+8
+The homogeneous solution ψ(hom)
+[2.ℓ]
+will be like (3.25).
+Also note that a particular solution needs to be found
+for the ℓ = 2 mode, only. We find ψ(part)
+[2.2]
+= −B2/(9Aλ4).
+Hence,
+ψ[2.ℓ](λ) =A
+�
+C[1.ℓ]P 2
+ℓ
+� 4λ
+A − 1
+�
++ C[2.ℓ]Q2
+ℓ
+� 4λ
+A − 1
+�
+2A − 4λ
+�
++
+�
+− B2
+9Aλ4
+�
+δ2
+ℓ.
+(3.52)
+It follows by the same regularity arguments like in the
+discussion for (3.25) that in order the Weyl curvature
+scalar Ψ0 does not blow up at the horizon of the un-
+perturbed solution and towards null infinity we must set
+C[1.ℓ] = C[2.ℓ] = 0.
+Consequently a solution for the ψ[2.ℓ]–modes is
+ψ[2.ℓ](λ) =
+�
+− B2
+9Aλ4
+�
+δ2
+ℓ .
+(3.53)
+Setting
+γ[2](λ, y) = γ[2.ℓ](λ)P 2
+ℓ (y),
+(3.54)
+we find after integration of (3.47)
+γ[2.ℓ](λ, y) = Cγ
+[0.ℓ] +
+Cγ
+[1.ℓ]
+λ
+−
+B2
+54Aλ3 δ2
+ℓ
+(3.55)
+Insertion of (3.55) and (3.45) into (3.44d) gives us
+0 =
+(λ4W y
+[2],r)
+2sλ2
++ sCR20,y
+λ2
++
+�dγ[2.ℓ]
+dλ
+� 1
+s
+d
+dy
+�
+s2P 2
+ℓ
+�
+.
+(3.56)
+using (A5) we find
+0 =
+�
+λ4 W y
+[2],r
+s
+�
+,λ
++ 2sCR20,y − 2λ2Kℓ
+�dγ[2.ℓ]
+dλ
+�
+P 1
+ℓ (y)
+(3.57)
+which indicates that the angular behaviour of W y
+[2]/s and
+sCR20,y are dictated by the associated Legendre polyno-
+mials P 1
+ℓ (y). As of (A7), we set (note Pℓ(y) = P 0
+ℓ (y))
+R[2](λ, y) =R[2.ℓ](λ)Pℓ(y) =
+�
+CR
+[20.ℓ] + CR
+[21.ℓ]λ
+�
+Pℓ(y),
+(3.58)
+W y
+[2](λ, y) =W y
+[2.ℓ](λ)s(y)P 1
+ℓ (y)
+(3.59)
+This gives us
+0 = d
+dλ
+�
+λ4 d
+dλW y
+[2.ℓ]
+�
+− 2CR
+[20.ℓ] − 2λ2Kℓ
+�dγ[2.ℓ]
+dλ
+�
+,
+(3.60)
+Integrating (3.60) yields
+W y
+[2.ℓ] =Cy
+[0.ℓ] +
+KℓCγ
+[1.ℓ] − CR
+[20.ℓ]
+λ2
+−
+Cy
+[3.ℓ]
+3λ3 −
+B2
+9Aλ4 δ2
+ℓ
+(3.61)
+where we set the integration constants Cy
+[0.0] = Cy
+[3.0] = 0,
+because P 1
+0 (y) = 0.
+Considering (3.7d) with (3.54),
+(3.59), (A8) and s2 = P 2
+ℓ (y)/3 gives us
+�
+λ2
+�
+1 − A
+2λ
+�
+γ[2.ℓ],r
+�
+,λ
+= 1
+2
+�
+λ2W y
+[2.ℓ]
+�
+,λ + B2
+12λ4 δ2
+ℓ
+(3.62)
+so that after insertion of (3.55) and (3.61), we obtain
+λCy
+[0.ℓ] =
+A
+2λ2
+�
+Cγ
+[1.ℓ] −
+Cy
+[3.ℓ]
+3A
+�
++
+B2
+9Aλ3
+�
+1 + Kℓ
+4
+�
+δ2
+ℓ
+(3.63)
+implying for any ℓ ≥ 2
+Cy
+[0.ℓ] = 0 , Cy
+[3.ℓ] = 3ACγ
+[1.ℓ]
+(3.64)
+Next, proceed with the hypersurface equation (3.44e) for
+W[2]. Insertion of (3.54), (3.58) and (3.59) into (3.44e)
+gives us
+(λW[2]),λ =
+�
+−
+��
+1 − A
+2λ
+�
+(λR[2.ℓ]),λ
+�
+,λ
++ ℓ(ℓ + 1)
+�
+R[2.ℓ]
+λ
++
+(λ4W y
+[2.ℓ]),λ
+2λ2
+− Kℓγ[2,ℓ]
+��
+P 0
+ℓ (y)
+− B2s2
+4λ4
+(3.65)
+Using
+s2 = 1 − y2 = 2
+3[P 0
+ℓ (y) − P 0
+2 (y)]
+(3.66)
+as well as setting
+W[2](λ, y) = W[2.ℓ](λ)P 0
+ℓ (y)
+(3.67)
+yields
+(λW[2.ℓ]),λ = −
+��
+1 − A
+2λ
+�
+(λR[2.ℓ]),λ
+�
+,λ
++ ℓ(ℓ + 1)
+�
+R[2.ℓ]
+λ
++
+(λ4W y
+[2.ℓ]),λ
+2λ2
+− Kℓγ[2,ℓ]
+�
+− B2
+6λ4 (δ0
+ℓ − δ2
+ℓ)
+(3.68)
+Since R[2.ℓ], W y
+[2.ℓ] and γ[2.ℓ] are known, we find after
+integration
+W[2.ℓ] = −KℓCR
+[21.ℓ] − ℓ(ℓ + 1)KℓCγ
+[0.ℓ] +
+CW
+[1.ℓ]
+λ
++
+ACR
+[20.ℓ]
+2λ2
++
+ℓ(ℓ + 1)Cy
+[3.ℓ]
+6λ2
++
+� 2B2
+9Aλ3 − B2
+18λ4
+�
+δ2
+ℓ + B2
+18λ4 δ0
+ℓ
+(3.69)
+
+9
+where CW
+[1.ℓ] are integration constants.
+Calculation of (3.44a) and (3.44b) while using (3.58),
+(3.59), (3.66), (A1) and (A10) gives us
+0 =
+�
+1 − A
+2λ
+� �
+λ2W[2.ℓ],r + AR[2.ℓ]
+λ
+�
+,λ
+− ℓ(ℓ + 1)
+�
+W[2.ℓ] + A
+2 W y
+[2,ℓ]
+�
+(3.70)
+0 =
+1
+2λ2
+�
+1 − A
+2λ
+�
+(λ4W y
+[2.ℓ]),λ − 1
+2W[2.ℓ],r + W y
+[2.ℓ]
+(3.71)
+and insertion of the respective coefficient solutions
+(3.58),(3.61) and (3.69) yields
+0 = ℓ(ℓ + 1)
+�
+−
+CW
+[1.ℓ]
+λ
++ Kℓ
+�
+Cγ
+[1.ℓ] − CR
+[21.ℓ]
+�
++
+Cy
+[3.ℓ] − 3ACγ
+[1.ℓ]
+6λ2
+�
+(3.72)
+0 =
+CW
+[1.ℓ]
+2λ2 +
+(Cy
+[3.ℓ] − 3ACγ
+[1.ℓ])Kℓ
+6λ3
+,
+∀ℓ ≥ 1
+(3.73)
+Therefore,
+CW
+[1.ℓ] =0 , ∀ℓ ≥ 1
+(3.74a)
+Cy
+[3.ℓ] =3ACγ
+[1.ℓ] , ∀ℓ ≥ 2
+(3.74b)
+CR
+[21.ℓ] =Cγ
+[1.ℓ] , ∀ℓ ≥ 2
+(3.74c)
+Note, (3.74b) is consistent with (3.64). The requirement
+of an asymptotic inertial observer leads to
+Cγ
+[0.ℓ] = CR
+[21.ℓ] = 0
+(3.75)
+which gives with (3.74) that Cγ
+[1.ℓ] = Cy
+[3.ℓ] = 0. Thus,
+redefining C := CW
+[1.1], the quadratic perturbations are
+γ[2](λ, y) =
+�
+−
+B2
+54Aλ3 δ2
+ℓ
+�
+P 2
+ℓ (y)
+(3.76)
+R[2](λ, y) = 0
+(3.77)
+W y
+2 (λ, y) =
+�
+− B2
+9Aλ4 δ2
+ℓ
+�
+s(y)P 1
+ℓ (y)
+(3.78)
+W[2](λ, y) = C
+λ + B2
+18λ4 +
+� 2B2
+9Aλ3 − B2
+18λ4
+�
+P 0
+2 (y) .
+(3.79)
+E.
+Third order perturbations
+Similarly, expressions for the higher order perturba-
+tions quantities f[i] can be obtained using the same pro-
+cedere as in the previous sections. In this and in the next
+subsection we show the fundamental results without re-
+peating intermediate steps.
+The relevant equations for the third perturbations are
+0 = ˆS3(δ[3], W φ
+[3]) − B3s4
+6Aλ6
+(3.80)
+0 = ˆH(δ)
+1 (δ[3], W φ
+[3]) − B3s4
+6Aλ6
+(3.81)
+0 = ˆH(δ)
+2 (δ[3], W φ
+[3]) + 2B3ys2
+3Aλ5
+(3.82)
+Similarily to (3.9b), we can deduce a master equation
+for δ[3]
+0 =M(δ[3]) − 40B3
+3Aλ6 s2y
+(3.83a)
+Using P 2
+3 (y) = 15ys2 and following the steps of Sec. III C,
+we find
+δ[3](λ, y) =
+�
+−
+B3
+162A2λ4
+�
+P 2
+3 (y)
+(3.84)
+W[3](λ, y) =
+�
+− D
+3λ3 −
+2B3
+135Aλ6
+� P 1
+1 (y)
+s(y)
++
+�
+B3
+405Aλ6 −
+4B3
+81A2λ5
+� P 1
+3 (y)
+s(y)
+(3.85)
+where D is the only free new remaining integration con-
+stant that appears at this order.
+F.
+Fourth order perturbations
+Here the relevant main equations are those containing
+γ[4], R[4], W y
+[4] and W[4] which are
+0 = ˆS1(W[4], W y
+4]) +
+� 14
+9Aλ −
+1
+12λ2 − 35
+6A2
+� B4s4
+3λ8
++
+��
+1 − A
+2λ
+�
+D − CB
+2A +
+� 16
+Aλ2 −
+7
+3λ3
+� B3
+9A
+� Bs2
+λ6
+−
+8B4
+27λ8A2 + CB2
+3Aλ6
+(3.86a)
+0 = ˆS2(W[4], W y
+[4]) +
+�(7A + 120λ)s2
+12λ
+− 8
+� B4ys
+9A2λ7
++ 2ysCB2
+3Aλ5
+(3.86b)
+0 = ˆH(γ)
+1
+(R[4]) −
+B4s4
+18A2λ8
+(3.86c)
+0 = ˆH(γ)
+2
+(R[2], γ[2], W y
+[2]) + B4ys3
+27A2λ7
+(3.86d)
+0 = ˆH(γ)
+3
+(W[2], R[2], γ[2], W y
+[2]) + (A − 14r)B4s4
+36A2λ7
++ 2B4s2
+9A2λ6 + DBs4
+2λ4
+(3.86e)
+0 = ˆH(γ)
+4
+(γ[2], W y
+[2]) +
+�
+14 + A
+2r
+� B4s4
+9A2λ6
++
+� B2C
+2Aλ4 − BD
+2λ4 −
+38B2
+27A2λ6
+�
+s2
+(3.86f)
+
+10
+The first hypersurface equation (3.86c) is readily inte-
+grated
+R[4](λ, y) =ER0(y) + ER1(y)λ −
+s4B4
+1080A2λ5
+(3.87)
+or expressing in terms of the Legendre polynomials P 0
+ℓ (y)
+R[4](λ, y) =
+�
+ER
+[0.ℓ] + ER
+[1.ℓ]λ
+�
+P 0
+ℓ (y)
+−
+B4
+135A2λ5
+�P 0
+0 (y)
+15
+− 2P 0
+2 (y)
+21
++ P 0
+4 (y)
+35
+�
+(3.88)
+Similarily to (3.9a) we can deduce a master equation
+for γ[4]
+0 =M(γ[4]) − s2R[4],λλyy − 5B2Cs2
+Aλ5
+− 5BDs2
+λ5
++
+�
+358 − s2
+�
+397 + A
+λ
+�� B4s2
+9A2λ7
+(3.89a)
+Using the methods of Sec. III D together with the in-
+verted Legendre relations
+1 =P 0
+0 (y) = −P 1
+1 (y)
+s
+(3.90a)
+y =P 0
+1 (y) = −P 1
+2 (y)
+3s
+(3.90b)
+y2 =1
+3 − 2
+3P 0
+2 (y) = 1 − 1
+3P 2
+2 (y)
+(3.90c)
+y3 = − 2P 1
+4 (y)
+35s
++ P 1
+2 (y)
+7s
+(3.90d)
+y4 =1
+5 − 4P 0
+2 (y)
+7
++ 8P 0
+4 (y)
+35
+= 1 − 8
+21P 2
+2 (y) −
+2
+105P 2
+4 (y)
+(3.90e)
+we deduce the following solution for the fourth order per-
+turbation
+R[4] = −
+B4
+135A2λ5
+�P 0
+0
+15 − 2P 0
+2
+21 + P 0
+4
+35
+�
+(3.91a)
+γ[4] =
+� BD
+27Aλ3 −
+B2C
+27A2λ3 −
+B4
+1134A2λ6
+�
+P 2
+2
++
+�
+B4
+405A3λ5 +
+B4
+17010A2λ6
+�
+P 2
+4
+(3.91b)
+W y
+[4] =
+� 2BD
+9Aλ4 − 2B2C
+9A2λ4 −
+2B4
+2835A2λ7
+�
+P 1
+2
++
+�
+2B4
+81A3λ6 +
+B4
+4725A2λ7
+�
+P 1
+4
+(3.91c)
+W[4] =E
+λ − BD
+9λ4 +
+4B4
+405A2λ6 −
+B4
+675Aλ7
++
+�
+2B4
+945Aλ7 −
+2B4
+81A2λ6 + 4B2C
+9A2λ3 − 4BD
+9Aλ3 + BD
+9λ4
+�
+P 0
+2
++
+�
+−
+8B4
+81A3λ5 +
+2B4
+135A2λ6 −
+B4
+1575Aλ7
+�
+P 0
+4
+(3.91d)
+Note that E is the only remaining new integration con-
+stant, all other vanish because of the reasons mentioned
+in Sec. III D.
+G.
+Perturbations in terms of Komar quantities
+The solution of the perturbation involve the free inte-
+gration constants A, B, C, D and E. These free constants
+determine the Komar mass, Km, and the Komar angular
+momentum, KL, which can be found by calculation of
+(2.7) and (2.8)
+m := Km = A
+4 − C
+2 ε2 − E
+2 ε4 + O(ε5)
+(3.92)
+L := KL = −B
+6 ε + D
+6 ε3 + O(ε5)
+(3.93)
+If ε = 0, Km = A/4 corresponds to the mass m0 of the
+unperturbed system. Furthermore, we can see that L =
+O(ε). This allows us relate ε with the angular momentum
+L of the system. To do that we have to solve the cubic
+equation
+0 = D
+6 ε3 − B
+6 ε + L
+(3.94)
+for ε. This equation also shows that in order to make the
+substitution of ε by L, we seek the solution ε(L) = O(L).
+The root of (3.94) which fulfils this requirement is
+ε = − 6
+B L − 216D
+B4 L3 + O(L5)
+(3.95)
+Subsequent insertion of this expansion into (3.92) and
+solving for A gives us
+A = 4m + 72C
+B2 L2 + 2592EB − 2CD
+B5
+L4 + O(L6) (3.96)
+The relations (3.95) and (3.96) allow us to substitute
+A and ε, by the physical quantities m and L. Insertion
+of (3.95) and (3.96) into the solution of the perturba-
+tions and subsequent expansion up to O(L4) allows us to
+eliminate the integration constants C, D and E from the
+perturbations. This gives us
+
+11
+R(λ, y) = λ −
+�P 0
+0
+5 − 2P 0
+2
+7
++ 3P 0
+4
+35
+�
+L4
+5m2λ5 + O(L6)
+(3.97a)
+W(λ, y) = 1 − 2m
+λ +
+� 2
+λ4 +
+�
+2
+mλ3 − 2
+λ4
+�
+P 0
+2
+�
+L2 +
+�
+4
+5m2λ6 −
+12
+25λ7 +
+�
+24
+35mλ7 − 2L4
+m2λ6
+�
+P 0
+2
++
+�
+−
+2
+m3λ5 +
+6
+5m2λ6 −
+36
+175mλ7
+�
+P 0
+4
+�
+L4 + O(L6)
+(3.97b)
+W y(λ, y) =
+�
+−
+1
+mλ4 (sP 1
+2 )
+�
+L2 +
+�
+−
+2
+35m2λ7 (sP 1
+2 ) +
+�
+1
+2m3λ6 +
+3
+175m2λ7
+�
+(sP 1
+4 )
+�
+L4 + O(L6)
+(3.97c)
+W φ(λ, y) =
+�
+− 2
+λ3
+P 1
+1
+s
+�
+L +
+�
+4
+5mλ6
+P 1
+1
+s +
+�
+2
+3m2λ5 −
+2
+15mλ6
+� P 1
+3
+s
+�
+L3 + O(L5)
+(3.97d)
+γ(λ, y) =
+�
+−
+1
+6mλ3 P 2
+2
+�
+L2 +
+�
+−
+1
+14m2λ6 P 2
+2 +
+�
+1
+20m3λ5 +
+1
+210m2λ6
+�
+P 2
+4
+�
+L4 + O(L6)
+(3.97e)
+δ(λ, y) =
+�
+1
+12m2λ4 P 2
+3
+�
+L3 + O(L5)
+(3.97f)
+We see in (3.97) that the perturbations are determined by the mass and angular momentum, i.e. the solution has two
+hairs. To show that this solutions represents the Kerr solution in affine-null coordinates, we introduce the specific
+angular momentum, a := L/m. In terms of a, (3.97) read after changing to the angular coordinate θ
+R(λ, θ) = λ − 3m2 sin4 θ
+40λ5
+a4 + O(a6)
+(3.98a)
+W(λ, θ) = 1 − 2m
+λ +
+�2m
+λ3 +
+�
+−3m
+λ3 + 3m2
+λ4
+�
+sin2 θ
+�
+a2
++
+�
+−2m
+λ5 +
+�10m
+λ5
+− 3m2
+λ6
+�
+sin2 θ +
+�
+−35m
+4λ5 + 21m2
+4λ6 − 9m3
+10λ7
+�
+sin4 θ
+�
+a4 + O(a6)
+(3.98b)
+W θ(λ, θ) =
+�
+−3m
+λ4 a2 +
+�5m
+λ6 −
+�35m
+4λ6 + 3m2
+10λ7
+�
+sin2 θ
+�
+a4
+�
+sin θ cos θ + O(a6)
+(3.98c)
+W φ(λ, θ) = 2m
+λ3 a +
+�
+−4m
+λ5 +
+�5m
+λ5 − m2
+λ6
+�
+sin2 θ
+�
+a3 + O(a5)
+(3.98d)
+γ(λ, θ) =
+�
+−m sin2 θ
+2λ3
+�
+a2 +
+�9m sin2 θ
+4λ5
++
+�
+−21m
+8λ5 − m2
+4λ6
+�
+sin4 θ
+�
+a4 + O(a6)
+(3.98e)
+δ(λ, θ) = −5m cosθ sin2 θ
+4λ4
+a3 + O(a5)
+(3.98f)
+Comparing with [28], we find agreement for R which corresponds to their areal coordinate r. Calculation of the metric
+
+12
+components gab using (3.98) gives us
+guu(λ, θ) = −1 + 2m
+λ +
+��3m
+λ3 + m2
+λ4
+�
+sin2 θ − 2m
+λ3
+�
+a2
++
+�2m
+λ5 −
+�10m
+λ5
++ 4m2
+λ6
+�
+sin2 θ +
+�35m
+4λ5 + 23m2
+4λ6 + 9m3
+10λ7
+�
+sin4 θ
+�
+a4 + O(a6)
+(3.99a)
+guλ(λ, θ) = −1
+(3.99b)
+guθ(λ, θ) =
+��3m
+λ2
+�
+a2 +
+�
+−5m
+λ4 +
+�35m
+λ4
++ 23m2
+10λ5
+�
+sin2 θ
+�
+a4
+�
+sin θ cos θ + O(a6)
+(3.99c)
+guφ(λ, θ) =
+��
+−2m
+λ
+�
+a +
+�4m
+λ3 −
+�5m
+λ3 + m2
+λ4
+�
+sin2 θ
+�
+a3
+�
+sin2 θ + O(a5)
+(3.99d)
+gθθ(λ, θ) = λ2 +
+�
+−m sin2 θ
+λ
+�
+a2 +
+� 9m
+2λ3 sin2 θ −
+�21m
+4λ3 + 3m2
+20λ4
+�
+sin4 θ
+�
+a4 + O(a6)
+(3.99e)
+gθφ(λ, θ) =
+�
+−5m sin3 θ cos θ
+2λ2
+�
+a3 + O(a5)
+(3.99f)
+gφφ(λ, θ) =
+�
+λ2 +
+�m sin2 θ
+λ
+�
+a2 +
+�
+− 9m
+2λ3 sin2 θ +
+�21m
+4λ3 + 17m2
+20λ4
+�
+sin4 θ
+�
+a4
+�
+sin2 θ + O(a6)
+(3.99g)
+Eqs.(3.99) constitute our final expression for the slowly
+rotating stationary and axially symmetric (Kerr) metric
+adapted to null coordinates which asymptotically match
+an inertial Bondi frame. At difference of all previous ap-
+proaches, it was obtained as an explicit solution of the
+Einstein equations. After comparison with [28], we find
+agreement up to a typo in their equation for gθφ. We also
+note care should be taken when comparing [28]’s expres-
+sions with ours. First, [28] present a Bondi-Sachs form of
+the metric, while we have an affine-null metric approach-
+ing a Bondi frame, the difference is in the choice of radial
+coordinate, and the two agree only up to O(λ−4) with one
+another. Second, [28] make a large λ expansion while we
+make a small a expansion, this results in powers of λ−k
+absorbed by order symbols in [28]. A slowly rotating ver-
+sion of the Kerr metric in null affine coordinates at second
+order in a was also obtained by Dozmorov who made a
+null tetrad rotations starting with the Kerr metric as ex-
+pressed in Boyer-Lindquist coordinates [35]. In the next
+section we show an alternative procedure to recover the
+slowly rotating Kerr metric components as expressed in
+(3.99) by doing appropriate coordinates transformations.
+IV.
+APPROXIMATED AFFINE-NULL METRIC
+DERIVED FROM THE KERR-METRIC
+Here, starting with the Kerr metric expressed in Boyer-
+Lindquist coordinates (BL) {ˆt, ˆr, ˆθ, ˆφ}, we present an
+explicit transformation to affine-null coordinates up to
+fourth order in a. The Kerr metric in BL coordinates
+reads:
+ds2 = gˆtˆtdˆt2+gˆtˆφdˆtdˆφ+gˆrˆrdˆr2+gˆrˆrdˆr2+gˆθˆθdˆθ2+g ˆφˆφdˆφ2;
+(4.1)
+with
+gˆtˆt = −
+�
+1 − 2mˆr
+Σ
+�
+,
+(4.2)
+gˆt ˆφ = −2maˆr sin2 ˆθ
+Σ
+,
+(4.3)
+gˆrˆr = Σ
+∆,
+(4.4)
+gˆθˆθ = Σ,
+(4.5)
+g ˆφˆφ =
+�
+ˆr2 + a2 + 2ma2ˆr sin2 ˆθ
+Σ
+�
+sin2 ˆθ,
+(4.6)
+with ∆ = ˆr2 − 2mˆr + a2 and Σ = ˆr2 + a2 cos2 ˆθ. The u
+null coordinate must satisfy the eikonal equation,
+gab∇au∇bu = 0,
+(4.7)
+Inspired by [28], we propose the following expansion for
+u,
+u = ˆt − ˆr − 2m ln
+� ˆr − 2m
+2m
+�
++
+∞
+�
+i=1
+fi(ˆr, ˆθ)ai.
+(4.8)
+Note that for a = 0 this expression reduces to the stan-
+dard outgoing Schwarzschild null coordinate. By replac-
+ing (4.8) into (4.7), we obtain a set of differential equa-
+tions for fi(ˆr, ˆθ) that can be solved iteratively. Conserv-
+ing terms up to fourth order in a we find that only the
+even coefficients f2n(ˆr, ˆθ) are non–vanishing with:
+
+13
+f2(ˆr, ˆθ) =
+5ˆr − 2m
+4ˆr(2m − ˆr) + cos 2ˆθ
+4ˆr
+− ln(1 − 2m
+ˆr )
+2m
+,(4.9)
+f4(ˆr, ˆθ) = (2ˆr + m)
+16ˆr4
+sin4(2ˆθ) − 3 ln(1 − 2m
+ˆr )
+8m3
+−4m2 − 9mˆr + 3ˆr2
+4m2ˆr(ˆr − 2m)2 ,
+(4.10)
+Similarly, affine-null coordinates {λ, θ, φ} can be ob-
+tained from the requirements
+gab∇au∇bλ = −1,
+(4.11a)
+gab∇au∇bθ = gab∇au∇bφ = 0,
+(4.11b)
+by assuming relations of the form:
+λ = ˆr +
+∞
+�
+i=1
+ˆΛi(ˆθ, ˆr)ai,
+(4.12)
+θ = ˆθ +
+∞
+�
+i=1
+ˆΘi(ˆθ, ˆr)ai,
+(4.13)
+φ = ˆφ +
+∞
+�
+i=1
+ˆΦi(ˆθ, ˆr)ai,
+(4.14)
+and replacing into the set (4.11), the coefficients functions
+ˆΛi, ˆΘi, ˆΦi can be obtained in the same way as u. After
+that, the resulting relations can be inverted in order to
+express the BL coordinates in terms of the affine-null
+coordinates. Following these steps up to fourth order,
+the final transformation coordinates reads:
+ˆt = u + λ + 2m ln( λ
+2m − 1) +
+�ln(1 − 2m
+λ )
+2m
++ 3 m cos(2 θ) + 4 λ − 3 m
+(4 λ − 8 m) λ
+�
+a2
++
+�
+−m
+�
+175 λ2 − 224 mλ − 72 m2�
+(cos (2 θ))2
+320 (λ − 2 m)2 λ4
+− m
+�
+25 λ2 + 64 mλ + 72 m2�
+cos (2 θ)
+160 (λ − 2 m)2 λ4
++3 ln
+�
+1 − 2 m
+λ
+�
+8 m3
++ 240 λ5 − 720 λ4m + 320 λ3m2 + 225 λ2m3 − 96 λ m4 + 72 m5
+320 m2λ4 (λ − 2 m)2
+�
+a4 + O(a6)
+(4.15)
+ˆr = λ − (λ + m) sin2 θ
+2λ2
+a2 +
+�sin2 θ(5 cos 2θ + 3))
+16λ3
++ m sin2 θ(7 cos 2θ + 1)
+16λ4
+− m2 sin4 θ
+5λ5
+�
+a4 + O(a6)
+(4.16)
+ˆθ = θ − sin (2 θ)
+4 λ2
+a2 + sin (2 θ) (3 λ cos (2 θ) + m cos(2 θ) − m)
+16 λ5
+a4 + O(a6)
+(4.17)
+ˆφ = φ +
+� 1
+λ + ln(1 − 2m
+λ )
+2m
+�
+a +
+�ln(1 − 2m
+λ )
+4m
++ m(2m + 5λ) cos 2θ
+8(λ − 2m)λ4
+−6m4 − m3λ + 8m2λ2 + 12mλ3 − 12λ4
+24m2(λ − 2m)λ4
+�
+a3 + O(a5)
+(4.18)
+Finally, with these transformations in hand, we obtain
+the same metric components in affine-null coordinates
+up to fourth order in a as given in (3.99) in the previous
+Section.
+V.
+LOCALIZING THE EVENT HORIZON AND
+ERGOSPHERE IN AFFINE-NULL CORDINATES
+In this Section we show that the affine-null coordinates
+for the slowly rotating Kerr metric cover the ergosphere
+and the (past) event horizon r+. In order to find them in
+a consistent way, they must be localized at O(a4). Recall
+that in BL coordinates the Kerr metric has the external
+
+14
+ergosphere placed at
+ˆrerg =m +
+�
+m2 − a2 cos2 ˆθ
+=2m − a2 cos2 ˆθ
+m
+− a4 cos4 ˆθ
+8m3
++ O(a6),
+(5.1)
+and the event horizon at
+r+ = m +
+�
+m2 − a2 = 2m − a2
+2m − a4
+8m3 + O(a6). (5.2)
+The boundary of the external ergosphere is obtained by
+looking for the timelike surface Γ where the stationary
+Killing vector field ∂u becomes a null vector field that is
+where
+guu|Γ = 0.
+(5.3)
+Taking into account the expression for guu as found in the
+first equation of (3.99), the ergosphere will be located at
+a given λ = λerg(θ), with
+λerg(θ) =
+2
+�
+i=0
+λerg[2i](θ)a2i + O(a6).
+(5.4)
+where the even expansion is a consequence of the symme-
+try assumption of Sec. II. Inserting (5.4) into (5.3), and
+after re-expanding in powers of a we find
+λerg(θ) =2 m −
+�
+7 cos2 θ − 3
+�
+8m
+a2
+−
+�
+51 cos4 θ − 2 cos2 θ + 31
+�
+640 m3
+a4 + O(a6),
+(5.5)
+which gives the location of the (external) ergosphere in
+affine-null coordinates.
+By replacing (5.5) into (4.16),
+and after a re-expansion in powers of a it can be checked
+that the standard fourth order expression for the BL ex-
+pression of the ergosphere as given by (5.1) is recovered.
+Similarly, for the (Killing) event horizon we search
+a null surface Σ described in affine-null coordinates by
+Σ(λ, θ) = λ − λH(θ) = 0.
+Hence, its normal vector
+Na = ∇aΣ must satisfy N aNa = 0 which implies the
+following differential equation for λH(θ) = 0,
+gabNaNb = W + 2W θ ∂λH(θ)
+∂θ
++ hφφ
+R2
+�∂λH(θ)
+∂θ
+�2
+= 0.
+(5.6)
+Let us assume an expansion for λH(θ) similar to (5.4),
+i.e.
+λH(θ) =
+2
+�
+i=0
+λH[2i](θ)a2i + O(a6);
+(5.7)
+with λH[0] = 2m (the Schwarzschild value for the loca-
+tion of the horizon).
+Introducing (5.7) into (5.6); re-
+expanding again in powers of a, we find (omitting the
+O(a6) term)
+0 =
+�λH[2]
+2m + 3 cos2 θ + 1
+16m2
+�
+a2
++
+�
+(λH[2],θ)2
+4m2
+− 3 sin θ cos θ
+8m3
+λH[2],θ −
+λ2
+H[2]
+4m2 − 3(cos2 θ + 1)
+16m3
+λH[2] − 127 cos4 θ − 320m3λH[4] − 84 cos2 θ − 3
+640m4
+�
+a4.
+(5.8)
+So that solving for the coefficient λH[2] and λH[4] gives
+us
+λH(θ) =2m − (1 + 3 cos2 θ)
+8m
+a2
++ (29 cos4 θ − 78 cos2 θ − 31)
+640m3
+a4 + O(a6),
+(5.9)
+which gives the location of the (past) event horizon in
+affine-null coordinates. By replacing into (4.16) and af-
+ter a reexpansion in a up to fourth order, the well known
+value (5.2) for the BL radial coordinate of the event hori-
+zon is recovered. At this location, the resulting compo-
+nents of the metric are regular.
+VI.
+SUMMARY
+We have derived high-order slow rotation approxima-
+tion of the Kerr metric in affine-null coordinates.
+To
+achieve this aim a metric in affine-null coordinates was
+expanded off a spherically symmetric background metric
+that corresponds to a Schwarzschild metric in outgoing
+Eddington Finkelstein coordinates. This quasi-spherical
+expansion was done with respect to a general smallness
+parameter ε. Subject to stationarity and axial symmetry
+
+15
+the perturbations did not depend on the u and φ coor-
+dinate.
+Moreover, requiring even parity of the Komar
+integral of stationary (giving the mass of the system)
+and odd parity of the Komar integral of axial symme-
+try (giving the angular momentum of the system), we
+argued that on the one hand the metric functions γ, R,
+W θ and W have only even perturbations in ε while on
+the other hand the metric fields δ and W φ have only odd
+perturbations in ε. This fact significantly simplifies the
+integration of the perturbation equations resulting form
+the quasi-spherical expansion of the Ricci tensor. In ad-
+dition, we find that the integration of the perturbation
+equations follows an alternative hierarchical structure.
+Meaning with the spherically symmetric background so-
+lution at hand, the linear perturbations only involve the
+functions δ and W φ and its integration provides (after
+application of the boundary condition of an asymptotic
+inertial observer) one free integration constant B.
+At
+next order, the quadratic perturbations turn out to be
+a linear combination of the derivatives of functions γ,
+R, W θ and W together with nonlinear terms containing
+the integration constants A of the background model and
+the free integration constant B of the linear perturbation.
+Their integration also yields a free integration constant,
+C. Following up the next order, there only differential
+equations involving the cubic perturbations of δ and W φ
+as well as the integration constants A, B and C charac-
+terizing the lower order perturbations. This alternative
+scheme between the perturbations of (δ, W φ) and those of
+(γ, R, W θ, W) continues up to any order and is in fact a
+result of the symmetry assumptions. A common feature
+in solving for the even and odd-parity modes of ε, is that
+at any order there is a fourth order master equation for ei-
+ther the perturbation in γ or the perturbation in δ. With
+the solution of this master equation, the remaining per-
+turbations can be solve by mere integration. After having
+obtained the perturbed solution and calculation of the
+Komar mass and Komar angular momentum, the arising
+free integration constants A, B, C, ... can expressed by the
+Komar mass and Komar angular momentum or by mass
+and specific angular momentum. Hence, the solution de-
+pends only on two free physical parameters. Since the
+Komar angular momentum is O(ε), it turns out that the
+formal expansion parameter ε relates to the specific angu-
+lar momentum and the previously made quasi-spherical
+approximation is in fact a slow rotation approximation,
+like those of Hartle and Thorne [30, 36]. By successively
+solving Einstein equations, we thus derived a slow rota-
+tion approximation of the Kerr-metric up to fourth order
+in the specific angular momentum. This solution is fur-
+ther verified for correctness using a ’standard’ approach
+by obtaining a different representation of a given metric
+in another coordinate chart via a coordinate transforma-
+tion. The slowly rotating Kerr metric presented here also
+obeys the peeling property, which can be seen considering
+the Weyl scalars in (6.1)
+Ψ0 =
+�3ma2
+λ5
++ i15ma3
+λ6
+cos θ
+�
+sin2 θ + O(λ−7)
+(6.1a)
+Ψ1 = i3
+√
+2ma
+2λ4
+sin θ + O(λ−5)
+(6.1b)
+Ψ2 = m
+λ3 + i3ma cosθ
+λ4
++ O(λ−5)
+(6.1c)
+Ψ3 = i3
+√
+2ma
+4λ4
+sin θ + O(λ−5)
+(6.1d)
+Ψ4 = 3
+√
+2ma2
+4λ5
+sin2 θ + O(λ−6)
+(6.1e)
+Moreover it is easily checked that the (only) conserved
+Newman Penrose constant [21] vanishes [28].
+What is is interesting to remark is that up to the con-
+sidered order of approximation of our work and those of
+[28], the small a expansion and the large λ expansion
+coincide. It would be interesting to see up until which
+order this is the case. Such analysis might give insight
+on the validity and universality of general small param-
+eter expansions of the Kerr spacetime in relation to null
+coordinates.
+It may also give insight if a closed form
+solution of the Kerr metric with a surface forming null
+coordinate can be obtained at all. The method presented
+here offers the possibility to calculate any type of approx-
+imate rotating null-metric solution that is stationary, ax-
+ially symmetric and has a known spherically symmetric
+background, like e.g.
+those to describe compact mat-
+ter systems or with a cosmological constant. Indeed, the
+study presented here (solving the characteristic equations
+in this affine-null, metric formulation for vacuum space-
+times) is the natural starting point for further studying
+matter system under the given symmetry assumptions.
+Some of such questions we are currently investigating.
+Acknowledgements
+The authors thanks J. Winicour, L. Lehner, N. Ster-
+gioulas, E. M¨uller and G. Dotti for discussions at (early)
+stages of the project. T.M acknowledges financial sup-
+port from the FONDECYT de iniciaci´on 2019 (Project
+No. 11190854) of the ”Agencia Nacional de Investigaci´on
+y Desarrollo” in Chile. E.G gratefully acknowledges the
+hospitality extended to him during his stay at the Fac-
+ultad de Ingenier´ıa, Universidad Diego Portales and the
+financial support from CONICET and SeCyT-UNC.
+Appendix A: Useful Relations between Legendre
+Polynomials
+For completeness, we list some properties of the Legen-
+dre differential equations and relations between the Leg-
+endre polynomials.
+
+16
+The Legendre differential equation for the Legendre
+polynomials Pℓ(y) is
+d
+dy
+�
+(1 − y2)dPℓ
+dy
+�
++ ℓ(ℓ + 1)Pℓ = 0
+(A1)
+where the Legendre Polynomials Pℓ(y) are defined
+Pℓ(y) =
+1
+2ℓℓ!
+dℓ
+dyℓ (y2 − 1)ℓ
+(A2)
+The associated Legendre differential equation is
+d
+dy
+�
+(1 − y2)dP m
+ℓ
+dy
+�
++
+�
+ℓ(ℓ + 1) −
+m2
+1 − y2
+�
+P m
+ℓ
+= 0 (A3)
+where P m
+ℓ (y) are the associated Legendre polynomials,
+defined via
+P m
+ℓ (y) =(−)m(1 − y2)m/2 dmPℓ(y)
+dxm
+=(−)m
+2ℓℓ! (1 − y2)m/2 dℓ+m
+dyℓ+m (y2 − 1)ℓ
+(A4)
+which also shows P 0
+ℓ (y) = Pℓ(y). From these definitions,
+some useful identities can be derived
+d
+dy
+�
+(1 − y2)P 2
+ℓ (y)
+�
+= [ℓ(ℓ+1)−2](1−y2)1/2P 1
+ℓ (y) (A5)
+d
+dy
+�
+(1 − y2)2 dP 2
+ℓ
+dy
+�
+1 − y2
+− 2P 2
+ℓ
+= ℓ(ℓ + 1)(ℓ + 2)(ℓ − 1)Pℓ(y)
+(A6)
+P 1
+ℓ
+= −(1 − y2)1/2 dP 0
+ℓ
+dy
+(A7)
+P 2
+ℓ
+= (1 − y2)d2P 0
+ℓ
+dy2
+(A8)
+d
+dy
+�
+(1 − y2)2 d
+dy
+P 1
+ℓ
+(1 − y2)1/2
+�
+= [2−ℓ(ℓ+1)](1−y2)1/2P 1
+ℓ
+(A9)
+d
+dy(1 − y2)1/2P 1
+ℓ = ℓ(ℓ + 1)P 0
+ℓ
+(A10)
+Appendix B: Komar charges
+Depending on the Killing vector Xa ∈ {∂u, ∂φ}, we
+take the Komar charges to be
+KX = −kX
+8π
+�
+∇[aXb]dΣab
+(B1)
+with kX = 1, −1/2 for a timelike ( e.g.
+∂u) or rota-
+tional Killing vector (e.g. ∂φ), respectively. Consider the
+general null metric with the nonzero contravariant com-
+ponents g01, g11, g1A and gAB. The corresponding line
+element is
+gabdxadxb = (g11 + gABg1Ag1B)
+�dx0
+g01
+�2
++ 2
+�dx0
+g01
+�
+dx1
+− 2gABg1AdxB
+�dx0
+g01
+�
++ gABdxAdxB
+(B2)
+where gACgCB = δB
+A. Defining the null vectors
+l = −dx0 = −g01∂1
+(B3)
+and
+n = −1
+2
+g11
+(g01)2 dx0 + dx1
+g01 = ∂0 + 1
+2
+g11
+g01 ∂1 + g1A
+g01 ∂A (B4)
+which obey lana + 1 = lala = nana = 0, the surface
+element follows as
+dΣab = 2k[anb]
+�
+det(gAB)dx2dx3
+(B5)
+with xA = (x2, x3) being any angular coordinates for the
+units sphere. Setting gAB = R2hAB with hAB having
+the determinant of the unit sphere metric qAB, q(xC) :=
+det(hAB). The corresponding volume element is defined
+as d2q := √qdx2dx3 and we have � d2q = 4π. Hence,
+dΣab = 2l[anb]R2d2q .
+This allows us to write the Komar integal as
+K(X) = −kX
+8π
+�
+(2lanb∂[aXb])R2d2q ,
+(B6)
+Since
+2lanb∂[aXb] = 2l[anb]Xb,a
+(B7)
+= (lanb − lbna)Xb,a
+(B8)
+= l1(nbXb,1) − l1(nbX1,b)
+(B9)
+= −g01[(nbXb,1) − (nbX1,b)] , (B10)
+we have
+K(X) = kX
+8π
+� �
+(nbXb,1) − (nbX1,b)
+�
+g01R2d2q, (B11)
+Taking the Killing vector to be X = Xa∂a and specifica-
+tion to an affine null metric
+g01 = ǫ , g1A = ǫW A , g11 = W ,
+g0A = −R2hABW B , gAB = R2hAB
+(B12)
+and ǫ2 = 1 gives us
+2lanb∂[aXb] = −
+�
+W,1 − R2hABW AW B
+,1
+�
+X0
++ R2�
+hABW B
+,1 XA − 2hABW BXA
+,1
+�
++ ǫ(X1
+,1 − X0
+,0) − WX0
+,1 − W AX0
+,A
+(B13)
+
+17
+Assuming the timelike Killing vector X = ∂0 gives us
+2lanb∂[aXb] = −
+�
+W,1 − R2hABW AW B
+,1 .
+�
+Thus for the above form of the Killing vector we have the
+related Komar charge using kX = 1
+K(∂0) = 1
+8π
+� �
+− ǫ
+�
+W,1 − R2hABW AW B
+,1
+��
+R2d2q.
+(B14)
+With the rotational Killing X = ∂3, we have
+2lanb∂[aXb] = R2h3BW B
+,1
+so that the Komar charge is with kX = − 1
+2
+K(∂3) = − ǫ
+16π
+� �
+R4h3BW B
+,1
+�
+d2q.
+(B15)
+[1] H. Bondi, Nature (London) 186, 535 (1960).
+[2] H. Bondi, M. G. J. van der Burg, and A. W. K. Metzner,
+Proceedings of the Royal Society of London Series A 269,
+21 (1962).
+[3] R. K. Sachs, Proceedings of the Royal Society of London
+Series A 270, 103 (1962).
+[4] R. P. Kerr, Phys. Rev. Lett. 11, 237 (1963).
+[5] E. T. Newman, E. Couch, K. Chinnapared, A. Exton,
+A. Prakash, and R. Torrence, Journal of Mathematical
+Physics 6, 918 (1965).
+[6] P. Jordan, J. Ehlers, and R. K. Sachs, General Relativity
+and Gravitation 45, 2691 (2013).
+[7] R. G´omez, L. Lehner, R. L. Marsa, and J. Winicour,
+Phys. Rev. D 57, 4778 (1998), gr-qc/9710138.
+[8] J. Winicour, Living Reviews in Relativity 15, 2 (2012).
+[9] T. M¨adler and J. Winicour, Scholarpedia 11, 33528
+(2016), 1609.01731.
+[10] G. Barnich and C. Troessaert, Journal of High Energy
+Physics 2010, 62 (2010), 1001.1541.
+[11] S. Pasterski, A. Strominger, and A. Zhiboedov, Journal
+of High Energy Physics 2016, 53 (2016), 1502.06120.
+[12] T. M¨adler and J. Winicour, Classical and Quantum
+Gravity 33, 175006 (2016), 1605.01273.
+[13] D. A. Nichols,
+Phys. Rev. D
+95,
+084048 (2017),
+1702.03300.
+[14] T. M¨adler and J. Winicour, Classical and Quantum
+Gravity 35, 035009 (2018), 1708.08774.
+[15] T. M¨adler and J. Winicour, Classical and Quantum
+Gravity 36, 095009 (2019), 1811.04711.
+[16] J. Winicour, Phys. Rev. D 87, 124027 (2013), 1303.6969.
+[17] T. M¨adler, Phys. Rev. D 99, 104048 (2019), 1810.04743.
+[18] E. Gallo, C. Kozameh, T. M¨adler, O. M. Moreschi, and
+A. Perez, Phys. Rev. D 104, 084048 (2021), 2107.10120.
+[19] J. A. Crespo, H. P. de Oliveira, and J. Winicour, Phys.
+Rev. D 100, 104017 (2019), 1910.03439.
+[20] O. Baake and T. M¨adler, in prep. (2023).
+[21] N. T. Bishop and L. R. Venter, Phys. Rev. D 73, 084023
+(2006), gr-qc/0506077.
+[22] M. A. Arga˜naraz and O. M. Moreschi, Phys. Rev. D 104,
+024049 (2021).
+[23] S. J. Fletcher and A. W. C. Lun, Classical and Quantum
+Gravity 20, 4153 (2003).
+[24] S. Jahanur Hoque and A. Virmani,
+arXiv e-prints
+arXiv:2108.01098 (2021), 2108.01098.
+[25] S. A. Hayward, Phys. Rev. Lett. 92, 191101 (2004), gr-
+qc/0401111.
+[26] M. A. Arga˜naraz and O. M. Moreschi, Phys. Rev. D 105,
+084012 (2022).
+[27] E. Gallo and O. M. Moreschi, Phys. Rev. D 89, 084009
+(2014), 1404.2475.
+[28] S. Bai, Z. Cao, X. Gong, Y. Shang, X. Wu, and Y. K.
+Lau, Phys. Rev. D 75, 044003 (2007), gr-qc/0701171.
+[29] X. Gong, Y. Shang, S. Bai, Z. Cao, Z. Luo, and Y. K.
+Lau, Phys. Rev. D 76, 107501 (2007).
+[30] J. B. Hartle, Astrophys. J. 150, 1005 (1967).
+[31] J.
+Tafel,
+Class.
+Quant.
+Grav.
+39,
+115013
+(2022),
+2105.09372.
+[32] R. G´omez, S. Husa, and J. Winicour, Phys. Rev. D 64,
+024010 (2001), gr-qc/0009092.
+[33] M. G. J. van der Burg, Proceedings of the Royal Society
+of London Series A 294, 112 (1966).
+[34] T. M¨adler, Phys. Rev. D 87, 104016 (2013), 1212.3316.
+[35] I. M. Dozmorov, Fizika 18, 95 (1975).
+[36] J. B. Hartle and K. S. Thorne, Astrophys. J. 153, 807
+(1968).
+

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+ 
+ 
+ 
+ 
+ 
+ 
+Investigation of radiation hardness of silicon semiconductor 
+detectors under irradiation with fission products of 252Cf 
+nuclide.  
+N V Bazlov1,2, A V Derbin1, I S Drachnev1, I M Kotina1, O I Konkov1,3, I S 
+Lomskaya1, M S Mikulich1, V N Muratova1, D A Semenov1, M V Trushin1  and  
+E V Unzhakov1 
+1 NRC "Kurchatov Institute" - PNPI, Gatchina, Russia 
+2 Saint-Petersburg State University, Universitetskaya nab. 7/9, St. Petersburg, Russia 
+3 Ioffe Physical-Technical Institute of the Russian Academy of Sciences, St. 
+Petersburg, Russia 
+ 
+e-mail: [email protected] 
+Abstract. Influence of the prolonged irradiation by fission products of 252Cf radionuclide on 
+the operational parameters of silicon-lithium Si(Li) p-i-n detectors, Si surface barrier detectors 
+and Si planar p+n detector was investigated. The obtained results revealed a linear shift of the 
+fission fragment peaks positions towards the lower energies with increase of the irradiation 
+dose for all investigated detectors. The rate of the peaks shift was found to depend strongly on 
+the detector type and the strength of the electric field in the detector’s active region, but not on 
+the temperature of irradiation (room or liquid nitrogen temperature). Based on the obtained 
+results, the possibility of integration of the investigated types of Si semiconductor detectors in 
+a radionuclide neutron calibration source is considered. 
+1.  Introduction 
+Heavy nuclides subjected to spontaneous fission decay accompanied by emission of several fast 
+neutrons can be utilized as a compact neutron calibration source. The most common spontaneous 
+fission source is 252Cf which undergoes α-decay and spontaneous fission with a branching ratio of 
+97:3, whereas each spontaneous fission event liberates 3.8 neutrons and 9.7 gamma-ray photons on 
+average [1]. The timing of the moment of neutron production can be fixed by detecting the fission 
+fragments signal with a semiconductor detector. 
+Semiconductor detectors possess sufficiently high energy resolution for detection of the high-
+energy heavy ions. The main obstacle for the integration of such detectors in the neutron calibration 
+source could be their limited lifetime under the influence of the nuclide radiation [2]. Degradation of 
+the detector’s operational parameters effectively proceeds just in case of irradiation by alpha particles 
+and fission fragments (FF), which are capable of transferring a significant fraction of their energy to 
+the atoms of the detector lattice. Therefore, the degradation of the semiconductor detector will limit 
+the maximum neutron source activity and/or the source expiration period.  
+This article is devoted to the investigations of degradation of the operational parameters of several 
+types of silicon semiconductor detectors under prolonged irradiation with fission products of 252Cf (-
+
+ 
+ 
+ 
+ 
+ 
+ 
+particles and fission fragments). The main issue was to study the rate of degradation of different 
+detector types under irradiation by 252Cf fission products at various irradiation conditions. Irradiation 
+was performed at room and liquid nitrogen temperatures as well as with different detector’s 
+operational biases, i.e. with different electric field strength in the detectors active regions. Results of 
+the preceding investigations were presented in previous articles [3-5].  
+2.  Detectors and experimental setup 
+Three types of silicon semiconductor detectors were under investigations. Detectors of the first type 
+are SiLi p-i-n detectors produced from p-type silicon ingot with resistivity of 2.5 kΩ×cm and carrier 
+lifetime of 1000 µs. Two similar detectors with a sensitive region of 20 mm in diameter and 4 mm 
+thick were produced using standard Li drift technology [6]. The thickness of the undrifted p-type layer 
+in these detectors (i.e. the entrance window thickness) usually amounts to 300-500 nm [7], which is 
+kept to suppress the excessive growth of the leakage current at high operation reverse voltage [8]. 
+Detectors of the second type were two surface-barrier (SB) detectors fabricated from p-type boron-
+doped silicon wafer of (111) orientation and 10 mm in diameter. The resistivity and the carrier lifetime 
+were 1 kΩ×cm and 1000 µs, respectively. The front side of the wafers was covered by a thin layer of 
+amorphous silicon which served as a passivation coating [9]. The ohmic contact was made by 
+sputtering of Pd layer on the whole rear side of the wafer, whereas the rectifying one – by evaporation 
+of Al dot with diameter of 7 mm in the center of the wafer’s front side. Detector of the third type was 
+p+n planar detector with the thickness of 300 m produced in Ioffe Physical-Technical Institute 
+(entrance window thickness was about 50 nm and the voltage of full depletion – nearly 150 V).  
+Irradiation by a 252Cf source was performed in vacuum cryostat typically during 10-20 days. The 
+source representing a stainless steel substrate covered by an active layer under the thin protective 
+coating was mounted 1 cm above the detector front surface that was collimated in order to exclude 
+side surface effects of incomplete charge collection. The spectra of the fission products of 252Cf were 
+recorded continuously during the whole irradiation period in short 1-hour series, what allowed us to 
+observe the spectra evolution directly. Detector reverse current was also monitored during the whole 
+irradiation period on 5-second basis with the following averaging on 1-hour measurement series. 
+Details of the measurement setup were presented in [3-5]. 
+ 
+ 
+ 
+ 
+Figure 1. (a) The first and the last spectra measured by SB2 detector in the beginning and at the end of 
+the prolonged irradiation period. The following peaks are marked: constant amplitude generator peak 
+(g), peak of -particles at 6.118 MeV (), peak at doubled energy of -particles (2) and the peaks 
+due to FFs of light (LF) and heavy (HF) groups. (b) Dependence of the light and heavy FF peaks 
+visible energies on exposure by FFs.  
+
+a)
+106
+α
+first spectrum
+last spectrum
+g
+105
+Counts per hour
+104
+103
+2α
+HF
+102
+101
+100
+0
+20
+40
+60
+80
+100
+Energy, MeVb)
+Heavy Fragments
+80
+Light Fragments
+Peak Position, MeV
+75
+70
+65
+60
+55
+0
+1
+2
+3
+4
+Exposure, FF*107 
+ 
+ 
+ 
+ 
+ 
+3.  Experimental results 
+In order to study the influence of temperature of irradiation on the degradation of the detector’s 
+parameters, the irradiation of identical SiLi detectors was performed at room (SiLi1 detector) and 
+liquid nitrogen (SiLi2 detector) temperature, respectively. To study the influence of external electric 
+field strength on the detector’s parameters degradation, two identical SB detectors were subjected to 
+the irradiation with different applied reverse biases, i.e. with different electric field strengths in their 
+active regions. The operating biases applied to the respective detector during the irradiation period, the 
+corresponding surface electric field strengths and the total exposures are collected in Table 1.  
+For all investigated detectors the similar signs of operational parameters degradation as a result of 
+the prolonged irradiation by 252Cf fission products were revealed. As an example, Figure 1a represents 
+the spectra recorded by SB2 detector at the beginning and at the end of the prolonged irradiation 
+period. The peak at 6.1 MeV corresponding to α-particles and another peak at doubled energy of the α-
+particles caused by their accidental coincidences were used as reference points for the calibration of 
+the energy scale. Two broad unresolved peaks appearing at higher energies correspond to fission 
+fragments of light and heavy groups, respectively.  
+The main effect of the detector degradation is a gradual shift of fission fragments visible energy 
+towards the lower values, see Figure 1a. The positions of the peaks corresponding to heavy (HF) and 
+light (LF) fission fragments were approximated using the Gaussian function for each 1-hour series. 
+The dependences of the peaks positions with exposure by fission fragments can be well described by 
+linear functions (Figure 1b) for any masses of fission fragments and for all investigated detectors. The 
+obtained slope coefficients are summarized in Table 1. It is interesting to note, that the obtained 
+coefficients for the peaks of light and heavy fission fragments groups differ approximately by the 
+factor of 2 – this holds for all types of investigated detectors and for all irradiation conditions. In more 
+details this fact will be discussed separately in the next paper. A similar approximation of the positions 
+of α-peaks didn’t reveal any measurable shift with the irradiation dose for all studied detectors [4-5].  
+Another sign of the detector’s operational parameters degradation under irradiation is the rapid 
+increase of the leakage current which proceeds linear with the number of absorbed fission products 
+[3]. The obtained slope coefficients of the leakage current growth are also collected in Table 1.  
+4.  Discussion  
+It could be noted in Figure 1 that the peak energies of light and heavy groups of fission fragments are 
+below the predicted values of 104 MeV and 79 MeV [10], respectively, even on the spectrum 
+measured by non-irradiated detectors. The same is true for all other investigated detectors. This effect 
+is known as pulse-height defect (PHD) in heavy charged particles spectroscopy by semiconductor 
+detectors implying that the measured pulse height amplitude for heavy charged particles is somewhat 
+lower than that for -particles of the same energy [1]. It is generally considered that PHD is caused by 
+a combination of energy losses (i) in the detector dead layer/entrance window, (ii) due to the atomic 
+collisions and (iii) due to recombination of the electron-hole pairs created by the incident heavy 
+particle. Whereas energy losses by (i) and (ii) mechanisms are well understood, the full understanding 
+of the charge losses due to recombination is still missing. Two models were suggested supposing that 
+enhanced carrier recombination proceeds either in the bulk region on the radiation-induced defects 
+created by incident FFs [11], or at the surface states of the semiconductor [12]. The later model is 
+consistent with the TRIM [13] simulation results (Figure 2) showing that the density of electron-hole 
+pairs generated by fission fragments reaches the maximum in the near-surface region of the detector 
+and then gradually drops down towards the bulk, suggesting therefore that decisive influence on PHD 
+would have the carrier recombination at the surface states.  
+Previously, the PHD of about 7-10 MeV was reported for 252Cf fission fragments detection by 
+semiconductor detectors not subjected to the prolonged irradiation [10]. These PHD values are close to 
+those ones obtained for the investigated planar and SB1 detectors operated at high reverse bias – see 
+Table 1. We believe, that higher PHD values in non-irradiated SiLi are related with rather thick 
+entrance window in these detectors. Whereas the increase of PHD for SB2 detector operated at lower 
+
+ 
+ 
+ 
+ 
+ 
+ 
+Table 1. Irradiation conditions and the degradation of the operational parameters of the investigated 
+detectors: Ub – applied bias during irradiation; Fs – surface electric field strength; PHDLF/ PHDHF – 
+pulse-height defects for light and heavy fragments peaks registered by non-irradiated detectors; NFF 
+and N – exposure by fission fragments and -particles, respectively; ∆EHF/∆NFF – slope coefficient 
+describing the linear shift of heavy fission fragment maximum; ∆ELF/∆NFF – slope coefficient 
+describing the linear shift of light fission fragment maximum; ∆I/∆N – rate of the reverse current 
+increase relative to the total number of the registered fission products (wasn’t measured for SiLi2 
+detector); NFFmax – maximal permissible exposure by fission fragments; t – expected active operation 
+period of the detector in a neutron source.  
+ 
+ 
+p+n planar 
+SB1 
+SB2 
+SiLi1 
+SiLi2 
+Ub, V 
+150 
+200 
+30 
+400 
+400 
+Fs, kV/cm 
+8.5 
+40 
+17 
+1.5 
+1.5 
+PHDLF/PHDHF, MeV 
+8/10 
+9/11 
+18/19 
+28/29 
+35/37 
+NFF *108 
+1.1 
+0.45 
+0.43 
+3.4 
+1 
+N* 
+0.5 
+0.20 
+0.19 
+1.5 
+0.44 
+∆EHF/∆NFF*10-5, keV/FF 
+-0.9 
+-1.8 
+-8.9 
+-3.6 
+-5.7 
+∆ELF/∆NFF*10-5, keV/FF 
+-1.9 
+-3.9 
+-20 
+-6.2 
+-12 
+∆I/∆N*10-16, A/ion
+8.9 
+14 
+8.0 
+4.4 
+- 
+NFFmax *108 
+22 
+12 
+2.2 
+6.9 
+4.7 
+t, years 
+11.6 
+6.3 
+1.2 
+3.6 
+2.5 
+ 
+ 
+electric field (Table 1) reflects the influence of the electric field strength on the charge carrier 
+collection efficiency, i.e. on the recombination of the generated electron-hole pairs (note that the 
+active layer thickness in SB2 detector exceeds the projection range of incident FFs even at 30V). 
+As a result of the prolonged irradiation by 252Cf fission products, the linear shift of FF peaks 
+positions, i.e. the linear increase of PHD for fission fragments peaks, was revealed. Since the task of 
+semiconductor detector operating as a part of neutron calibration source is the reliable detection of 
+fission fragments signal, the irradiated detector could be considered to be "degraded" when the 
+spectrum of the heavy fission fragment overlaps with much more intense signal at double energy of α-
+peak, what prevents us from discrimination between them [3]. The values of maximal “permissible” 
+exposure by fission fragments NFFmax corresponding to the beginning of the peaks overlap at three 
+standard deviations from their maxima were estimated for each detector using the corresponding slope 
+coefficients derived for HF peak and the results are presented in Table 1.  
+ 
+ 
+ 
+Figure 2. TRIM simulated vacancies 
+distribution profiles (solid lines) and 
+linear densities of electron-hole pairs 
+(dashed lines) generated by light and 
+heavy FFs with mean energies and 
+masses of 104 MeV and 79 MeV, 106 
+amu and 142 amu, respectively.  
+
+Heavy Fragments
+300
+2
+Light Fragments
+250
+200
+150
+100
+50
+0
+0
+0
+5
+10
+15
+20
+Depth, μm 
+ 
+ 
+ 
+ 
+ 
+Permissible exposure values NFFmax for the investigated detectors appeared to vary approximately 
+by one order of magnitude. The highest NFFmax values were found for planar and SB1 detector operated 
+at 200V. Reduction of the operating bias and thus the electric field strength in case of SB2 detector has 
+led to considerable decrease of the expected permissible exposure value. Therefore, the electric field 
+strength affects not only the PHD on non-irradiated detector, but also the value of the expected 
+maximal exposure. However the NFFmax exposure values for SiLi detectors – which operated with 
+lowest electric field as compared with other detectors – are significantly higher than that for SB2 
+detector. Thus the expected maximal exposure appeared to be more sensitive to the electric field 
+strength in the surface barrier detectors and less sensitive in SiLi and planar detectors. It follows then 
+that not only the electric field strength, but also a detector’s internal structure defines the PHD growth 
+under irradiation and the maximal permissible exposure.  
+According to TRIM simulations, irradiation of Si detectors by fission fragments will lead to the 
+creation of vacancy-interstitial pairs and therefore to the formation of high density of radiation-
+induced defects in the region from detector surface till the depth of 17 μm with the maxima at 14-16 
+μm (Figure 2). Additionally TRIM indicates, that the energy of FFs is high enough to damage the 
+detector surface by sputtering. Therefore, prolonged irradiation with fission fragments will lead to an 
+increase of the carrier recombination rate both in Si bulk and on the surface of the semiconductor, thus 
+contributing to the PHD growth.  
+The transition region in the detectors produced by planar and by SiLi technology (p+n and p-i 
+transition regions, respectively) is located inside the crystalline matrix at the typical depths of 50-500 
+nm from the surface. Apparently, the contribution of the surface recombination to the charge carrier 
+losses will be more significant for surface-barrier detectors than for SiLi and planar ones, whereas the 
+contribution of bulk defects – approximately similar in all detectors, what may be the reason for 
+different sensitivity of NFFmax exposure to the electric field strength in these detectors. Additional 
+investigations are needed to determine the dominant charge loss channel. 
+Suggested neutron calibration source should operate also at cryogenic temperatures (liquid nitrogen 
+or slightly above). Performed irradiation of SiLi2 detector at liquid nitrogen temperature has shown, 
+that in contrast to the electric field, temperature of irradiation seems to have no or only minor 
+influence on the expected value of maximal exposure as it could be concluded from the comparable 
+NFFmax values obtained for SiLi detectors irradiated at different temperatures. Somewhat smaller NFFmax 
+exposure obtained for SiLi2 detector is probably related with thicker entrance window in this detector.  
+Knowing the maximal expected exposure values NFFmax, it is possible to estimate the duration of 
+active “lifetime” of neutron calibration source. For the operation of neutron calibration source the 
+reasonable neutron activity would be the around 20 neutrons/s and taking into account that each 
+spontaneous fission releases in average 3.7 fast neutrons, the activity of 20 neutrons/s would 
+correspond to ~6 spontaneous fissions per second. Therefore, considering the maximal exposure value 
+from Table 1, the duration of active “lifetime” of such neutron calibration source will be 1.2-11.6 
+years (without taking into account the decay of the radiation source).  
+During this operation period, a significant increase of leakage current up to ~100 μA can be 
+expected at room temperature, as can be calculated from the obtained coefficients of leakage current 
+growth (Table 1). Such high reverse current is unacceptable and therefore the detector cooling in order 
+to reduce the reverse current during the neutron source operation will be required. The coefficients of 
+current growth upon irradiation by fission products of 252Cf appeared to be an order of magnitude 
+higher than the corresponding coefficients of 7-17×10–17 A/α determined by us earlier for the identical 
+detectors subjected to long-term irradiation by -particles [4]. This fact confirms that prolonged 
+irradiation by FFs leads to the creation of the effective recombination-generation defect centers 
+participating in charge carrier recombination and the reverse current growth.  
+5.  Conclusions  
+Prolonged irradiation of three different types of Si semiconductor detectors by fission products of 252Cf 
+nuclide has led to a gradual increase of pulse-height defect for the fission fragments peaks in all 
+
+ 
+ 
+ 
+ 
+ 
+ 
+investigated detectors. This will eventually lead to the overlap with more intense -peak and therefore 
+to the impossibility of further reliable detection of fission fragments by the semiconductor detector and 
+thus to the limitation of the operation period of neutron calibration source. Obtained experimental 
+results suggest, that in order to assure the longest operation period of the neutron calibration source it 
+is worth to use the semiconductor detectors with lowest surface recombination rate and with highest 
+possible electric field strength in their active region. Among the investigated detectors, the planar one 
+most fully meets these requirements, whereas in relatively thick SiLi detectors it is difficult to achieve 
+the high electric field strength and surface-barrier detectors may suffer from high surface 
+recombination. With properly chosen semiconductor detector the expected active operation period of 
+252Cf-based neutron calibration source may reach up to 12 years.  
+ 
+Acknowledgements  
+The reported study was funded by RFBR, project number 20-02-00571 
+References 
+[1] 
+Knoll G F 2000 Radiation Detection and Measurement, 3rd ed. (New York: John Wiley and 
+Sons) ISBN 978-0-471-07338-3, 978-0-471-07338-3 
+[2] 
+Moll M 2018 IEEE Transactions on Nuclear Science 65 1561–1582 
+[3] 
+Bakhlanov S V, Derbin A V, Drachnev I S, Konkov O I, Kotina I M, Kuzmichev A M, 
+Lomskaya I S, Mikulich M S, Muratova N V, Niyazova N V, Semenov D A, Trushin M V 
+and Unzhakov E V 2021 Journal of Physics: Conference Series 2103 012138 
+[4] 
+Bakhlanov S V, Bazlov N V, Chernobrovkin I D, Derbin A V, Drachnev I S, Kotina I M, 
+Konkov O I, Kuzmichev A M, Mikulich M S, Muratova N V, Trushin M V and Unzhakov E 
+V 2021 Journal of Physics: Conference Series 2103 012139 
+[5] 
+Bazlov N V, Bakhlanov S V, Derbin A V, Drachnev I S, Eremin V K, Kotina I M, Muratova V 
+N, Pilipenko N V, Semenov D A, Unzhakov E V and Chmel E A 2018 Instruments and 
+Experimental Techniques 61 323 
+[6] 
+Bazlov N V, Bakhlanov S V, Derbin A V, Drachnev I S, Izegov G A, Kotina I M, Muratova V 
+N, Niyazova N V, Semenov D A, Trushin M V, Unzhakov E V, Chmel E A 2020 Instrum. 
+Exp. Tech. 63(1) 25 
+[7] 
+Alekseev I E, Bakhlanov S V, Derbin A V, Drachnev I S, Kotina I M, Lomskaya I S, Muratova 
+V N, Niyazova N V, Semenov D A, Trushin M V, Unzhakov E V 2020 Physical Review C 
+102 064329 
+[8] 
+Kozai M, Fuke H, Yamada M, Perez K, Erjavec T, Hailey C J, Madden N, Rogers F, Saffold N, 
+Seyler D, Shimizu Y, Tokuda K, Xiao M 2019 Nuclear Inst. and Methods in Physics 
+Research A 947 162695 
+[9] 
+Kotina I M, Danishevskii A M, Konkov O I, Terukov E I, Tuhkonen L M 2014 Semiconductors 
+48(9) 1167 
+[10] Paasch K, Krause H, Scobel W 1984 Nuclear Inst. and Methods in Physics Research 221 558 
+[11] Eremin V K, Il’yashenko I N, Strokan N B, Shmidt B 1995 Fiz. Tekh. Poluprovodn. 29(1) 79 
+[in Russian] 
+[12] Tsyganov Y S 2013 Physics of Particles and Nuclei 44(1) 92 
+[13] Ziegler J F, Biersack J P, Ziegler M D SRIM – Stopping and Range of Ions in Matter 
+www.srim.org (accessed: May 2022) 
+ 
+

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+A Systematic Mapping Study on Responsible AI
+Risk Assessment
+Boming Xia∗†, Qinghua Lu∗†, Harsha Perera∗, Liming Zhu∗†, Zhenchang Xing∗, Yue Liu∗†, Jon Whittle∗
+∗CSIRO’s Data61, Sydney, Australia
+†University of New South Wales, Sydney, Australia
+Abstract—The rapid development of artificial intelligence (AI)
+has led to increasing concerns about the capability of AI systems
+to make decisions and behave responsibly. Responsible AI (RAI)
+refers to the development and use of AI systems that benefit
+humans, society, and the environment while minimising the risk
+of negative consequences. To ensure responsible AI, the risks as-
+sociated with AI systems’ development and use must be identified,
+assessed and mitigated. Various AI risk assessment frameworks
+have been released recently by governments, organisations, and
+companies. However, it can be challenging for AI stakeholders to
+have a clear picture of the available frameworks and determine
+the most suitable ones for a specific context. Additionally, there
+is a need to identify areas that require further research or
+development of new frameworks. To fill the gap, we present a
+mapping study of 16 existing RAI risk assessment frameworks
+from the industry, governments, and non-government organiza-
+tions (NGOs). We identify key characteristics of each framework
+and analyse them in terms of RAI principles, stakeholders,
+system lifecycle stages, geographical locations, targeted domains,
+and assessment methods. Our study provides a comprehensive
+analysis of the current state of the frameworks and highlights
+areas of convergence and divergence among them. We also
+identify the deficiencies in existing frameworks and outlines the
+essential characteristics a concrete framework should possess.
+Our findings and insights can help relevant stakeholders choose
+suitable RAI risk assessment frameworks and guide the design
+of future frameworks towards concreteness.
+Index Terms—artificial intelligence, machine learning, risk
+assessment, impact assessment, responsible AI, risk mitigation,
+pattern
+I. INTRODUCTION
+The adoption of artificial intelligence (AI) in various appli-
+cation domains has led to numerous advantages, such as im-
+proved efficiency and reduced cost in manufacturing. However,
+the risks associated with AI systems have also attracted signif-
+icant attention from both industry and academia [1]–[3]. For
+example, an AI system may make biased decisions that lead
+to unintended discrimination [4]–[6]. Also, the AI system’s
+dataset may contain sensitive information, risking violation of
+laws such as EU General Data Protection Regulation (GDPR)1
+and EU AI Act (proposed)2. The AI incident database3 has
+collected over 2200 (as of January 2023) reported real-world
+incidents caused by AI systems.
+Responsible AI (RAI) is developing and applying AI sys-
+tems that benefit humans, society, and the environment (HSE)
+1https://gdprinfo.eu/
+2https://artificialintelligenceact.eu/
+3https://incidentdatabase.ai/
+while minimizing the associated risks. A number of RAI
+principle frameworks that AI systems and stakeholders should
+adhere to have been released recently, such as Australia’s
+AI Ethics Principles4 and European Commission’s Ethics
+guidelines for trustworthy AI5. Many organizations have de-
+veloped principle-driven RAI risk assessment frameworks to
+implement RAI based on the RAI principles, m (e.g., US NIST
+AI risk management framework [7], EU Assessment List for
+Trust AI framework [8]). These frameworks are designed to
+help organizations and individuals systematically assess and
+mitigate potential risks associated with AI systems. Despite
+the availability of these RAI risk assessment frameworks, AI
+system stakeholders need to gain a holistic view of the existing
+frameworks to choose the most appropriate one for their
+context. Also, it is unclear how effective these frameworks
+are at assessing and mitigating RAI risks.
+To bridge the gaps, we have performed a systematic map-
+ping study on the existing RAI risk assessment frameworks.
+The main objectives of this study are: 1) to provide a summary
+of the current available higher-quality AI risk frameworks to
+which researchers and practitioners can refer; 2) to investigate
+the capabilities and limitations of the RAI risk assessment
+frameworks; and 3) to provide insights for future research and
+development on concrete AI risk assessment frameworks.
+The main contributions of this study are:
+• We present a comprehensive qualitative and quantitative
+analysis and synthesis of 16 state-of-practice RAI risk
+assessment frameworks selected from the grey literature.
+• We provide empirical findings and insights on the capa-
+bilities and limitations of the existing frameworks and
+highlight the essentials for developing concrete RAI risk
+assessment frameworks.
+The remainder of this paper is organized as follows: Sec-
+tion II presents the methodology and research questions (RQ)
+followed by the results and findings for each RQ in section
+III. Section IV discusses RAI risk assessment framework
+“concreteness” and threats to validity. Then, section V lists
+related work while section VI concludes the paper with a
+summary and the future work of this study.
+4https://www.industry.gov.au/publications/australias-artificial-intelligence-
+ethics-framework/australias-ai-ethics-principles
+5https://ec.europa.eu/digital-single-market/en/news/ethics-guidelines-
+trustworthy-ai
+arXiv:2301.11616v1  [cs.SE]  27 Jan 2023
+
+Fig. 1: Methodology overview.
+II. METHODOLOGY
+We perform the systematic mapping study following
+Kitchenham’s guideline [9] on conducting literature reviews
+in software engineering. The overall methodology is presented
+in Figure 1. To investigate the capabilities of the existing AI
+risk assessment frameworks, we derived the following RQs:
+• RQ1: Who have published RAI risk assessment frame-
+works?
+• RQ2: What are the characteristics of the existing RAI
+risk assessment frameworks?
+– RQ2.1 What RAI principles are addressed?
+– RQ2.2 Who are the stakeholders?
+∗ RQ2.2.1 Who conducts the assessment?
+∗ RQ2.2.2 Whose activities are assessed?
+– RQ2.3 What is the scope of the frameworks?
+∗ RQ2.3.1 Which development stages are covered
+by the frameworks?
+∗ RQ2.3.2 Where can the frameworks be applied?
+∗ RQ2.3.3 Which domains/sectors are the frame-
+works designed for?
+• RQ3: How are the RAI risks assessed?
+– RQ3.1: What are the inputs?
+– RQ3.2: What is the assessment process?
+– RQ3.3: What are the outputs?
+The data sources include ACM, IEEE, Science Direct,
+Springer, Google scholar for academic papers and Google
+Search for industrial frameworks. The search was conducted
+in November 2022 and the search term is (“artificial in-
+telligence” OR “machine learning” OR AI OR ML) AND
+(impact OR risk) AND (assess OR assessment OR assessing
+OR evaluate OR evaluation OR evaluating OR measure OR
+measurement OR measuring OR mitigate OR mitigation OR
+mitigating OR manage OR managing OR management). The
+study only includes frameworks with relatively concrete AI
+risk assessment solutions and excludes frameworks discussing
+high-level AI risks.
+Although the literature search included academic databases,
+academic papers and frameworks are excluded considering the
+following reasons: a) one of our inclusion criteria is to include
+frameworks that are currently being used in practice (e.g.,
+governmental/industrial/international organizations consulting
+extensively with practitioners and extracting proven uses); b)
+based on the inclusion and exclusion criteria, the academic pa-
+pers we collected are either discussing the identified industrial
+frameworks, or lack of details on concrete AI risk assessment
+solutions. In the end, we selected 16 industrial frameworks to
+be investigated. The complete research protocol is available
+online6. We adopt Australia’s AI ethics principles in this
+study: Human, societal and environmental (HSE) wellbeing,
+Human-centered values, Fairness, Privacy protection and se-
+curity, Reliability and safety, Transparency and explainability,
+Contestability, and Accountability.
+III. RESEARCH RESULTS
+This section presents the results and findings of each RQ.
+A. RQ1: Who have published RAI risk assessment frame-
+works?
+As illustrated in Table I and Fig. 2, we identified 16 frame-
+works to be included in this study. These frameworks have
+been published by organizations based in the United States
+(US), the United Kingdom (UK), the European Union (EU),
+Canada (CA), Australia (AU), Singapore (SA), the Netherlands
+(NL), and Germany (DE). Additionally, one framework has
+been released by the World Economic Forum, an international
+(INT) organization. Although we did not set a specific time
+limit for the literature search, as shown in Fig. 2b, the majority
+of the frameworks (10 out of 16, 62.5%) were published
+or last updated (some frameworks tend to be updated over
+time) in 2022, followed by 1 framework (6.25%) published in
+2021, 3 frameworks (18.75%) published in 2020, 0 framework
+published in 2019, and 2 (12.5%) frameworks published in
+2018. Also, a significant proportion of the frameworks (9
+out of 16, 56.25%) were published by government agencies
+worldwide, with 6 of them last updated in 2022. This suggests
+that the issue of RAI risks has been gaining significant
+attention worldwide, particularly among government agencies.
+In terms of the number of published frameworks (Fig. 2a
+and Fig. 2c), the US leads in research and development on RAI
+risk assessment and published 6 related frameworks, including
+3 frameworks from US government agencies (i.e., I1 by
+National Institute of Standards and Technology (NIST), and I8
+6https://docs.google.com/document/d/1F sAmRI7zvJBYyiF96cn5oNtG3O8
+psyB/edit?usp=sharing&ouid=111846093034327217492&rtpof=true&sd=true
+
+Retrieved
+Academic: 10988
+Grey: 3160
+Tentative
+Academic: 104
+Grey: 198
+Tentative
+Academic: 104+39=143
+Grey: 198+24=222
+Selected
+Research protocol
+Academic: 0
+Grey: 16
+ReportingTABLE I: Industrial AI risk assessment frameworks (collected in November 2022).
+Demographics (RQ1)
+Characteristics (RQ2)
+Processes (RQ3)
+No.
+Frameworks
+Region
+Affiliation
+Affiliation type First release Last update
+RAI Princples
+Stakeholders
+Stages
+Region
+Sector
+Type
+Mitigation
+*Risk factors
+I1
+AI risk management framework
+(AI RMF) [7]
+US
+National Institute of Standards
+and technology (NIST)
+Government
+2022.05
+2022.08
+All principles
+Specified
+All stages
+Region-
+agnostic
+Sector-
+agnostic
+Descriptive
+*Yes
+Hazard,
+exposure,
+vulnerability
+I2
+Assessment list for trustworthy AI
+(ALTAI) [8]
+EU
+European Commission High-
+Level Expert Group on AI
+Government
+2019.06
+2020.07
+All principles
+Specified
+All stages
+Region-
+agnostic
+Sector-
+agnostic
+Procedural
+Yes
+Hazard,
+exposure,
+vulnerability
+I3
+Algorithm Impact Assessment
+tool (AIA) [10]
+CA
+Government of Canada
+Government
+2019
+2022.11
+Not specified
+Not specified
+Planning &
+requirements
+analysis,
+design, testing
+Region-
+agnostic
+Sector-
+agnostic
+Procedural
+Yes
+Hazard,
+exposure,
+vulnerability
+I4
+Fundamental rights and algorithm
+impact assessment (FRAIA) [11]
+NL
+Ministry of the Interior and
+Kingdom Relations (BZK)
+Government
+2022.03
+N/A
+Not specified
+Specified
+All stages
+Region-
+agnostic
+Public
+sectors
+Procedural
+*Yes
+Hazard,
+exposure,
+vulnerability
+I5
+AI and data protection risk toolkit
+[12]
+UK
+Information Commissioner’s
+Office (ICO)
+Government
+2021
+2022.05
+HSE wellbeing, human-
+centered values, fairness,
+privacy protection &
+security, reliability &
+safety, transparency &
+explainability,
+accountability
+Specified
+All stages
+Reusable
+anywhere
+with
+adjustments
+Sector-
+agnostic
+Procedural
+No
+Hazard,
+exposure,
+vulnerability
+I6
+Model AI governance framework
+[13]
+SA
+Personal Data Protection
+Commission (PDPC)
+Government
+2019.01
+2020.01
+HSE wellbeing, human-
+centered values, fairness,
+transparency &
+explainability, reliability
+& safety
+Not specified
+Not specified
+Region-
+agnostic
+Sector-
+agnostic
+Descriptive
+Yes
+Hazard,
+exposure,
+vulnerability
+I7
+NSW artificial intelligence
+assurance framework [14]
+AU
+NSW Government
+Government
+2022.03
+N/A
+HSE wellbeing, human-
+centered values, fairness,
+privacy protection &
+security, reliability &
+safety, transparency &
+explainability,
+accountability
+Specified
+All stages
+Reusable
+anywhere
+with
+adjustments
+Sector-
+agnostic
+Procedural
+*Yes
+Hazard,
+exposure,
+vulnerability,
+mitigation risk
+I8
+Ethics & algorithms toolkit [15]
+US
+GovEX, the City and County
+of San Francisco, Harvard
+DataSmart, and Data
+Community DC
+Government
+involved
+2018
+N/A
+HSE wellbeing, human-
+centered values, fairness,
+privacy protection
+& security, reliability &
+safety, transparency &
+explainability,
+accountability
+Specified
+Not specified
+Region-
+agnostic
+Sector-
+agnostic
+Procedural
+No
+Hazard,
+exposure,
+vulnerability
+I9
+RFD-BUS012A artificial
+intelligence assessment tool [16]
+US
+Pennsylvania Office of
+Administration
+Government
+2018.09
+2022.08
+Not specified
+Not specified
+Planning &
+requirements
+analysis, design
+Region-
+agnostic
+Sector-
+agnostic
+Procedural
+No
+Vulnerability
+I10
+Model rules on impact assessment
+of algorithmic decision-making
+systems used by public
+administration [17]
+EU
+European Law Institute
+(ELI)
+NGO
+2022.01
+N/A
+Not specified
+Not specified
+Not specified
+Region-
+agnostic
+Public
+sectors
+Procedural
+Yes
+Hazard,
+exposure,
+vulnerability
+I11 Artificial intelligence for children
+toolkit [18]
+INT
+World Economic Forum
+(WEF)
+NGO
+2022.03
+N/A
+HSE wellbeing, human-
+centered values, fairness,
+privacy protection &
+security, reliability &
+safety, transparency &
+explainability,
+contestability
+Specified
+Not specified
+Region-
+agnostic
+Children
+& youth
+Procedural
+Yes
+Hazard,
+exposure,
+vulnerability
+I12
+Recommended practices for
+assessing the impact of
+autonomous and intelligent
+systems on human well-being [19]
+US
+IEEE
+NGO
+2020.05
+N/A
+HSE wellbeing, human-
+centered values
+Specified
+All stages
+Region-
+agnostic
+Sector-
+agnostic
+Descriptive
+*Yes
+Hazard,
+exposure,
+vulnerability
+I13 Responsible AI impact assessment
+template [20]
+US
+Microsoft
+Industry
+2022.06
+N/A
+All principles
+Not specified
+Not specified
+Region-
+agnostic
+Sector-
+agnostic
+Procedural
+No
+Hazard,
+exposure,
+vulnerability
+I14 Algorithmic impact assessments
+(AIAs) in healthcare [21]
+UK
+Ada Lovelace Institute
+NGO
+2022.01
+N/A
+HSE wellbeing, human
+centered values, fairness,
+privacy protection &
+security, reliability &
+safety, transparency &
+explainability
+Specified
+Planning &
+requirements
+analysis,
+design, tetsing,
+deployment,
+monitoring
+UK only
+UK
+healthcare Procedural
+*Yes
+Hazard,
+exposure,
+vulnerability
+I15 Artificial intelligence impact
+assessment [22]
+NL
+ECP, Platform for the
+Information Society
+NGO
+2018
+N/A
+Not specified
+Specified
+Not specified
+Region-
+agnostic
+Sector-
+agnostic
+Procedural
+*Yes
+Hazard,
+exposure,
+vulnerability,
+mitigation risk
+I16
+Automated decision-making
+systems in the public sector: an
+impact assessment
+tool for public authorities [23]
+DE
+Algorithm Watch
+NGO
+2021.06
+N/A
+All principles
+Not specified
+Not specified
+Region-
+agnostic
+Public
+sectors
+Procedural
+*Yes
+Hazard,
+exposure,
+vulnerability
+
+(a) Percentage by region.
+(b) Number by year.
+(c) Number by year and region.
+Fig. 2: Demographics of collected frameworks.
+and I9 by two state/city government agencies) and 2 from US-
+based organizations (i.e., I12 by IEEE and I13 by Microsoft).
+The UK, EU, and Netherlands rank second with 2 frameworks
+developed. In the UK, one framework was developed by a
+government agency (Information Commissioner’s Office, ICO)
+and Ada Lovelace Institute published the other specifically
+for the proposed National Medical Imaging Platform of the
+National Health Service (NHS). The EU published 2 frame-
+works on RAI risk assessment in 2020 (last updated) and
+2022, respectively. Notably, the EU has drafted its AI Act,
+which marks a significant step towards operationalizing RAI
+by legislation. The Netherlands published 2 frameworks, one
+in 2018 by a non-government organization (ECP) and the other
+in 2022 by its Ministry of the Interior and Kingdom Relations
+(BZK). Australia’s New South Wales government published
+the nation’s first AI Assurance Framework in 2022. Singapore
+had its AI governance framework published in 2019, while
+it launched AI Verify in May 2022 to objectively assess AI
+systems in a verifiable way. The Canadian government released
+its Algorithm impact assessment tool in 2021. The World
+Economic Forum (WEF) published a toolkit for managing
+RAI risks to children. Lastly, one framework published by
+a German-based organization, Algorithm Watch, is identified
+in this study.
+Finding to RQ1: The growing number of RAI risk
+assessment frameworks worldwide indicates increasing
+global concern about the risks associated with the
+development and use of AI systems and a growing
+recognition of RAI approaches to assess and mitigate
+RAI risks.
+B. RQ2: What are the characteristics of the existing AI risk
+assessment frameworks?
+This subsection discusses the characteristics of the collected
+frameworks based on the following aspects: RAI principles,
+stakeholders, software development lifecycle stages, geograph-
+ical locations, and targeted sectors.
+To improve presentation, we first classify the frameworks
+based on whether they have clear specifications on differ-
+ent characteristics (e.g., whether RAI principles/stakeholder-
+s/stages are specified). Then, we further categorize them to see
+whether they have formulated the assessment and mitigation
+based on different sub-categories of those characteristics (e.g.,
+different RAI principles).
+1) RQ2.1 What RAI principles are addressed?: This sub-
+RQ aims to investigate the RAI principles (i.e., the correspond-
+ing risk category) addressed by the identified frameworks. We
+have mapped the various principles from different frameworks
+to Australia’s AI ethics principles (see Table I).
+As illustrated in Fig. 3a, among the 16 identified frame-
+works, 11 frameworks (I1, I2, I5-I8, I11-I14, I16) have speci-
+fied their guiding principles. 5 frameworks (I3, I4, I9, I10, I15)
+do not explicitly state their corresponding principles, although
+they may implicitly encompass these principles through their
+framework description and introduction (e.g., I3) or references
+to other existing frameworks, standards, and guidelines (e.g.,
+I4). Among the 11 frameworks with specified guiding princi-
+ples, only 5 frameworks (i.e., I1, I2, I11, I13, I16) organize
+their sets of RAI risk assessment questions or checklists based
+on different RAI principles.
+All the 11 frameworks that explicitly specify the guid-
+ing principles or targeted risks consider HSE wellbeing and
+human-centred values. Out of these 11 frameworks, 10 frame-
+works cover fairness, reliability & safety, transparency &
+explainability. The only exception is framework I12, which
+focuses mainly on HSE wellbeing. Privacy protection & se-
+curity is covered by 9 (I1, I2, I5, I7, I8, I11, I13, I14, I16)
+and accountability is covered by 8 frameworks (I1, I2, I5, I7,
+I8, I11, I13, I16). Only 5 frameworks (I1, I2, I11, I13, I16)
+include contestability (Fig. 3b).
+2) RQ2.2: Who are the stakeholders?: This subsection
+examines the stakeholders involved in the frameworks from
+two perspectives: the framework user(s) who are responsible
+for conducting the risk assessment (i.e., assessor), and those
+whose activities are being assessed (i.e., assessee). The stake-
+holders classification is based on [24], where the stakeholders
+are categorized into three levels: industry-level, organization-
+level, and team-level (see Fig. 4).
+
+INT, 1
+DE,1
+SA, 1
+US, 5
+CA, 1
+AU,1
+NL, 2
+EU,2
+UK, 21
+3
+6
+1
+1
+2
+11
+1
+1
+1
+2
+1
+1
+1
+1
+3
+1
+1
+1(a) Principle overview.
+(b) Principle coverage.
+Fig. 3: Ethical principles covered by the identified frameworks.
+Fig. 4: Stakeholders classification [24].
+Fig. 5 shows that 10 of the collected frameworks (I1, I2,
+I4, I5, I7, I8, I11, I12, I14, I15) have mentioned their targeted
+stakeholders. For example, NIST’s AI RMF (I1) specifies
+the framework is intended for “AI actors” defined by the
+Organisation for Economic Co-operation and Development
+(OECD), while EU’s ALTAI (I2) has listed the example
+stakeholders in its guide on “How to complete ALTAI7”.
+However, only the Netherlands BZK’s FRAIA (I4) has clearly
+specified the different stakeholders associated with different
+assessment stages to answer stage-specific questions.
+The data synthesis results show that all 16 frameworks
+involve the participation of RAI governors as the assessors and
+development teams (e.g., data scientists, system developers) as
+the assessee. RAI governors are those who set and enforce RAI
+7https://altai.insight-centre.org/Home/HowToComplete
+Fig. 5: Stakeholders of collected frameworks.
+policies within an organization or community, and they can be
+internal or external.
+One issue identified through the data synthesis process is
+the lack of consideration of more diverse and inclusive (i.e.,
+comprehensive) roles of stakeholders from different levels. For
+example, industry-level procurers are largely neglected, with
+only I1, I2 and I7 considering this aspect. Team-level speaking,
+all 10 frameworks with identified stakeholders require input
+from AI system development teams (i.e., assessees) on infor-
+mation such as intended use, data source, data privacy, and
+algorithm transparency. The assessees typically include prod-
+uct managers, project managers, team leaders, data scientists,
+and system developers. However, 9 out of 10 frameworks fail
+to consider more diverse roles of assessees (e.g., architects,
+UI/UX designers [7], [24]). The US NIST’s AI RMF (I1) is
+distinguished by its inclusion of a broader range of stake-
+holders involved in various stages of AI system development
+and post-development, such as procurement, deployment, and
+operations. However, I1 does not explicitly present categorized
+assessments and mitigations based on different stakeholders.
+Finding to RQ2.1 & RQ2.2: The current RAI risk as-
+sessment frameworks are developed with ad-hoc scope
+and focus, making them difficult for organizations to
+use effectively in practice. This can be seen in the lack
+of consideration for certain key stakeholders, lifecycle
+stages, or ethical principles in their assessment and
+mitigation, failing to identify and mitigate important
+risks.
+3) RQ2.3 What is the scope of the frameworks?: With the
+RQ, we aim to explore the scope of the existing AI risk
+assessment frameworks.
+a) RQ2.3.1: Which development stages are covered by the
+frameworks?
+This RQ aims to investigate the stages covered by the
+collected frameworks in the AI system development lifecycle
+(AI-SDLC). By referencing several existing sources with AI-
+SDLC [1], [7], [12], [25], we first summarized the typical
+stages included in (AI) SDLC (i.e., planning & requirement
+analysis, design, implementation, testing, deployment, opera-
+tion & monitoring). Then, we adapted the tasks in each stage
+
+uestions
+tlassified,
+Principles
+Principles
+5
+not
+specified, 11
+Questions
+specified,
+not
+5
+classified
+611
+11
+10
+9
+10
+10
+5
+8Industry-level stakeholders
+ AI technology producers/procurers
+Al impacted subjects
+AI solution producers/procurers
+RAI governors
+AI users/consumers
+ RAI tool producers/procurers
+Organization-level stakeholders
+Employees
+Board members ·Executives
+Managers·
+Team-level stakeholders
+ Product managers · Project managers · Team leaders
+Business analysts · Architects
+·UX/UI designers
+Data scientists · Developers · Testers · OperatorsQuestions
+not
+Stakeholder
+classified, 9
+Stakeholder
+specified, 10
+not
+specified, 6with additional AI-specific context and derived an AI-SDLC
+(Fig. 6). The detailed results of the AI-SDLC stages covered
+by the collected frameworks are presented in Table I.
+Fig. 7 shows that 7 of the collected frameworks do not
+specify AI system lifecycle stages. Although the other 9
+frameworks (I1-I5, I7, I9, I12, I14) have clarified when they
+can be applied during the AI system lifecycle, The UK ICO’s
+AI and Data Protection Risk toolkit (I5) is the only one that has
+categorized AI risk assessment and evaluation processes based
+on different stages of the AI system lifecycle. Netherlands
+BZK’s FRAIA (I4) is similarly structured in a more coarse-
+grained way in that the assessment is conducted based on three
+stages: input, throughput, and output.
+6 (I1, I2, I4, I5, I7, I12) out of 9 frameworks with specified
+AI system lifecycle stages can be used to evaluate potential
+risks throughout the entire AI system lifecycle. The other 3
+frameworks (I3, I9, I14) focus on the initial stages of ideation
+(i.e., planning & requirements analysis, design). In addition,
+I3 covers the testing stage, while I14 covers the testing,
+deployment and follow-up monitoring stages.
+b) RQ2.3.2: Where can the frameworks be applied?
+With the RQ, we aim to explore whether there are geograph-
+ical constraints to applying the existing AI risk assessment
+frameworks.
+The government-developed frameworks (i.e., I1-I9) can be
+applied anywhere, although some of them may require adjust-
+ments considering region-specific elements in the frameworks.
+For example, The UK ICO’s AI and Data Protection Risk
+toolkit (I5) is aligned with the UK’s General Data Protection
+Regulation (GDPR), and the AU NSW’s AI Assurance Frame-
+work (I7) references relevant policies in the Australian state
+of New South Wales. 6 out of 7 frameworks developed by
+NGOs and industrial companies (I10-I13, I15-I16) are region-
+agnostic. At the same time, I14 is specially designed for UK
+NHS’s planned National Medical Imaging platform.
+c) RQ2.3.2 Which domains/sectors are the frameworks
+designed for?
+This RQ intends to investigate the domains and sectors
+where the frameworks can be applied.
+Most of the collected frameworks are generally designed
+across various domains. However, 5 frameworks have been
+designed for specific purposes. FRAIA (I4), Model Rules (I10)
+and Impact Assessment Tool for Public Authorities (I16) are
+intended for evaluating the development and deployment of
+AI systems in the public sector. The AI for Children toolkit
+(I11) is specifically designed for AI systems that may impact
+children and youth as potential users. AIAs in Healthcare
+(I15) is intended to assess risks associated with designing and
+developing AI systems that require access to the UK National
+Medical Imaging Platform.
+Finding to RQ2.3: The current RAI risk assessment
+frameworks generally consider the entire lifecycle of
+AI systems rather than focusing only on the AI model
+pipeline. However, these frameworks do not provide
+clear guidance on extending or adapting them to fit
+diverse contexts. This limitation restricts the effective-
+ness of RAI risk assessment frameworks as the risks
+and mitigation may vary depending on the context (e.g.
+different organizations, sectors, or regions) in which AI
+systems are used.
+C. RQ3: How are risks assessed?
+This section presents the assessment processes of the col-
+lection frameworks.
+The frameworks are categorized into two types: procedural
+and descriptive. The descriptive frameworks are less concrete
+by providing general non-prescriptive assessment and mitiga-
+tion and not referring to more specific and concrete solutions.
+In contrast, procedural frameworks are more structured and in-
+clude more detailed steps (e.g., inputs, processes, outputs) for
+conducting AI risk assessments. The procedural frameworks
+can also contain suggested mitigation solutions, assessment
+templates, or checklists.
+The collected frameworks examine underlying risks and/or
+corresponding mitigation plans. To better present the results,
+we summarize the different types of risks (i.e., risk factors)
+the frameworks take into account. We adapted the risk cate-
+gorization from a traditional risk management framework [26]
+and added AI-specific context. The adapted risk factors are
+categorized as follows:
+• Hazard: A hazard refers to any dangerous situation or
+condition arising from AI systems or related activities/ar-
+tifacts that can result in harm to HSE wellbeing. Hazards
+are sources of harm or exploit external to AI systems.
+• Exposure: Exposure refers to individuals, property, sys-
+tems, or other elements located within zones affected by
+AI-related hazards that are therefore at risk of potential
+losses.
+• Vulnerability: Vulnerability pertains to the characteris-
+tics and circumstances of an AI system or related artifacts
+that make it susceptible to the detrimental effects of a
+hazard. Compared to hazards, vulnerabilities are internal
+weaknesses/issues of AI systems.
+• Risks by/after mitigation (Mitigation risk): Mitigation
+risks refer to the potential newly introduced risks brought
+about by the implementation of specific mitigation, re-
+silience, or control measures, or residual risks that persist
+even after the implementation of mitigation measures.
+For each of the collected frameworks, we summarized their
+types (i.e., descriptive or procedural) and examined mitigation
+measures and risk factors in Table I.
+We only articulate the answers to RQ3.1 (framework inputs)
+and RQ3.3 (framework outputs) for the procedural framework,
+as the descriptive frameworks do not have direct inputs or
+outputs. RQ3.2 (assessment processes) fits all frameworks, and
+the answer is thus presented based on all frameworks collected.
+
+Fig. 6: AI system lifecycle (adapted from [1], [7], [12], [25]).
+Fig. 7: Stages covered by collected frameworks.
+1) RQ3.1: What are the inputs?: This RQ investigate the
+inputs and the forms of inputs of the procedural frameworks.
+The procedural frameworks are all based on certain forms of
+questionnaires (e.g., self-assessment template, checklist etc.).
+The inputs to these frameworks are answers to predefined
+questions provided by relevant stakeholders (e.g., development
+teams including system developers, data scientists etc.).
+2 frameworks (EU ALTAI, I2 and CA AIA, I3) are de-
+signed as interactive online tools. Users can input the required
+information about their AI systems and get instant feedback
+based on their inputs. Similarly, I5 and I9 are based on
+excel sheets where users can fill in system details or check
+if the recommended practices for minimizing potentials are
+met. I13 and I14 provide self-assessment templates where
+predefined questions regarding the AI system (e.g., intended
+use, stakeholders, benefits/harms) need to be answered. The
+other seven procedural frameworks (I4, I7, I8, I10, I11, I15,
+I16) are available as published reports, where more detailed
+descriptions of the contexts are given. In these reports, AI
+risk/impact assessment questionnaires/checklists are given. It
+is important to note that in Q&A-style assessments, both the
+quality of the answers and the underpinning methodology used
+to generate them are crucial factors, rather than relying solely
+on subjective inputs from the assessors.
+Finding to RQ3.1: The current RAI risk assessment
+frameworks primarily rely on subjective evaluation
+from the assessors via a series of questions or check-
+lists without the support of more objective tools and
+techniques, leading to potentially biased results.
+2) RQ3.2: What are the processes?: This section discusses
+how risk assessments are conducted in both descriptive and
+procedural frameworks.
+The descriptive industrial frameworks include AI RMF (I1)
+by US NIST, Model AI governance framework (I6) by Sin-
+gapore, and recommended practices for assessing the impact
+of autonomous and intelligent systems on human well-being
+(I12) by IEEE.
+AI RMF (I1) is a framework with four components (map,
+measure, manage, and govern) that gives organizations rec-
+ommendations to adopt and adapt to their specific needs. AI
+RMF (I1) is a non-prescriptive framework that aims to identify,
+assess, and manage context-related risks by presenting desired
+outcomes and general approaches for risk management. It pro-
+motes the development of a culture of active risk management
+through its recommendations and non-exhaustive solutions
+presented in its companion playbook8. AI RMF (I1) is a
+non-prescriptive framework that aims to identify, assess, and
+manage context-related risks by presenting desired outcomes
+and general approaches for risk management. It promotes the
+development of a culture of active risk management through
+its recommendations and non-exhaustive solutions presented
+in its companion playbook. Similarly, Singapore’s Model
+Framework (I6) and IEEE’s standard on AI impact assessment
+(I12) are designed to be flexible by providing higher-level
+guidance on the assessment processes.
+8https://pages.nist.gov/AIRMF/
+
+Questions
+not
+Stage not
+Stage
+classified, 8
+specified, 7
+specified, 9Planning &
+Business and ethical requirements analysis: identification of the system's concept and objectives, stakeholders (and possible impacts to
+Requirements analysis
+stakeholders), intended uses (application domain) etc. Ethical considerations (ethics application) etc
+Architectural/structural design (e.g., software architecture design, AI/ML paradigm design (e.g., centralized/distributed/decentralized),
+Design
+detailed design of desired behavior of AI/non-AI components (e.g., UI design, data source identification, model/algorithm selection)
+System construction of both AI (e.g., data collection and processing, existing/new model/algorithm creation/selection) and non-Al
+Implementation
+components, including unit testing and integration testing.
+Implemented system tested against a finite set of test cases. AI/ML model(s) verified & validated on test data, model output calibrated
+Testing
+and interpreted
+Deployment
+Deploying (e.g., canary/blue-green/shadow deployment) the tested system, and verifying regulatory/ethical compliance
+Operation &
+Operating, continuously monitoring (assess both intended and unintended system/model output and impacts), feedback gathering, and
+Monitoring
+maintenance of the deployed systemAs for the procedural frameworks, the assessment pro-
+cesses are based on the input answers, where potential risks
+are identified through the Q&A processes. The assessment
+and evaluation processes of various procedural frameworks
+can be grouped into four categories: risk/principle-based (I2,
+I5, I7, I8, I11, I16), system development process-based (I4,
+I5), essential system component-based (I3, I9), and sys-
+tem description- and requirements-based (I10, I13, I14, I15).
+The risk/principle-based assessments include questions de-
+signed for each of the different risks/principles. The process-
+based assessments include questions throughout different AI-
+SDLC stages, from planning to monitoring & operations.
+The component-based assessments are formulated based on
+essential components (e.g., algorithms, data). The system
+description- and requirements-based solutions offer mecha-
+nisms for the assessee to provide information about their AI
+systems and reflect on compliance with specific requirements.
+For the more developed tools and frameworks, such as
+I2 and I3, the risk scores and potential risks are calculated
+automatically based on the selections/inputs. As for other
+procedural frameworks, such as report- or excel-based ones,
+they identify and assess risks by the assessment conductors
+through a more manual process. The assessors evaluate the
+system’s details, such as intended and unintended uses, stake-
+holders, data integrity, algorithmic explainability, and consult
+with external or internal stakeholders if necessary. This process
+enables a seemingly systematic analysis of an AI system to
+evaluate its impact and risks.
+A valuable part of the AI risk assessment is the mitigation
+plans suggested by some frameworks. In Table I, we summa-
+rize whether clear mitigation considerations are included in the
+frameworks by examining the questions/recommendations in-
+cluded in each of the 16 frameworks (Yes: Mitigation specified.
+*Yes: Mitigation included but not specified. No: Mitigation not
+included). Only 5 (I2, I3, I6, I10, I11) out of 16 frameworks
+have specified mitigation-related aspects. 7 frameworks (I1,
+I4, I7, I12, I14-I16) have more or less included risk mitigation
+measures without clearly specifying them. 4 frameworks (I5,
+I8, I9, I13) do not cover mitigation.
+As for the risk factors, none of the frameworks specified
+the different risk factors they considered. However, given the
+potential value of such categorization in helping organizations
+better triage and prioritize risks, we examined the frameworks
+and their questions/recommendations and extracted the risk
+factors each framework takes into account (see Table I and Fig.
+8). Despite their respective focus (e.g., I14 focuses on hazards
+while touching vulnerability and exposure), all 16 frameworks
+consider potential vulnerability, and 15 frameworks cover haz-
+ard and exposure. However, mitigation risks are significantly
+underemphasized and only covered by AU NSW AI Assurance
+Framework (I7) and ECP’s AI impact assessment framework
+(I15). Even for these two frameworks that consider mitigation
+risks, they do not provide a comprehensive assessment rather
+briefly mention such risks. For example, in I7: “Are there any
+residual risks?”, and in I15: “Considering planned mitigations,
+could the AI system cause significant or irreversible harms?”.
+Fig. 8: Risk factors considered by collected frameworks.
+Finding 1 to RQ3.2: RAI risk assessment frameworks
+need to distinguish among risk factors (i.e., hazard,
+exposure, vulnerability, and mitigation risk). Although
+collected frameworks categorically encompass these
+factors to some degree, they may focus on particular
+factors while briefly touching on others. Further, mit-
+igation risks are significantly neglected. This can lead
+to potential failure to identify and mitigate crucial RAI
+risks.
+Finding 2 to RQ3.2: Existing RAI risk assessment
+frameworks provide some information on assessment
+procedures but fail to clearly specify inputs/outputs,
+stakeholders, and resources needed at each step.
+Finding 3 to RQ3.2: Many RAI risk assessment frame-
+works plainly list assessment measures (e.g., questions,
+checklists) without considering their interconnections
+or dependencies, leading to an inefficient assessment
+process.
+3) RQ3.3: What are the outputs?: This section discuss the
+outputs of the procedural frameworks.
+Whether the output of a documented report is specified or
+not, the outputs of the procedural frameworks are, or at least
+should be, risk/impact assessment reports. Some frameworks,
+such as I2, which creates a visualization of the risk level
+correlated to the RAI principles, and I3, which calculates
+risk and mitigation scores for various risk areas and gener-
+ates the level of impact, generate reports automatically. For
+I10, assessors must manually generate a report based on the
+questionnaire and their answers to the questions. The other
+procedural frameworks (I4, I5, I7-I9, I11, I13-I16) serve as
+(self-)assessment tools to guide assessors in identifying risks
+and do not require the preparation of a report. However, since
+assessors should clearly document all the answers and the
+related questions when using the procedural frameworks, the
+processes result in documented assessment reports.
+
+15
+15
+16
+2Finding to RQ3: While organizations may rely on RAI
+risk assessment frameworks for potential mitigation
+solutions, current frameworks either fail to provide
+concrete mitigation solutions, or lack a structured way
+to present the solutions. This makes it challenging for
+organizations to address identified risks effectively.
+IV. DISCUSSION
+A. On the concreteness of RAI risk assessment frameworks
+1) Relative concreteness: A risk assessment framework
+may appear concrete at one level but too abstract for the
+next. For example, management teams may consider certain
+assessment questions concrete, while development teams may
+find them not doable. Additionally, even seemingly concrete
+checklists or templates for RAI risk assessment may only
+be effective if assessors have a standardized and trustworthy
+approach to completing each item. Therefore, it is essential
+to have well-structured, concrete, and reusable solutions (e.g.,
+design patterns [24], [27]) in the lower level that align/connect
+with higher-level practices such as governance guidelines to
+ensure a comprehensive and effective risk assessment.
+2) Trivialized concreteness: Our mapping study reveals that
+many frameworks trivialize the concept of “concreteness” by:
+• Applying existing assessment concepts to new AI-specific
+artifacts/processes without further specifying potential
+solutions. Examples include acknowledging the existence
+of RAI risks and broadly mentioning that they need to
+be identified, documented, and mitigated.
+• Identifying new concepts in AI and providing some
+sub-categorization without providing potential solutions.
+Examples include acknowledging bias as a common issue
+in AI systems and listing different sources of bias (e.g.,
+data, algorithm), but not providing specified solutions to
+different biases.
+• Identifying important RAI risks and referring to poten-
+tially stale non-AI frameworks, which may not be suitable
+for addressing RAI risks.
+It’s important to note that while higher-level frameworks
+may seem abstract, they do not always trivialize concreteness.
+These frameworks are generally more abstract because they
+need to be widely applicable and less prone to obsolescence.
+They can be helpful, particularly for management teams,
+as they point out areas where organizations can uplift their
+practices. The key criteria for determining if a higher-level
+framework is concrete or not include:
+• Whether high-level abstractions of potential assessment
+and/or mitigation measures are underpinned (specified or
+reasonably inferable) by lower-level concrete assessment
+techniques.
+• Whether there is a clear understanding among higher-
+level stakeholders (e.g., management) about the inputs,
+processes, outputs, as well as required personnel and
+resources to complete the assessment. This understanding
+may not necessarily require technical expertise but rather
+an understanding of the trust placed in the lower-level
+concrete assessment techniques utilized.
+B. Essentials to “Concreteness”
+We summarize the essential qualities, elements, and pro-
+cesses that a concrete RAI risk assessment framework should
+process in Fig. 9.
+A concrete RAI risk assessment framework should have the
+following characteristics: 1) The assessment and/or mitigation
+(e.g., questions/checklists/recommendations) proposed at one
+level/stage are reasonably underpinned/aligned/connected to
+other level/stage even if the measure itself is narrow-scoped
+and not directly covering different levels/stages. 2) The or-
+ganization of assessment and mitigation should be layered,
+considering their dependencies on each other. This allows
+for a clearer assessment logic and more efficient assessment
+processes. Currently, only EU’s ALTAI (I2) achieves a certain
+level of interconnectivity by providing an interactive online
+assessment tool where the following questions may vary
+depending on the previous answers. 3) The framework should
+be extensible, dynamic, and adaptive in that it can be adapted
+and extended to more specific contexts. All the qualities above
+result in enhanced assessment efficiency.
+The elements to be covered by a comprehensive RAI
+risk assessment framework should include different dimen-
+sions, contexts, measurements, and mitigations. Assessment
+and mitigation should be organized based on different RAI
+principles, RAI stakeholders, and AI-SDLC stages. Existing
+organizational governance structures and measures should also
+be considered. Even if a framework focuses on a specific
+aspect (e.g., stage/principle-specific, designed for assessment
+instead of mitigation), it needs to be well connected (i.e., in-
+terconnected) with other aspects. Additionally, the framework
+should consider and specify contextual elements such as appli-
+cable regions, sectors, and compatibility with an organization’s
+existing risk management processes and structures. Moreover,
+different risk factors should be considered, and corresponding
+assessment and mitigation be presented. Especially, mitiga-
+tion risks are significantly neglected by existing frameworks.
+Reusable mitigation plans should be suggested in a structured
+way, along with their pros and cons considered.
+Specifications on the procedures required to conduct the
+RAI risk assessment can help assessors and assessees from
+different levels better understand the inputs/processes/outputs
+and required stakeholders and resources (e.g., data, tools,
+funds) for each step. However, only half (I1, I3, I4, I7, I10, I12,
+I14, I15) of the 16 frameworks provided such specifications
+to a certain extent. Furthermore, 7 out of the 8 frameworks
+merely stated the steps needed to conduct the assessment, with
+only I4 specifying stakeholders involved in each step. None
+of the frameworks provides details on the resources required
+to complete each step.
+C. Threats to validity
+External. The term “AI risk assessment” along with a set
+of other terms such as “AI risk management” and “AI impact
+
+Fig. 9: Essentials to building a concrete RAI risk assessment framework.
+assessment” has been used to mean largely the same topic:
+identification, assessment/measurement, and mitigation of RAI
+risks. We extended our search terms with a set of supportive
+terms that are being used interchangeably in the search string
+to ensure that all the relevant work were covered to mitigate
+this issue. Another issue is that we only include the publicly
+accessible RAI risk assessment frameworks, although some
+organizations have their own frameworks for internal use.
+Internal. To mitigate the threat of not finding all rele-
+vant studies, we conducted a rigorous search using defined
+keywords with support terms and conducted snowballing to
+recover the missing studies from the literature. To address
+the bias from data collection and synthesis, one researcher
+performed the tasks and the other researcher reviewed and
+double-checked the results. The two researchers discussed the
+inconsistency and reached a common ground.
+V. RELATED WORK
+The pressing need to manage RAI risks has attracted
+significant attention in both industry and academia. Many
+studies on RAI risks have been published in recent years.
+However, they heavily focus on AI risk conceptualization and
+taxonomy (e.g., [28]–[31]) and provide no concrete solutions
+(i.e., assessment/mitigation techniques) to RAI risks. With the
+increasing interest in managing RAI risks, more actionable
+solutions to managing RAI risks have been proposed. Zhang
+et al. [32] proposed to evaluate model risks by inspecting their
+behavior on counterfactuals. Schwee et al. [33] introduced a
+toolchain for assessing privacy risks. The toolchain takes in
+a model trained from the dataset to be shared and creates a
+privacy risk report. Yajima et al. [34] showcased their work in
+progress on assessing machine learning security risks. Failure
+mode and effect analysis (FMEA) has been adopted/extended
+for assessing RAI risks in [35]–[37].
+Notably, EY and Trilateral Research published a survey of
+AI risk assessment methodologies in January 2022 [38]. With
+the objective of providing RAI governors with noteworthy
+practices and regulations in the field, this survey presents
+a high-level overview of the global landscape of AI risk
+assessment. The report discusses: 1) regulations and legislation
+worldwide containing AI risk assessment related elements;
+2) solutions to RAI risk assessment by several international
+organizations; 3) standards related to AI risk management
+and governance; 4) a brief overview of part of the proposed
+approaches in industry and academia. While this report is cate-
+gorically comprehensive, it mainly aims to help RAI governors
+grasp the worldwide outline of the field. Furthermore, it lacks
+detailed and systematic analysis of the existing frameworks.
+In contrast, the objective of this study is to provide RAI
+practitioners with a systematic summary of the existing RAI
+risk assessment frameworks, and shed light on the future
+
+Interconnected
+Layered
+Industry-level
+Efficient
+Organization-level
+Different levels
+Qualities
+Extensible
+Team-level
+Dynamic
+RAI principles
+Assessors
+Adaptive
+RAI Stakeholders
+Different Roles
+Assessees
+Dimension
+AI software
+lifecycle stages
+Existing (other) risk
+assessment frameworks
+Region
+Governance structure
+Essentials to
+Elements
+Context
+concreteness
+Sector
+Hazard
+Organization
+Exposure
+Measurement
+Vulnerability
+Risk factors
+Tools and techniques
+Mitigation risk
+for assessment
+Control
+Mitigation
+Reusable solutions
+Steps
+Stakeholders
+Processes
+Inputs & outputs
+Resourcesdevelopment of concrete RAI risk assessment frameworks.
+VI. CONCLUSION AND FUTURE WORK
+This paper conducts a systematic mapping study to evaluate
+the capabilities and limitations of existing RAI risk assessment
+frameworks. We examine key characteristics of a concrete
+framework, including specified RAI principles, stakeholders,
+AI system lifecycle stages, applicable regions and sectors, risk
+factors, and reusable mitigations. We provide insights to help
+facilitating the development of concrete RAI risk assessment
+frameworks. These includes presenting the assessment and
+mitigation measures in an interconnected and layered way and
+specifying the assessment procedures as well as associated
+inputs/outputs, stakeholders, and resources. For future work,
+we are developing a question bank with questions clearly
+labelled with respect to different characteristics, mitigations,
+and risk factors etc. Based on the question bank, we plan to
+develop a concrete RAI risk assessment framework.
+REFERENCES
+[1] Q. Lu, L. Zhu, X. Xu, J. Whittle, and Z. Xing, “Towards a
+roadmap on software engineering for responsible ai,” in Proceedings
+of the 1st International Conference on AI Engineering: Software
+Engineering for AI, ser. CAIN ’22.
+New York, NY, USA: Association
+for Computing Machinery, 2022, p. 101–112. [Online]. Available:
+https://doi.org/10.1145/3522664.3528607
+[2] N. Sambasivan, S. Kapania, H. Highfill, D. Akrong, P. Paritosh, and
+L. M. Aroyo, ““everyone wants to do the model work, not the data
+work”: Data cascades in high-stakes ai,” in proceedings of the 2021
+CHI Conference on Human Factors in Computing Systems, 2021, pp.
+1–15.
+[3] J. Gao, J. Yao, and Y. Shao, “Towards reliable learning for high stakes
+applications,” in Proceedings of the AAAI Conference on Artificial
+Intelligence, vol. 33, no. 01, 2019, pp. 3614–3621.
+[4] X. Ferrer, T. van Nuenen, J. M. Such, M. Cot´e, and N. Criado, “Bias and
+discrimination in ai: a cross-disciplinary perspective,” IEEE Technology
+and Society Magazine, vol. 40, no. 2, pp. 72–80, 2021.
+[5] S. F. Nasim, M. R. Ali, and U. Kulsoom, “Artificial intelligence incidents
+& ethics a narrative review,” International Journal of Technology,
+Innovation and Management (IJTIM), vol. 2, no. 2, 2022.
+[6] P. R. Nagbøl, O. M¨uller, and O. Krancher, “Designing a risk assessment
+tool for artificial intelligence systems,” in International Conference
+on Design Science Research in Information Systems and Technology.
+Springer, 2021, pp. 328–339.
+[7] US National Institute of Standards and Technology, NIST, “AI
+Risk Management Framework: Second Draft.” [Online]. Available:
+https://https://www.nist.gov/system/files/documents/2022/08/18/AI
+RMF 2nd draft.pdf
+[8] European
+Commission
+High-Level
+Expert
+Group
+on
+Artificial
+Intelligence,
+“The
+Assessment
+List
+for
+Trustworthy
+Artificial
+Intelligence.” [Online]. Available: https://altai.insight-centre.org/
+[9] B. Kitchenham, O. P. Brereton, D. Budgen, M. Turner, J. Bailey, and
+S. Linkman, “Systematic literature reviews in software engineering–
+a systematic literature review,” Information and software technology,
+vol. 51, no. 1, pp. 7–15, 2009.
+[10] Government of Canada, “Algorithmic Impact Assessment tool .”
+[Online].
+Available:
+https://www.canada.ca/en/government/system/
+digital-government/digital-government-innovations/responsible-use-ai/
+algorithmic-impact-assessment.html
+[11] Government
+of
+the
+Netherlands,
+Ministry
+of
+the
+Interior
+and
+Kingdom Relations, “Fundamental Rights and Algorithms Impact
+Assessment (FRAIA) .” [Online]. Available: https://www.government.nl/
+documents/reports/2021/07/31/impact-assessment-fundamental-rights-
+and- algorithms#: ∼:text=The%20Fundamental%20Rights%20and%
+20Algorithm,deployment%20of%20an%20algorithmic%20system.
+[12] UK Information Commissioner’s Office, “AI and data protection risk
+toolkit.” [Online]. Available: https://ico.org.uk/for-organisations/guide-
+to-data-protection/key-dp-themes/guidance-on-ai-and-data-protection/
+ai-and-data-protection-risk-toolkit/
+[13] Personal
+Data
+Protection
+Commission
+Singapore,
+“Model
+AI
+Governance Framework.” [Online]. Available: https://www.pdpc.gov.sg/
+help-and-resources/2020/01/model-ai-governance-framework
+[14] Australia
+NSW
+Government,
+“NSW
+Artificial
+Intelli-
+gence
+Assurance
+Framework.”
+[Online].
+Available:
+https:
+//www.digital.nsw.gov.au/policy/artificial- intelligence/nsw- artificial-
+intelligence-assurance-framework
+[15] GovEx, the City and County of San Francisco, Harvard DataSmart, Data
+Community DC, “Ethics & Algorithms Toolkit.” [Online]. Available:
+http://ethicstoolkit.ai/
+[16] US
+Pennsylvania
+Office
+of
+Administration,
+“RFD BUS012A
+Artificial
+Intelligence
+Assessment
+Too.”
+[Online].
+Available:
+https://www.oa.pa.gov/Policies/Documents/rfd bus012a.xlsx
+[17] European Law Institute, “Model Rules on Impact Assessment of Algo-
+rithmic Decision-Making Systems Used by Public Administration.” [On-
+line]. Available: https://www.europeanlawinstitute.eu/fileadmin/user
+upload/p eli/Publications/ELI Model Rules on Impact Assessment
+of ADMSs Used by Public Administration.pdf
+[18] World Economic Forum, “Artificial Intelligence for Children Toolkit.”
+[Online]. Available: https://www3.weforum.org/docs/WEF Artificial
+Intelligence for Children 2022.pdf
+[19] “Ieee recommended practice for assessing the impact of autonomous
+and intelligent systems on human well-being,” IEEE Std 7010-2020, pp.
+1–96, 2020.
+[20] Microsoft, “Microsoft Responsible AI Impact Assessment Template.”
+[Online]. Available: https://blogs.microsoft.com/wp-content/uploads/
+prod/sites/5/2022/06/Microsoft-RAI-Impact-Assessment-Template.pdf
+[21] Ada
+Lovelace
+Institute,
+“Algorithmic
+impact
+assessment
+in
+healthcare.” [Online]. Available: https://www.adalovelaceinstitute.org/
+project/algorithmic-impact-assessment-healthcare/
+[22] ECP, Platform for the Information Society, “Artificial Intelligence Impact
+Assessment (English version).” [Online]. Available: https://ecp.nl/
+publicatie/artificial-intelligence-impact-assessment-english-version/
+[23] Algorithm Watch, “Automated Decision-Making Systems in the Public
+Sector: An Impact Assessment Tool for Public Authorities.” [Online].
+Available:
+https://uploads- ssl.webflow.com/5f57d40eb1c2ef22d8a8
+ca7e/61a8efeb70b677a50f9c24cb 2021 AW Decision Public Sector
+EN v5.pdf
+[24] Q. Lu, L. Zhu, X. Xu, J. Whittle, D. Zowghi, and A. Jacquet, “Respon-
+sible ai pattern catalogue: a multivocal literature review,” arXiv preprint
+arXiv:2209.04963, 2022.
+[25] S. Amershi, A. Begel, C. Bird, R. DeLine, H. Gall, E. Kamar, N. Na-
+gappan, B. Nushi, and T. Zimmermann, “Software engineering for
+machine learning: A case study,” in 2019 IEEE/ACM 41st International
+Conference on Software Engineering: Software Engineering in Practice
+(ICSE-SEIP).
+IEEE, 2019, pp. 291–300.
+[26] Emergency Management Victoria, “Colac Otway Shire Municipal
+Emergency Management Plan: 4.3 Hazard, Exposure, Vulnerability and
+Resilience.” [Online]. Available: http://memp.colacotway.vic.gov.au/
+ch01s04s03.php
+[27] Q. Lu, L. Zhu, X. Xu, and J. Whittle, “Responsible-ai-by-design: a
+pattern collection for designing responsible ai systems,” IEEE Software,
+2023.
+[28] B. Perry and R. Uuk, “Ai governance and the policymaking process: key
+considerations for reducing ai risk,” Big data and cognitive computing,
+vol. 3, no. 2, p. 26, 2019.
+[29] K. Sotala, “Disjunctive scenarios of catastrophic ai risk,” in Artificial
+Intelligence Safety and Security.
+Chapman and Hall/CRC, 2018, pp.
+315–337.
+[30] M. Ciupa and K. Abney, “Conceptualizing ai risk,” in Proceedings of
+the 8th International Conference on Computer Science, Engineering and
+Applications (CCSEA 2018), 2018, pp. 73–81.
+[31] C. Garvey, “Ai risk mitigation through democratic governance: Intro-
+ducing the 7-dimensional ai risk horizon,” in Proceedings of the 2018
+AAAI/ACM Conference on AI, Ethics, and Society, 2018, pp. 366–367.
+[32] Z. Zhang, V. Setty, and A. Anand, “Sparcassist: A model risk assessment
+assistant based on sparse generated counterfactuals,” in Proceedings
+of the 45th International ACM SIGIR Conference on Research and
+Development in Information Retrieval, ser. SIGIR ’22.
+New York,
+NY, USA: Association for Computing Machinery, 2022, p. 3219–3223.
+
+[33] J. H. Schwee, F. C. Sangogboye, F. D. Salim, and M. B. Kjærgaard,
+“Tool-chain for supporting privacy risk assessments,” in Proceedings of
+the 7th ACM International Conference on Systems for Energy-Efficient
+Buildings, Cities, and Transportation, 2020, pp. 140–149.
+[34] J. Yajima, M. Inui, T. Oikawa, F. Kasahara, I. Morikawa, and
+N. Yoshioka, “A new approach for machine learning security risk
+assessment: Work in progress,” in Proceedings of the 1st International
+Conference on AI Engineering: Software Engineering for AI, ser. CAIN
+’22. New York, NY, USA: Association for Computing Machinery, 2022,
+p. 52–53. [Online]. Available: https://doi.org/10.1145/3522664.3528613
+[35] G. A. Dominguez, K. Kawaai, and H. Maruyama, “Fails: a tool for
+assessing risk in ml systems,” in 2021 28th Asia-Pacific Software
+Engineering Conference Workshops (APSEC Workshops).
+IEEE, 2021,
+pp. 1–4.
+[36] S. Rismani and A. Moon, “How do ai systems fail socially?: an engi-
+neering risk analysis approach,” in 2021 IEEE International Symposium
+on Ethics in Engineering, Science and Technology (ETHICS).
+IEEE,
+2021, pp. 1–8.
+[37] J. Li and M. Chignell, “Fmea-ai: Ai fairness impact assessment using
+failure mode and effects analysis,” AI and Ethics, pp. 1–14, 2022.
+[38] EY,
+Trilateral
+Research,
+“A
+survey
+of
+artificial
+intelligence
+risk
+assessment
+methodologies:
+The
+global
+state
+of
+play
+and
+leading
+practices
+identified.”
+[Online].
+Available:
+https:
+//www.trilateralresearch.com/wp-content/uploads/2022/01/A-survey-of-
+AI-Risk-Assessment-Methodologies-full-report.pdf
+

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@@ -0,0 +1,3951 @@
+Quantum Langevin theory for two coupled phase-conjugated electromagnetic waves
+Yue Jiang,1, 2, ∗ Yefeng Mei,3, † and Shengwang Du4, ‡
+1JILA, National Institute of Standards and Technology and the University of Colorado, Boulder, Colorado 80309, USA
+2Department of Physics, University of Colorado, Boulder, Colorado 80309, USA
+3Department of Physics, University of Michigan, Ann Arbor, Michigan 48109, USA
+4Department of Physics, The University of Texas at Dallas, Richardson, Texas 75080, USA
+(Dated: January 31, 2023)
+While loss-gain-induced Langevin noises have been intensively studied in quantum optics, the ef-
+fect of a complex-valued nonlinear coupling coefficient on the noises of two coupled phase-conjugated
+optical fields has never been questioned before. Here, we provide a general macroscopic phenomeno-
+logical formula of quantum Langevin equations for two coupled phase-conjugated fields with linear
+loss (gain) and complex nonlinear coupling coefficient. The macroscopic phenomenological formula
+is obtained from the coupling matrix to preserve the field commutation relations and correlations,
+which does not require knowing the microscopic details of light-matter interaction and internal
+atomic structures. To validate this phenomenological formula, we take spontaneous four-wave mix-
+ing in a double-Λ four-level atomic system as an example to numerically confirm that our macroscopic
+phenomenological result is consistent with that obtained from the microscopic Heisenberg-Langevin
+theory. Finally, we apply the quantum Langevin equations to study the effects of linear gain and
+loss, complex phase mismatching, as well as complex nonlinear coupling coefficient in entangled
+photon pair (biphoton) generation, particularly to their temporal quantum correlations.
+I.
+INTRODUCTION
+Quantum Langevin equations is a common approach
+to studying an open quantum system involving loss or
+gain, where the stochastic coupling between the system
+and its environment is molded as a set of Langevin noise
+operators [1–5]. For example, in the parametric down-
+conversion (PDC) process, a pump laser beam passes
+through a χ(2) nonlinear crystal and is down-converted
+into a pair of phase-conjugated electromagnetic (EM)
+waves.
+In the simplest case with the perfect phase-
+matching condition and an undepleted pump beam, with-
+out linear loss or gain, the two phase-conjugated single-
+mode fields are governed by the following coupled equa-
+tions [6]
+∂
+∂z
+�ˆa1
+ˆa†
+2
+�
+= M
+�ˆa1
+ˆa†
+2
+�
+=
+�
+0
+iκ
+−iκ
+0
+� �ˆa1
+ˆa†
+2
+�
+,
+(1)
+where ˆam and ˆa†
+m (m = 1, 2) are the field annihilation
+and creation operators, M is the 2 × 2 coupling matrix,
+and κ is the (real) nonlinear coupling coefficient. Here
+we consider only the forward-wave case with both fields
+propagating along the same +z direction. If losses are
+presented during the propagation of the two fields, the
+coupling matrix is
+M =
+�
+−α1
+iκ
+−iκ −α2
+�
+,
+(2)
+∗ [email protected]
+† [email protected]
+‡ [email protected]
+and their coupled equations become [3, 7]
+∂
+∂z
+�ˆa1
+ˆa†
+2
+�
+=
+�
+−α1
+iκ
+−iκ −α2
+� �ˆa1
+ˆa†
+2
+�
++
+�√2α1 ˆf1
+√2α2 ˆf †
+2
+�
+,
+(3)
+where αm > 0 are the loss (absorption) coefficients,
+and ˆfm are the associated Langevin noise operators sat-
+isfying [ ˆfm(ω, z), ˆf †
+n(ω′, z′)] = δmnδ(ω − ω′)δ(z − z′).
+If there is linear gain instead of loss, for example in
+channel 1, i.e., α1 < 0, equation (3) can be modi-
+fied by taking √2α1 ˆf1 → √−2α1 ˆf †
+1.
+One can show
+that these Langevin noise operators are necessary to pre-
+serve the commutation relations during propagation, i.e.
+[ˆam(ω, z), ˆa†
+n(ω′, z)] = [ˆam(ω, 0), ˆa†
+n(ω′, 0)] = δmnδ(ω −
+ω′).
+Equation (3) has been widely applied for PDC pro-
+cesses where the nonlinear coupling coefficient κ is real
+[3, 7–9].
+However, in a more general case of cou-
+pled phase-conjugated fields, such as four-wave mixing
+(FWM) near atomic resonances [10–12], the nonlinear
+coupling coefficient κ can take a complex value involving
+complicated atomic transitions.
+In this case, equation
+(3) is not valid and its solution does not preserve com-
+mutation relations of the fields. What are the general
+quantum Langevin coupled equations accounting for the
+complex nonlinear coupling coefficient?
+To answer the question, the common approach is to
+derive quantum Langevin equations by solving the light-
+matter coupled Heisenberg equations, which requires
+knowing microscopic details of light-matter interaction
+such as atomic populations and transitions [11–13]. The
+complexity of this approach increases dramatically as
+more atomic transitions are involved and it is extremely
+difficult for experimentalists to follow, particularly in
+some situations where it is impossible to obtain full mi-
+croscopic details. Then our reduced question becomes:
+arXiv:2301.11993v1  [quant-ph]  27 Jan 2023
+
+2
+Is it possible to obtain self-consistent quantum Langevin
+coupled equations from the general expression of the cou-
+pling matrix? We call this the macroscopic phenomeno-
+logical approach. To our best knowledge, there has been
+no published work in investigating Langevin noises in-
+duced by a complex nonlinear coupling coefficient κ.
+In this article, for the first time, we provide a gen-
+eral macroscopic phenomenological formula of quantum
+Langevin equations for two coupled phase-conjugated
+fields with linear loss (gain) and complex nonlinear cou-
+pling coefficient, in both forward- and backward-wave
+configurations. The macroscopic phenomenological for-
+mula is obtained from the coupling matrix by preserv-
+ing commutation relations and correlations of the fields,
+which does not require knowing the microscopic details of
+light-matter interaction and internal atomic structures.
+We aim to make it readable and accessible for experi-
+mental researchers in the quantum optics community.
+This article is structured as follows. In Sec. II, to ful-
+fill the requirement of preserving commutation relations,
+we formulate the general macroscopic phenomenologi-
+cal quantum Langevin coupled equations and their solu-
+tions from the coupling matrix taking into account linear
+loss (gain) and complex nonlinear coupling coefficient,
+in both forward- and backward-wave configurations. In
+Sec. III, taking spontaneous four-wave mixing (SFWM)
+in a double-Λ four-level atomic system as an example,
+we derive the coupled Langevin equations from micro-
+scopic light-atom Heisenberg interaction for this special
+case. We numerically confirm that the macroscopic phe-
+nomenological solution in Sec. II agrees well with the
+microscopic approach. In Sec. IV, we apply the quan-
+tum Langevin theory to study effects of linear gain and
+loss, complex phase mismatching, and complex nonlinear
+coupling coefficient in entangled photon pair (biphoton)
+generation, particularly to their temporal quantum cor-
+relations. We conclude in the last section V.
+II.
+QUANTUM LANGEVIN EQUATIONS
+Here we consider the two coupled single-mode phase-
+conjugated fields in either forward-wave or backward-
+wave configuration, as illustrated in Fig. 1.
+In the
+forward-wave configuration [Fig. 1(a)], both fields prop-
+agate along +z direction through a nonlinear medium
+with a length L.
+In the backward-wave configuration
+[Fig. 1(b)], the two fields propagate in opposing direc-
+tions. The field annihilation operators ˆam(t, z) can be
+expressed as
+ˆa1(t, z) =
+1
+√
+2π
+�
+dωˆa1(ω, z)ei( ω
+c z−ωt),
+ˆa2(t, z) =
+1
+√
+2π
+�
+dωˆa2(ω, z)ei(± ω
+c z−ωt),
+(4)
+where ± represents that field 2 propagates along +z or
+−z direction, for the forward-wave or backward-wave
+configuration, respectively.
+The filed operators satisfy
+the following commutation relations
+�
+ˆam (t, z) , ˆa†
+n (t′, z)
+�
+= δmnδ(t − t′),
+�
+ˆam (ω, z) , ˆa†
+n (ω′, z)
+�
+= δmnδ(ω − ω′).
+(5)
+In the forward-wave configuration, both fields are input
+at z = 0, or ˆa1(0) and ˆa2(0) are the “initial” boundary
+conditions. The general coupling matrix is [14]
+MF =
+�
+−α1 + i ∆k
+2
+iκ
+−iκ
+−α∗
+2 − i ∆k
+2
+�
+,
+(6)
+where αm = −i ωm
+2c χm with χm being linear suscepti-
+bility, and ∆k (real) is the phase mismatching in vac-
+uum. In general, αm is complex valued, whose real part
+Re{αm} > 0 represents loss (or gain for Re{αm} < 0)
+and imaginary part represents phase velocity dispersion.
+The nonlinear coupling coefficient κ can also be complex-
+valued. In the backward-wave configuration, the general
+coupling matrix becomes [12, 15]
+MB =
+�
+−α1 + i ∆k
+2
+iκ
+iκ
+α∗
+2 − i ∆k
+2
+�
+,
+(7)
+and the “initial” boundary conditions are ˆa1(0) and
+ˆa2(L): field 1 is input at z = 0 and field 2 is input at
+z = L.
+One can show that, under the following unitary gauge
+transformation
+�ˆa1
+ˆa†
+2
+�
+=
+�
+eiθ/2
+0
+0
+e−iθ/2
+� �ˆa1
+ˆa†
+2
+�
+= U
+�ˆa1
+ˆa†
+2
+�
+=
+� ˆa1eiθ/2
+ˆa†
+2e−iθ/2
+�
+,
+(8)
+the corresponding coupling matrix become
+MF(θ) = UMFU† =
+�
+−α1 + i ∆k
+2
+iκeiθ
+−iκe−iθ
+−α∗
+2 − i ∆k
+2
+�
+,
+(9)
+and
+MB(θ) = UMBU† =
+�
+−α1 + i ∆k
+2
+iκeiθ
+iκe−iθ
+α∗
+2 − i ∆k
+2
+�
+.
+(10)
+As physics is preserved and unchanged under the above
+gauge transformation, we take θ = 0 throughout this
+article for convenience and simplification.
+In presence of linear loss or gain, i.e., Re{αm} ̸= 0, or
+complex nonlinear coupling coefficient, κ ̸= κ∗, the two-
+mode coupled equations must include Langevin noise op-
+erators to preserve the commutation relations of the field
+operators in Eq. (5). The noise operators should only
+be related to Re{αm} and Im{κ}. As κ is real, the cou-
+pled equations in forward-wave configuration should be
+reduced to the known Eq. (3). For both forward- and
+backward-wave configurations in the same nonlinear ma-
+terial, the noise origin is the same except field 2 prop-
+agates along ±z direction for different configurations.
+With these guidelines, we provide quantum Langevin
+equations for the two phase-conjugated fields from their
+coupling matrix in the following subsections.
+
+3
+𝑧
+𝑎��
+𝑎��
+�
+Medium
+0
+𝐿
+𝜅
+𝑧
+𝑎��
+𝑎��
+�
+Medium
+0
+𝐿
+𝜅
+(b)
+(a)
+Figure 1.
+Schematics of two coupled phase-conjugated electromagnetic waves:
+(a) forward-wave configuration, and (b)
+backward-wave configuration. κ is the nonlinear coupling coefficient between the two modes.
+A.
+Forward-Wave Configuration
+In
+the
+forward-wave
+configuration
+as
+shown
+in
+Fig. 1(a), we find that its quantum Langevin coupled
+equations can be expressed in the following general form
+∂
+∂z
+�ˆa1
+ˆa†
+2
+�
+= MF
+�ˆa1
+ˆa†
+2
+�
++ NFR
+� ˆf1
+ˆf †
+2
+�
++ NFI
+� ˆf †
+1ˆf2
+�
+(11)
+with the “initial” condition at z = 0:
+�
+ˆam(ω, 0), ˆa†
+n(ω′, 0)
+�
+= δmnδ(ω − ω′).
+(12)
+The Langevin noise operators satisfy
+�
+ˆfm(ω, z), ˆf †
+n(ω′, z′)
+�
+= δmnδ(ω − ω′)δ(z − z′)
+(13)
+and have the following correlations
+�
+ˆf †
+m(ω, z) ˆfn(ω′, z′)
+�
+= 0,
+�
+ˆfm(ω, z) ˆf †
+n(ω′, z′)
+�
+= δmnδ(ω − ω′)δ(z − z′),
+�
+ˆfm(ω, z) ˆfn(ω′, z′)
+�
+=
+�
+ˆf †
+m(ω, z) ˆf †
+n(ω′, z′)
+�
+= 0.
+(14)
+The Langevin noise matrix is given by
+NF ≡
+�
+−(MF + MF
+∗) = NFR + iNFI,
+(15)
+where NFR and NFI are the real and imaginary parts of
+the matrix NF (i.e., NFmn = NFRmn + iNFImn), respec-
+tively.
+We obtain the solution of Eq. (11) at the output sur-
+face z = L as the following
+�ˆa1 (L)
+ˆa†
+2 (L)
+�
+= eMFL
+�ˆa1 (0)
+ˆa†
+2 (0)
+�
++
+� L
+0
+eMF(L−z)
+�
+NFR
+� ˆf1 (z)
+ˆf †
+2 (z)
+�
++ NFI
+� ˆf †
+1 (z)
+ˆf2 (z)
+��
+dz.
+(16)
+Defining
+eMFL ≡
+�
+A B
+C D
+�
+,
+(17)
+eMF(L−z) ≡
+�
+A1 (z) B1 (z)
+C1 (z) D1 (z)
+�
+,
+(18)
+we rewrite Eq. (16) as
+�ˆa1 (L)
+ˆa†
+2 (L)
+�
+=
+�
+A B
+C D
+� �ˆa1 (0)
+ˆa†
+2 (0)
+�
++
+� L
+0
+�
+A1 (z) B1 (z)
+C1 (z) D1 (z)
+� �
+NFR
+� ˆf1 (z)
+ˆf †
+2 (z)
+�
++ NFI
+� ˆf †
+1 (z)
+ˆf2 (z)
+��
+dz.
+(19)
+We numerically confirm that the solution preserves the
+commutation relations
+�
+ˆam(ω, L), ˆa†
+n(ω′, L)
+�
+=
+�
+ˆam(ω, 0), ˆa†
+n(ω′, 0)
+�
+= δmnδ(ω − ω′).
+(20)
+Now we examine some special cases.
+Case 1: We first consider the coupling matrix MF in Eq.
+(6) where the nonlinear coupling coefficient κ is real and
+both modes have losses (Re{αm} ≥ 0) . This works for
+most PDC processes [3, 7]. Under such a condition, we
+have the following diagonalized noise matrix
+NF = NFR =
+��
+2Re{α1}
+0
+0
+�
+2Re{α2}
+�
+,
+(21)
+and the coupled Langevin equations
+∂
+∂z
+�ˆa1
+ˆa†
+2
+�
+= MF
+�ˆa1
+ˆa†
+2
+�
++
+��
+2Re{α1} ˆf1
+�
+2Re{α2} ˆf †
+2
+�
+,
+(22)
+which is the well-known result in literature [3, 7].
+Case 2: κ is real, the mode 1 has linear loss (Re{α1} =
+α ≥ 0), and the mode 2 has linear gain (Re{α2} = −g ≤
+0). The noise matrix becomes
+NF =
+�√
+2α
+0
+0
+i√2g
+�
+.
+(23)
+We have the following coupled Langevin equations
+∂
+∂z
+�ˆa1
+ˆa†
+2
+�
+= MF
+�ˆa1
+ˆa†
+2
+�
++
+�√
+2α ˆf1
+√2g ˆf2
+�
+.
+(24)
+
+4
+Case 3: The two modes are perfectly phase-matched
+without linear gain or loss: ∆k = 0, α1 = α2 = 0, but
+the nonlinear coupling coefficient is complex-valued κ =
+η + iζ. In this case, the coupled matrix is
+MF =
+�
+0
+−ζ + iη
+ζ − iη
+0
+�
+.
+(25)
+The noise matrix becomes
+NF = Θ(ζ)
+�
+ζ
+�
+1
+1
+−1 1
+�
++ iΘ(−ζ)
+�
+−ζ
+�
+1
+1
+−1 1
+�
+,
+(26)
+where Θ(ζ) is Heaviside step function, Θ(ζ) = 1 if ζ > 0,
+Θ(ζ) = 0 if ζ ≤ 0. The Langevin coupled equations are
+∂
+∂z
+�ˆa1
+ˆa†
+2
+�
+=MF
+�ˆa1
+ˆa†
+2
+�
++ Θ(ζ)
+�
+ζ
+�
+1
+1
+−1 1
+� � ˆf1
+ˆf †
+2
+�
++ Θ(−ζ)
+�
+−ζ
+�
+1
+1
+−1 1
+� � ˆf †
+1ˆf2
+�
+.
+(27)
+Eq. (27) shows that a complex-valued nonlinear coupling
+coefficient also leads to Langevin noises even when there
+is no linear gain or loss. This is revealed by this article
+for the first time.
+Case 4: As κ is real and there is no linear loss or gain
+(α1 = α2 = 0), the coupled equations can be written as
+i ∂
+∂z
+�ˆa1
+ˆa†
+2
+�
+=
+�
+− ∆k
+2
+−κ
+κ
+∆k
+2
+� �ˆa1
+ˆa†
+2
+�
+= ˆH
+�ˆa1
+ˆa†
+2
+�
+.
+(28)
+The effective Hamiltonian ˆH has anti-parity-time (APT)
+symmetry, which has been demonstrated in FWM in cold
+atoms [14, 16].
+B.
+Backward-Wave Configuration
+In the back-wave configuration as shown in Fig. 1(b),
+the quantum Langevin coupled equations can be ex-
+pressed in the following general form
+∂
+∂z
+�ˆa1
+ˆa†
+2
+�
+= MB
+�ˆa1
+ˆa†
+2
+�
++ NBR
+� ˆf1
+ˆf †
+2
+�
++ NBI
+� ˆf †
+1ˆf2
+�
+.
+(29)
+Different
+from
+the
+forward-wave
+configuration,
+the
+“boundary” condition is
+�
+ˆa1(ω, 0), ˆa†
+1(ω′, 0)
+�
+=
+�
+ˆa2(ω, L), ˆa†
+2(ω′, L)
+�
+= δ(ω − ω′).
+(30)
+The Langevin noise operators satisfy the same commu-
+tation relations and correlations in Eqs. (13) and (14).
+The Langevin noise matrix is given by
+NB ≡
+�
+1
+0
+0 −1
+� ��
+−MB11 −MB12
+MB21
+MB22
+�
++
+�
+−MB11 −MB12
+MB21
+MB22
+�∗
+= NBR + iNBI,
+(31)
+where NBR and NBI are the real and imaginary parts of
+the matrix NB, respectively. One can show that the noise
+matrix defined in Eq. (31) has the same origin as that
+in the forward-wave configuration in the same nonlinear
+material:
+NB =
+�
+1
+0
+0 −1
+�
+NF.
+(32)
+We note that the choice of noise matrix is not unique.
+For example, transformation ˆf1 → − ˆf1 or/and ˆf2 → − ˆf2
+does not affect computing any physical observable. We
+elaborate on this more in Appendix A.
+We obtain the solution of Eq. (29) at z = L as follow-
+ing
+�ˆa1 (L)
+ˆa†
+2 (L)
+�
+= eMBL
+�ˆa1 (0)
+ˆa†
+2 (0)
+�
++
+� L
+0
+eMB(L−z)
+�
+NBR
+� ˆf1 (z)
+ˆf †
+2 (z)
+�
++ NBI
+� ˆf †
+1 (z)
+ˆf2 (z)
+��
+dz.
+(33)
+We define
+eMBL ≡
+� ¯A
+¯B
+¯C
+¯D
+�
+,
+(34)
+eMB(L−z) ≡
+� ¯A1 (z)
+¯B1 (z)
+¯C1 (z)
+¯D1 (z)
+�
+.
+(35)
+Different from the forward-wave case, in the backward-
+wave configuration, the mode 1 input is at z = 0 and the
+mode 2 input is at z = L. With known ˆa1(0) and ˆa2(L),
+we rearrange Eq. (33) and obtain solutions for ˆa1(L) and
+ˆa2(0):
+�ˆa1 (L)
+ˆa†
+2 (0)
+�
+=
+�
+A B
+C D
+� �ˆa1 (0)
+ˆa†
+2 (L)
+�
++
+�
+1 −B
+0 −D
+� � L
+0
+� ¯A1 (z)
+¯B1 (z)
+¯C1 (z)
+¯D1 (z)
+� �
+NBR
+� ˆf1 (z)
+ˆf †
+2 (z)
+�
++ NBI
+� ˆf †
+1 (z)
+ˆf2 (z)
+��
+dz,
+(36)
+
+5
+where
+A = ¯A −
+¯B ¯C
+¯D ,
+B =
+¯B
+¯D,
+C = −
+¯C
+¯D,
+D = 1
+¯D.
+(37)
+We numerically confirm that Eq. (36) preserves the com-
+mutation relations
+�
+ˆa1(ω, L), ˆa†
+1(ω′, L)
+�
+=
+�
+ˆa1(ω, 0), ˆa†
+1(ω′, 0)
+�
+,
+�
+ˆa2(ω, 0), ˆa†
+2(ω′, 0)
+�
+=
+�
+ˆa2(ω, L), ˆa†
+2(ω′, L)
+�
+.
+(38)
+Similarly to the forward-wave configuration, we exam-
+ine the following four special cases.
+Case 1: We assume the nonlinear coupling coefficient κ
+is real and both modes have losses (Re{αm} ≥ 0). Under
+such a condition, we have the following diagonalized noise
+matrix
+NB =
+��
+2Re{α1}
+0
+0
+−
+�
+2Re{α2}
+�
+,
+(39)
+and the coupled Langevin equations
+∂
+∂z
+�ˆa1
+ˆa†
+2
+�
+= MB
+�ˆa1
+ˆa†
+2
+�
++
+� �
+2Re{α1} ˆf1
+−
+�
+2Re{α2} ˆf †
+2
+�
+.
+(40)
+Case 2: κ is real, mode 1 has linear loss (Re{α1} = α ≥
+0), and mode 2 has linear gain (Re{α2} = −g ≤ 0). The
+noise matrix becomes
+NF =
+�√
+2α
+0
+0
+−i√2g
+�
+.
+(41)
+We have the following coupled Langevin equations
+∂
+∂z
+�ˆa1
+ˆa†
+2
+�
+= MB
+�ˆa1
+ˆa†
+2
+�
++
+� √
+2α ˆf1
+−√2g ˆf2
+�
+.
+(42)
+Case 3: The two modes are perfectly phase-matched
+without linear gain and loss: ∆k = 0, α1 = α2 = 0,
+but the nonlinear coupling coefficient is complex-valued
+κ = η + iζ. In this case, the coupled matrix is
+MB =
+�
+0
+−ζ + iη
+−ζ + iη
+0
+�
+.
+(43)
+The noise matrix becomes
+NB = Θ(ζ)
+�
+ζ
+�
+1
+1
+1 −1
+�
++ iΘ(−ζ)
+�
+−ζ
+�
+1
+1
+1 −1
+�
+.
+(44)
+The Langevin coupled equations are
+∂
+∂z
+�ˆa1
+ˆa†
+2
+�
+=MB
+�ˆa1
+ˆa†
+2
+�
++ Θ(ζ)
+�
+ζ
+�
+1
+1
+1 −1
+� � ˆf1
+ˆf †
+2
+�
++ Θ(−ζ)
+�
+−ζ
+�
+1
+1
+1 −1
+� � ˆf †
+1ˆf2
+�
+.
+(45)
+Eq. (45) shows that in the backward-wave configuration,
+a complex-valued nonlinear coupling coefficient also leads
+to Langevin noises even though there is no linear gain or
+loss.
+Case 4: As κ is real and there are equal losses in both
+modes (α1 = α2 = α > 0) with perfect phase matching
+(∆k = 0), the coupled equations can be written as
+i ∂
+∂z
+�ˆa1
+ˆa†
+2
+�
+=
+�
+−iα −κ
+−κ
+iα
+� �ˆa1
+ˆa†
+2
+�
+= ˆH
+�ˆa1
+ˆa†
+2
+�
+.
+(46)
+Interestingly, the effective Hamiltonian ˆH here follows
+parity-time (PT) symmetry [17, 18].
+III.
+MICROSCOPIC ORIGIN OF LANGEVIN
+NOISES: SFWM
+One could validate the above phenomenological ap-
+proach of quantum Langevin coupled equations by con-
+firming the microscopic origin of the Langevin noises.
+However,
+for two systems with the same quantum
+Langevin equations, their microscopic structures may be
+quite different. Therefore it is impossible to sort all mi-
+croscopic systems. In this section, we focus on SFWM in
+a double-Λ four-level atomic system [10–12, 19, 20] with
+electromagnetically induced transparency (EIT) [21, 22],
+and show that the phenomenological approach in the
+above section agrees with the numerical results from the
+microscopic quantum theory of light-atom interaction.
+We start from a single-atom picture, considering an
+EM wave couples the atomic transition |j⟩ and |k⟩. The
+induced single atom polarization ˆpjk ∝ µjkˆσjk, where
+µjk is the electric dipole moment matrix element, ˆσjk =
+|j ⟩⟨ k| is single atom transition operator from state |k⟩ to
+|j⟩. In the Heisenberg-Langevin picture, the single-atom
+transition operator can be expressed as
+ˆσjk = ˆσ(0)
+jk +
+�
+µν
+βµν ˆf (σ)
+µν ,
+(47)
+where ˆσ(0)
+jk = ⟨ˆσjk⟩ is the zeroth-order steady state so-
+lution. The single atom noise operator between atomic
+transition |ν⟩ → |µ⟩ is represented by ˆf (σ)
+µν , which satisfies
+the following correlations:
+⟨ ˆf (σ)
+µν (ω) ˆf (σ)†
+µ′ν′ (ω′)⟩ = ⟨ ˆf (σ)
+µν (ω) ˆf (σ)
+ν′µ′(ω′)⟩
+= Dµν,ν′µ′δ(ω − ω′),
+⟨ ˆf (σ)†
+µν (ω) ˆf (σ)
+µ′ν′(ω′)⟩ = ⟨ ˆf (σ)
+νµ (ω) ˆf (σ)
+µ′ν′(ω′)⟩
+= Dνµ,µ′ν′δ(ω − ω′),
+(48)
+where Dµν,ν′µ′ and Dνµ,µ′ν′ are diffusion coefficients.
+In a continuous medium with atomic number density
+n, the noises from different atoms are uncorrelated. We
+have the spatially averaged atomic operator
+ˆ¯σjk ≡ ˆσ(0)
+jk +
+1
+√
+nA
+�
+µν
+βµν ˆ¯f (σ)
+µν ,
+(49)
+
+6
+|1⟩
+|2⟩
+Δ�
+𝑎���
+𝑧
+0
+𝐿
+(a)
+(b)
+|3⟩
+𝜔��
+𝜔�
+𝐸�
+𝐸�
+𝑎��
+|4⟩
+𝜔�
+𝜔�
+𝜛
+𝜛
+Figure 2. Spontaneous four-wave mixing (SFWM) in a double-Λ four-level cold atomic medium. (a) Backward-wave geometry
+of SFWM optical configuration. Driven by counter-propagating pump (Ep) and coupling (Ec) beams, phase-matched backward
+Stokes (ˆas) and anti-Stokes (ˆaas) are spontaneously generated from a laser-cooled atomic medium. (b) Atomic energy-level
+diagram. The pump (ωp) laser is detuned with ∆p from transition |1⟩ → |4⟩, and the coupling (ωc) laser is on-resonant with
+transition |2⟩ → |3⟩. Stokes (ωs) photons are spontaneously generated from transition |4⟩ → |2⟩, and anti-Stokes (ωas) photons
+from transition |3⟩ → |1⟩. ϖ = ωas − ω13 is the anti-Stokes photon frequency detuning from transition |1⟩ → |3⟩.
+where A is the single-mode cross-section area, and the
+spatially averaged atomic noise operators ˆ¯f (σ)
+µν satisfy the
+following modified correlations
+⟨ ˆ¯f (σ)
+µν (ω, z) ˆ¯f (σ)†
+µ′ν′ (ω′, z′)⟩ = ⟨ ˆ¯f (σ)
+µν (ω, z) ˆ¯f (σ)
+ν′µ′(ω′, z′)⟩
+= Dµν,ν′µ′δ(ω − ω′)δ(z − z′),
+⟨ ˆ¯f (σ)†
+µν (ω, z) ˆ¯f (σ)
+µ′ν′(ω′, z′)⟩ = ⟨ ˆ¯f (σ)
+νµ (ω, z) ˆ¯f (σ)
+µ′ν′(ω′, z′)⟩
+= Dνµ,µ′ν′δ(ω − ω′)δ(z − z′),
+(50)
+where the diffusion coefficients are the same as those from
+the single-atom picture.
+The electric field and polarization are described as
+ˆE(t, z) = 1
+2
+�
+ˆE(+)(t, z) + ˆE(−)(t, z)
+�
+,
+ˆP(t, z) = 1
+2
+�
+ˆP (+)(t, z) + ˆP (−)(t, z)
+�
+,
+(51)
+Where ˆE(+), ˆP (+) and ˆE(−), ˆP (−) are positive and nega-
+tive frequency parts. We take the following Fourier trans-
+form
+ˆE(+)(t, z) =
+1
+√
+2π
+�
+dω ˆE(ω, z)ei(± ω
+c z−ωt),
+ˆP (+)(t, z) =
+1
+√
+2π
+�
+dω ˆP(ω, z)ei(± ω
+c z−ωt),
+(52)
+where ˆE(ω, z), ˆP(ω, z) are complex amplitudes in fre-
+quency domain.
+The Maxwell equation under slowly
+varying envelope approximation (SVEA) can be written
+as
+±∂ ˆE(ω, z)
+∂z
+= i
+2ωη ˆP(ω, z),
+(53)
+where ± represents for propagation direction along ±z,
+and free space impedance η = 1/(cε0) = 377 Ohm, with
+c being the speed of light in vacuum, and ε0 the vacuum
+permittivity. With quantized electric field
+ˆE(ω, z) =
+�
+2ℏω
+cε0Aˆa(ω, z),
+(54)
+and
+ˆP(ω, z) = 2nµjkˆ¯σjk(ω, z),
+(55)
+we obtain the Langevin equation for the EM field in the
+atomic medium
+±∂ˆa(ω, z)
+∂z
+= i nAgjkˆ¯σjk(ω, z)
+= i nAgjkˆσ(0)
+jk (ω, z) + ˆ¯F(ω, z),
+(56)
+where
+gjk = µjk
+�
+ωjk
+2cε0ℏA,
+ˆ¯F(ω, z) = i
+√
+nAgjk
+�
+µν
+βµν ˆ¯f (σ)
+µν (ω, z)
+= iµjk
+� nωjk
+2cε0ℏ
+�
+µν
+βµν ˆ¯f (σ)
+µν (ω, z).
+(57)
+Here gjk = g∗
+kj is single photon-atom coupling strength.
+Now we turn to the backward-wave SFWM in a double-
+Λ four-level atomic system as illustrated in Fig. 2. In
+presence of counter-propagating pump (Ep, ωp) and cou-
+pling (Ec, ωc) laser beams, phase-matched Stokes (ωs)
+and anti-Stokes (ωas) are spontaneously generated and
+propagate through the medium in opposing directions.
+In the rotating reference frame, the interaction Hamilto-
+nian for a single atom is
+ˆV = − ℏ
+�
+g31ˆaasˆσ31 + g13ˆa†
+asˆσ13
+�
+− ℏ
+�
+g42ˆasˆσ42 + g24ˆa†
+sˆσ24
+�
+− 1
+2ℏ (Ωcˆσ32 + Ω∗
+c ˆσ23) − 1
+2ℏ
+�
+Ωpˆσ41 + Ω∗
+pˆσ14
+�
+− ℏ∆pˆσ44 − ℏϖˆσ33 − ℏϖˆσ22,
+(58)
+where Ωc = µ32Ec/ℏ is coupling Rabi frequency. The
+coupling laser is on-resonant with transition |2⟩ → |3⟩.
+Ωp = µ41Ep/ℏ is pump Rabi frequency.
+The pump
+laser is far detuned from the transition |1⟩ → |4⟩ with
+∆p = ωp − ω14 so that the atomic population mainly oc-
+cupies the ground state |1⟩. We take this ground-state
+
+7
+approximation through this section.
+With continuous-
+wave pump and coupling driving fields, the energy con-
+servation leads to ωas+ωs = ωc+ωp. Here ϖ = ωas−ω13
+is the anti-Stokes frequency detuning and thus the Stokes
+frequency detuning is ωs − ωs0 = −ϖ.
+The atomic evolution is governed by the following
+Heisenberg-Langevin equation [11]
+∂
+∂t ˆσjk = i
+ℏ[ ˆV , ˆσjk] − γjkˆσjk + rA
+jk + ˆf (σ)
+jk ,
+(59)
+where γjk = γkj (nonzero only as j ̸= k) are dephasing
+rates, rA
+jk (nonzero only as j = k) are the population
+transfer resulting from spontaneous emission decay. The
+full equation of motion can be found in Appendix B. The
+diffusion coefficients Djk,j′k′ can be obtained through the
+Einstein relation
+Djk,j′k′ = ∂
+∂t ⟨ˆσjkˆσj′k′⟩
+−
+�
+ˆAjkˆσj′k′
+�
+−
+�
+ˆσjk ˆAj′k′
+�
+,
+(60)
+where ˆAjk =
+∂
+∂t ˆσjk − ˆf (σ)
+jk . For the SFWM governed by
+Eq. (59), we have [11, 12]
+�
+��
+D12,21 D12,24
+D42,21 D42,24
+D12,31 D12,34
+D42,31 D42,34
+D13,21 D13,24
+D43,21 D43,24
+D13,31 D13,34
+D43,31 D43,34
+�
+��
+=
+�
+��
+2γ12 ⟨ˆσ11⟩ + Γ31 ⟨ˆσ33⟩ + Γ41 ⟨ˆσ44⟩ γ12 ⟨ˆσ14⟩
+0
+0
+γ12 ⟨ˆσ41⟩
+0
+0
+0
+0
+0
+Γ3 ⟨ˆσ11⟩ + Γ31 ⟨ˆσ33⟩ + Γ41 ⟨ˆσ44⟩ Γ3 ⟨ˆσ14⟩
+0
+0
+Γ3 ⟨ˆσ41⟩
+Γ3 ⟨ˆσ44⟩
+�
+�� ,
+(61)
+�
+��
+D21,12 D21,42
+D24,12 D24,42
+D21,13 D21,43
+D24,13 D24,43
+D31,12 D31,42
+D34,12 D34,42
+D31,13 D31,43
+D34,13 D34,43
+�
+��
+=
+�
+��
+2γ12 ⟨ˆσ22⟩ + Γ32 ⟨ˆσ33⟩ + Γ42 ⟨ˆσ44⟩
+0
+γ12 ⟨ˆσ23⟩
+0
+0
+Γ4 ⟨ˆσ22⟩ + Γ32 ⟨ˆσ33⟩ + Γ42 ⟨ˆσ44⟩
+0
+Γ4 ⟨ˆσ23⟩
+γ12 ⟨ˆσ32⟩
+0
+0
+0
+0
+Γ4 ⟨ˆσ32⟩
+0
+Γ4 ⟨ˆσ33⟩
+�
+�� .
+(62)
+Solving Eq. (59) under the ground-state approximation
+⟨ˆσ11⟩ ∼= 1 with weak pump excitation ∆p ≫ {Ωp, Γ4}, we
+get the single-atom steady-state solutions (with µν =
+12, 13, 42, 43)
+ˆσ13 = ˆσ(0)
+13 +
+�
+µν
+βas
+µν ˆf (σ)
+µν ,
+ˆσ42 = ˆσ(0)
+42 +
+�
+µν
+βs
+µν ˆf (σ)
+µν ,
+(63)
+where
+ˆσ(0)
+13 =4 (ϖ + iγ12)
+T (ϖ)
+g31ˆaas
++
+ΩcΩp
+T (ϖ) (∆p + iγ14)g24ˆa†
+s,
+ˆσ(0)
+42 =(ϖ + iγ13)
+T (ϖ)
+|Ωp|2
+(∆p − iγ24)
+1
+(∆p + iγ14)g24ˆa†
+s
++
+Ω∗
+pΩ∗
+c
+T (ϖ) (∆p − iγ24)g31ˆaas,
+(64)
+βas
+12 = i2Ωc
+T (ϖ),
+βas
+13 = −i4 (ϖ + iγ12)
+T (ϖ)
+,
+βas
+42 = −
+iΩcΩp
+T (ϖ) (∆p − iγ24),
+βas
+43 =
+i2Ωp (ϖ + iγ12)
+T (ϖ) (∆p − iγ34),
+βs
+12 = i2 (ϖ + iγ13)
+T (ϖ)
+Ω∗
+p
+(∆p − iγ24),
+βs
+13 = −
+iΩ∗
+pΩ∗
+c
+T (ϖ) (∆p − iγ24),
+βs
+42 = −
+i
+(∆p − iγ24),
+βs
+43 = −
+iΩ∗
+c
+2 (∆p − iγ24) (∆p − iγ34),
+(65)
+where T(ϖ) ≡ |Ωc|2 − 4 (ϖ + iγ13) (ϖ + iγ12). We then
+obtain the ensemble spatially averaged atomic operators
+
+8
+-50
+0
+50
+0
+0.2
+0.4
+0.6
+0.8
+1
+-50
+0
+50
+-10
+-8
+-6
+-4
+-2
+0
+10-7
+-50
+0
+50
+0
+0.2
+0.4
+0.6
+0.8
+1
+Macro
+Micro
+NLN
+-50
+0
+50
+-2
+-1
+0
+1
+2
+3
+10-8
+(
+-
+)
+(
+-
+)
+(
+-
+)
+(
+-
+)
+(a)
+(b)
+(c)
+(d)
+Figure 3. Comparison of commutation relations between the macroscopic (“Macro”, blue solid lines) and microscopic (“Micro”,
+red dashed lines) approaches in the group delay regime: (a) [ˆaas(L), ˆa†
+as(L)], (b) [ˆaas(L), ˆa†
+as(L)] − δ(ϖ − ϖ′), (c) [ˆas(0), ˆa†
+s(0)],
+and (d)[ˆas(0), ˆa†
+s(0)] − δ(ϖ − ϖ′). The results with no Langevin noise operators (“NLN”) are shown as black dotted lines in
+(a) and (c).
+for generating anti-Stokes and Stokes fields from Eq. (49)
+ˆ¯σ13 = ˆσ(0)
+13 +
+1
+√
+nA
+�
+µν
+βas
+µν ˆ¯f
+(σ)
+µν ,
+ˆ¯σ42 = ˆσ(0)
+42 +
+1
+√
+nA
+�
+µν
+βs
+µν ˆ¯f
+(σ)
+µν .
+(66)
+Following the procedures in Eqs. (56) and (57),
+∂ˆaas(ω, z)
+∂z
+= i nAg13ˆ¯σ13(ω, z),
+∂ˆa†
+s(ω, z)
+∂z
+= i nAg42ˆ¯σ42(ω, z),
+(67)
+we get coupled equations for counter-propagating anti-
+Stokes (propagating along +z) and Stokes (propagating
+along −z) fields in the backward-wave configuration
+∂
+∂z
+�ˆaas
+ˆa†
+s
+�
+=
+�
+−αas + i ∆k
+2
+iκas
+iκs
+α∗
+s − i ∆k
+2
+� �ˆaas
+ˆa†
+s
+�
++
+� ˆ¯Fas
+− ˆ¯F †
+s
+�
+,
+(68)
+where
+ˆ¯Fas = ig13
+√
+nA
+�
+βas
+12 ˆ¯f (σ)
+12 + βas
+13 ˆ¯f (σ)
+13 + βas
+42 ˆ¯f (σ)
+42 + βas
+43 ˆ¯f (σ)
+43
+�
+,
+ˆ¯F †
+s = −ig42
+√
+nA
+�
+βs
+12 ˆ¯f (σ)
+12 + βs
+13 ˆ¯f (σ)
+13 + βs
+42 ˆ¯f (σ)
+42 + βs
+43 ˆ¯f (σ)
+43
+�
+,
+(69)
+and
+αas = −iωas
+2c χas,
+αs = −iωs
+2c χs,
+κas =
+√ωasωs
+2c
+χ(3)
+as EpEc,
+κs =
+√ωsωas
+2c
+χ(3)∗
+s
+E∗
+pE∗
+c ,
+χas = 4n |µ13|2
+ε0ℏ
+(ϖ + iγ12)
+T (ϖ)
+,
+χs = n |µ24|2
+ε0ℏ
+(ϖ − iγ13)
+T ∗ (ϖ)
+|Ωp|2
+∆2p + γ2
+14
+,
+χ(3)
+as = nµ13µ32µ24µ41
+ε0ℏ3
+1
+T (ϖ)
+1
+(∆p + iγ14),
+χ(3)
+s
+= nµ13µ32µ24µ41
+ε0ℏ3
+1
+T ∗ (ϖ)
+1
+(∆p + iγ14),
+(70)
+
+9
+-10
+-5
+0
+5
+10
+1
+1.0001
+1.0002
+1.0003
+1.0004
+Macro
+Micro
+-10
+-5
+0
+5
+10
+0
+1
+2
+3
+4
+10-4
+-10
+-5
+0
+5
+10
+1
+1.0001
+1.0002
+1.0003
+1.0004
+-10
+-5
+0
+5
+10
+0
+1
+2
+3
+4
+10-4
+(
+-
+)
+(
+-
+)
+(
+-
+)
+(
+-
+)
+(a)
+(b)
+(c)
+(d)
+Figure 4.
+Four real correlations of Stokes and anti-Stokes fields in the group delay regime:
+(a) ⟨ˆaas(L)ˆa†
+as(L)⟩, (b)
+⟨ˆa†
+as(L)ˆaas(L)⟩, (c) ⟨ˆas(0)ˆa†
+s(0)⟩, and (d) ⟨ˆa†
+s(0)ˆas(0)⟩.
+The macroscopic (“Macro”) and microscopic (“Micro”) approaches
+are shown as blue solid and red dashed lines, respectively.
+The expressions for βas
+µν and βs
+µν are listed in Eqs. (65).
+∆k = (ωas−ωs)/c−(⃗kc+⃗kp)· ˆz is the phase mismatching
+in vacuum.
+Here the complex αas represents the EIT
+loss and phase dispersion.
+α∗
+s is the Raman gain and
+dispersion along −z propagation direction. One can show
+that the nonlinear coupling coefficients can be expressed
+as κas = κeiθ and κs = κe−iθ, where
+κ =
+√ωasωs
+2c
+nµ13µ24
+ε0ℏ
+����
+ΩpΩc
+∆p + iγ14
+����
+1
+T(ϖ),
+(71)
+and θ is the phase of ΩpΩc/(∆p + iγ14). As a result, κas
+and κs fulfill the gauge transformation discussed in Sec.
+II. Therefore, to be consistent with the treatment in Sec.
+II, we rewrite Eq. (68) to
+∂
+∂z
+�ˆaas
+ˆa†
+s
+�
+= MB
+�ˆaas
+ˆa†
+s
+�
++
+� ˆFas
+− ˆF †
+s
+�
+,
+(72)
+where
+MB =
+�
+−αas + i ∆k
+2
+iκ
+iκ
+α∗
+s − i ∆k
+2
+�
+,
+ˆFas = ˆ¯Fase−iθ/2,
+ˆF †
+s = ˆ¯F †
+s eiθ/2.
+(73)
+Similarly, we rewrite the SFWM quantum Langevin
+equations in the forward-wave configuration in Ap-
+pendix C.
+We now turn to compare Eq.
+(72) with Eq.
+(29)
+from the phenomenological approach in Sec. II, where
+we take mode 1 as anti-Stokes and mode 2 as Stokes in
+the backward-wave configuration.
+From Eq.
+(29), we
+have
+ˆFas = NBR11 ˆf1 + NBI11 ˆf †
+1 + NBI12 ˆf2 + NBR12 ˆf †
+2,
+ˆF †
+s = −NBR21 ˆf1 − NBI21 ˆf †
+1 − NBI22 ˆf2 − NBR22 ˆf †
+2.
+(74)
+Therefore, we obtain ˆFas and ˆF †
+s from two different ap-
+proaches: Eq. (69) from the microscopic photon-atom
+interaction, and Eq. (74) from the macroscopic phe-
+nomenological approach.
+Although we remark that
+the atomic noise operators ˆ¯f (σ)
+µν
+are different from the
+field noise operators ˆfm, the correlations of ˆFas and ˆFs
+uniquely determine the system performance. While we
+find it difficult to analytically prove the two approaches
+are equivalent, we could numerically compute and com-
+pare the commutation relations and correlations of ˆaas,
+ˆa†
+as, ˆas, and ˆa†
+s.
+We consider here the backward-wave SFWM in laser-
+cooled
+85Rb atoms with relevant atomic energy lev-
+els being |1⟩ =
+��52S1/2, F = 2
+�
+, |2⟩ =
+��52S1/2, F = 3
+�
+,
+
+10
+-2
+-1
+0
+1
+2
+Macro
+Micro
+-2
+-1
+0
+1
+2
+-2
+-1
+0
+1
+2
+-10
+-5
+0
+5
+10
+-2
+-1
+0
+1
+2
+-10
+0
+10
+-2
+-1
+0
+1
+2
+-10
+-5
+0
+5
+10
+-2
+-1
+0
+1
+2
+-2
+-1
+0
+1
+2
+-2
+-1
+0
+1
+2
+-2
+-1
+0
+1
+2
+-10
+-5
+0
+5
+10
+-2
+-1
+0
+1
+2
+-10
+0
+10
+-2
+-1
+0
+1
+2
+-10
+-5
+0
+5
+10
+-2
+-1
+0
+1
+2
+10-2 (
+-
+)
+10-2 (
+-
+)
+10-2 (
+-
+)
+10-2 (
+-
+)
+10-2 (
+-
+)
+10-2 (
+-
+)
+(a)
+(b)
+(c)
+(d)
+(e)
+(f)
+Figure 5.
+Twelve complex correlations of Stokes and anti-Stokes fields in the group delay regime: (a) ⟨ˆaas(L)ˆaas(L)⟩ =
+⟨ˆa†
+as(L)ˆa†
+as(L)⟩∗, (b) ⟨ˆaas(L)ˆas(0)⟩ = ⟨ˆa†
+s(0)ˆa†
+as(L)⟩∗, (c) ⟨ˆaas(L)ˆa†
+s(0)⟩ = ⟨ˆas(0)ˆa†
+as(L)⟩∗, (d) ⟨ˆa†
+as(L)ˆas(0)⟩ = ⟨ˆa†
+s(0)ˆaas(L)⟩∗,
+(e) ⟨ˆas(0)ˆaas(L)⟩ = ⟨ˆa†
+as(L)ˆa†
+s(0)⟩∗, and (f) ⟨ˆas(0)ˆas(0)⟩ = ⟨ˆa†
+s(0)ˆa†
+s(0)⟩∗. The macroscopic (“Macro”) and microscopic (“Mi-
+cro”) approaches are shown as blue solid and red dashed lines, respectively.
+|3⟩ =
+��52P1/2, F = 3
+�
+, |4⟩ =
+��52P3/2, F = 3
+�
+. The decay
+and dephasing rates for corresponding energy levels are
+Γ3 = Γ4 = 2π × 6 MHz, Γ31 = 5
+9Γ3, Γ32 = 4
+9Γ3, Γ41 =
+4
+9Γ4, Γ42 = 5
+9Γ4, γ13 = γ23 = γ14 = γ24 = 2π × 3 MHz,
+and γ12 = 2π × 0.03 MHz. With vacuum inputs in both
+Stokes (z = L) and anti-Stokes (z = 0) modes, we have
+⟨ˆaas(ϖ, 0)ˆa†
+as(ϖ′, 0)⟩ = ⟨ˆas(ϖ, L)ˆa†
+s(ϖ′, L)⟩ = δ(ϖ − ϖ′)
+and ⟨ˆa†
+as(ϖ, 0)ˆaas(ϖ′, 0)⟩ = ⟨ˆa†
+s(ϖ, L)ˆas(ϖ′, L)⟩ = 0.
+There is also no correlation between Stokes and anti-
+Stokes fields at their inputs.
+We numerically compute SFWM in two different
+regimes to confirm the consistency between the macro-
+scopic and microscopic theories. i) The first is the group
+delay regime, where the SFWM spectrum bandwidth is
+determined by the EIT slow-light induced phase mis-
+matching [10].
+The working parameters are:
+Ωp =
+2π × 1.2 MHz, Ωc = 2π × 12 MHz, ∆p = 2π × 500 MHz.
+The cold atomic medium with length L = 2 cm has den-
+sity n = 5.1 × 1016 m−3, corresponding to an atomic
+optical depth OD = 80 on the anti-Stokes resonance
+transition. ii) The second is the Rabi oscillation regime,
+where biphoton correlation reveals single-atom dynamics
+[10]. The working parameters are: Ωp = 2π × 1.2 MHz,
+Ωc = 2π ×24 MHz, ∆p = ωp −ω14 = 2π ×500 MHz. The
+cold atomic medium with length L = 0.2 cm has density
+n = 6.4×1014 m−3, corresponding to OD = 0.1. In both
+cases, we take ∆k = 127 rad/m.
+The numerical results in the group delay regime are
+
+11
+-50
+0
+50
+0
+0.2
+0.4
+0.6
+0.8
+1
+-50
+0
+50
+-10
+-5
+0
+10-9
+-50
+0
+50
+0
+0.2
+0.4
+0.6
+0.8
+1
+Macro
+Micro
+NLN
+-50
+0
+50
+-1
+0
+1
+2
+10-11
+(
+-
+)
+(
+-
+)
+(
+-
+)
+(
+-
+)
+(a)
+(b)
+(c)
+(d)
+Figure 6. Comparison of commutation relations between the macroscopic (“Macro”, blue solid lines) and microscopic (“Micro”,
+red dashed lines) approaches in the damped Rabi oscillation regime: (a) [ˆaas(L), ˆa†
+as(L)], (b) [ˆaas(L), ˆa†
+as(L)] − δ(ϖ − ϖ′), (c)
+[ˆas(0), ˆa†
+s(0)], and (d)[ˆas(0), ˆa†
+s(0)] − δ(ϖ − ϖ′). The results with no Langevin noise operators (“NLN”) are shown as black
+dotted lines in (a) and (c).
+plotted in Figs. 3, 4, and 5.
+The commutation re-
+lations [ˆaas(L), ˆa†
+as(L)] and [ˆas(0), ˆa†
+s(0)] are shown in
+Fig. 3.
+Both macroscopic and microscopic approaches
+agree well with each other [Figs. 3(a) and (c)], with neg-
+ligible relative small difference < 1.0 × 10−6 [Figs. 3(b)
+and (d)]. As expected, the macroscopic phenomenologi-
+cal results give perfect flat lines at [ˆaas(L,ϖ),ˆa†
+as(L,ϖ′)]
+δ(ϖ−ϖ′)
+=
+[ˆas(0,ϖ),ˆa†
+s(0,ϖ′)]
+δ(ϖ−ϖ′)
+= 1 which is the starting point of Sec.
+II. The microscopic results of field commutations are
+consistent with the macroscopic approach, but with <
+1.0 × 10−6 deviation at some spectra points.
+As we
+understand, these small spectra discrepancies may be
+caused by the ground-state and zeroth-order approxi-
+mations we take for solving the microscopic Heisenberg-
+Langevin equations (59). If the Langevin noise operators
+are not taken into account, as shown in the black dotted
+curves in Figs. 3(a) and (c), the anti-Stokes commuta-
+tion relation is not preserved and displays EIT transmis-
+sion spectrum, while Stokes commutation relation still
+approximately holds due to the negligible gain or loss in
+Stokes channel under the ground-state approximation.
+Figure 4 displays four real-valued correlations of
+Stokes and anti-Stokes fields:
+(a )⟨ˆaas(L)ˆa†
+as(L)⟩, (b)
+⟨ˆa†
+as(L)ˆaas(L)⟩, (c) ⟨ˆas(0)ˆa†
+s(0)⟩, and (d) ⟨ˆa†
+s(0)ˆas(0)⟩.
+Figure 5 shows the twelve (six pairs) complex-valued
+correlations
+of
+Stokes
+and
+anti-Stokes
+fields:
+(a)
+⟨ˆaas(L)ˆaas(L)⟩ = ⟨ˆa†
+as(L)ˆa†
+as(L)⟩∗, (b) ⟨ˆaas(L)ˆas(0)⟩ =
+⟨ˆa†
+s(0)ˆa†
+as(L)⟩∗, (c) ⟨ˆaas(L)ˆa†
+s(0)⟩ = ⟨ˆas(0)ˆa†
+as(L)⟩∗, (d)
+⟨ˆa†
+as(L)ˆas(0)⟩ = ⟨ˆa†
+s(0)ˆaas(L)⟩∗, (e) ⟨ˆas(0)ˆaas(L)⟩ =
+⟨ˆa†
+as(L)ˆa†
+s(0)⟩∗, and (f) ⟨ˆas(0)ˆas(0)⟩ = ⟨ˆa†
+s(0)ˆa†
+s(0)⟩∗.
+The macroscopic solutions agree well with those obtained
+from the microscopic approach.
+The numerical results in the Rabi oscillation regime are
+plotted in Figs. 6, 7, and 8. The macroscopic phenomeno-
+logical results also agree remarkably well with those from
+the microscopic theory.
+IV.
+BIPHOTON GENERATION
+We now turn to apply the quantum Langevin the-
+ory to study time-frequency entangled photon pair
+(biphoton) generation through spontaneous four-wave
+mixing
+process,
+especially
+in
+a
+variety
+of
+situa-
+tions involving gain, loss, and/or complex nonlinear
+coupling
+coefficient.
+We
+consider
+continuous-wave
+pumping
+whose
+time
+translation
+symmetry
+leads
+to
+frequency
+anti-correlation
+ω1 + ω2
+=constant
+between the paired photons.
+In the spontaneous
+
+12
+-50
+0
+50
+1
+1+0.5E-7
+1+1.0E-7
+1+1.5E-7
+Macro
+Micro
+-50
+0
+50
+0
+5
+10
+15
+10-8
+-50
+0
+50
+1
+1+0.2E-7
+1+0.4E-7
+1+0.6E-7
+1+0.8E-7
+1+1.0E-7
+-50
+0
+50
+0
+2
+4
+6
+8
+10
+10-8
+(
+-
+)
+(
+-
+)
+(
+-
+)
+(
+-
+)
+(a)
+(b)
+(c)
+(d)
+Figure 7. Four real correlations of Stokes and anti-Stokes fields in the damped Rabi oscillation regime: (a) ⟨ˆaas(L)ˆa†
+as(L)⟩, (b)
+⟨ˆa†
+as(L)ˆaas(L)⟩, (c) ⟨ˆas(0)ˆa†
+s(0)⟩, and (d) ⟨ˆa†
+s(0)ˆas(0)⟩. The macroscopic (“Macro”) and microscopic (“Micro”) approaches are
+shown as blue solid and red dashed lines, respectively.
+four-wave mixing process, both input states are vac-
+uum:
+⟨ˆa†
+1(ϖ, 0)ˆa1(ϖ′, 0)⟩ = ⟨ˆa†
+2(ϖ, 0)ˆa2(ϖ′, 0)⟩ = 0,
+⟨ˆa1(ϖ′, 0)ˆa†
+1(ϖ, 0)⟩
+=
+⟨ˆa2(ϖ′, 0)ˆa†
+2(ϖ, 0)⟩
+=
+δ(ϖ
+− ϖ′)
+for
+the
+forward-wave
+configuration,
+and ⟨ˆa†
+1(ϖ, 0)ˆa1(ϖ′, 0)⟩
+=
+⟨ˆa†
+2(ϖ, L)ˆa2(ϖ′, L)⟩
+=
+0,
+⟨ˆa1(ϖ, 0)ˆa†
+1(ϖ′, 0)⟩ = ⟨ˆa2(ϖ, L)ˆa†
+2(ϖ′, L)⟩ = δ(ϖ − ϖ′)
+for the backward-wave configuration. From Eq. (4), with
+ω1 = ω10 + ϖ and ω2 = ω20 − ϖ, we have
+ˆa1(t, z1) = eiω10( z1
+c −t)
+√
+2π
+�
+dϖˆa1(ϖ, z1)eiϖ( z1
+c −t)e−i ∆k
+2 z1,
+ˆa2(t, z2) = eiω20(± z2
+c −t)
+√
+2π
+�
+dϖˆa2(ϖ, z2)eiϖ(± z2
+c −t)e−i ∆k
+2 z2,
+(75)
+where ± represents the forward-wave (+) or backward-
+wave (−) configuration, z = z1 and z = z2 are the
+output positions of channels 1 and 2, respectively. For
+the forward-wave configuration, z1 = z2 = L. For the
+backward-wave configuration, z1 = L and z2 = 0. The
+phase mismatching in vacuum ∆k = (ωas ±ωs)/c−(⃗kc +
+⃗kp)· ˆz ≃ (ωas0 ±ωs0)/c−(⃗kc +⃗kp)· ˆz is nearly a constant.
+The vacuum time delay zi/c constants are usually very
+small in usual experimental conditions, from now on we
+ignore these constants for simplification and rewrite the
+above equations to (otherwise one just needs to make a
+time translation t → t − zi/c)
+ˆa1(t, z1) = e−iω10t
+√
+2π
+�
+dϖˆa1(ϖ, z1)e−iϖt,
+ˆa2(t, z2) = e−iω20t
+√
+2π
+�
+dϖˆa2(ϖ, z2)eiϖt.
+(76)
+The photon rate in channel m can be computed from
+Rm ≡
+�
+ˆa†
+m (t, zm) ˆam (t, zm)
+�
+= 1
+2π
+�� ∞
+−∞
+dϖdϖ′e−iϖteiϖ′t �
+ˆa†
+m (ϖ′, zm) ˆam (ϖ, zm)
+�
+.
+(77)
+Here we are particularly interested in the two-photon
+Glauber correlation in the time domain, which can be
+computed from the following two different orders
+G(2)
+2,1 (t2, t1)
+≡⟨ˆa†
+1 (t1, z1) ˆa†
+2 (t2, z2) ˆa2 (t2, z2) ˆa1 (t1, z1)⟩
+=|⟨ˆa2 (t2, z2) ˆa1 (t1, z1)⟩|2
++ |⟨ˆa†
+2 (t2, z2) ˆa1 (t1, z1)⟩|2 + R1R2,
+(78)
+
+13
+-5
+0
+5
+Macro
+Micro
+-5
+0
+5
+-5
+0
+5
+-50
+0
+50
+-5
+0
+5
+-50
+0
+50
+-5
+0
+5
+-50
+0
+50
+-5
+0
+5
+-5
+0
+5
+-5
+0
+5
+-5
+0
+5
+-50
+0
+50
+-5
+0
+5
+-50
+0
+50
+-5
+0
+5
+-50
+0
+50
+-5
+0
+5
+10-5 (
+-
+)
+10-5 (
+-
+)
+10-5 (
+-
+)
+10-5 (
+-
+)
+10-5 (
+-
+)
+10-5 (
+-
+)
+(a)
+(b)
+(c)
+(d)
+(e)
+(f)
+Figure 8.
+Twelve complex correlations of Stokes and anti-Stokes fields in the damped Rabi oscillation regime:
+(a)
+⟨ˆaas(L)ˆaas(L)⟩ = ⟨ˆa†
+as(L)ˆa†
+as(L)⟩∗, (b) ⟨ˆaas(L)ˆas(0)⟩ = ⟨ˆa†
+s(0)ˆa†
+as(L)⟩∗, (c) ⟨ˆaas(L)ˆa†
+s(0)⟩ = ⟨ˆas(0)ˆa†
+as(L)⟩∗, (d) ⟨ˆa†
+as(L)ˆas(0)⟩ =
+⟨ˆa†
+s(0)ˆaas(L)⟩∗, (e) ⟨ˆas(0)ˆaas(L)⟩ = ⟨ˆa†
+as(L)ˆa†
+s(0)⟩∗, and (f) ⟨ˆas(0)ˆas(0)⟩ = ⟨ˆa†
+s(0)ˆa†
+s(0)⟩∗. The macroscopic (“Macro”) and mi-
+croscopic (“Micro”) approaches are shown as blue solid and red dashed lines, respectively.
+G(2)
+1,2 (t1, t2)
+≡⟨ˆa†
+2 (t2, z2) ˆa†
+1 (t1, z1) ˆa1 (t1, z1) ˆa2 (t2, z2)⟩
+=|⟨ˆa1 (t1, z1) ˆa2 (t2, z2)⟩|2
++ |⟨ˆa†
+2 (t2, z2) ˆa1 (t1, z1)⟩|2 + R1R2,
+(79)
+where we have applied the Gaussian moment theorem
+[23, 24] to decompose the fourth-order field correlations
+to the sum of the products of second-order field corre-
+lations.
+The first term in Eqs.
+(78) and (79) can be
+expressed as |Ψ2,1(t2, t1)|2 and |Ψ1,2(t1, t2)|2, where
+Ψ2,1(t2, t1) = ⟨ˆa2 (t2, z2) ˆa1 (t1, z1)⟩
+= e−iω20t2e−iω10t1ψ2,1(t1 − t2),
+(80)
+Ψ1,2(t1, t2) = ⟨ˆa1 (t1, z1) ˆa2 (t2, z2)⟩
+= e−iω20t2e−iω10t1ψ1,2(t1 − t2),
+(81)
+are the two-photon wavefunctions with the relative parts
+ψ2,1(t1 − t2)
+= 1
+2π
+��
+dϖdϖ′⟨ˆa2(ϖ′, z2)ˆa1(ϖ, z1)⟩e−iϖ(t1−t2). (82)
+ψ1,2(t1 − t2)
+= 1
+2π
+��
+dϖdϖ′⟨ˆa1(ϖ, z1)ˆa2(ϖ′, z2)⟩e−iϖ(t1−t2). (83)
+One can show that the second term in Eqs. (78) and (79)
+is zero if the nonlinear coupling coefficient is real-valued,
+
+14
+and it is usually very small as compared to other terms.
+The third term in Eqs. (78) and (79) is the accidental
+coincidence counts. The two-photon wavefunction and
+Glauber correlation satisfy the following exchange sym-
+metry
+ψ21(t1 − t2) = ψ2,1(t1 − t2) = ψ1,2(t1 − t2),
+Ψ21(t2, t1) = Ψ2,1(t2, t1) = Ψ1,2(t1, t2),
+G(2)
+21 (t2, t1) = G(2)
+2,1 (t2, t1) = G(2)
+1,2 (t1, t2) .
+(84)
+The normalized two-photon correlation is defined as
+g(2)
+21 (t2, t1) ≡ G(2)
+21 (t2, t1)
+R1R2
+.
+(85)
+As the system has time translation symmetry with
+continuous-wave pumping, G(2)
+21 (t2, t1) = G(2)
+21 (t1 − t2)
+depends only on the relative time t1 − t2.
+A.
+Loss and Gain
+To simplify and unify the descriptions for account-
+ing both forward- and backward-wave cases, we define
+“input-output” fields:
+ˆa1,in ≡ ˆa1(0), ˆa2,in ≡ ˆa2(0),
+ˆa1,out ≡ ˆa1(L), and ˆa2,out ≡ ˆa2(L) for the forward-wave
+case; ˆa1,in ≡ ˆa1(0), ˆa2,in ≡ ˆa2(L), ˆa1,out ≡ ˆa1(L), and
+ˆa2,out ≡ ˆa2(0) for the backward-wave case. In this sub-
+section, we aim to investigate the roles of loss and gain
+in biphoton generation, considering linear loss in mode 1
+(Re{α1} = α ≥ 0) and linear gain (Re{α2} = −g ≤ 0)
+in mode 2. We also assume κ is real, or its contribution
+to Langevin noises is much smaller than the linear gain
+and loss, i.e., Im{κ} ≪ {α, g}. In this case, for forward-
+and backward-wave configurations, the noise matrix is
+reduced to
+NF,B =
+�√
+2α
+0
+0
+±i√2g
+�
+.
+(86)
+Hence, the output fields in Eqs. (19) and (36) can be
+rewritten as
+�ˆa1,out
+ˆa†
+2,out
+�
+=
+�
+A B
+C D
+� �ˆa1,in
+ˆa†
+2,in
+�
++
+� L
+0
+�
+X11 X12
+X21 X22
+� � ˆf1 (z)
+ˆf2 (z)
+�
+dz.
+(87)
+where Xmn are combined coefficients. We further rewrite
+Eq. (87) as
+ˆa1,out = Aˆa1,in + Bˆa†
+2,in +
+� L
+0
+�
+X11 ˆf1(z) + X12 ˆf2(z)
+�
+,
+ˆa2,out = C∗ˆa†
+1,in + D∗ˆa2,in +
+� L
+0
+�
+X∗
+21 ˆf †
+1(z) + X∗
+22 ˆf †
+2(z)
+�
+.
+(88)
+As shown in Eq.
+(84), there are two different orders
+[⟨: ˆa2ˆa1 :⟩ or ⟨: ˆa1ˆa2 :⟩] to compute the two-photon wave-
+function and Galuber correlation. Although these two
+orders are equivalent, the numerical computation com-
+plexity may be significantly different. Computing bipho-
+ton wavefunction in Eq. (83) in the order ⟨: ˆa1ˆa2 :⟩ in-
+volves nonzero noise field correlations ⟨ ˆfm ˆf †
+m⟩, while in
+the order ⟨: ˆa2ˆa1 :⟩ [Eq.
+(82)] these noise field corre-
+lations disappear because of ⟨ ˆf †
+m ˆfm⟩ = 0.
+These field
+correlations in the frequency domain can be expressed as
+⟨ˆa2out (ϖ′) ˆa1out (ϖ)⟩ = δ(ϖ − ϖ′) [BD∗] ,
+(89)
+⟨ˆa1out (ϖ) ˆa2out (ϖ′)⟩
+= δ(ϖ − ϖ′)
+�
+AC∗ +
+� L
+0
+dz (X11X∗
+21 + X12X∗
+22)
+�
+.
+(90)
+Therefore, we obtain the biphoton wavefunction follow-
+ing the order ⟨: ˆa2ˆa1 :⟩
+ψ21(τ) =
+��
+dϖdϖ′⟨ˆa2,out(ϖ′)ˆa1,out(ϖ)⟩e−iϖτ
+=
+�
+dϖBD∗e−iϖτ.
+(91)
+where τ = t1 − t2. If following the order ⟨: ˆa1ˆa2 :⟩, we
+have
+ψ12(τ) =
+��
+dϖdϖ′⟨ˆa1,out(ϖ)ˆa2,out(ϖ′)⟩e−iϖτ
+=
+�
+dϖ
+�
+AC∗ +
+� L
+0
+dz (X11X∗
+21 + X12X∗
+22)
+�
+e−iϖτ.
+(92)
+One can show that the second term in Eqs. (78) and (79)
+is zero in this loss-gain configuration. The single-channel
+photon rates can be obtained as
+R1 = 1
+2π
+�
+|B|2dϖ,
+R2 = 1
+2π
+� �
+|C|2 +
+� L
+0
+dz
+�
+|X21|2 + |X22|2�
+�
+dϖ.
+(93)
+It is interesting to remark that, in the loss-gain config-
+uration, the biphoton field correlation following the order
+⟨: ˆagainˆaloss :⟩ does not involve noise field correlations as
+shown in Eqs. (89) and (91), which dramatically reduces
+the computation complexity. On the other side, taking
+the order ⟨: ˆalossˆagain :⟩ must include noise field corre-
+lations as shown in Eqs. (90) and (92). This may be
+understood in the heralded photon picture [25]: When
+a photon in a lossy channel is detected (annihilated) by
+a detector, we can always ensure there is its partner (or
+paired) photon in another channel; On the other side,
+when a photon is detected in a gain channel which pro-
+duces multiple photons, we can not always ensure it has
+a partner photon in another channel. The exchange sym-
+metry can only be preserved by taking into account the
+Langevin noises.
+
+15
+0
+1
+2
+109
+Macro
+Micro
+NLN
+-0.5
+0
+0.5
+1
+0
+1
+2
+(a)
+(b)
+Figure 9. Two-photon Glauber correlation in time domain in
+the group delay regime: (a) G(2)
+s,as(τ) and (b) G(2)
+as,s(τ). The
+simulation conditions are the same as that in Figs. 3, 4, and
+5. NLN: no Langevin noise included.
+In the SFWM described in Sec. III, the anti-Stokes
+photons experience finite EIT loss due to the ground
+state dephasing (γ12 ̸= 0), and the Stokes photons prop-
+agate with negligible but small Raman gain.
+Figure
+9 displays the two-photon Glauber correlation in the
+group delay regime with the same parameters as those
+in Figs. 3, 4 and 5. As shown in Fig. 9(a) and (b), both
+macroscopic and microscopic approaches with Langevin
+noises give consistent results. As expected, the compu-
+tation of G(2)
+s,as(τ) (following the order ⟨: ˆasˆaas :⟩) with-
+out Langevin noise operators (black dotted line: NLN)
+agrees with the exact results obtained from both macro-
+scopic (blue solid line) and microscopic (red dashed line)
+approaches, shown in Fig. 9(a).
+On the contrary, the
+computation of G(2)
+as,s(τ) (following the order ⟨: ˆaasˆas :⟩)
+without Langevin noise operators deviates significantly
+from the exact results, as shown in Fig. 9(b).
+B.
+Complex Phase Mismatching
+Different from the Heisenberg picture where the evo-
+lution of field operators is governed by their Langevin
+coupled equations, reference [10] provides a perturbation
+theory to describe biphoton state in the interaction pic-
+ture. The solution from Heisenberg-Langevin theory may
+contain correlations of more than two photons, while the
+perturbation theory focuses only on the two-photon state
+by ignoring higher-order terms. These two treatments are
+expected to give the same results in the limit of small pa-
+rameter gain. Although the perturbation theory in the
+interaction picture provides a much clear physics picture
+of two-photon state, treating loss and gain requires a
+proper justification. In the perturbation theory, linear
+loss and gain are included in the complex phase mis-
+matching ∆˜k(ϖ) [10]. For the SFWM described in Sec.
+III, Ref. [10] derives the biphoton relative wavefunction
+with perturbation theory as
+ψ(τ) = iL
+2π
+�
+dϖκ(ϖ)Φ(ϖ)e−iϖτ,
+(94)
+where the longitudinal detuning function is
+Φ(ϖ) = sinc
+�
+∆˜kL
+2
+�
+ei(kas+ks)L,
+(95)
+There is a statement in Ref. [10]: “It is found that to be
+consistent with the Heisenberg–Langevin theory in the
+low-gain limit, the argument in Φ should be replaced by
+∆˜k =
+�
+⃗kas + ⃗k∗
+s − ⃗kc − ⃗kp
+�
+· ˆz, where ⃗k∗
+s is the conjugate
+of ⃗ks.” For the SFWM in the double-Λ four-level atomic
+system, there is small Raman gain in the Stokes chan-
+nel. What happens if there is loss in the Stokes channel?
+Should we take ⃗k∗
+s or ⃗ks in the complex phase mismatch-
+ing ∆˜k(ϖ)? Although Ref. [10] takes ⃗k∗
+s for Stokes pho-
+tons with gain, it is not clear whether it still holds for the
+case with loss. In this subsection, we do not only provide
+a justification for the above statement in Ref. [10] from
+the quantum Langevin theory by taking small parametric
+gain approximation, but also extend the complex phase
+mismatching to the case with loss in the Stokes channel.
+We take the same backward-wave configuration in
+Ref. [10]. We assume anti-Stokes photons in mode 1 are
+lossless with EIT and there is gain (or loss) in Stokes
+mode 2. The small parametric gain fulfills |κ| ≪ {α, g}.
+In the backward-wave configuration, using Eq.
+(7),
+(34), and (37), we obtain analytical expressions of
+A, B, C, and D as
+A =
+�
+q2 − 4κ2e−(α1−α∗
+2)L/2
+qsinh
+�
+L
+2
+�
+q2 − 4κ2
+�
++
+�
+q2 − 4κ2cosh
+�
+L
+2
+�
+q2 − 4κ2
+�,
+B =
+2iκ
+q +
+�
+q2 − 4κ2coth( L
+2
+�
+q2 − 4κ2)
+,
+C =
+−2iκ
+q +
+�
+q2 − 4κ2coth( L
+2
+�
+q2 − 4κ2)
+,
+D =
+�
+q2 − 4κ2e(α1−α∗
+2)L/2
+qsinh
+�
+L
+2
+�
+q2 − 4κ2
+�
++
+�
+q2 − 4κ2cosh
+�
+L
+2
+�
+q2 − 4κ2
+�,
+(96)
+where q ≡ α1 + α∗
+2 − i∆k. In the small parametric gain
+approximation, we have
+�
+q2 − 4κ2 ≈ q
+= α1 + α∗
+2 − i∆k = −i(∆k1 − ∆k2
+∗ + ∆k),
+(97)
+and
+α1 − α∗
+2 = −i(∆k1 + ∆k2
+∗).
+(98)
+
+16
+where ∆km = ωm
+2c χm is the wavenumber difference from
+that in vacuum. Hence, we simplify A, B, C, and D to
+A =exp [i∆k1L] exp
+�i∆kL
+2
+�
+,
+B =iκLsinc
+�(∆k1 − ∆k∗
+2 + ∆k)L
+2
+�
+× exp
+�i(∆k1 − ∆k∗
+2 + ∆k)L
+2
+�
+,
+C = − iκLsinc
+�(∆k1 − ∆k∗
+2 + ∆k)L
+2
+�
+× exp
+�i(∆k1 − ∆k∗
+2 + ∆k)L
+2
+�
+,
+D =exp [−i∆k∗
+2L] exp
+�i∆kL
+2
+�
+.
+(99)
+We first look at the case with gain in the Stokes (mode
+2). As discussed in Sec. IV A, we take the order ⟨: ˆa2ˆa1 :⟩
+ψ21(τ) =
+��
+dϖdϖ′⟨ˆa2,out(ϖ′)ˆa1,out(ϖ)⟩e−iϖτ
+=
+�
+dϖBD∗e−iϖτ,
+(100)
+where
+BD∗ = iκLsinc
+�(∆k1 − ∆k∗
+2 + ∆k)L
+2
+�
+× exp
+�i(∆k1 − ∆k∗
+2 + 2∆k2)L
+2
+�
+.
+(101)
+Comparing Eqs. (100) and (101) with Eqs. (94) and (95),
+particularly for the argument in the sinc function, we
+have ∆˜k = ∆k1 − ∆k∗
+2 + ∆k = k1 − k∗
+2 − kc + kp =
+kas − k∗
+s − kc + kp which is consistent with the statement
+in Ref. [10].
+We now look at the case with loss in the Stokes (mode
+2). We take the order ⟨: ˆa1ˆa2 :⟩ and have
+ψ12(τ) =
+��
+dϖdϖ′⟨ˆa1,out(ϖ)ˆa2,out(ϖ′)⟩e−iϖτ
+=
+�
+dϖAC∗e−iϖτ,
+(102)
+where
+AC∗ = iκ∗Lsinc
+�(∆k∗
+1 − ∆k2 + ∆k)L
+2
+�
+exp
+�i(2∆k1 − ∆k∗
+1 + ∆k2)L
+2
+�
+.
+(103)
+Comparing Eqs. (102) and (103) with Eqs. (94) and (95),
+we have ∆˜k = ∆k∗
+1 − ∆k2 + ∆k = k1 − k2 − kc + kp =
+kas − ks − kc + kp, which is different from the case with
+gain. Here we have taken k1 ≃ k∗
+1 for lossless mode 1.
+Although our discussion is based on the backward-
+wave configuration, the conclusion can be extended to
+the forward-wave configuration, which is derived in de-
+tail in Appendix D. Therefore, in the case with gain in
+the Stokes mode 2, the complex phase mismatching is
+∆˜k =
+�
+⃗kas + ⃗k∗
+s − ⃗kc − ⃗kp
+�
+· ˆz. In the case with loss in
+the Stokes mode 2, the complex phase mismatching be-
+comes ∆˜k =
+�
+⃗kas + ⃗ks − ⃗kc − ⃗kp
+�
+· ˆz.
+C.
+Complex Nonlinear Coupling Coefficient and
+Rabi Oscillation
+As illustrated in Fig. 2, we can understand the SFWM
+process in the following picture. After a Stoke and anti-
+Stokes photon pair is born from a single atom follow-
+ing the atomic transitions [Fig. 2(b)], the paired pho-
+tons then propagate through the medium [Fig. 2(a)]. As
+the photon pair can be generated at any atom inside the
+medium, the overall two-photon wavefunction (or prob-
+ability amplitude) is a superposition of all possible such
+generation-propagation two-photon Feynman paths. Fol-
+lowing this picture, when the propagation effect can be
+ignored, the biphoton state reveals the single atom dy-
+namics, which is connected to the nonlinear coupling co-
+efficient. In the following, we consider SFWM in the limit
+of small optical depth (OD) where the linear propaga-
+tion effect is small and show how the complex spectrum
+of nonlinear coupling coefficient reveals single-atom Rabi
+oscillation.
+We
+rewrite
+the
+nonlinear
+coupling
+coefficient
+in
+Eq. (71) as:
+κ(ϖ) = J
+�
+1
+(ϖ − Ωe/2 + iγe) −
+1
+(ϖ + Ωe/2 + iγe)
+�
+,
+(104)
+where
+J = −
+√ωasωsnµ13µ24
+8cε0ℏΩe
+����
+ΩpΩc
+∆p + iγ14
+���� .
+(105)
+Here Ωe =
+�
+|Ωc|2 − (γ13 − γ12)2 is the effective coupling
+Rabi frequency, and γe = (γ12 + γ13)/2 is the effective
+dephasing rate. Obviously, the nonlinear coupling coeffi-
+cient κ(ϖ) has a complex spectrum, with two resonances
+separated by the effective coupling Rabi frequency Ωe. In
+the ground-state approximation with major atomic pop-
+ulation in state |1⟩, the undepleted pump laser beam is
+far detuned from the transition |1⟩ → |4⟩ and its exci-
+tation is weak such that we can take χs ≃ 0. On the
+other side, from Eq.
+(70) we have the complex linear
+susceptibility for anti-Stokes photons
+χas(ϖ) = −n |µ13|2
+ε0ℏ
+(ϖ + iγ12)
+(ϖ − Ωe/2 + iγe)(ϖ + Ωe/2 + iγe)
+(106)
+Although the anti-Stokes photon absorption at ϖ = 0
+is suppressed by the EIT effect, there are two absorp-
+tion resonances appearing at ϖ = ±Ωe/2 which coin-
+cide with the two resonances of nonlinear coupling coef-
+ficient in Eq. (104). We take the pump laser with weak
+
+17
+intensity (∝ |Ωp|2) and large detuning (∆p) such that
+Re{αas(ϖ = ±Ωe/2)}>Im{κ(ϖ = ±Ωe/2)}, which are
+usually satisfied in the ground state condition. As the
+propagation effect is small and the phase matching is not
+important, the paired photons are mostly generated from
+the two resonances (ϖ = ±Ωe/2) of the nonlinear cou-
+pling coefficient.
+In the forward-wave configuration, with the coupling
+matrix
+MF =
+�
+−αas + i ∆k
+2
+iκ
+−iκ
+−i ∆k
+2
+�
+,
+(107)
+and short medium length L satisfying |MFL| ≪ 1, we
+have approximately
+�
+A B
+C D
+�
+= eMFL ∼= 1 + MFL
+=
+�
+1 − αasL + i ∆k
+2 L
+iκL
+−iκL
+1 − i ∆k
+2 L
+�
+.
+(108)
+As discussed in Sec. IV A, the biphoton field correlation
+following the order ⟨: ˆasˆaas :⟩ does not need count the
+Langevin noise operators:
+⟨ˆas(ϖ′, L)ˆaas(ϖ, L)⟩ = BD∗δ(ϖ − ϖ′)
+= iκL(1 + i∆k
+2 L)δ(ϖ − ϖ′)
+∼= iκ(ϖ)Lδ(ϖ − ϖ′),
+(109)
+where we have neglected higher order terms O(L2). From
+Eq. (82), we have the relative biphoton wavefunction
+ψs−as(τ) = iL
+2π
+�
+dϖκ(ϖ)e−iϖτ,
+(110)
+which is the Fourier transform of the nonlinear coupling
+coefficient with τ = tas − ts. Substituting Eq. (104) into
+Eq. (110) we obtain
+ψs−as(τ) = LJe−γeτ[e−iΩeτ/2 − eiΩeτ/2]Θ(τ)
+= −2iLJe−γeτ sin
+�Ωeτ
+2
+�
+Θ(τ),
+(111)
+where Θ(τ) is the Heaviside function.
+Equation (111)
+shows a damped Rabi oscillation, resulting from the beat-
+ing between biphotons generated from the two resonances
+at ϖ = ±Ωe/2. The Heaviside function shows the anti-
+Stokes photon is always generated after its paired Stokes
+photon following the time order of atomic transitions
+|1⟩ → |4⟩ → |2⟩ → |3⟩ → |1⟩ in an SFWM cycle shown in
+Fig. 2(b).
+In the backward-wave configuration, the coupling ma-
+trix becomes
+MB =
+�
+−αas + i ∆k
+2
+iκ
+iκ
+−i ∆k
+2
+�
+.
+(112)
+0
+2
+4
+6
+105
+Macro
+Micro
+NLN
+0
+2
+4
+6
+-0.4
+-0.2
+0
+0.2
+0.4
+0
+2
+4
+6
+|
+s-as|2
+(a)
+(b)
+(c)
+Figure 10. Two-photon Glauber correlation in time domain
+in the damped Rabi oscillation regime: (a) G(2)
+s,as(τ) and (b)
+G(2)
+as,s(τ). The simulation conditions are the same as that in
+Figs. 6, 7, and 8. (c) shows the analytic solution for the bipho-
+ton waveform |ψs−as(τ)|2. NLN: no Langevin noise included.
+With |MBL| ≪ 1 we have
+� ¯A
+¯B
+¯C
+¯D
+�
+= eMBL ∼= 1 + MBL
+=
+�
+1 − αasL + i ∆k
+2 L
+iκL
+iκL
+1 − i ∆k
+2 L
+�
+,
+(113)
+and
+�
+A B
+C D
+�
+=
+�
+1 − αasL + i ∆k
+2 L
+iκL
+−iκL
+1 + i ∆k
+2 L
+�
+,
+(114)
+where we have neglect higher order terms O(L2). Simi-
+larly, we have
+⟨ˆas(ϖ′, 0)ˆaas(ϖ, L)⟩ ∼= iκ(ϖ)Lδ(ϖ − ϖ′),
+(115)
+which is the same as Eq. (109) of the forward-wave con-
+figuration. Therefore, we obtain Rabi oscillations in both
+forward- and backward-wave configurations.
+Equation
+(111) is identical to the result derived from the pertur-
+bation theory in the interaction picture [10].
+
+18
+Figure 10 displays the two-photon Glauber correlation
+in the damped Rabi oscillation regime with the same pa-
+rameters as those in Figs.
+6, 7 and 8.
+As illustrated
+in Fig. 10(a) and (b), both macroscopic and microscopic
+approaches with Langevin noises give consistent results.
+As expected, the computation of G(2)
+s,as(τ) (following the
+order ⟨: ˆasˆaas :⟩) without Langevin noise operators (dot
+points) agrees with the exact results obtained from both
+microscopic (red dashed line) and macroscopic (blue solid
+line) approaches, shown in Fig. 10(a).
+On the con-
+trary, the computation of G(2)
+as,s(τ) (following the order
+⟨: ˆaasˆas :⟩) without Langevin noise operators (dot points:
+NLN) deviates significantly from the exact results and vi-
+olates the causality, as shown in Fig. 10(b). Fig. 10(c)
+shows the result from the analytic solution in Eq. (111)
+which agree well with the exact results in Figs. 10(a) and
+(b).
+It is interesting to examine a system without gain and
+loss whose Langevin noises are purely contributed by the
+complex nonlinear coupling coefficient. In this case, the
+above approximation and conclusion do not hold. Let’s
+now consider the case 3 with the forward-wave config-
+uration in Sec. II A, where α1 = α2 = ∆k = 0, and
+κ = η + iζ. As shown in Sec. II A, the noise matrix is
+different as ζ is positive or negative. We first consider
+ζ > 0, the Langevin coupled equations (27) becomes
+∂
+∂z
+�ˆa1
+ˆa†
+2
+�
+=
+�
+0
+iκ
+−iκ
+0
+� �ˆa1
+ˆa†
+2
+�
++
+�
+ζ
+�
+1
+1
+−1 1
+� � ˆf1
+ˆf †
+2
+�
+.
+(116)
+Under the condition |MFL| ≪ 1, we solve Eq.
+(116) to
+the first order of L and have
+ˆa1(L) ∼= ˆa1(0) + iκLˆa†
+2(0) +
+�
+ζ
+� L
+0
+dz
+�
+ˆf1 + ˆf †
+2
+�
+,
+ˆa2(L) ∼= ˆa2(0) + iκ∗Lˆa†
+1(0) +
+�
+ζ
+� L
+0
+dz
+�
+− ˆf †
+1 + ˆf2
+�
+.
+(117)
+The two-photon field correlations are
+⟨ˆa1(L)ˆa2(L)⟩ = ⟨ˆa2(L)ˆa1(L)⟩ ∼= i
+2(κ + κ∗)Lδ(ϖ − ϖ′).
+(118)
+As ζ < 0, the Langevin coupled equations (27) becomes
+∂
+∂z
+�ˆa1
+ˆa†
+2
+�
+=
+�
+0
+iκ
+−iκ
+0
+� �ˆa1
+ˆa†
+2
+�
++
+�
+−ζ
+�
+1
+1
+−1 1
+� � ˆf †
+1ˆf2
+�
+. (119)
+Under the condition |MFL| ≪ 1, we solve Eq.
+(119) to
+the first order of L and have
+ˆa1(L) ∼= ˆa1(0) + iκLˆa†
+2(0) +
+�
+−ζ
+� L
+0
+dz
+�
+ˆf †
+1 + ˆf2
+�
+,
+ˆa2(L) ∼= ˆa2(0) + iκ∗Lˆa†
+1(0) +
+�
+−ζ
+� L
+0
+dz
+�
+− ˆf1 + ˆf †
+2
+�
+.
+(120)
+The two-photon field correlations are
+⟨ˆa1(L)ˆa2(L)⟩ = ⟨ˆa2(L)ˆa1(L)⟩ ∼= i
+2(k + k∗)Lδ(ϖ − ϖ′),
+(121)
+which is the same as Eq. (118). The biphoton relative
+wavefunction is
+ψ21(τ) = ψ∗
+21(−τ) = iL
+2π
+�
+dϖ1
+2(k + k∗)e−iϖτ.
+(122)
+One can prove that under the same limit |MBL| ≪ 1, the
+backward-wave configuration gives the same two-photon
+field correlation [Eqs.
+(118) and (121)] and temporal
+wavefunction [Eq. (122)]. Equation (122) suggests the
+biphoton temporal wavefunction has time reversal sym-
+metry when there is no linear gain and loss.
+V.
+CONCLUSION
+In summary, we provide a macroscopic phenomenolog-
+ical formula of quantum Langevin equations for two cou-
+pled phase-conjugated fields with linear loss (gain) and
+complex nonlinear coupling coefficient, in both forward-
+and backward-wave configurations.
+The macroscopic
+phenomenological formula, obtained from the coupling
+matrix and the requirement of preserving commutation
+relations of field operators during propagation, does not
+require knowing microscopic details of light-matter inter-
+action and internal atomic structures. To validate this
+phenomenological formula, we take SFWM in a double-
+Λ four-level atomic system as an example to numeri-
+cally confirm that our macroscopic phenomenological re-
+sult is consistent with that obtained from microscopic
+Heisenberg-Langevin theory. As compared to the com-
+plicated microscopic theory which varies from system
+to system, the macroscopic coupled equations are much
+more friendly to experimentalists. We apply the quantum
+Langevin equations to study the effects of gain and/or
+loss as well as complex nonlinear coupling coefficient in
+biphoton generation, particularly to the temporal quan-
+tum correlations. We show that the computation com-
+plexity can be dramatically reduced by taking a proper
+order of field operators based on loss and gain. Making
+a comparison between the quantum Langevin theory (in
+the Heisenberg picture) and the perturbation theory (in
+the interaction picture [10]), we extend the expression of
+complex phase mismatching to account for loss and gain.
+At last, we reveal Rabi oscillation in SFWM biphoton
+temporal correlation when the propagation effect is small.
+Although in this article we focus on biphoton generation
+from the spontaneous parametric process, the quantum
+Langevin coupled equations can also be used to study
+two-mode squeezing, parametric oscillation, and other
+quantum light state generation.
+ACKNOWLEDGMENTS
+S.D.
+acknowledges
+support
+from
+DOE
+(DE-
+SC0022069),
+AFOSR
+(FA9550-22-1-0043)
+and
+NSF
+(CNS-2114076, 2228725).
+
+19
+Appendix A: Noise Matrix in Backward-Wave
+Configuration
+In the macroscopic quantum Langevin equations, the
+requirement of preserving commutation relations allows
+multiple choices of the noise matrix. For example, ˆf1 →
+− ˆf1 or/and ˆf2 → − ˆf2 do not affect any computation re-
+sults of physical observables involving pairs of Langevin
+noise operators. As an example, here we provide several
+equivalent noise matrices for backward-wave configura-
+tion:
+NB1 ≡
+�
+1
+0
+0 −1
+� ��
+−MB11 −MB12
+MB21
+MB22
+�
++
+�
+−MB11 −MB12
+MB21
+MB22
+�∗
+=
+�
+1
+0
+0 −1
+�
+NF,
+NB2 ≡ NB1
+�
+1
+0
+0 −1
+�
+=
+��
+−MB11 MB12
+−MB21 MB22
+�
++
+�
+−MB11 MB12
+−MB21 MB22
+�∗
+,
+NB3 ≡ NB1
+�
+−1 0
+0
+1
+�
+,
+NB4 ≡ NB1
+�
+−1
+0
+0
+−1
+�
+= −NB1.
+(A1)
+We take the first choice NB1 in the main text [see Eq. (31)
+in Sec. II B] so that it is consistent with the microscopic
+treatment in Sec. III.
+Appendix B: Heisenberg-Langevin Equations of
+SFWM
+The full Heisenberg equation of motion can be written
+as
+˙ˆS = i( ˆO ˆS − ˆS ˆO) + ˆG + ˆF,
+(B1)
+where
+ˆS =
+�
+��
+ˆσ11 ˆσ12 ˆσ13 ˆσ14
+ˆσ21 ˆσ22 ˆσ23 ˆσ24
+ˆσ31 ˆσ32 ˆσ33 ˆσ34
+ˆσ41 ˆσ42 ˆσ43 ˆσ44
+�
+�� ,
+(B2)
+ˆO = −
+�
+��
+0
+0
+g31ˆaas Ωp/2
+0
+ϖ
+Ωc/2
+g42ˆas
+g13ˆa∗
+as Ω∗
+c/2
+ϖ
+0
+Ω∗
+p/2
+g24ˆa∗
+s
+0
+∆p
+�
+�� ,
+(B3)
+ˆG =
+�
+��
+Γ31ˆσ33 + Γ41ˆσ44
+−γ12ˆσ12
+−γ13ˆσ13 −γ14ˆσ14
+−γ12ˆσ21
+Γ32ˆσ33 + Γ42ˆσ44 −γ23ˆσ23 −γ24ˆσ24
+−γ13ˆσ31
+−γ23ˆσ32
+−Γ3ˆσ33 −γ34ˆσ34
+−γ14ˆσ41
+−γ24ˆσ42
+−γ34ˆσ43 −Γ4ˆσ44
+�
+�� ,
+(B4)
+ˆF =
+�
+����
+ˆf (σ)
+11
+ˆf (σ)
+12
+ˆf (σ)
+13
+ˆf (σ)
+14
+ˆf (σ)
+21
+ˆf (σ)
+22
+ˆf (σ)
+23
+ˆf (σ)
+24
+ˆf (σ)
+31
+ˆf (σ)
+32
+ˆf (σ)
+33
+ˆf (σ)
+34
+ˆf (σ)
+41
+ˆf (σ)
+42
+ˆf (σ)
+43
+ˆf (σ)
+44
+�
+���� .
+(B5)
+Γm = Γm1 + Γm2 is the total spontaneous decay rate of
+excited state |m⟩, where m = 3, or 4, and Γmj is the decay
+rate from state |m⟩ to |j⟩. For the two hyperfine ground
+states, there are Γ1 = Γ2 = 0.
+For cold atoms with
+only spontaneous emisson decay, the dephasing rates γjk
+(j ̸= k) between states |k⟩ and |j⟩ are γ13 = γ23 = Γ3/2,
+γ14 = γ24 = Γ4/2, γ34 = (Γ3+Γ4)/2. γ12 is the dephasing
+rate between two hyperfine ground states |1⟩ and |2⟩.
+Appendix C: Microscopic SFWM Quantum
+Langevin Equations in Forward-Wave Configuration
+Although Sec. III focuses on numerical confirmation
+of backward-wave SFWM, we remark that it may be
+helpful for general readers to write the SFWM quantum
+Langevin equations in the forward-wave configuration as
+well.
+In the forward-wave configuration with both Stokes
+and anti-Stokes fields propagating along +z direction,
+the coupled Langevin equations become
+∂
+∂z
+�ˆaas
+ˆa†
+s
+�
+= MF
+�ˆaas
+ˆa†
+s
+�
++
+� ˆFas
+ˆF †
+s
+�
+,
+(C1)
+where
+MF =
+�
+−αas + i ∆k
+2
+iκ
+−iκ
+−α∗
+s − i ∆k
+2
+�
+,
+(C2)
+with ∆k = (ωas+ωs)/c−(⃗kc+⃗kp)·ˆz. The noise operators
+ˆFas and ˆF †
+s , defined in Eq. (69), originate from micro-
+scopic atom-light interaction. To compare Eq. (C1) with
+Eq. (11) from the phenomenological approach in Sec. II,
+we take mode 1 as anti-Stokes and mode 2 as Stokes in
+the forward-wave configuration. From Eq. (11), we can
+also obtain ˆFas and ˆF †
+s from the noise matrix:
+ˆFas = NFR11 ˆf1 + NFI11 ˆf †
+1 + NFI12 ˆf2 + NFR12 ˆf †
+2,
+ˆF †
+s = NFR21 ˆf1 + NFI21 ˆf †
+1 + NFI22 ˆf2 + NFR22 ˆf †
+2.
+(C3)
+Appendix D: Complex Phase Mismatching in
+Forward-Wave Configuration
+In the forward-wave configuration, similar to the
+backward-wave configuration in Sec. IV B, we assume
+anti-Stokes photons in mode 1 are lossless with EIT and
+there is gain (or loss) in Stokes mode 2. The small para-
+metric gain fulfills |κ| ≪ {α, g}. Using Eq. (6) and (17),
+
+20
+we obtain analytical expressions of A, B, C, and D as
+A =
+�
+q2 + 4κ2cosh
+�
+L
+2
+�
+q2 + 4κ2
+�
+− qsinh
+�
+L
+2
+�
+q2 + 4κ2
+�
+�
+q2 + 4κ2e(α1+α∗
+2)L/2
+,
+B =
+2iκsinh
+�
+L
+2
+�
+q2 + 4κ2
+�
+�
+q2 + 4κ2e(α1+α∗
+2)L/2 ,
+C =
+−2iκsinh
+�
+L
+2
+�
+q2 + 4κ2
+�
+�
+q2 + 4κ2e(α1+α∗
+2)L/2 ,
+D =
+�
+q2 + 4κ2cosh
+�
+L
+2
+�
+q2 + 4κ2
+�
++ qsinh
+�
+L
+2
+�
+q2 + 4κ2
+�
+�
+q2 + 4κ2e(α1+α∗
+2)L/2
+,
+(D1)
+where q ≡ α1 − α∗
+2 − i∆k. In the small parametric gain
+approximation, we have
+�
+q2 − 4κ2 ≈ q
+= α1 − α∗
+2 − i∆k = −i(∆k1 + ∆k∗
+2 + ∆k),
+(D2)
+and
+α1 + α∗
+2 = −i(∆k1 − ∆k∗
+2),
+(D3)
+where ∆km = ωm
+2c χm is the wavenumber difference from
+that in vacuum. Hence, we simplify A, B, C, and D to
+A =exp [i∆k1L] exp
+�i∆kL
+2
+�
+,
+B =iκLsinc
+�(∆k1 + ∆k∗
+2 + ∆k)L
+2
+�
+× exp
+�i(∆k1 − ∆k∗
+2)L
+2
+�
+,
+C = − iκLsinc
+�(∆k1 + ∆k∗
+2 + ∆k)L
+2
+�
+× exp
+�i(∆k1 − ∆k∗
+2)L
+2
+�
+,
+D =exp [−i∆k∗
+2L] exp
+�−i∆kL
+2
+�
+.
+(D4)
+We first look at the case with gain in the Stokes (mode
+2). As discussed in Sec. IV A, we take the order ⟨: ˆa2ˆa1 :⟩
+ψ21(τ) =
+��
+dϖdϖ′⟨ˆa2,out(ϖ′)ˆa1,out(ϖ)⟩e−iϖτ
+=
+�
+dϖBD∗e−iϖτ,
+(D5)
+where
+BD∗ = iκLsinc
+�(∆k1 + ∆k∗
+2 + ∆k)L
+2
+�
+× exp
+�i(∆k1 − ∆k∗
+2 + 2∆k2 + ∆k)L
+2
+�
+.
+(D6)
+Comparing Eqs. (D5) and (D6) with Eqs. (94) and (95),
+particularly for the argument in the sinc function, we
+have ∆˜k = ∆k1 + ∆k∗
+2 + ∆k = k1 + k∗
+2 − kc − kp =
+kas + k∗
+s − kc − kp which is consistent with the statement
+in Ref. [10].
+We now look at the case with loss in the Stokes (mode
+2). We take the order ⟨: ˆa1ˆa2 :⟩ and have
+ψ12(τ) =
+��
+dϖdϖ′⟨ˆa1,out(ϖ)ˆa2,out(ϖ′)⟩e−iϖτ
+=
+�
+dϖAC∗e−iϖτ,
+(D7)
+where
+AC∗ = iκ∗Lsinc
+�(∆k∗
+1 + ∆k2 + ∆k)L
+2
+�
+× exp
+�i(2∆k1 − ∆k∗
+1 + ∆k2 + ∆k)L
+2
+�
+.
+(D8)
+Comparing Eqs. (D7) and (D8) with Eqs. (94) and (95),
+we have ∆˜k = ∆k∗
+1 + ∆k2 + ∆k = k1 + k2 − kc +
+kp = kas + ks − kc − kp, which is different from the
+case with gain.
+Here we have taken k1 ≃ k∗
+1 for loss-
+less mode 1.
+Therefore, in the case with loss in the
+Stokes mode 2, the complex phase mismatching becomes
+∆˜k =
+�
+⃗kas + ⃗ks − ⃗kc − ⃗kp
+�
+· ˆz.
+[1] C. W. Gardiner and M. J. Collett, Input and output in
+damped quantum systems: Quantum stochastic differen-
+tial equations and the master equation, Phys. Rev. A 31,
+3761 (1985).
+[2] M. O. Scully and M. S. Zubairy, Quantum optics (1999).
+[3] Y. Yamamoto and A. Imamoglu, Mesoscopic quantum
+optics, Mesoscopic Quantum Optics (1999).
+[4] C. Gardiner, P. Zoller, and P. Zoller, Quantum noise:
+a handbook of Markovian and non-Markovian quantum
+stochastic methods with applications to quantum optics
+(Springer Science & Business Media, 2004).
+[5] R. Benguria and M. Kac, Quantum langevin equation,
+Phys. Rev. Lett. 46, 1 (1981).
+[6] R. W. Boyd, Nonlinear optics (Academic press, 2020).
+[7] S. Shwartz, R. N. Coffee, J. M. Feldkamp, Y. Feng, J. B.
+Hastings, G. Y. Yin, and S. E. Harris, X-ray parametric
+down-conversion in the langevin regime, Phys. Rev. Lett.
+109, 013602 (2012).
+[8] U. A. Javid and Q. Lin, Quantum correlations from
+dynamically modulated optical nonlinear interactions,
+Phys. Rev. A 100, 043811 (2019).
+[9] G. Shafiee, D. V. Strekalov, A. Otterpohl, F. Sedlmeir,
+
+21
+G. Schunk, U. Vogl, H. G. L. Schwefel, G. Leuchs, and
+C. Marquardt, Nonlinear power dependence of the spec-
+tral properties of an optical parametric oscillator below
+threshold in the quantum regime, New Journal of Physics
+22, 073045 (2020).
+[10] S. Du, J. Wen, and M. H. Rubin, Narrowband biphoton
+generation near atomic resonance, J. Opt. Soc. Am. B
+25, C98 (2008).
+[11] P. Kolchin, Electromagnetically-induced-transparency-
+based paired photon generation, Phys. Rev. A 75, 033814
+(2007).
+[12] L. Zhao, Y. Su, and S. Du, Narrowband biphoton gener-
+ation in the group delay regime, Phys. Rev. A 93, 033815
+(2016).
+[13] C. H. Raymond Ooi, Q. Sun, M. S. Zubairy, and M. O.
+Scully, Correlation of photon pairs from the double ra-
+man amplifier: Generalized analytical quantum langevin
+theory, Phys. Rev. A 75, 013820 (2007).
+[14] Y. Jiang, Y. Mei, Y. Zuo, Y. Zhai, J. Li, J. Wen, and
+S. Du, Anti-parity-time symmetric optical four-wave mix-
+ing in cold atoms, Phys. Rev. Lett. 123, 193604 (2019).
+[15] Y. Mei, X. Guo, L. Zhao, and S. Du, Mirrorless opti-
+cal parametric oscillation with tunable threshold in cold
+atoms, Phys. Rev. Lett. 119, 150406 (2017).
+[16] X.-W. Luo, C. Zhang, and S. Du, Quantum squeez-
+ing and sensing with pseudo-anti-parity-time symmetry,
+Phys. Rev. Lett. 128, 173602 (2022).
+[17] C. M. Bender and S. Boettcher, Real spectra in non-
+Hermitian Hamiltonians having PT symmetry, Phys.
+Rev. Lett. 80, 5243 (1998).
+[18] M.-A. Miri and A. Al`u, Exceptional points in optics and
+photonics, Science 363, 10.1126/science.aar7709 (2019).
+[19] D. A. Braje, V. Bali´c, S. Goda, G. Y. Yin, and S. E. Har-
+ris, Frequency mixing using electromagnetically induced
+transparency in cold atoms, Phys. Rev. Lett. 93, 183601
+(2004).
+[20] V. Bali´c, D. A. Braje, P. Kolchin, G. Y. Yin, and S. E.
+Harris, Generation of paired photons with controllable
+waveforms, Phys. Rev. Lett. 94, 183601 (2005).
+[21] S. E. Harris, Electromagnetically induced transparency,
+Phys. Today 50, 36 (1997).
+[22] M. Fleischhauer, A. Imamoglu, and J. P. Marangos, Elec-
+tromagnetically induced transparency: Optics in coher-
+ent media, Rev. Mod. Phys. 77, 633 (2005).
+[23] L. Mandel and E. Wolf, Optical Coherence and Quantum
+Optics (Cambridge University Press, Cambridge Eng-
+land, New York, 1994).
+[24] W. H. Louisell, Optical Coherence and Quantum Optics
+(Wiley, New York, 1973).
+[25] S. Du, Quantum-state purity of heralded single photons
+produced from frequency-anticorrelated biphotons, Phys.
+Rev. A 92, 043836 (2015).
+

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@@ -0,0 +1,1865 @@
+arXiv:2301.00104v1  [cs.CR]  31 Dec 2022
+Separating Computational and Statistical Differential Privacy
+(Under Plausible Assumptions)
+Badih Ghazi∗
+Rahul Ilango†
+Pritish Kamath‡
+Ravi Kumar§
+Pasin Manurangsi¶
+Abstract
+Computational differential privacy (CDP) is a natural relaxation of the standard notion of
+(statistical) differential privacy (SDP) proposed by Beimel, Nissim, and Omri (CRYPTO 2008)
+and Mironov, Pandey, Reingold, and Vadhan (CRYPTO 2009). In contrast to SDP, CDP only
+requires privacy guarantees to hold against computationally-bounded adversaries rather than
+computationally-unbounded statistical adversaries. Despite the question being raised explicitly
+in several works (e.g., Bun, Chen, and Vadhan, TCC 2016), it has remained tantalizingly open
+whether there is any task achievable with the CDP notion but not the SDP notion. Even a
+candidate such task is unknown. Indeed, it is even unclear what the truth could be!
+In this work, we give the first construction of a task achievable with the CDP notion but not
+the SDP notion. More specifically, under strong but plausible cryptographic assumptions, we
+construct a task for which there exists an ε-CDP mechanism with ε = O(1) achieving 1 − o(1)
+utility, but any (ε, δ)-SDP mechanism, including computationally unbounded ones, that achieves
+a constant utility must use either a super-constant ε or a non-negligible δ. To prove this, we
+introduce a new approach for showing that a mechanism satisfies CDP: first we show that a
+mechanism is “private” against a certain class of decision tree adversaries, and then we use
+cryptographic constructions to “lift” this into privacy against computational adversaries. We
+believe this approach could be useful to devise further tasks separating CDP from SDP.
+∗Google Research, Mountain View. [email protected].
+†MIT. Part of this work was done during an internship at Google Research. [email protected].
+‡Google Research, Mountain View. [email protected].
+§Google Research, Mountain View. [email protected].
+¶Google Research, Thailand. [email protected].
+
+Contents
+1
+Introduction
+1
+2
+Overview of the Results
+3
+2.1
+The d-Distance Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+3
+2.2
+SDP Lower Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+4
+2.3
+A CDP Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+4
+2.4
+Final Steps
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+6
+2.5
+On the Plausiblility of the Cryptographic Assumptions . . . . . . . . . . . . . . . . .
+6
+3
+Preliminaries
+7
+3.1
+Dataset and Adjacency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+8
+3.2
+Mechanism, Utility Function, and Usefulness
+. . . . . . . . . . . . . . . . . . . . . .
+8
+3.3
+Notions of Differential Privacy
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+8
+4
+Low Diameter Set Problem and Nearby Point Problem
+9
+4.1
+Simplification of Input Representation . . . . . . . . . . . . . . . . . . . . . . . . . .
+10
+4.2
+Nearby Point Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+10
+4.3
+Verifiable Low Diameter Set Problem . . . . . . . . . . . . . . . . . . . . . . . . . . .
+10
+4.4
+From Low Diameter Set Problem to Nearby Point Problem
+. . . . . . . . . . . . . .
+11
+5
+CDP Mechanism for Verifiable Low Diameter Set Problem
+11
+5.1
+CDP Mechanism without Verifiability
+. . . . . . . . . . . . . . . . . . . . . . . . . .
+12
+5.1.1
+Additional Preliminaries: Cryptographic Primitives . . . . . . . . . . . . . . .
+12
+5.1.2
+Public-Coin Differing-Inputs Circuits from CRKHFs . . . . . . . . . . . . . .
+13
+5.1.3
+From Differing-Inputs Circuits to CDP . . . . . . . . . . . . . . . . . . . . . .
+15
+5.2
+CDP Mechanism for VLDS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+17
+5.2.1
+Witness-Indistinguishable Proofs . . . . . . . . . . . . . . . . . . . . . . . . .
+17
+5.2.2
+Making Utility Function Efficient Using Witness-Indistinguishable Proofs
+. .
+17
+6
+SDP Lower Bounds for the Nearby Point Problem
+19
+6.1
+Additional Preliminaries: Tools from Differential Privacy
+. . . . . . . . . . . . . . .
+20
+6.2
+Weak Hardness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+21
+6.3
+Boosting the Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+22
+6.4
+Boosting the Failure Probability
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+23
+6.5
+Putting Things Together: Proof of Theorem 30 . . . . . . . . . . . . . . . . . . . . .
+24
+7
+Putting Things Together: Proof of Theorem 5
+24
+8
+Conclusion and Discussion
+24
+A Comparison of various diO assumptions
+29
+
+1
+Introduction
+The framework of differential privacy (DP) [DMNS06, DKM+06] gives formal privacy guarantees
+on the outputs of randomized algorithms.
+It has been the subject of a significant body of re-
+search, leading to numerous practical deployments including the US census [Abo18], and industrial
+applications [EPK14, Sha14, Gre16, App17, DKY17, KT18, RSP+21].
+The definition of DP requires privacy against computationally unbounded, i.e., statistical, adver-
+saries. A natural modification is to instead only require privacy against computationally bounded
+adversaries. In cryptography, considering computationally bounded adversaries instead of statisti-
+cal ones enables a vast array of applications, like public-key cryptography. Could the same be true
+for DP? A good survey of the area can be found in Vadhan’s monograph [Vad17, Section 10]. De-
+spite Beimel, Nissim, and Omri [BNO08] defining computational differential privacy (CDP) in 2008
+(definitions that were further extended by Mironov, Pandey, Reingold, and Vadhan [MPRV09]),
+the central question of separating it from statistical differential privacy (SDP)1, in the standard
+client-server model, remains open:
+Question 1. [Vad17, Open Problem 10.7]
+Is there a computational task solvable by a single cura-
+tor with computational differential privacy but is impossible to achieve with information-theoretic
+differential privacy?2
+There have been several positive and negative results towards resolving this question. In the
+positive direction, it is known that in the multi-party setting, CDP is stronger than SDP [MMP+10,
+MPRV09]. Roughly speaking, this is because secure multi-party computation enables many data cu-
+rators to simulate acting as a single central curator, without compromising privacy. Still, the multi-
+party setting seems very different than the single-curator (aka central) setting. Indeed, [MMP+10]
+remark3 that their “strong separation between (information-theoretic) differential privacy and com-
+putational differential privacy ... stands in sharp contrast with the client-server setting where ...
+there are not even candidates for a separation.”
+In the central setting, Bun, Chen, and Vadhan [BCV16] show there is a task for which there
+is a CDP mechanism, but any SDP mechanism for this task must be inefficient (modulo certain
+cryptographic assumptions). We stress that the task they consider does have an inefficient SDP
+mechanism (with parameters that match their CDP mechanism), so it does not resolve Question 1.
+While this may seem like a minor technical point, we emphasize that it is of crucial importance.
+Perhaps the main practical motivation behind studying CDP is the hope that there are CDP mech-
+anisms for natural tasks with parameters that beat the lower bounds against SDP mechanisms. But
+if, as in the case of the result in [BCV16], there exists (even an inefficient) SDP mechanism match-
+ing the parameters of the CDP mechanism, then clearly there is no hope of the CDP mechanism’s
+parameters beating SDP lower bounds.
+In the negative direction, Mironov, Pandey, Reingold, and Vadhan [MPRV09] (building on
+Green and Tao [GT08], Tao and Ziegler [TZ08], and Reingold, Trevisan, Tulsiani, and Vad-
+han [RTTV08]) show a “dense model theorem” for pairs of random variables with “pseudodensity”
+with each other. [MPRV09] note that (roughly speaking) extending this dense model theorem to
+handle multiple pairs of random variables would prove that any CDP mechanism could be converted
+into an SDP mechanism; such an extension is still open [Vad17, Open Problem 10.8].
+1For the formal definitions of CDP and SDP, we refer the reader to Section 3.
+2We state this verbatim from [Vad17].
+3This remark is also quoted by Groce, Katz, and Yerukhimovich [GKY11].
+1
+
+Groce, Katz, and Yerukhimovich [GKY11] show that CDP mechanisms for certain tasks where
+the output is low-dimensional actually do imply SDP mechanisms. Many natural statistical tasks
+fall into this category, and consequently, such tasks cannot separate CDP from SDP. This result
+was further strengthened by [BCV16]. Furthermore, [GKY11] show that CDP mechanisms con-
+structed in a black-box way from a variety of cryptographic objects, such as one-way functions,
+random oracles, trapdoor permutations, and cryptographic hash functions, cannot separate CDP
+from SDP.
+In summary, there are at least two barriers to separate CDP from SDP:
+1. High-dimensionality: One needs to consider (perhaps non-natural) tasks with high dimen-
+sional outputs;
+2. Exotic cryptography: One needs to use cryptography somewhat specially (perhaps either
+an exotic primitive or in a non-black-box manner).
+In light of these both positive and negative results as well as the lack of a candidate separation, it
+was not even clear what the truth could be: is there any task for which there is a CDP mechanism
+but not an SDP one?
+Our Contributions.
+We show, under plausible cryptographic hypotheses, that there are indeed
+tasks for which there exist CDP mechanisms but no SDP mechanisms. This not only positively
+answers Question 1 but also negatively answers the dense model extension question [Vad17, Open
+Problem 10.8]. We state this result now informally and formalize it later in Section 2. We also delay
+discussing our precise cryptographic assumptions to Section 2.5, where we discuss their plausibility
+in detail.
+Theorem 2. [Informal version of Theorem 5] Under cryptographic assumptions, there exists a task
+for which there is a CDP mechanism but no SDP mechanism.
+Let us take a step back to discuss the implications of Theorem 2. Although (as we will see in
+a moment) our task is specifically constructed for the purpose of separating CDP and SDP, the
+fact that we can separate them at all opens up a possibility that such a separation even holds for
+some “natural” tasks. Indeed, some of the current lower bound techniques for SDP—such as the
+ubiquitous “packing lower bounds”4 (see [HT10])—do not necessarily rule out CDP mechanisms.
+It seems prudent to carefully reexamine the current lower bound techniques to see whether they
+also apply to CDP. The ultimate hope for this program would be to employ CDP to overcome the
+known SDP lower bounds for some more “natural” tasks. (Of course, such tasks would also give a
+more “natural” separation of CDP and SDP.)
+In fact, the technical approach we use in our construction already suggests a general approach
+for constructing non-trivial CDP mechanisms that could apply to more tasks. We discuss this in
+more detail in Section 2, but the idea is as follows. In order to show a task has a CDP mechanism,
+first show there is a mechanism for that task that is “private” against a certain class of decision
+tree adversaries. Then, second, use cryptographic assumptions to “lift” this into privacy against
+computational adversaries.
+4Specifically, when the packing lower bound requires the use of super-polynomially many datasets, the correspond-
+ing adversary does not necessarily run in polynomial time.
+2
+
+Organization.
+The rest of the paper is organized as follows.
+Section 2 provides a high-level
+overview of our techniques as well as a discussion of our cryptographic assumptions and their plau-
+sibility. Section 3 contains the background material and Section 4 formally defines the problems.
+We provide our CDP mechanism in Section 5, and prove lower bounds against SDP mechanisms in
+Section 6. These two components are put together to prove the main result in Section 7. Finally,
+we discuss the open problems and future directions in Section 8.
+2
+Overview of the Results
+We will next discuss the high-level overview of our techniques.
+We will sometimes have to be
+informal here, but all details are formalized later in the paper.
+Let us quickly recall how the
+“task” is defined5.
+Following [GKY11, BCV16], a task is defined by an efficiently computable
+utility function u that takes in an input dataset D and a response y such that u(D, y) = 1 if y is
+considered “useful” for D and u(D, y) = 0 otherwise. A mechanism M is said to be α-useful for u
+iff E[u(D, M(D))] ≥ α for all input datasets D. We remark that many well-studied problems—such
+as linear queries with various error metrics—can be written in this form. We will refer to α as the
+usefulness of the mechanism.
+One of our main conceptual contributions is to define a class of tasks that seems to naturally
+circumvent the two earlier-mentioned barriers—tasks where one needs to output a circuit.
+2.1
+The d-Distance Problem
+Before we detail why tasks that output a circuit might evade the two barriers, let us describe a
+concrete example. We call the following the d-distance problem (where d ∈ N is a parameter):
+◮ Given: dataset D that consists of n bits
+◮ Output: circuit C mapping n bits to 1 bit
+◮ Utility: C is considered useful6 if it outputs
+⊲ 1 on D, and
+⊲ 0 on all points at distance greater than d from D.
+Informally, this problem asks to output a circuit that checks if its input is “close” to D. Looking
+ahead, we will ultimately separate CDP from SDP under cryptographic assumptions by considering
+a version of this problem where we only care about datasets in a cryptographically special set.
+We now revisit the two barriers and discuss how the distance problem might circumvent them.
+1. High-dimensionality: The output of this task is a circuit, which is high-dimensional.
+2. Exotic cryptography: Because the output of the task is a circuit, it lends itself to a powerful
+class of cryptographic objects: circuit obfuscators [BGI+12]. Roughly speaking, circuit obfusca-
+tors take as input a circuit C and output a scrambled, obfuscated circuit C′ that computes the
+same function as C but which, ideally, has the property that “anything you could do with access
+to the circuit C′, you could do with only black-box access to the function the circuit computes.”
+Importantly, obfuscation is not in the list of primitives ruled out by the barrier in [GKY11].
+5Please refer to Section 3 for a more formal definition.
+6One might be concerned about whether this utility function is actually efficiently computable. We will address
+this in Section 2.3 after we describe our final construction.
+3
+
+2.2
+SDP Lower Bound
+Our starting point for separating CDP from SDP is the d-distance problem described above. Indeed,
+we show that there is no SDP mechanism for this problem for most settings of d.
+Lemma 3. If 0 < d ≤ n.99, then there is no (ε, δ)-SDP mechanism for d-distance that is 0.01-useful
+for ε = O(1) and δ negligible in n.
+In fact, this lower bound is straightforward (Lemma 15) from the well-known blatant non-privacy
+notion (see, e.g., [De12]): no DP algorithm can output a dataset that is (with large probability)
+close to the input dataset. Crucially, our lower bounds are non-constructive, and do not yield an
+efficient adversary (which would imply a similar lower bound against CDP mechanisms). Thus, to
+separate CDP from SDP it suffices to come up with a CDP mechanism M for, say, n.99-distance.
+2.3
+A CDP Mechanism
+One of our main ideas to help construct a CDP mechanism M is to use obfuscation. In particular,
+we will consider mechanisms where the returned circuit is obfuscated. Recall that in order to prove
+a mechanism M that outputs a circuit C is CDP, one needs to argue that no efficient adversary
+that gets C as input can break the privacy guarantee. By considering mechanisms that return
+obfuscated circuits, we can drastically simplify the type of adversaries we need to prove privacy
+against.
+Instead of proving privacy against adversaries that see the circuit C (i.e., white-box
+setting), sufficiently strong obfuscation means we only need to prove privacy against decision tree
+adversaries that can query the function computed by the circuit (i.e., black-box setting). In other
+words, if we have a mechanism that satisfies DP against black-box adversaries (decision trees) with
+a polynomial number of queries, we can then hope to use sufficiently strong obfuscation to “lift”
+this into a mechanism that is secure against (white-box) computational adversaries with polynomial
+running time.
+Of course, one needs to be careful about whether such “sufficiently strong obfuscation” is even
+possible, but, putting that aside for the moment, the question of whether there is a CDP mechanism
+for n.99-distance (Question 4 below) appears to reduce to whether there is a mechanism for n.99-
+distance that is DP against query (a.k.a. decision-tree) adversaries.
+Question 4. Let ε = O(1) and 0 ≤ d ≤ n.99. Does there exist an ε-CDP mechanism for d-distance
+with constant usefulness?
+While we do not resolve Question 4, we (roughly speaking) show that there is a mechanism that
+is DP against non-adaptive decision tree adversaries, whose queries are fixed a priori. It turns out
+a relatively simple mechanism based on randomized response [War65] works for these less powerful
+adversaries.
+From Non-Adaptive Lower Bound to Computational Lower Bound.
+This switch from
+the usual adaptive query adversaries to non-adaptive query adversaries comes at a price however. It
+is not clear how to use obfuscation to lift a mechanism that is private against non-adaptive queries
+into one that is private against computational adversaries. Indeed, a polynomial-time algorithm
+with even black-box access to a function seems to be an inherently adaptive adversary!
+Surprisingly, we manage to get around this by using another cryptographic object introduced
+by Bitansky, Kalai, and Paneth [BKP18]: collision-resistant keyless hash functions. Informally
+4
+
+speaking, a hash function being collision-resistant and keyless means that “any efficient adversary
+can only generate a number of hash collisions that is at most polynomially larger than the advice
+the adversary gets.”
+We then modify the d-distance problem to only consider datasets that hash to, say, the all
+zeroes string. Formally, zero hash d-distance is the following problem. Let R ⊆ {0, 1}n be the set
+of strings that hash to the all zeroes string.
+◮ Given: dataset D that consists of n bits
+◮ Output: circuit C mapping n bits to 1 bit
+◮ Utility: C is considered useful if D /∈ R or both of the following hold:
+⊲ it outputs 1 on D
+⊲ it outputs 0 on all points in R at distance greater than d from D
+In other words, the utility function now completely ignores all points outside of R.
+The high-level intuition behind this change is the following:
+1. Our CDP mechanism can output a circuit C such that the only inputs where C(x) reveals
+information are those x in the set R (i.e., that hash to zero).
+2. Any polynomial-time adversary A can only generate fixed polynomial number of elements of
+R by the collision-resistance property of the hash function.
+3. Combining the above effectively makes the inputs A can query C on “non-adaptive”.
+Finally, in order to “lift” the query separation into the computational realm we use another cryp-
+tographic tool: differing-inputs obfuscation (diO) [BGI+01, BGI+12, ABG+13]. Roughly speaking,
+diO is an obfuscator with the following guarantee: if any efficient adversary can distinguish the
+obfuscation of two circuits C1 and C2, then an efficient adversary can find an input x on which
+C1(x) ̸= C2(x). In particular, the assumption we use is even weaker than public-coin diO [IPS15],
+which is already considered to more plausible than general diO.7
+In summary, diO allows us to reduce computational adversaries to adaptive query adversaries
+and collision-resistant keyless hash functions allows us to reduce adaptive query adversaries to
+non-adaptive query adversaries. Interestingly, to the best of our knowledge, this is the first time
+collision-resistant keyless hash functions are being used together with any obfuscation assumption.
+Making the Utility Function Efficiently Computable.
+Observant readers may have already
+noticed an issue: utility functions that we have considered so far are not necessarily efficiently
+computable. Specifically, a trivial way to implement the utility function would be to enumerate all
+points at distance at least d, feed it into the circuit, and check that the output is as expected; this
+would take 2nΩ(1) time.
+To overcome the above problem, we restrict circuits to only those that are relatively simple, so
+that there is a small “witness” w that certifies that the circuit outputs zero at all points that are
+d-far from D. A naive idea is then to let the CDP mechanism output the circuit C together with
+such a witness w. The utility function can then just efficiently check that w is a valid witness (and
+that C(D) = 0 or x ∈ R). This makes the utility function efficient but unfortunately compromises
+privacy because the witness w itself can leak additional information. To avoid this, we instead use
+non-interactive witness indistinguishable (NIWI) proofs (e.g., [BOV07]). Roughly speaking, this
+allows us to produce a proof π from w (and C and diO), which does not leak any information about
+w (against computationally bounded adversaries), but at the same time still allows us to verify
+7See Assumption 22 for formal statement of the assumption and Appendix A for comparison with other diO
+assumptions in literature.
+5
+
+that the underlying witness w is valid. The former is sufficient for CDP, while the latter ensures
+that the utility function can be computed efficiently.
+This completes the high-level overview of the constructed task and our CDP mechanism. The
+cryptographic primitives needed for our mechanism are formalized in Assumptions 18, 22 and 26.
+2.4
+Final Steps
+Finally, we remark that since our problem is now not exactly the original d-distance problem
+anymore, as the utility guarantees are only now meaningful for datasets in R. This means that
+we cannot use the lower bound in Lemma 3 for the d-distance problem directly. Fortunately, we
+can still adapt its proof—a “packing-style” lower bound on each coordinate—to one which applies
+a packing-style argument on each block of coordinates instead. With this, we can prove the lower
+bound for zero hash d-distance as long as the set R has sufficiently large density (≈ 1/n−o(log n)).
+Putting all the ingredients together, we arrive at the following8:
+Theorem 5 (Main Result). Under Assumptions 18, 22 and 26, for any constant εCDP > 0, there
+exists an ensemble u = {un}n∈N of polynomial time computable utility functions such that
+◮ There is an εCDP-CDP mechanism that is (1 − on(1))-useful for u.
+◮ For any constant εSDP > 0, there is no εSDP-SDP mechanism that is 0.01-useful for u.
+2.5
+On the Plausiblility of the Cryptographic Assumptions
+We now discuss the plausiblility of the three cryptographic assumptions we use for our result:
+(i) NIWI: Non-interactive Witness Indistinguishable Proofs (formally, Assumption 26)
+(ii) CRKHF: Collision-Resistant Keyless Hash Functions (formally, Assumption 18)
+(iii) diO-for-pcS: Differing-Inputs Obfuscation for Public-coin Samplers (formally, Assumption 22)
+Regarding (i), NIWI.
+Bitansky and Paneth [BP15a] show that NIWIs exist assuming one-
+way permutations exist and indistinguishability obfuscation (iO) exists. Recently, Jain, Lin, and
+Sahai [JLS21] show that the existence of iO follows from well-founded assumptions; consequently,
+NIWIs exist based on widely-believed assumptions. (We note that other previous works have also
+constructed NIWIs based on other more specific assumptions [BOV07, GOS12].)
+Regarding (ii), CRKHF.
+Bitansky, Kalai, and Paneth [BKP18] defined CRKHFs to model the
+properties of existing hash functions like SHA-2 used in practice. They suggest several candidates
+for CRKHFs, such as hash functions based on AES and Goldreich’s one-way functions. They also
+note that CRKHFs exist in the Random Oracle model, as a random function is a CRKHF. Still,
+it is an open question to base the security of a CRKHF on a standard cryptographic assumption.
+Part of the difficulty of doing this, as [BKP18] describe, is that most cryptographic assumptions
+involve some sort of structure that is useful for constructing cryptographic objects. In contrast,
+the goal of a CRKHF is to have no structure at all. In summary, given the various CRKHF candi-
+dates, the existence in the Random Oracle model, and the fact that CRKHFs exist “in practice,”
+this assumption is quite plausible. For our specific construction, we need a different hash length
+8We remark that εSDP-SDP mechanism here refers to an ensemble of mechanisms {Mn} which are (εSDP, negl)-SDP.
+(See Definition 7.)
+6
+
+(equivalently, different compression rate) than that used in [BKP18]; please refer to the discussion
+preceding Assumption 18 for the parameters and justification.
+Finally, we remark that, even though the existence of CRKHFs is not known to reduce to any
+“well-founded” assumption, even refuting their existence would answer a longstanding question in
+cryptography: giving non-contrived separations between the Random Oracle model [BR93] and the
+standard model. In the words of Bitansky, Kalai, and Paneth [BKP18]
+“Any attack on the multi-collision resistance of a [keyless] cryptographic hash function
+would constitute a strong and natural separation between the hash and random oracles.
+For several cryptographic hash functions used in practice, the only known separations
+from random oracles are highly contrived [CGH04].”
+Regarding (iii), diO-for-pcS.
+One can think of diO [BGI+01, BGI+12] as an “extractable”
+strengthening of iO. While iO has now become a widely-believed assumption [JLS21], the exis-
+tence of diO is controversial. Several papers (e.g., [BP15b, GGHW17, BSW16]) cast doubt on the
+existence of diO, especially in the case where an arbitrary auxillary input is allowed; we stress that
+all the negative results for diO hold for contrived auxillary inputs and/or distributions. On the
+positive side, [BCP14] show that diO reduces to iO in special cases, such as when the number of
+differing-inputs is bounded by a polynomial. More related to our result, [IPS15] gives a definition
+of public-coin diO that avoids the difficulties presented by earlier negative results regarding auxil-
+iary inputs, although [BP15b] presented some evidence against this definition in special cases. Our
+specific assumption of diO-for-pcS is in fact weaker than the assumption of public-coin diO. In the
+definition of public-coin diO, as in [IPS15], we start with any public-coin sampler (pcS), for which it
+is hard to find an input on which two circuits differ, even given the knowledge of all the randomness
+that underlies the circuits. The security of the obfuscation is required to hold even against adver-
+saries that know all the randomness that underlies the generation of the two circuits. However,
+in our definition, the security of the obfuscation is required to hold only against adversaries that
+observes a single obfuscated circuit, which makes the assumption weaker. See Appendix A for a
+more detailed discussion on comparison of this assumption with other diO assumptions in literature.
+Finally, we only use the existence of diO-for-pcS for a simple circuit family for our result, so even if
+general purpose diO-for-pcS does not exist, we think it is plausible that diO-for-pcS exists for the
+specific family of circuits we need for our result. (See Assumption 22 for the exact pcS family for
+which we require a diO.)
+Final thoughts on our assumptions.
+In conclusion, we view each of our three assumptions
+as plausible. Moreover, each of assumptions has at least some evidence that is hard to refute:
+NIWIs exist based on a widely-believed assumption, refuting CRKHFs would require giving the
+first non-contrived separation between the standard and the Random Oracle model, and despite
+many attempts (e.g., [BP15b, GGHW17, BSW16]) to refute diO, the question is still open, especially
+for the particular diO-for-pcS version.
+3
+Preliminaries
+A function g : N → R≥0 is said to be negligible if g(n) = n−ω(1). Let PPT be an abbreviation for
+probabilistic polynomial-time Turing machine.
+7
+
+For x ∈ {0, 1}n and r ∈ N, we use Br(x) to denote the (Hamming) ball of radius r around x,
+i.e., {z ∈ {0, 1}n | ∥x − z∥1 ≤ r}. Furthermore, we use diam(S) for a set S ⊆ {0, 1}n to denote the
+(Hamming) diameter of S, i.e., maxx,x′ ∈S ∥x − x′∥1.
+3.1
+Dataset and Adjacency
+For a domain X, we view a dataset D as a histogram over the domain X, i.e., D ∈ ZX
+≥0 where Dx
+denotes the number of times x ∈ X appears in the dataset. The size of the dataset is defined as
+∥D∥1 := �
+x∈X Dx. We write X m as a shorthand for the set of all datasets of size m, and X ∗ for
+the set of all datasets over domain X. Two datasets are adjacent iff ∥D − D′∥1 = 1, i.e., one of the
+datasets is a result of adding or removing a single row from the other dataset.
+3.2
+Mechanism, Utility Function, and Usefulness
+A mechanism M is a randomized algorithm that takes in a dataset D ∈ X ∗ and outputs an element
+from a set Y. The utility of a mechanism is measured by a utility function u, which is a polynomial-
+time deterministic algorithm that takes in a dataset D ∈ X ∗ together with a response y ∈ Y and
+outputs 0 or 1 (whether the response is good for the dataset). We say that the mechanism M is
+α-useful for utility u iff Pr[u(D, M(D)) = 1] ≥ α.
+Below, we will often discuss an ensemble M = {Mn}n∈N of mechanisms where9 Mn : X ∗
+n → Yn.
+We say that an ensemble of mechanisms is efficient if Mn on input D ∈ X m
+n runs in time poly(n, m).
+For an ensemble u = {un}n∈N of utility functions and α = {αn ∈ [0, 1]}n∈N, we say that M is α-
+useful with respect to u iff Mn is αn-useful with respect to un for all n ∈ N.
+For brevity, we will sometimes refer to “ensemble of mechanisms” simply as “mechanism” and
+“ensemble of utility functions” simply as “utility function” when there is no ambiguity.
+3.3
+Notions of Differential Privacy
+We now define the notions of DP that will be used throughout the paper.
+(Statistical) Differential Privacy.
+The standard (statistical) notion of DP can be defined in
+terms of the following notion of indistinguishability.
+Definition 6 (Statistical Indistinguishability). Distributions P, Q are said to be (ε, δ)-indistinguishable,
+denoted P ≈ε,δ Q, if for all events (measurable sets) E, it holds that
+Pr
+X∼P[X ∈ E] ≤ eε · Pr
+X∼Q[X ∈ E] + δ,
+and
+Pr
+X∼Q[X ∈ E] ≤ eε · Pr
+X∼P[X ∈ E] + δ.
+For simplicity, we use ≈ε to denote ≈ε,0.
+Definition 7 (Statistical Differential Privacy (SDP) [DMNS06, DKM+06]). For ε, δ > 0, a mecha-
+nism M is said to be (ε, δ)-SDP if and only if for every pair D, D′ of adjacent datasets, we have that
+M(D) ≈ε,δ M(D′). We say that an ensemble M = {Mn}n∈N is ε-SDP for a sequence ε = {εn}n∈N
+if there exists a negligible sequence {δn}n∈N such that Mn is (εn, δn)-SDP for all n ∈ N.
+We note that the above notation, which omits explicit δ for an ensemble of mechanisms, was also
+used by [BCV16].
+9It is always implicitly assumed that Xn, Yn are of size poly(n).
+8
+
+Computational Differential Privacy.
+The notion of computational DP relaxes the notion
+of indistinguishability to a computational version, where the privacy holds only with respect to
+computationally bounded adversaries.
+Definition 8 (Computational Indistinguishability). Two ensembles of distributions P = {Pn}n∈N
+and Q = {Qn}n∈N, where Pn and Qn are supported over {0, 1}p(n) for some polynomial p(·), are
+said to be ε-computationally-indistinguishable for a sequence ε = {εn}n∈N, denoted P ≈c
+ε Q, if there
+exists a negligible function negl(·) such that for any PPT adversary A, it holds that
+Pr
+X∼Pn[A(X) = 1] ≤ eεn ·
+Pr
+X∼Qn[A(X) = 1] + negl(n), and
+Pr
+X∼Qn[A(X) = 1] ≤ eεn ·
+Pr
+X∼Pn[A(X) = 1] + negl(n).
+In the special case of ε = 0, we suppress the subscript and simply write P ≈c Q.
+Throughout, when we refer to a sequence {(Dn, D′
+n)}n∈N of adjacent datasets, it is always assumed
+that Dn ∈ X mn
+n
+, D′
+n ∈ X m′
+n
+n
+are of sizes mn, m′
+n = poly(n).
+Definition 9 (Computational Differential Privacy (CDP) [MPRV09]). An ensemble M = {Mn}n∈N
+of mechanisms is said to be ε-CDP for a sequence ε = {εn}n∈N, if for any sequence {(Dn, D′
+n)}n∈N
+of adjacent datasets, it holds that {Mn(Dn)}n∈N ≈c
+εn {Mn(D′
+n)}n∈N.
+This definition is often referred to as indistinguishability-based CDP (IND-CDP) in previous
+works [MPRV09, GKY11, BCV16]. Since we only use this notion for our main result, we refer to
+it simply as CDP. The other definition of CDP used in previous works is simulation-based:
+Definition 10 (SIM-CDP [MPRV09]). An ensemble M = (Mn)n∈N of mechanisms is said to be ε-
+SIM-CDP if there exists an (εn, 0)-SDP ensemble {M′
+n}n∈N of mechanisms such that for any sequence
+{Dn ∈ X ∗
+n}n∈N of datasets, with size of Dn being at most poly(n), it holds that Mn(Dn) ≈c M′
+n(Dn).
+It should be noted that SIM-CDP cannot be used for the separation we are looking for. Specif-
+ically, if {Mn}n∈N is ε-SIM-CDP, we may use {M′
+n}n∈N as our ε-SDP mechanism. Since the utility
+function runs in polynomial time, it follows immediately that, if {Mn}n∈N is α-useful, then {M′
+n}n∈N
+is also (α − o(1))-useful. Due to this, we will not consider SIM-CDP again in this paper.
+Calculus of ≈ and ≈c.
+The following properties are well-known.
+Fact 11. The notions of (ε, δ)-indistinguishability and ε-computational-indistinguishability satisfy:
+◮ Basic Composition: If P0 ≈ε,δ P1 and P1 ≈ε′,δ′ P2, then P0 ≈ε+ε′,δ+δ′ P2.
+Similarly, if
+P0 ≈c
+ε P1 and P1 ≈c
+ε′ P2, then P0 ≈c
+ε+ε′ P2.
+◮ Post-processing: If P ≈ε,δ Q, then for all (randomized) functions f, it holds that f(P) ≈ε,δ
+f(Q). Similarly, if P ≈c
+ε Q, then for all PPT algorithms A, it holds that A(P) ≈c
+ε A(Q).
+4
+Low Diameter Set Problem and Nearby Point Problem
+In this section, we introduce the problems that we will use in our separation. Before that, we will
+describe a simplifying assumption that we can make about the inputs.
+9
+
+4.1
+Simplification of Input Representation
+Recall that so far a dataset may contain multiple copies of an element. Below, however, it will be
+more convenient to only discuss the case where each element appears only once, i.e., D ∈ {0, 1}X .
+This is sufficient since if we have a utility function u : {0, 1}X × Y → {0, 1} defined only on
+D ∈ {0, 1}X , we can easily define the utility function u : NX × Y → {0, 1} by
+u(D, r) =
+�
+u(D, r)
+if D ∈ {0, 1}X ,
+1
+otherwise.
+In other words, the utility function considers any response good for datasets with repetition. Clearly,
+if u is efficiently computable, then so is u. Furthermore, suppose that we have an ε-CDP mechanism
+M = {Mn}n∈N for u = {un}n∈N. For every dataset D, let D be defined by Di = min
+�
+Di, 1
+�
+.
+Then, we may define M =
+�
+Mn
+�
+n∈N by M(D) = M(D). It is simple to check that M remains
+ε-CDP. Furthermore, if M is α-useful for u, then M remains α-useful for u.
+Finally, note that a lower bound for DP algorithms restricted to non-repeated datasets trivially
+implies a lower bound against all datasets.
+Due to this, we will henceforth focus our attention only on the datasets D ∈ {0, 1}X . Further-
+more, throughout the remainder of this paper, we will always pick Xn = [n]. This further simplifies
+the input representation to be just a bit vector x ∈ {0, 1}n. We will define an input of our problem
+in this way. Furthermore, we will henceforth use x instead of D to denote the input dataset.
+4.2
+Nearby Point Problem
+We will start by defining our first problem, which asks to output a point that is close to the input
+point if the latter belongs to some set R. As we noted in the introduction, when R is the set
+of all points (i.e., Rn = {0, 1}n), this is exactly the same as the problem considered in blatant
+non-privacy [DN03, DMT07]. As we will see later, the presence of the set R is due to our use of
+hashing, which is required in our proof for the CDP mechanism.
+Definition 12 (τ-Nearby R-Point Problem). The nearby point problem parameterized by sequences
+{τn ∈ N}n∈N and {Rn ⊆ {0, 1}n}n∈N is denoted by NBPτ,R.
+For input x ∈ {0, 1}n and output
+y ∈ Yn = {0, 1}n, the utility is defined as:
+uNBP
+τn,Rn(x, y) := 1 {∥x − y∥1 ≤ τn or x /∈ Rn}
+For brevity, we will assume throughout that Rn is efficiently recognizable and henceforth we do
+not state this explicitly. Note that this assumption implies that the utility function defined above
+is efficiently computable. The nearby point problem will be primarily used for proving the lower
+bounds against SDP.
+4.3
+Verifiable Low Diameter Set Problem
+Next, we define circuit-based tasks for which we will give CDP mechanisms. To do so, we need to
+first define a “τ-diameter verifier”.
+Definition 13 (τ-Diameter Verifier). For a sequence τ = {τn}n∈N of integers, we say that an
+efficiently computable (deterministic) verifier V = {Vn}n∈N is a τ-diameter verifier for circuits of
+size s(n) if it takes as input a circuit C : {0, 1}n → {0, 1} of (polynomial) size s(n) and a proof π
+of size poly(n), and outputs Vn(C, π) = 1 only if diam(C−1(1)) ≤ τn.
+10
+
+We can now define the (verifiable) low diameter set problem as follows:
+Definition 14 (Verifiable τ-Diameter R-Set Problem). The verifiable low diameter set problem
+parameterized by sequences τ = {τn}n∈N, R = {Rn ⊆ {0, 1}n}n∈N, and τ-diameter verifier V =
+{Vn}n∈N is denoted by VLDSτ,R,V . The input, output, and utility are defined as follows:
+◮ Input: x ∈ {0, 1}n.
+◮ Output: circuit C and a proof π, both of size poly(n).
+◮ Utility: uVLDS
+τn,Rn,Vn(x, (C, π)) := 1 {C(x) = 1 or x /∈ Rn} and 1 {Vn(C, π) = 1}.
+For convenience, we also define the following utility function
+ueval
+R (x, C) := 1 {C(x) = 1 or x /∈ R} .
+Note that this does not correspond to a hard task, because a circuit that always outputs one is
+1-useful. Nonetheless, it will be convenient to state usefulness of some intermediate algorithms via
+this utility function.
+4.4
+From Low Diameter Set Problem to Nearby Point Problem
+Below we provide a simple observation that reduces the task of proving an SDP lower bound for
+the verifiable low diameter set problem to that of the nearby point problem. (Note here that the
+SDP mechanisms considered below can be computationally inefficient.)
+Lemma 15. If there is an (ε, δ)-SDP α-useful mechanism for the VLDSτ,R,V problem, then there
+is an (ε, δ)-SDP α-useful mechanism for the NBPτ,R problem.
+Proof. Let M be an (ε, δ)-SDP α-useful mechansim for the VLDSτ,R,V problem. We will construct
+an (ε, δ)-SDP α-useful mechanism M′ for the NBPτ,R problem.
+The mechanism M′
+n on input dataset x ∈ {0, 1}n works as follows. First, let (C, π) ← Mn(x).
+If Vn(C, π) = 1, then output the lexicographically first element of C−1(1) (else, output 0n). This
+completes our description of M′.
+Since M is (ε, δ)-SDP, we have that M′ is also (ε, δ)-SDP by post-processing. It remains to
+show that M′ is α-useful. Fix some input x ∈ {0, 1}n. If x /∈ Rn, then any output satisfies utility.
+Thus, it suffices to consider the case where x ∈ Rn. With probability α, we have that Vn(C, π) = 1
+(which implies that C−1(1) has diameter at most τn), and x ∈ C−1(1). Consequently, the distance
+between x and the lexicographically first element of C−1(1) is at most τn. So with probability at
+least α, the output of M′ is useful for x, as desired.
+5
+CDP Mechanism for Verifiable Low Diameter Set Problem
+In this section we build a CDP mechanism for the verifiable low diameter set problem. We establish
+the following result:
+Theorem 16. Suppose that Assumptions 18, 22 and 26 hold. Then, for all constant εCDP > 0 and
+τ =
+�
+τn = n0.9�
+n∈N, there exists a τ-diameter verifier V and a sequence R = {Rn}n∈N of sets of
+sizes |Rn| ≥ 2n/no(log n), such that there exists an εCDP-CDP mechanism that is (1 − on(1))-useful
+for uVLDS
+τ,R,V .
+As discussed in the overview, we first build a mechanism that is CDP but without verifiability
+using collision-resistant keyless hash functions and differing-inputs obfuscators (Section 5.1). We
+then turn it into a verifiable one using non-interactive witness indistinguishable proofs (Section 5.2).
+11
+
+5.1
+CDP Mechanism without Verifiability
+In this section, we construct our first CDP mechanism (Algorithm 3). We depart from the overview
+in Section 2 slightly and do not prove a non-adaptive query lower bound explicitly. Instead, we
+directly show in Section 5.1.2 how to sample the appropriate differing-inputs circuit family. This
+can be then easily turned into our CDP mechanism via diO in Section 5.1.3.
+5.1.1
+Additional Preliminaries: Cryptographic Primitives
+Throughout this section, we will repeatedly use the so-called randomized response (RR) mecha-
+nism [War65]. Specifically, RRε is an algorithm that takes in x ∈ {0, 1}n and outputs ˜x ∈ {0, 1}n,
+where ˜xi = xi with probability
+eε
+1+eε independently for each i ∈ [n]. It is well-known (and very
+simple to verify) that RRε is ε-SDP.
+Collision-Resistant Keyless Hash Functions.
+In our construction, we will use the Collision-
+Resistant Keyless Hash Functions (CRKHFs) [BKP18]. The formal definition is as given below.
+Definition 17 (Collision-Resistant Keyless Hash Functions [BKP18]). A sequence of hash func-
+tions
+�
+Hn : {0, 1}n → {0, 1}γ(n)�
+n∈N is K-collision resistant for advice length ζ for sequences K =
+{Kn}n∈N, ζ = {ζn}n∈N if, for any PPT A and a sequence {zn}n∈N of advices where |zn| = ζn, we
+must have
+Pr
+(Y1,...,YKn)←A(1n;zn) [Y1, . . . , YKn are distinct and Hn(Y1) = · · · = Hn(YKn)] ≤ negl(n).
+We will skip the subscript n whenever it is clear from context.
+In [BKP18], the hash value length γ(n) is assumed to be either linear, i.e., γ(n) = Ω(n), or
+polynomial, i.e., γ(n) = nΘ(1). However, we need a collision-resistant hash function with a much
+smaller γ(n), namely O(log2 n).
+We remark that this is still very much plausible: as long as
+γ(n) is ω(log n), the “guess-and-check” algorithm will only produce a collision with only negligible
+probability. A more precise statement of our assumption is stated below.
+Assumption 18. There is an efficiently computable sequence H = {Hn}n∈N of hash functions with
+hash value length γ(n) = o(log2 n) such that, for any constant c1 > 0, there exists a constant c2 > 0
+such that the hash function sequence is K-collision resistant for advice length ζ where K(n) = nc2
+and ζ(n) = nc1.
+We remark that, for the existence of CDP mechanism (shown in this section), we will only use
+the multi-collision-resistance without relying on the assumption on the value of γ. The latter is
+only used to show that no SDP mechanism exists for the problem (Section 7).
+Differing-Inputs Obfuscators for Public-Coin Samplers.
+For any two circuits C0 and C1,
+a differing-inputs obfuscator diO [BGI+12] guarantees that the non-existence of an efficient ad-
+versary that can find an input on which C0 and C1 differ implies that diO(C0) and diO(C1) are
+computationally indistinguishable. For our application, it even suffices to assume a weaker notion,
+namely that of differing-inputs obfuscator for public-coin samplers, as defined below.
+12
+
+Definition 19 (Public-Coin Differing-Inputs Circuit Sampler). An efficient non-uniform sampling
+algorithm Sampler = {Samplern} is a public-coin differing-inputs sampler for the parameterized
+collection C = {Cn} of circuits if the output of Samplern is distributed over Cn × Cn and for every
+efficient non-uniform algorithm A = {An}, there exists a negligible function negl(·) such that for
+all n ∈ N:
+Pr
+θ [C0(y) ̸= C1(y) : (C0, C1) ← Samplern(θ), y ← An(θ)] ≤ negl(n).
+Here, Samplern is a deterministic algorithm and the only source of randomness is the seed θ.
+Definition 20 (Differing-Inputs Obfuscator for Public-Coin Samplers (cf. [IPS15])). A uniform
+PPT diO is a differing-inputs obfuscator for public-coin samplers for the parameterized circuit
+family C = {Cn} if the following conditions are satisfied:
+◮ Correctness: For all n ∈ N, for all C ∈ Cn, for all inputs y, we have that
+Pr[C′(y) = C(y) : C′ ← diO(1n, C)] = 1.
+◮ Polynomial slowdown: There exists a universal polynomial p(·) such that for all C ∈ Cn, it
+holds that
+Pr[|C′| ≤ p(|C|) : C′ ← diO(1n, C)] = 1.
+◮ Differing-inputs: For every public-coin differing inputs sampler Sampler = {Samplern} for
+C, and every (not necessarily uniform) PPT distinguisher D = {Dn}, there exists a negligible
+function negl such that the following holds for all n ∈ N: For (C0, C1) ← Samplern(θ)
+| Pr
+θ [Dn(diO(1n, C0)) = 1] − Pr
+θ [Dn(diO(1n, C1)) = 1]| ≤ negl(n).
+We note that the notion of diO-for-pcS is in fact weaker than the notion of general public-coin diO
+as given by [IPS15]. We elaborate on this comparison in Appendix A. Whenever n is clear from
+context, we use diO(C) to denote diO(1n, C) for simplicity. When we want to be explicit about the
+randomness ρ (of poly(n) bit length) used by diO we will denote it as diOρ(C).
+We only need the existence of a differing-inputs obfuscator for a specific family of circuits. This cir-
+cuit family will be defined later and therefore we defer formalizing our assumption to Section 5.1.3.
+5.1.2
+Public-Coin Differing-Inputs Circuits from CRKHFs
+The first step of our proof is to construct a differing-inputs circuit family based on CRKHFs. Our
+sampler is described in Algorithm 1.
+We next prove that the above sampler is a public-coin differing-inputs sampler, which means
+that any efficient adversary, even with the knowledge of ˜x (which is the only source of randomness),
+cannot find an input on which C0 and C1 differ. The proof starts by noticing that any input that
+differentiates C0, C1 must, by definition of the circuits, have hash value υn. Therefore, if there were
+an adversary that can find a differing input, then we could run it multiple times to get Y1, . . . , YK
+that have the same hash value. (See Algorithm 2 below.) However, our proof is not finished yet,
+since it is possible that Y1, . . . , YK are not distinct. Indeed, the crux of the construction is that,
+due to how we select ˜x and define the circuits, a fixed Y will be a differing input with negligible
+probability10. It follows that Y1, . . . , YK must be distinct w.h.p. This is formalized below.
+10It is also simple to see that this property suffices to prove a non-adaptive query lower bound as discussed in
+Section 2.
+13
+
+Algorithm 1 Differing-Inputs Circuit Family Sampler LDS-Samplern.
+Parameters:
+Adjacent datasets x, x′ ∈ {0, 1}n, hash value υn ∈ {0, 1}γ(n), privacy parameter
+ε > 0, radius r, ˜r > 0.
+Randomness: θ ∼ RRε(0n).
+Output: Circuits C0, C1.
+˜x ← x ⊕ θ (bit-wise XOR; equivalent to RRε(x))
+C0 ← circuit that takes in z and computes 1
+�
+z ∈ Br(x) ∩ B˜r(˜x) ∩ H−1
+n (υn)
+�
+C1 ← circuit that takes in z and computes 1
+�
+z ∈ Br(x′) ∩ B˜r(˜x) ∩ H−1
+n (υn)
+�
+return (C0, C1)
+Lemma 21. Let H be as in Assumption 18. For any constant ε > 0, choosing r = 0.5n0.9 and
+˜r =
+1
+1+eε n + n0.6 makes LDS-Samplern (Algorithm 1) a public-coin differing-inputs sampler.
+Proof. Suppose for the sake of contradiction that for some adjacent x, x′ ∈ {0, 1}n, there exists a
+PPT ADI such that
+Pr
+θ [C0(y) ̸= C1(y) : (C0, C1) ← LDS-Samplern(θ), y ← ADI
+n (θ)] ≥ n−c,
+(1)
+for some constant c > 0. Furthermore, let c1 be such that the total size of the descriptions of
+ADI
+n , LDS-Samplern is at most nc1. Finally, let c2 > 0 be as in Assumption 18 and K = nc2.
+Algorithm 2 Collision-Resistant Hash Function Adversary ACRH
+n
+.
+Parameter:
+The target number of collisions K ∈ N, constant c > 0.
+Advice:
+Descriptions of ADI
+n , LDS-Samplern.
+Output: Y1, . . . , YK ∈ {0, 1}n or ⊥.
+i ← 0
+for j = 1, . . . , K · nc+1 do
+θj ← RRε(0n)
+(Cj
+0, Cj
+1) ← LDS-Samplern(θj)
+yj ← ADI
+n (θj)
+if Cj
+0(yj) ̸= Cj
+1(yj) then
+i ← i + 1
+Yi ← yj
+if i ≥ K then
+break
+if i < K then
+return ⊥
+else
+return Y1, . . . , YK
+Consider the adversary ACRH
+n
+for collision-resistant hash function described in Algorithm 2.
+First, note that by (1) and a standard concentration inequality, the probability that ACRH
+n
+outputs
+⊥ is on(1). Furthermore, notice that C0, C1 can differ on y only if Hn(y) = υn, meaning that
+Hn(Yi) = υn always. Therefore, it suffices for us to show that the probability that Y1, . . . , YK are
+14
+
+distinct is 1 − on(1). By a union bound, we have that ACRH
+n
+violates the collision-resistance of H
+as desired.
+Thus, we are only left to show that Y1, . . . , YK are not distinct with probability o(1). To see
+that this is the case, notice that
+Pr[Y1, . . . , YK are not distinct] ≤
+�
+1≤i1<i2≤K
+Pr[Yi1 = Yi2].
+(2)
+Let us now bound Pr[Yi1 = Yi2] for a fixed pair i1 < i2. Suppose that we fix a value of Yi1 and
+suppose that Yi1 is assigned at step j1 ∈ [1, . . . , K ·nc+1]. Conditioned on these, notice further that
+Pr[Yi2 = Yi1] ≤ Pr[∃j > j1, yj = Yi1]
+≤ Pr[∃j > j1, Cj
+0(Yi1) ̸= Cj
+1(Yi1)]
+≤
+�
+j>j1
+Pr[Cj
+0(Yi1) ̸= Cj
+1(Yi1)].
+(3)
+Now, let us bound the RHS probability for a fixed j > j1. To see this, first observe that Yi1 must
+belong to the symmetric difference Br(x)△Br(x′); otherwise, we must have Cj1
+0 (Yi1) = Cj1
+1 (Yi1), a
+contradiction to our definition of Yi1.
+Now, let ˜xj denote the ˜x selected by LDS-Sampler when constructing Cj
+0, Cj
+1. We have
+Pr[Cj
+0(Yi1) ̸= Cj
+1(Yi1)] ≤ Pr[Yi1 ∈ B˜r(˜xj)].
+(4)
+Let d := ∥Yi1 − x∥1 and ˜d := ∥Yi1 − ˜xj∥1. Since Yi1 ∈ Br(x)△Br(x′), it holds that d ∈ {r, r + 1}.
+Thus, ˜d is distributed as Bin(d,
+eε
+1+eε ) + Bin(n − d,
+1
+1+eε ). We have E˜xj∼RRε(x) ˜d =
+1
+1+eε n + eε−1
+eε+1d.
+By Bernstein’s inequality,
+Pr[ ˜d ≤ ˜r] ≤ exp
+�
+−
+t2
+eε
+(1+eε)2 n + 2
+3t
+�
+≤ exp(−Ω(n0.8)),
+where t = E˜xj∼RRε(x) ˜d − ˜r ≥ eε−1
+eε+1(0.5n0.9 − 1) − n0.6. Plugging into (4), we have
+Pr[Cj
+0(Yi1) ̸= Cj
+1(Yi1)] ≤ exp(−Ω(n0.8)).
+(5)
+Combing (2), (3), (5), we have
+Pr[Y1, . . . , YK are not distinct] ≤ K3nc+1 · exp(−Ω(n0.8)) ≤ exp(−Ω(n0.8)),
+where the last inequality follows from K = nO(1).
+5.1.3
+From Differing-Inputs Circuits to CDP
+We will next construct CDP mechanism from the previously constructed differing-inputs circuit
+family. First, let us state the assumption we need here:
+Assumption 22. For H as in Assumption 18, any constant ε > 0 and r = 0.5n0.9, ˜r =
+1
+1+eε n+n0.6,
+there exists a differing-inputs obfuscator diO for the sampler LDS-Sampler.
+15
+
+Algorithm 3 CDP mechanism MdiO.
+Parameter:
+Differing-inputs obfuscator diO, hash function H, parameters ε, r, ˜r (as in
+Assumption 22), and a hash value υn ∈ {0, 1}γ(n).
+Input: Dataset x ∈ {0, 1}n.
+Output: Circuit : {0, 1}n → {0, 1}.
+˜x ← RRε(x).
+C ← circuit that takes in z and compute 1
+�
+z ∈ Br(x) ∩ B˜r(˜x) ∩ H−1
+n (υn)
+�
+�C ← diOρ(C) for randomness ρ
+return
+�C
+Distribution H0:
+˜x ← RRε(x)
+C(z) := 1
+�
+z ∈ Br(x) ∩ B˜r(˜x) ∩ H−1
+n (υn)
+�
+return diOρ(C)
+Distribution H1:
+˜x ← RRε(x)
+C(z) := 1
+�
+z ∈ Br(x′) ∩ B˜r(˜x) ∩ H−1
+n (υn)
+�
+return diOρ(C)
+Distribution H2:
+˜x ← RRε(x′)
+C(z) := 1
+�
+z ∈ Br(x′) ∩ B˜r(˜x) ∩ H−1
+n (υn)
+�
+return diOρ(C)
+Figure 1: Hybrids in proof of Theorem 23. H0 is precisely MdiO(x) and H2 is precisely MdiO(x′).
+Our mechanism can then be defined by simply applying obfuscator to the circuit generated
+in the same way as C1 in LDS-Samplern. This mechanism MdiO is described more formally in
+Algorithm 3. The CDP property of the mechanism follows rather simply from the definition of diO
+and the fact that RRε is ε-SDP.
+Theorem 23. Under Assumptions 18 and 22, MdiO is ε-CDP.
+Proof. For any adjacent datasets x, x′, we want to show that MdiO(x) ≈c
+ε MdiO(x′). We show this
+using an intermediate hybrid, as shown in Figure 1, where changes from one hybrid to next are
+highlighted in red.
+◮ Distribution H0 is precisely MdiO(x).
+◮ Distribution H1 is a variant of H0, where we change x to x′ in the definition of C, but continue
+to sample ˜x ∼ RRε(x).
+◮ Distribution H2 is a variant of H1, where we sample ˜x ∼ RRε(x′). Note that this is exactly
+MdiO(x′).
+We show that H0 ≈c
+ε H2 by showing that H0 ≈c H1 and H1 ≈ε,0 H2 and using basic
+composition (Fact 11).
+We have from Lemma 21, that under Assumption 18, the joint distri-
+bution of ˜x ∼ RRε(x), and circuits C in H0 and H1 is precisely the output of LDS-Sampler.
+Thus, from Assumption 22, it follows that H0 ≈c H1 by post-processing (Fact 11).
+Next, we
+have that H1 ≈(ε,0) H2, since the only difference between the two is the distribution of ˜x, and
+RRε(x) ≈(ε,0) RRε(x′) (again by post-processing).
+Finally, its utility also follows simply from a standard concentration inequality.
+16
+
+Theorem 24. When choosing ˜r =
+1
+1+eε n + n0.6, MdiO is (1 − o(1))-useful for ueval
+H−1
+n (υn).
+Proof. Consider any dataset x. If x /∈ H−1
+n (υn), then, by definition of uLDS
+H−1
+n (υn), the utility always
+evaluates to one. Therefore, we may only consider the case where x ∈ H−1
+n (υn).
+In this case, observe that Pr
+�
+ueval
+H−1
+n (υn)(x, MdiO(x)) = 1
+�
+= Pr˜x∼RRε(x)[x ∈ B˜r(˜x)]. Notice that
+∥x − ˜x∥1 is distributed as Bin(n,
+1
+1+eε ). Therefore, applying Bernstein’s inequality, we have
+Pr
+˜x∼RRε(x)[x /∈ B˜r(˜x)] ≤ exp
+�
+−
+t2
+eε
+(1+eε)2 n + 2
+3t
+�
+≤ exp(−Ω(n0.2)),
+where t = ˜r−
+n
+1+eε = n0.6. This means that Pr
+�
+ueval
+H−1
+n (υn)(x, MdiO(x)) = 1
+�
+= 1−o(1) as desired.
+5.2
+CDP Mechanism for VLDS
+5.2.1
+Witness-Indistinguishable Proofs
+For any NP language L with associated verifier VL, let RL denote the corresponding relation
+{(x, w) : x ∈ L and VL(x, w) = 1}. Let RL(x) := {w : (x, w) ∈ RL}.
+Definition 25 (NIWI Proof System). A pair (P, V ) of PPT algorithms is a non-interactive witness
+indistinguishable (NIWI) proof system for an NP relation RL if it satisfies:
+◮ Correctness: for every (x, w) ∈ RL
+Pr[V (x, π) = 1 : π ← P(x, w)] = 1.
+◮ Soundness: there exists a negligible function negl such that for all x /∈ L and π ∈ {0, 1}∗:
+Pr[V (x, π) = 1] ≤ negl(|x|).
+◮ Witness Indistinguishability: There exists a polynomial ζ(·) and a negligible function negl(·),
+such that for any sequence I = {(x, w0, w1) : w0, w1 ∈ RL(x)} and for all circuits C of size at
+most ζ(|x|):
+����
+Pr
+π0←P (x,w0)[C(x, π0) = 1] −
+Pr
+π1←P (x,w1)[C(x, π1) = 1]
+���� ≤ negl(|x|).
+Assumption 26 ([BOV07, GOS12, BP15a]). There exists a NIWI proof system for any language
+in NP.
+5.2.2
+Making Utility Function Efficient Using Witness-Indistinguishable Proofs
+We consider the NP language �L defined below, and use the corresponding NIWI verifier to define
+the utility for VLDS.
+Definition 27. Language �L consists of all circuits �C with a top AND gate, namely of the form
+�C0 ∧ �C1 such that there exists some x, ˜x and ρ, such that at least one of �C0 or �C1 can be obtained
+as diOρ(C) where C is a circuit that takes in z and computes 1
+�
+z ∈ Br(x) ∩ B˜r(˜x) ∩ H−1(υ)
+�
+.
+A “witness” for �C ∈ �L is given by w = (b, x, ˜x, ρ), where b ∈ {0, 1} indicates whether the witness
+is provided for �C0 or for �C1. Let ( �P, �V ) denote the NIWI proof system for L (guaranteed to exist
+by Assumption 26).
+17
+
+Algorithm 4 Sub-routine Maux
+diO.
+Parameter:
+Differing-inputs obfuscator diO, hash function H, parameters ε, r, ˜r (as in
+Assumption 22), and a hash value υ ∈ {0, 1}γ(n).
+Input: Dataset x ∈ {0, 1}n.
+Output: Circuit : {0, 1}n → {0, 1}.
+˜x ← RRε(x).
+C ← circuit that takes in z and compute 1
+�
+z ∈ Br(x) ∩ B˜r(˜x) ∩ H−1
+n (υ)
+�
+�C ← diOρ(C) for randomness ρ
+return
+�C, ˜x, ρ
+Algorithm 5 CDP mechanism Mcdp.
+Input: Dataset x ∈ {0, 1}n, radius parameters r, ˜r > 0 and privacy parameter ε.
+Output: Circuit C and a proof string π.
+�C0, ˜x0, ρ0 ← Maux
+diO(x)
+�C1, ˜x1, ρ1 ← Maux
+diO(x)
+�C = �C0 ∧ �C1
+π ← �P( �C, (0, x, ˜x0, ρ0)) (NIWI proof for �C ∈ �L using witness (0, x, ˜x0, ρ0)).
+return
+�C, π
+We consider the verifiable low diameter set problem VLDSτ,H−1(υ),�V . Note that �C ∈ �L auto-
+matically implies that �C encodes a τ-diameter set (since �C = �C0 ∧ �C1, it suffices to certify that at
+least one of �C0 or �C1 encodes a τ-diameter set) where τ = 2r = n0.9.
+Theorem 28. Mcdp (described in Algorithm 5) is 2ε-CDP.
+Proof. For any adjacent datasets x, x′, we want to show that Mcdp(x) ≈c
+2ε Mcdp(x′). We show this
+through the means of intermediate hybrids, as shown in Figure 2, where changes from one hybrid
+to next are highlighted in red.
+◮ Distribution H0 is precisely Mcdp(x).
+◮ Distribution H1 is a variant of H0, where �C1 is generated through x′ instead of x.
+◮ Distribution H2 is a variant of H1, where we switch π from corresponding to witness (0, x, ˜x0, ρ0)
+to the witness (1, x′, ˜x1, ρ1).
+◮ Distribution H3 is a variant of H2, where �C0 is also generated through x′ instead of x.
+◮ Distribution H4 is a variant of H3, where we switch π from corresponding to witness (1, x′, ˜x1, ρ1)
+to the witness (0, x′, ˜x0, ρ0). Note that this is exactly Mcdp(x′).
+From Assumption 26 and post-processing (Fact 11), we have that H1 ≈c H2, and similarly H3 ≈c
+H4.
+Next, we show that H0 ≈c
+ε H1. Note that the output of H0 and H1 do not depend on ˜x1
+and ρ1. Thus the only material change between H0 and H1 is that �C1 ∼ MdiO(x) in H0 versus
+�C1 ∼ MdiO(x′) in H1. From Theorem 23, we have that MdiO(x) ≈c
+ε MdiO(x′). Thus, it follows
+that H0 ≈c
+ε H1 by post-processing (Fact 11). Similarly, it follows that H2 ≈c
+ε H3 (here we use that
+˜x0 and ρ0 are immaterial to the final output of H2 and H3).
+Combining these using basic composition (Fact 11), we get that H0 ≈c
+2ε H4, thus implying that
+Mcdp is 2ε-CDP.
+18
+
+Distribution H0:
+�C0, ˜x0, ρ0 ← Maux
+diO(x)
+�C1, ˜x1, ρ1 ← Maux
+diO(x)
+�C = �C0 ∧ �C1
+π ← �P( �C, (0, x, ˜x0, ρ0))
+return
+�C, π
+Distribution H1:
+�C0, ˜x0, ρ0 ← Maux
+diO(x)
+�C1, ˜x1, ρ1 ← Maux
+diO(x′)
+�C = �C0 ∧ �C1
+π ← �P( �C, (0, x, ˜x0, ρ0))
+return
+�C, π
+Distribution H2:
+�C0, ˜x0, ρ0 ← Maux
+diO(x)
+�C1, ˜x1, ρ1 ← Maux
+diO(x′)
+�C = �C0 ∧ �C1
+π ← �P( �C, (1, x′, ˜x1, ρ1)).
+return
+�C, π
+Distribution H3:
+�C0, ˜x0, ρ0 ← Maux
+diO(x′)
+�C1, ˜x1, ρ1 ← Maux
+diO(x′)
+�C = �C0 ∧ �C1
+π ← �P( �C, (1, x′, ˜x1, ρ1)).
+return
+�C, π
+Distribution H4:
+�C0, ˜x0, ρ0 ← Maux
+diO(x′)
+�C1, ˜x1, ρ1 ← Maux
+diO(x′)
+�C = �C0 ∧ �C1
+π ← �P( �C, (0, x′, ˜x0, ρ0))
+return
+�C, π
+Figure 2: Hybrids in proof of Theorem 28. H0 is precisely Mcdp(x) and H4 is precisely Mcdp(x′).
+Corollary 29. Mcdp is (1 − o(1))-useful for uVLDS
+τ,H−1(υ),�V .
+Proof. The utility for x /∈ H−1(υ) is trivially 1. Consider x ∈ H−1(υ). Suppose the mechanism
+MdiO is (1 − η)-useful for ueval
+H−1(υ). Since we sample �C0 and �C1 from MdiO independently we have
+that �C(x) = 1 with probability at least 1 − 2η. Finally, note that the proof π in the output of
+Mcdp is always accepted by �V . From Theorem 24, we have that η = o(1), and hence Mcdp is
+1 − 2η = 1 − o(1) useful for uVLDS
+τ,H−1(υ),�V .
+We end this section by proving Theorem 16. The proof is essentially a straightforward combina-
+tion of the previous two results. The only choice left to make is to select the hash value υ; we select
+it so that the size of the preimage H−1(υ) is maximized. This ensures that the set R = H−1(υ)
+has enough density as required in Theorem 16. (Note: the density requirement in Theorem 16 is
+not important for showing the existence of a CDP mechanism, but instead is later used to show the
+non-existence of SDP mechanisms.)
+Proof of Theorem 16. Let H, τ, �V be as defined above. Furthermore, let υ be such that H−1(υ) is
+maximized and ε = εCDP/2. The fact that there exists an εCDP-CDP mechanism that is (1 − o(1))-
+useful for uVLDS
+τ,R,�V follows immediately from Theorem 28 and Corollary 29.
+Furthermore, by our
+choice of υ, notice that |R| = |H−1(υ)| ≤ 2n/2γ(n) ≥ 2n/no(log n), where the latter comes from our
+assumption on γ in Assumption 18.
+6
+SDP Lower Bounds for the Nearby Point Problem
+In this section, we will show that there is no O(1)-SDP algorithm for the nearby point problem
+with target threshold n0.99 as long as the set Rn is fairly dense, as formalized below.
+Theorem 30. Let τ = {τn}n∈N and R = {Rn ⊆ {0, 1}n}n∈N be such that τn ≤ n0.99 and |Rn| ≥
+2n/no(log n). Then, for any constant ε > 0 and any negligible function negl, there exists a sufficiently
+large n ∈ N such that there is no (ε, negl(n))-SDP algorithm that is 0.01-useful for unear
+τ,R .
+19
+
+To prove Theorem 30, let us first recall the standard “blatant non-privacy implies non-DP”
+proof11, which corresponds to the case Rn = {0, 1}n. At a high-level, these proofs proceed by
+showing that the error in each coordinate is large by “matching” each x ∈ {0, 1}n with another
+point x′ which is the same as x except with the i-th bit flipped; a basic calculation then shows that
+(on average) the i-th bit is predicted incorrectly with large probability. Summing this up over all
+the coordinates yield the desired bound.
+As we are in the case where Rn ̸= {0, 1}n, we cannot use the proof above directly. Nonetheless,
+we can still adapt the above proof. More specifically, instead of looking at each coordinate at a
+time, we look at a block of coordinates. For each block, we try to find a matching in the same spirit
+as above, but we now allow the x, x′ to have a larger distance; simple calculations give us a lower
+bound on being incorrect in this block (Section 6.2). We then “sum up” across all blocks to get
+a large distance (Section 6.3). Even though we get a large distance τ via this approach, the error
+probability (i.e. one minus usefulness) is small (i.e. o(1)). Fortunately, we can overcome this using
+the so-called DP hyperparameter tuning algorithm [LT19, PS21] (Section 6.4). This concludes our
+proof overview.
+6.1
+Additional Preliminaries: Tools from Differential Privacy
+We will require several additional tools from DP literature, which we list below for completeness.
+Laplace Mechanism.
+The Laplace distribution with scale parameter b > 0, denoted by Lap(b),
+is the probability distribution over R with probability mass function z �→ 1
+2b exp(−|z|/b).
+Given a function f : X ∗ → R, its sensitivity is defined as ∆(f) := maxD,D′ |f(D) − f(D′)|,
+where the maximum is over all pair D, D′ of adjacent datasets.
+The Laplace mechanism [DMNS06] is an ε-SDP mechanism that simply outputs f(X)+Lap(∆(f)/ǫ).
+Basic Composition.
+We will also use the so-called basic composition theorem: an algorithm that
+just runs an (ε1, δ1)-SDP and an (ε2, δ2)-SDP algorithms as subroutines, is (ε1 + ε2, δ1 + δ2)-SDP.
+Group Privacy.
+The following fact is well-known and is often referred to as group privacy.
+Fact 31 (Group Privacy (e.g., [SU16])). Let M : X ∗ → Y be an (ε, δ)-SDP mechanism and let
+D, D′ ∈ X ∗ be such that ∥D−D′∥ ≤ t, then, for all S ⊆ Y we have Pr[M(D) ∈ S] ≤ eε′·Pr[M(D′) ∈
+S] + δ′, where ε′ = tε and δ′ = etε−1
+eε−1 · δ.
+DP Hyperparameter Tuning.
+We will also use the following result of Liu and Talwar [LT19] on
+DP hyperparameter tuning. We remark that some improvements in the constants has been made in
+[PS21], by using a different distribution of the number of repetitions. Nonetheless, since we are only
+interested in an asymptotic bound, we choose to work with the slightly simpler hyperparameter
+tuning algorithm from [LT19].
+The hyperparameter tuning algorithm from [LT19] allows us to take any DP “base” mechanism
+Mbase, which outputs a candidate y and a score q ∈ R, run it multiple times and output a candidate
+with score that is above a certain threshold. The precise description is in Algorithm 6.
+We will use the following DP guarantee of Mtuning, which was shown in [LT19]12.
+11Here we follow the proofs in [Sur19, Man22].
+12Note that this is a simplified version of [LT19, Theorem 3.1] where we simply set ε0 = 1.
+20
+
+Algorithm 6 DP Hyperparameter Tuning Mtuning.
+Parameters: Mechanism Mbase, Threshold s, Number of Steps T, Stopping Probability γ.
+Input: Dataset D
+for j = 1, . . . , T do
+Let (y, q) ← Mbase(D).
+if q ≥ s then
+return x (and halt)
+With probability γ:
+return ⊥ (and halt)
+Theorem 32 (DP Hyperparameter Tuning [LT19]). Let ε > 0 and δ, γ ∈ [0, 1]. Suppose that Mbase
+is (ε, δ)-SDP and T ≥ 2/γ. Then, the DP Hyperparameter Tuning mechanism Mtuning defined in
+Algorithm 6 is (2ε + 1, 10e2ε · δ/γ).
+6.2
+Weak Hardness
+We start with a relatively weak hardness for the case of τ = 0, i.e., the answer is considered correct
+iff it is the same as the input. To prove this, we recall a couple of facts.
+The first is a simple relation between independent set and maximum matching. Let ind(G)
+denote the size of the maximum independent set of G.
+Fact 33. For any graph G = (V, E), there exists matching of size at least (|V | − ind(G))/2.
+Let Hd denote the distance-d graph on the hypercube, i.e., Hd = ({0, 1}n, E) where (x, x′) ∈ E
+iff ∥x − x′∥1 ≤ d. Let
+� n
+≤d
+�
+= �d
+i=0
+�n
+i
+�
+. The following is the “packing” lower bound.
+Fact 34. For any d ∈ N, ind(H2d+1) ≤ 2n/
+� n
+≤d
+�
+.
+We are now ready to prove a lower bound for the nearby problem.
+Theorem 35. For any R ⊆ {0, 1}n, d, ε, δ, let ε′ = (2d + 1)ε and δ′ = eε′−1
+eε−1 δ. Then, for any
+(ε, δ)-SDP algorithm M, we have
+�
+x∈R
+Pr[M(x) ̸= x] ≥ 0.5e−ε′(1 − δ′)
+�
+|R| − 2n
+� n
+≤d
+�
+�
+.
+Proof. Let H2d+1[R] denote the subgraph of H2d+1 induced on R. Notice that ind(H2d+1[R]) ≤
+ind(H2d+1). Therefore, by Fact 33 and Fact 34, we can conclude H2d+1[R] contains a matching of
+size at least m ≥
+�
+|R| − 2n/
+� n
+≤d
+��
+/2. Let the matching be (x1, ˜x1), . . . , (xm, ˜xm).
+For each i ∈ [m], we have
+Pr[M(xi) ̸= xi] + Pr[M(˜xi) ̸= ˜xi]
+≥ Pr[M(xi) = ˜xi] + Pr[M(˜xi) ̸= ˜xi]
+(Group privacy, Fact 31)
+≥ e−ε′(Pr[M(˜xi) = ˜xi] − δ′) + Pr[M(˜xi) ̸= ˜xi]
+≥ e−ε′(Pr[M(˜xi) = ˜xi] + Pr[M(˜xi) ̸= ˜xi] − δ′)
+= e−ε′(1 − δ′).
+Adding this over all i ∈ [m] yields the claimed bound.
+21
+
+6.3
+Boosting the Distance
+We can now prove a hardness for larger τ by dividing the coordinates into groups and applying the
+previously derived weak hardness result on each group. We note that the “non-usefulness” we get
+on the right hand side is still insufficient for Theorem 30; this will be dealt with in Section 6.4.
+Theorem 36. Let n = n′ · b′ for some n′, b′ ∈ N. For any R ⊆ {0, 1}n, d, ε, δ, ζ, let ε′ = (2d + 1)ε
+and δ′ = eε′−1
+eε−1 δ. Then, for any (ε, δ)-SDP algorithm M, there exists x ∈ R such that
+Pr[unear
+ζ·b′,R(M(x), x) = 0] ≥
+�
+0.5e−ε′(1 − δ′)
+�
+1 −
+2n
+|R| ·
+� n′
+≤d
+�
+��
+− ζ.
+Proof. Let Bi := {(i − 1)n′ + 1, . . . , in′} for all i ∈ [b′]. Furthermore, let R(Bi,z−Bi) denote the set
+the set of all x ∈ R such that x−Bi = z−Bi.
+First, notice that
+�
+x∈R
+Pr[unear
+ζ·b′,R(M(x), x) = 0] =
+�
+x∈R
+Ey←M(x) 1
+�|{i ∈ [n] | yi ̸= xi}|
+b′
+> ζ
+�
+≥
+�
+x∈R
+Ey←M(x) 1
+�|{i ∈ [b′] | yBi ̸= xBi}|
+b′
+> ζ
+�
+≥
+�
+x∈R
+Ey←M(x)
+�
+Pr
+i∈[b′][yBi ̸= xBi] − ζ
+�
+=
+
+ 1
+b′
+�
+i∈[b′]
+�
+x∈R
+Pr[M(x)Bi ̸= xBi]
+
+ − ζ|R|
+≥
+
+
+ 1
+b′
+�
+i∈[b′]
+�
+z−Bi∈{0,1}[n]\Bi
+�
+x∈R(Bi,z−Bi )
+Pr[M(x)Bi ̸= xBi]
+
+
+ − ζ|R|.
+For each fixed z−Bi ∈ {0, 1}[n]\Bi, consider the mechanism M′ : {0, 1}Bi → {0, 1}Bi defined by
+M′(xBi) := Mi(xBi ◦ z−Bi)|Bi.
+It is clear that M′ is (ε, δ)-SDP.
+Furthermore, observe that
+Pr[M(x)Bi ̸= xBi] = Pr[M′(x) ̸= xBi] for all x ∈ R(Bi,z−Bi). Therefore, by applying Theorem 35
+and plugging it back into the above, we get
+�
+x∈R
+Pr[unear
+ζ·b′,R(M(x), x) = 0]
+≥
+
+
+ 1
+b′
+�
+i∈[b′]
+�
+z−Bi∈{0,1}[n]\Bi
+0.5e−ε′(1 − δ′)
+�
+|R(Bi,z−Bi)| − 2n′
+� n′
+≤d
+�
+�
+
+ − ζ|R|
+=
+
+ 1
+b′
+�
+i∈[b′]
+0.5e−ε′(1 − δ′)
+�
+|R| − 2n−n′ · 2n′
+� n′
+≤d
+�
+�
+ − ζ|R|
+=
+�
+0.5e−ε′(1 − δ′)
+�
+|R| − 2n
+� n′
+≤d
+�
+��
+− ζ|R|.
+Dividing by |R| then gives us the claimed bound.
+22
+
+6.4
+Boosting the Failure Probability
+We will now prove the last part of the lower bound, which is to show that the existence of even
+slightly useful mechanism also leads to an existence of a highly useful mechanism, albeit at a slight
+increanse in the distance threshold. The formal statement and its proof are given below; the proof
+uses the DP hyperparameter tuning algorithm (Theorem 32).
+Theorem 37. Suppose that there exists an (ε, δ)-SDP mechanism M : {0, 1}n → {0, 1}n that is
+α-useful for unear
+τ,R . Then, there exists an (ε′, δ′)-SDP mechanism M′ : {0, 1}n → {0, 1}n that is
+(1 − 1/n1000)-useful for unear
+τ ′,R where ε′ = 4ε + 1, δ′ = O
+�
+n11e2ε
+α
+· δ
+�
+and τ ′ = τ + O
+� ln n
+α
+�
+.
+Proof. First, let us construct the mechanism Mbase : {0, 1}n → {0, 1}n × R as follows:
+◮ On input x ∈ {0, 1}n, first let y ← M(x).
+◮ Then, let q = ∥x − y∥1 + z where z ∼ Lap(1/ε).
+◮ Output (x, q).
+Since M is (ε, δ)-SDP and the Laplace mechanism is ε-SDP, the basic composition theorem implies
+that the entire Mbase mechanism is (2ε, δ)-SDP.
+Let �T = ln(5n1000)/α. Let τ ′ = τ − log(10n1000 �T)/ε. We now apply Algorithm 6 with γ =
+0.5/(n1000 �T), T = 2/γ and threshold s = τ ′ − log(10n1000 �T)/ε.
+Theorem 32 ensures that the
+resulting algorithm Mtuning is (4ε + 1, 10e2εδ/γ)-SDP. Our final mechanism �
+M is the mechanism
+that runs Mtuning.
+If the output is not ⊥, �
+M returns that output.
+Otherwise, �
+M returns an
+arbritrary element of {0, 1}n. Since �
+M is simple a post-processing of Mtuning, we have �
+M is also
+(4ε + 1, 10e2εδ/γ)-SDP.
+We will next show that Mtuning is (1 − 1/n1000)-useful for unear
+τ ′,R. By definition of the utility
+function, this immediately holds for any x /∈ R. Therefore, we may only consider any x ∈ R.
+Consider Mtuning on such an x. Let yi, zi, qi denote the corresponding values of y, z, q in the ith
+run of Mbase.
+We will consider the following three events:
+◮ Let E1 denote the event that |∥xi − yi∥1 − qi| > log(10n1000 �T)/ε for some i ∈ [ �T].
+◮ Let E2 denote the event that uτ,R(yi) = 0 for all i ∈ [ �T].
+◮ Let E3 denote the event that Mtuning halts in the first �T steps.
+Before we bound the probability of each events, notice that, if none of E1, E2, E3 occurs, we must
+have unear
+τ ′,R(y) = 1 (where y denote the output of �
+M), since s − τ, τ ′ − s ≥ log(10n1000 �T)/ε. That is,
+Pr
+y←�
+M(x)
+[unear
+τ ′,R(y) = 0] ≤ Pr[E1 ∨ E2 ∨ E3] ≤ Pr[E1] + Pr[E2] + Pr[E3].
+We will now bound the probability for each event. For E1, it immediately follows from the
+Laplace tail bound together with a union bound that
+Pr[E1] ≤ �T · 2/(10n1000 �T) = 0.2/n1000.
+For E2, the α-usefulness of M implies that
+Pr[E2] ≤ (1 − α)
+�T ≤ 0.2/n1000.
+Finally, for E3, we may simply use a union bound, which gives
+Pr[E3] ≤ γ · �T ≤ 0.5/n1000.
+23
+
+By combining the four inequalities above, we have
+Pr
+y←�
+M(x)
+[unear
+τ ′,R(y) = 0] < 1/n1000,
+as desired.
+6.5
+Putting Things Together: Proof of Theorem 30
+Proof of Theorem 30. Suppose for the sake of contradiction that, for some constant ε > 0 and
+negligible function negl, there exists an (ε, negl(n))-SDP mechanism Mn that is 0.01-useful for
+unear
+τn,Rn for every n ∈ N.
+By Theorem 37, there is a (4ε + 1, δ′(n)) mechanism M′
+n that is (1 − 1/n1000)-useful for unear
+τ ′n,Rn
+where δ′(n) is a negligible function and τ ′
+n = τn + O(log n) = O(n0.99).
+Plugging this into
+Theorem 36 with R = Rn, n′ = n0.005, b′ = n0.995, ζ = τ ′
+n/b′ ≤ O(n−0.005), ε = 4ε + 1, δ = δ′(n), d =
+(log n0.004)/3ε (which gives ε′ ≤ log(2n0.004) and δ′ = 1/nω(1) in Theorem 36), we have
+1
+n1000 ≥
+�
+0.5 · e− log(2n0.004)(1 − n−ω(1)) (1 − o(1))
+�
+− O(n−0.005)
+= O(n−0.004) · (1 − o(1)) − O(n−0.005),
+which is a contradiction for any sufficiently large n.
+7
+Putting Things Together: Proof of Theorem 5
+Our main theorem follows from trivially combining the main results from the previous two sections.
+Proof of Theorem 5. Let u = uVLDS
+τ,R,V be as given in Theorem 16, which immediately yields the
+existence of an εCDP-CDP mechanism that is (1 − o(1))-useful. Furthermore, by |R| ≥ 2n/no(n),
+Theorem 30 implies that there is no εSDP-SDP mechanism that is 0.01-useful for {unear
+τ,R }. Finally,
+applying Lemma 15, we can conclude that there is no εSDP-SDP mechanism that is 0.01-useful for
+{uVLDS
+τ,R,V }. This concludes our proof.
+8
+Conclusion and Discussion
+In this work, we give a first task that, under certain assumptions, admits an efficient CDP algorithm
+but does not admit an (even inefficient) SDP algorithm. As mentioned in Section 1, perhaps the
+most intriguing next direction would be to see if there are more “natural” tasks for which CDP
+algorithms can go beyond known SDP lower bounds.
+On the technical front, there are also a few interesting directions. For example, it would be
+interesting to see if the three assumptions in our paper can be removed, relaxed, or replaced (by
+perhaps more widely believed assumptions). Alternatively, we can ask the opposite question: what
+are the (cryptographic) assumptions necessary for separating CDP and SDP?
+Such a question
+has been extensively studied in the multiparty model [HMST22, GMPS13, GKM+16, HMSS19,
+HNO+18]; for example, it is known that key-agreement is necessary and sufficient to get better-
+than-local-DP protocol for inner product in the two-party setting [HMST22]. Achieving such a
+24
+
+result in our setting would significantly deepen our understanding of the CDP-vs-SDP question in
+the central model.
+Another possible improvement is to strengthen the hardness of the adversary. In this paper,
+we only consider polynomial-time adversaries. Indeed, our CDP mechanism does not remain CDP
+against quasi-polynomial adversary. The reason is that we choose the hash value length to be only
+o(log2 λ) in Assumption 18, so a trivial “guess-and-check” algorithm can break this assumption in
+time λO(log λ). However, as far as we are aware, there is no inherent barrier in proving a separation
+with CDP that holds even against, e.g., sub-exponential time adversaries. Achieving such a result
+(potentially under stronger or different assumptions) would definitely be interesting.
+Furthermore, our task (or more precisely the utility function) is non-uniform (through the choice
+of υn). It would also be interesting to have a uniform task.
+Acknowledgments
+We thank Prabhanjan Ananth for helpful discussions about differing-inputs obfuscation, and anony-
+mous reviewers for helpful comments.
+25
+
+References
+[ABG+13]
+Prabhanjan Ananth, Dan Boneh, Sanjam Garg, Amit Sahai, and Mark Zhandry.
+Differing-inputs obfuscation and applications. IACR Cryptol. ePrint Arch., page 689,
+2013.
+[Abo18]
+John M Abowd. The US Census Bureau adopts differential privacy. In KDD, pages
+2867–2867, 2018.
+[App17]
+Apple Differential Privacy Team.
+Learning with privacy at scale.
+Apple Machine
+Learning Journal, 2017.
+[BCP14]
+Elette Boyle, Kai-Min Chung, and Rafael Pass. On extractability obfuscation. In TCC,
+pages 52–73, 2014.
+[BCV16]
+Mark Bun, Yi-Hsiu Chen, and Salil P. Vadhan. Separating computational and statis-
+tical differential privacy in the client-server model. In TCC, pages 607–634, 2016.
+[BGI+01]
+Boaz Barak, Oded Goldreich, Russell Impagliazzo, Steven Rudich, Amit Sahai, Salil P.
+Vadhan, and Ke Yang. On the (im)possibility of obfuscating programs. In CRYPTO,
+pages 1–18, 2001.
+[BGI+12]
+Boaz Barak, Oded Goldreich, Russell Impagliazzo, Steven Rudich, Amit Sahai, Salil P.
+Vadhan, and Ke Yang.
+On the (im)possibility of obfuscating programs.
+J. ACM,
+59(2):6:1–6:48, 2012.
+[BKP18]
+Nir Bitansky, Yael Tauman Kalai, and Omer Paneth.
+Multi-collision resistance: a
+paradigm for keyless hash functions. In STOC, pages 671–684, 2018.
+[BNO08]
+Amos Beimel, Kobbi Nissim, and Eran Omri. Distributed private data analysis: Si-
+multaneously solving how and what. In CRYPTO, pages 451–468, 2008.
+[BOV07]
+Boaz Barak, Shien Jin Ong, and Salil P. Vadhan. Derandomization in cryptography.
+SIAM J. Comput., 37(2):380–400, 2007.
+[BP15a]
+Nir Bitansky and Omer Paneth. Zaps and non-interactive witness indistinguishability
+from indistinguishability obfuscation. In TCC, pages 401–427, 2015.
+[BP15b]
+Elette Boyle and Rafael Pass. Limits of extractability assumptions with distributional
+auxiliary input. In ASIACRYPT, pages 236–261, 2015.
+[BR93]
+Mihir Bellare and Phillip Rogaway. Ccs. pages 62–73, 1993.
+[BSW16]
+Mihir Bellare, Igors Stepanovs, and Brent Waters. New negative results on differing-
+inputs obfuscation. In EUROCRYPT, pages 792–821, 2016.
+[CGH04]
+Ran Canetti, Oded Goldreich, and Shai Halevi.
+The random oracle methodology,
+revisited. J. ACM, 51(4):557–594, 2004.
+[De12]
+Anindya De. Lower bounds in differential privacy. In TCC, pages 321–338, 2012.
+26
+
+[DKM+06] Cynthia Dwork, Krishnaram Kenthapadi, Frank McSherry, Ilya Mironov, and Moni
+Naor. Our data, ourselves: Privacy via distributed noise generation. In EUROCRYPT,
+pages 486–503, 2006.
+[DKY17]
+Bolin Ding, Janardhan Kulkarni, and Sergey Yekhanin.
+Collecting telemetry data
+privately. In NeurIPS, pages 3571–3580, 2017.
+[DMNS06]
+Cynthia Dwork, Frank McSherry, Kobbi Nissim, and Adam D. Smith. Calibrating
+noise to sensitivity in private data analysis. In TCC, pages 265–284, 2006.
+[DMT07]
+Cynthia Dwork, Frank McSherry, and Kunal Talwar. The price of privacy and the
+limits of LP decoding. In STOC, pages 85–94, 2007.
+[DN03]
+Irit Dinur and Kobbi Nissim. Revealing information while preserving privacy. In PODS,
+pages 202–210, 2003.
+[EPK14]
+´Ulfar Erlingsson, Vasyl Pihur, and Aleksandra Korolova. RAPPOR: Randomized ag-
+gregatable privacy-preserving ordinal response. In CCS, pages 1054–1067, 2014.
+[GGHW17] Sanjam Garg, Craig Gentry, Shai Halevi, and Daniel Wichs. On the implausibility of
+differing-inputs obfuscation and extractable witness encryption with auxiliary input.
+Algorithmica, 79(4):1353–1373, 2017.
+[GKM+16] Vipul Goyal, Dakshita Khurana, Ilya Mironov, Omkant Pandey, and Amit Sahai. Do
+distributed differentially-private protocols require oblivious transfer? In ICALP, pages
+29:1–29:15, 2016.
+[GKY11]
+Adam Groce, Jonathan Katz, and Arkady Yerukhimovich. Limits of computational
+differential privacy in the client/server setting. In TCC, pages 417–431, 2011.
+[GMPS13]
+Vipul Goyal, Ilya Mironov, Omkant Pandey, and Amit Sahai. Accuracy-privacy trade-
+offs for two-party differentially private protocols. In CRYPTO, pages 298–315, 2013.
+[GOS12]
+Jens Groth, Rafail Ostrovsky, and Amit Sahai.
+New techniques for noninteractive
+zero-knowledge. J. ACM, 59(3):11:1–11:35, 2012.
+[Gre16]
+Andy Greenberg. Apple’s “differential privacy” is about collecting your data – but not
+your data. Wired, June, 13, 2016.
+[GT08]
+Ben Green and Terence Tao. The primes contain arbitrarily long arithmetic progres-
+sions. Annals of Mathematics, 167(2):481–547, 2008.
+[HMSS19]
+Iftach Haitner, Noam Mazor, Ronen Shaltiel, and Jad Silbak. Channels of small log-
+ratio leakage and characterization of two-party differentially private computation. In
+TCC, pages 531–560, 2019.
+[HMST22]
+Iftach Haitner, Noam Mazor, Jad Silbak, and Eliad Tsfadia. On the complexity of
+two-party differential privacy. In STOC, pages 1392–1405, 2022.
+[HNO+18]
+Iftach Haitner, Kobbi Nissim, Eran Omri, Ronen Shaltiel, and Jad Silbak. Compu-
+tational two-party correlation: A dichotomy for key-agreement protocols. In FOCS,
+pages 136–147, 2018.
+27
+
+[HT10]
+Moritz Hardt and Kunal Talwar. On the geometry of differential privacy. In STOC,
+pages 705–714, 2010.
+[IPS15]
+Yuval Ishai, Omkant Pandey, and Amit Sahai. Public-coin differing-inputs obfuscation
+and its applications. In TCC, pages 668–697, 2015.
+[JLS21]
+Aayush Jain, Huijia Lin, and Amit Sahai. Indistinguishability obfuscation from well-
+founded assumptions. In STOC, pages 60–73, 2021.
+[KT18]
+Krishnaram Kenthapadi and Thanh T. L. Tran. PriPeARL: A framework for privacy-
+preserving analytics and reporting at LinkedIn. In CIKM, pages 2183–2191, 2018.
+[LT19]
+Jingcheng Liu and Kunal Talwar. Private selection from private candidates. In STOC,
+pages 298–309, 2019.
+[Man22]
+Pasin Manurangsi. Tight bounds for differentially private anonymized histograms. In
+SOSA, pages 203–213, 2022.
+[MMP+10] Andrew McGregor, Ilya Mironov, Toniann Pitassi, Omer Reingold, Kunal Talwar, and
+Salil P. Vadhan. The limits of two-party differential privacy. In FOCS, pages 81–90,
+2010.
+[MPRV09]
+Ilya Mironov, Omkant Pandey, Omer Reingold, and Salil P. Vadhan. Computational
+differential privacy. In CRYPTO, pages 126–142, 2009.
+[PS21]
+Nicolas Papernot and Thomas Steinke. Hyperparameter tuning with renyi differential
+privacy. CoRR, abs/2110.03620, 2021.
+[RSP+21]
+Ryan Rogers, Subbu Subramaniam, Sean Peng, David Durfee, Seunghyun Lee, San-
+tosh Kumar Kancha, Shraddha Sahay, and Parvez Ahammad. LinkedIn’s audience
+engagements API: A privacy preserving data analytics system at scale. J. Priv. Con-
+fiden., 11(3), 2021.
+[RTTV08]
+Omer Reingold, Luca Trevisan, Madhur Tulsiani, and Salil P. Vadhan. Dense subsets
+of pseudorandom sets. In FOCS, pages 76–85, 2008.
+[Sha14]
+Stephen Shankland. How Google tricks itself to protect Chrome user privacy. CNET,
+October, 2014.
+[SU16]
+Thomas Steinke and Jonathan R. Ullman. Between pure and approximate differential
+privacy. J. Priv. Confidentiality, 7(2), 2016.
+[Sur19]
+Ananda Theertha Suresh. Differentially private anonymized histograms. In NeurIPS,
+pages 7969–7979, 2019.
+[TZ08]
+Terence Tao and Tamar Ziegler. The primes contain arbitrarily long polynomial pro-
+gressions. Acta Mathematica, 201(2):213 – 305, 2008.
+[Vad17]
+Salil P. Vadhan. The complexity of differential privacy. In Tutorials on the Foundations
+of Cryptography, pages 347–450. Springer International Publishing, 2017.
+[War65]
+Stanley L Warner. Randomized response: A survey technique for eliminating evasive
+answer bias. JASA, 60(309):63–69, 1965.
+28
+
+A
+Comparison of various diO assumptions
+We review and compare the various notions of differing inputs obfuscation, showing that the notion
+of diO-for-pcS (Definition 20) is in fact weaker (or at least, no stronger) than all notions of differing
+inputs obfuscation studied in literature.
+The definition of diO as given by [BGI+12] did not include the notion of a sampler. Instead for
+any circuits C0 and C1, if an adversary A can distinguish diO(C0) and diO(C1) with non-negligible
+advantage, then there exists an adversary A′ that, given any circuits C′
+0 and C′
+1, where C′
+b is
+functionally equivalent to Cb for b ∈ {0, 1}, A′(C′
+0, C′
+1) can return x such that C0(x) ̸= C1(x).
+This notion is stronger than the corresponding notion involving samplers. Since most applica-
+tions of differing-inputs obfuscation in literature are stated using differing-inputs samplers, we will
+only refer to diO notions that involve these.
+Definition 38 (Differing-Inputs Circuit Sampler [ABG+13]). An efficient non-uniform sampling
+algorithm Sampler = {Samplern} is a differing-inputs sampler for the parameterized collection
+C = {Cn} of circuits if the output of Samplern is distributed over Cn × Cn × {0, 1}∗ and for every
+efficient non-uniform algorithm A = {An}, there exists a negligible function negl(·) such that for
+all n ∈ N:
+Pr
+θ [C0(y) ̸= C1(y) : (C0, C1, aux) ← Samplern(θ), y ← An(C0, C1, aux)] ≤ negl(n).
+Plain Sampler. We call a differing-inputs sampler as a Plain Sampler if aux is always ⊥.
+Public-Coin Sampler. We call a differing-inputs sampler as Public-Coin Sampler if aux is equal
+to θ (precisely Definition 19).
+General Sampler. We call a differing-inputs sampler as a General Sampler whenever we want to
+emphasize that aux is allowed to be any function of θ. In particular, plain and public-coin
+samplers are special cases of general samplers.
+Note that, the more information that aux is allowed to contain, the more restricted the distribution
+over circuit pairs (C0, C1) gets. In particular, any public-coin Sampler remains a differing-inputs
+Sampler if we set aux to be some function of θ (instead of being all of θ), and similarly, any general
+differing-inputs Sampler can be converted to a plain-Sampler by simply setting aux = ⊥.
+We can consider two notions of security of differing inputs obfuscators, depending on whether or
+not the distinguisher has access to aux. Recall that the “differing-inputs” condition in Definition 20
+was
+| Pr
+θ [Dn(diO(1n, C0)) = 1] − Pr
+θ [Dn(diO(1n, C1)) = 1]| ≤ negl(n).
+(6)
+On the other hand, we could consider a different notion where for any general sampler Sampler, for
+(C0, C1, aux) ← Samplern(θ), we replace the “differing-inputs” condition with
+| Pr
+θ [Dn(diO(1n, C0), aux) = 1] − Pr
+θ [Dn(diO(1n, C1), aux) = 1]| ≤ negl(n).
+(7)
+Depending on the type of sampler (plain or public-coin or general) and the notion of security
+for differing inputs obfuscators ((6) or (7)), we get various kinds of diO assumptions, which we list
+below.
+29
+
+plain-diO
+pc-diO
+gen-diO
+diO-for-genS
+diO-for-pcS
+Figure 3: Comparisons between different diO assumptions, where A → B denotes that existence of
+A implies existence of B, or in other words, existence of A is a stronger assumption than existence of
+B. Existence of diO-for-pcS (assumption used in this paper) is the weakest among all the notions.
+Plain diO. We refer to plain-diO as the notion of diO that holds only against plain samplers. Note,
+there is no difference here between the security notions of (6) and (7), since aux = ⊥ anyway.
+Public-Coin diO. We refer to pc-diO, as the notion of public-coin diO defined by [IPS15], cor-
+responding to the notion of diO that holds only against public-coin samplers, where the
+distinguisher also has access to aux = θ, as in (7).
+General diO. We refer to gen-diO, as the notion of general diO defined by [ABG+13], correspond-
+ing to the notion of diO that holds for general samplers, and where the distinguisher also has
+access to aux, as in (7).
+diO for General Samplers. We define diO-for-genS as the notion of diO that holds only against
+general samplers, but where the distinguisher does not have access to aux = θ, as in (6).
+diO for Public-Coin Samplers. This is precisely Definition 20, where the security of diO holds
+only for public-coin samplers, where the distinguisher does not have access to aux, as in (6).
+Comparison between different diO assumptions.
+The comparison between the assumptions
+asserting existence of each type of diO is illustrated in Figure 3, with justification for each arrow
+given as follows:
+◮ Existence of gen-diO implies existence of plain-diO and pc-diO, since both are special cases
+corresponding to plain samplers and public-coin samplers respectively.
+◮ To the best of knowledge, it is unknown whether the assumptions of existence of plain-diO
+and the existence of pc-diO are comparable or not.
+◮ Existence of plain-diO implies existence of diO-for-genS since any general sampler can be
+converted to a plain sampler by simply setting aux = ⊥; note that the distinguisher (in the
+definition of diO) does not have access to aux in either case.
+30
+
+◮ Existence of diO-for-genS implies existence of plain-diO and diO-for-pcS since both are special
+cases corresponding to plain samplers and public-coin samplers respectively.
+◮ Existence of pc-diO implies existence of diO-for-pcS, since the distinguisher in the definition
+of diO-for-pcS does not have access to θ and hence is less powerful.
+Finally, one may wonder, what was special about the application of diO in this paper that only
+required diO-for-pcS and not gen-diO or pc-diO as in prior work in cryptography. The main reason is
+that, in cryptographic applications, an aux is provided to adversaries to enable certain cryptographic
+functionality (such as by revealing some public key parameters), and thus, it is required that the diO
+is secure even given knowledge of this aux information. In applications of pc-diO, the distinguisher
+typically does not have access to all of θ (such as some secret key parameters may be hidden), but
+security given knowledge of entire θ implies security given partial knowledge of θ. In the setting of
+this paper, there wasn’t any particular functionality that needed to be enabled, other than basic
+circuit evaluation, and the particular circuit samplers of interest were public-coin differing inputs
+samplers, which is why it suffices to only assume diO-for-pcS.
+31
+

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+Multiorbital effects in high-order harmonic emission from CO2
+Andres Mora, Lauren Bauerle, Yuqing Xia, Agnieszka Jaron1
+1JILA and Department of Physics, University of Colorado, Boulder, CO 80309-0440, USA
+We study the ellipticity of high-order harmonics emitted from CO2 molecule driven by linearly po-
+larized laser fields using numerical simulations within the time-dependent density functional theory.
+We find that the overall ellipticity of the harmonics is small, which is in agreement with experimen-
+tal data. On the other hand, our analysis of the numerical results indicates that several valence
+orbitals contribute significantly to the harmonic emission and some of these contributions show a
+strong ellipticity of the harmonics. The small ellipticity in the total harmonics signal arises from a
+combination of factors, namely, the fact that harmonic emission from different orbitals is strongest
+at different alignment angles of the molecular axis with respect to the laser polarization direction,
+as well as interference effects and a strong laser coupling between several inner valence orbitals.
+PACS numbers: 32.80.Fb,32.80.Wr
+I.
+INTRODUCTION
+High-order harmonic generation (HHG) is one of the
+highly nonlinear, nonperturbative processes that occur
+when an atom or molecule is irradiated by an intense
+laser field [1, 2]. It results from the distortion of the elec-
+tron density in the presence of the strong electromagnetic
+field of the laser and the power spectrum of the emitted
+harmonic radiation corresponds to the Fourier transform
+of the electron dipole acceleration. The intensity spectra
+of the emitted high harmonics shows some general char-
+acteristic features, such as a fast decrease of the signal
+over the first few harmonics followed by a region with
+fairly constant plateau harmonic intensities ending by a
+sharp cutoff, beyond which the harmonic intensity drops
+quickly.
+Over the past few decades HHG has been an active area
+of research since it provides a source for coherent short-
+wavelength light, extending into the soft-X-ray regime
+[3], and for ultrashort laser pulses and waveforms in the
+attosecond [4, 5]. Furthermore, it has been shown that
+HHG spectra contain information about the atomic and
+electronic structure of the target (e.g., [6–11]), ultrafast
+molecular and intra-molecular electron dynamics (e.g.,
+[12–15]) as well as time resolution of chemical processes
+(e.g., [16]).
+Basic intuitive picture of HHG is provided by the semi-
+classical three-step model [17–19], according to which an
+electron tunnels through the barrier created by the laser
+field and the Coulomb field into the continuum, is accel-
+erated by the electric field of the laser first away from
+and then back to the parent ion.
+Upon return it re-
+combines, emitting excess energy in the form of high-
+order harmonic radiation. Since the process occurs every
+half cycle of the driving laser field, an attosecond pulse
+train is produced.
+In recent years, it has been shown
+that, in particular for high-order harmonic generation
+from molecules, the generated harmonic spectra incorpo-
+rate more features than predicted by the basic three-step
+single-active-electron model. One example are polarime-
+try measurements of high-order harmonic emission from
+aligned diatomic and linear triatomic molecules driven by
+linearly polarized laser fields. Surprisingly, strong ellipti-
+cally polarized harmonics were observed for N2 [20, 21],
+while in contrast CO2 exhibited a much lower ellipticity
+in the harmonic emission [20]. Structural effects [22–26],
+such as the symmetry of the Highest Occupied Molecular
+Orbital (HOMO) as well as interference effects, and ultra-
+fast multielectron dynamics involving lower-lying orbitals
+in the molecule [27] or in the molecular ion [21] have been
+put forward as potential origins for the observed elliptic-
+ity.
+In this article we focus on the role of multielectron
+and multiorbital effects in the neutral CO2 molecule on
+the polarization state of high-order harmonics. We have
+shown previously [27], that results based on the time-
+dependent density functional theory (TDDFT) are in
+excellent agreement with the experimental data for N2
+[20, 21], if contributions from at least three Kohn-Sham
+orbitals are taken into account. Similar strong influence
+of inner shell contributions has been observed and pre-
+dicted for other strong-field processes as well [28–37].
+Our results of numerical TDDFT simulations show
+that indeed the contributions from several valence or-
+bitals contribute to the higher-order harmonic emission
+from CO2. Moreover, we find that the emission from each
+of the orbitals is elliptically polarized. However, our re-
+sults for the total high-order harmonic spectrum, which
+includes the contributions of up to six orbitals, surpris-
+ingly shows, in agreement with the experimental data
+[20], almost no ellipticity.
+Thus, despite the fact that
+high-order harmonic generation from CO2 appears to be
+a multielectron process with several orbitals actively in-
+volved, signatures in the ellipticity of the harmonic emis-
+sion from the different orbitals fade away in the total
+signal.
+The article is organized as follows: In the next sec-
+tion we briefly outline the basics of the time-dependent
+density functional approach used for our numerical sim-
+ulations. We then discuss the application to calculations
+of the ellipticity of high-order harmonic generation of
+molecules, including the proper account of the distribu-
+tion of alignment in the molecular ensemble. Next, we
+arXiv:2301.00356v1  [physics.atm-clus]  1 Jan 2023
+
+2
+compare the results of our calculations with the experi-
+mental data and analyze the contributions from the dif-
+ferent valence orbitals to the total harmonic spectra. We
+end with a brief summary of our results.
+II.
+THEORY
+In the nonperturbative intensity regime the theoret-
+ical study of the interaction of multielectron targets,
+e.g. molecules, with ultrashort laser pulses is challenging.
+An approximative approach to analyze multielectron and
+multiorbital effects in strong-field processes utilizes the
+framework of the time-dependent density functional the-
+ory (TDDFT). In this section we outline the application
+of TDDFT to the calculation of high-harmonic genera-
+tion in molecules, focusing in particular on the evaluation
+of the ellipticity of the radiation in an ensemble of aligned
+molecules.
+A.
+TDDFT for strong-field induced molecular
+processes
+The TDDFT approach is based on the one-to-one cor-
+respondence between the time-dependent electron den-
+sity ρ(r, t) and the time-dependent potential in multi-
+electron Schr¨odinger equation [38]. The density is calcu-
+lated from the time-dependent multielectron Schr¨odinger
+equation expressed as system of auxiliary time-dependent
+noninteracting single-electron Kohn-Sham equations:
+i ∂
+∂tφk(r, t) =
+�
+−∇2
+2 + VKS(r, t)
+�
+φk(r, t)
+(1)
+with
+ρ(r, t) =
+n
+�
+k=1
+fk|φk(r, t)|2
+(2)
+where r is the electronic coordinate, fk is the electron
+population in the k-th Kohn-Sham orbital φk(r, t) and
+n is the number of orbitals. For a molecule interacting
+with a time-dependent intense laser field the Kohn-Sham
+potential
+VKS(r, t) = Vext(r, t) +
+�
+ρ(r′, t)
+|r − r′|dr′ + Vxc(r)
+(3)
+includes the external potential due to the interaction of
+the electron with the N nuclei in the molecule and with
+the time-dependent electric field:
+Vext(r, t) =
+N
+�
+i=1
+−
+Zi
+|Ri − r| + E0(t) sin(ωt)
+n
+�
+k=1
+rk · ˆϵ (4)
+where Zi is the charge of the ith nucleus, ˆϵ is the polar-
+ization direction, ω and E0(t) are the angular frequency
+and the time-dependent amplitude of the laser field. In
+the present calculations we considered a sin2-shaped en-
+velope.
+The exact form of the exchange-correlation potential
+Vxc, which takes account of the multielectron effects, is
+unknown. To use TDDFT for practical calculations, dif-
+ferent approaches have been proposed to design density
+functionals for the exchange-correlation energy (for an
+overview, see e.g., [39]). For the present calculations, we
+have performed systematic studies with various function-
+als and found that functionals based on the local density
+approximation (LDA),
+ELDA
+xc
+[ρ] =
+�
+ρ(r)Vxc(r)dr ,
+(5)
+provide, in general, good results.
+An improvement
+is to take into account the correct asymptotic behav-
+ior (1/r), which can be done, for example, via the
+exchange-correlation potential proposed by van Leeuwen
+and Baerends [40],
+V LB
+xc (α, β; r) = αV LDA
+x
+(r) + βV LDA
+c
+(r)
+(6)
+−
+βx2(r)ρ1/3(r)
+1 + 3βx(r) ln[x2(r) + (x2(r) + 1)1/2],
+where V LDA
+x
+and V LDA
+c
+are the LDA exchange and cor-
+relation potentials and x(r) = |∇ρ(r)|/[ρ(r)]4/3. α and
+β are parameters obtained by fit to the exact exchange-
+correlation function of a certain atomic or molecular sys-
+tem. A similar TDDFT approach for the interaction of
+molecules with strong fields has been used recently by
+Chu and co-workers [41, 42].
+In order to solve the Kohn-Sham equations, Eq. (1),
+we have discretized the wavefunction in space and time
+with uniform step ∆x = 0.03 a.u. and ∆t = 0.03 a.u.,
+which converts the ansatz into a matrix equation using
+the Octopus code [43, 44]. The initial wavefunctions for
+the molecules considered in our study have been obtained
+by solving the eigenvalue problem self-consistently using
+an initial guess and geometry optimized using Octopus
+code as well (this ensures consistency and minimizes risk
+for errors). The wavefunction for each orbital is prop-
+agated forward in time using the enforced time-reversal
+symmetry method. We used grids that extend over 120
+a.u. in polarization direction and 36 a.u. in the trans-
+verse directions. To suppress reflection of the wavefunc-
+tions at the boundary of the grid an imaginary absorbing
+potential has been applied.
+B.
+High-order harmonic generation from an
+ensemble of aligned molecules
+High-order harmonic generation is determined through
+the Fourier transform of the laser induced dipole moment
+in the target. Within the TDDFT formalism, the laser
+
+3
+FIG. 1: Configuration of pump (aligning) pulse in the y − z
+plane, probe (driver) pulse along the ˆz-direction and molecu-
+lar axis.
+induced dipole moment is given by:
+dtot =
+n
+�
+k=1
+dk,
+(7)
+where dk is the contribution to the dipole moment from
+the kth Kohn-Sham orbital,
+dk = ⟨φk(r, t)|r|φk(r, t)⟩ .
+(8)
+The HHG spectrum is then found using the Fourier trans-
+form of the dipole moment, d(ω):
+P(ω) =
+ω4
+12πϵ0c3 d(ω) · d∗(ω) .
+(9)
+For the molecules studied below, the laser induced dipole
+moment has two components, parallel (d||) and perpen-
+dicular (d⊥) with respect to the direction of the electric
+field of the driving laser. The ellipticity of a given har-
+monic is then determined by:
+ϵ =
+�
+1 + r2 −
+�
+1 + 2r2 cos(2δ) + r4
+1 + r2 +
+�
+1 + 2r2 cos(2δ) + r4
+(10)
+where
+r = |d⊥(ω)|
+|d||(ω)|
+(11)
+is the amplitude ratio and
+δ = arg[d⊥(ω)] − arg[d||(ω)]
+(12)
+is the relative phase between the two components. Maxi-
+mum ellipticity, i.e. circular polarization, occurs for r = 1
+and δ = π.
+In the experimental observations of the ellipticity in
+high-order harmonic generation of linear molecules, the
+molecules are often aligned by a pump laser pulse. The
+distribution of the alignment, achieved in the experi-
+ments, is typically measured via ⟨cos2(θ)⟩, where θ is
+the angle between the polarization direction of the pump
+laser and the internuclear axis of the molecule (see Fig.
+1).
+In our simulations we have accounted for the ex-
+perimental alignment of molecular ensemble by solving
+the Kohn-Sham equations for different alignment angles.
+For each angle, we obtained the parallel and perpendicu-
+lar components of the dipole moment and then averaged
+them using the reported alignment distributions.
+III.
+RESULTS
+In this section we present our results for the po-
+larization and ellipticity of high-order harmonics from
+molecules H+
+2 , H2, and CO2.
+The data for the differ-
+ent molecules provide us with the opportunity to com-
+pare our results with those from other theoretical anal-
+ysis (for the one-electron system H+
+2 ) and demonstrate
+how multielectron effects and inner valence shell contri-
+butions influence the harmonics’ ellipticity for the larger
+molecules.
+A.
+Harmonic generation from H+
+2 and H2
+In order to test our numerical calculations, we first
+present results for the one-electron system H+
+2 . In Fig.
+2 we show results for the amplitudes (upper panel) and
+the phase difference (lower panel) for the 57th harmonics
+emitted from H+
+2 as a function of the alignment angle be-
+tween the molecular axis and the polarization direction of
+a driving laser pulse at 800 nm and 3×1014 W/cm2 with
+a pulse duration of 30 fs. The laser parameters are cho-
+sen to be the same as in a recent work by Son et al. [26],
+who studied the ellipticity of high-order harmonic gen-
+eration from H+
+2 using the time-dependent generalized
+pseudospectral method. Our results are in good agree-
+ment with those previously reported for the overall shape
+of the components with a minimum at about 50o for the
+parallel component and a phase jump at the same align-
+ment angle. It has been shown before [24–26], that these
+characteristic features are related to the two-center in-
+terference effect occurring in the parallel component.
+In order to get an impression of the influence of multi-
+electron effects on the ellipticity of high-order harmonics,
+we compare results for H+
+2 (Fig. 3, (a-c)) and H2 (Fig. 3,
+(d-f)) obtained at the same set of laser parameters (800
+nm, 3×1014 W/cm2). In each case we present theoretical
+predictions for four consecutive odd harmonics. For the
+single-electron molecule we observe, in agreement with
+our results in Fig. 2, a maximum close to 1 in the ratio
+of the amplitude in parallel and perpendicular direction
+(a), a rapid change in the phase difference (b) and corre-
+
+Probe
+Pump
+aser
+a
+molecule
+1
+m
+x4
+FIG. 2: Amplitudes of parallel and perpendicular components
+(a) and phase difference (b) of 57th harmonic order of H+
+2 as
+a function of the alignment angle. Laser parameters: 800 nm,
+3 × 1014 W/cm2 and 30 fs.
+spondingly a maximum in the ellipticity (c) around the
+alignment angle, at which the interference minimum in
+the specific harmonic occurs. For H2, one would expect
+a similar pattern for the amplitude and the phase dif-
+ference, since both electrons are in the same molecular
+orbital as in the case of H+
+2 . Indeed, some features in
+the overall trend of the results in Fig. 3 are similar, in
+particular we still note a maximum amplitude ratio (d)
+and a quick phase change (e) at about the same angles
+as for H+
+2 .
+However, for the ratio we observe a much
+narrower structure and for the lowest harmonic a second
+maximum. On the other hand, the data for the phase dif-
+ference are not as smooth as those for the single-electron
+molecule.
+As a result, we observe a much more com-
+plex pattern for the ellipticity of the harmonics generated
+from H2 (f), although some maxima in the structures still
+occur near the alignment angle for the interference min-
+imum. Thus, the comparison for the simplest molecules
+indicates that the ellipticity of high-order harmonics can
+be strongly influenced by multielectron effects. For larger
+molecules we may therefore expect even more complex
+features in the overall ellipticity patterns, since interfer-
+ences from orbitals with different symmetry as well as
+coupling between different orbitals [27, 36] may play ad-
+ditional role.
+FIG. 3: Comparison of amplitude ratio r (a, d), phase differ-
+ence δ (b, e), and ellipticity (c, f) of high order harmonics from
+H+
+2 (a-c) and H2 (d-f) as a function of the alignment angle:
+27th (solid lines), 29th (dashed lines), 31st (dashed-dotted
+lines), and 33th harmonic (dotted lines). Laser parameters as
+in Fig. 2.
+FIG. 4:
+TDDFT results for the intensity ratio of perpendic-
+ular to parallel component of four consecutive harmonics in
+CO2 as a function of the angle between the pump and the 30
+fs probe laser pulse at 800 nm and 1.5 × 1014 W/cm2: 17th
+(red line), 19th (blue line), 21st (green line) and 23rd har-
+monic (black line). For each angle, the experimental reported
+alignment distribution [20] was considered in the calculations.
+B.
+Harmonic generation from CO2
+Next, we analyze the results of our calculations for the
+ellipticity in the harmonic generation from the more com-
+plex but linear triatomic molecule CO2, which has been
+also studied experimentally [20].
+In order to compare
+with the experimental data, we have obtained the inten-
+
+0.008
+Amplitude [arb. units]
+0.007
+(a)
+0.006
+0.005
+0.004
+0.003
+0.002
+0.001
+0.000
+10
+30
+50
+70
+90
+Alignment angle [degree]
+1.0
+(b)
+Phase difference [π]
+0.5
+0.0
+0.5
+-1.0
+10
+30
+50
+70
+90
+Alignment angle [degree]2.5
+16
+(a)
+(d)
+amplitude ratio
+2.0
+amplitude ratio
+12
+1.5
+8
+1.0
+4
+0.5
+0.0
+0
+10
+30
+50
+70
+90
+10
+30
+50
+70
+90
+1.0
+1.0
+(b)
+(e)
+0.5
+0.5
+0.0
+0.0
+-0.5
+-0.5
+-1.0
+-1.0
+10
+30
+50
+70
+90
+10
+30
+50
+70
+90
+1.0
+1.0
+(c)
+(f)
+0.8
+0.8
+ellipticity
+ellipticity
+0.6
+0.6
+0.4
+0.4
+0.2
+0.2
+0.0
+0.0
+10
+30
+50
+70
+90
+10
+30
+50
+70
+90
+alignment angle (degree)
+alignment angle (degree)0.075
+H17
+-H19
+H21
+Intensity Ratio
+0.05
+一H23
+0.025
+-100
+-50
+0
+50
+100
+Pump-probe 
+angle (degree)5
+FIG. 5:
+Comparison of results for the ellipticity of high-order
+harmonics as a function of alignment angle for CO2: without
+(a) and with averaging (b). Laser parameters as in Fig. 4.
+sity ratio of the perpendicular to parallel component of
+the harmonic emission as a function of the angle between
+the pump and probe laser pulse.
+For each orientation
+angle considered, we have taken into account the exper-
+imentally reported alignment distribution by performing
+an average over the simulation results for the respective
+alignment angles in the distribution. Our results in Fig. 4
+show a rather small intensity ratio and, hence, relatively
+small ellipticity with a maximum at about a relative an-
+gle of about 60o between polarization direction of pump
+and probe pulse for each of the harmonics studied exper-
+imentally. The absolute values as well as the position of
+the maxima are in very good agreement with the obser-
+vations by Zhao et al. [20]. The observed and calculated
+rather weak perpendicular component of the harmonics
+in CO2 is in contrast to results for N2, for which both
+experiment [20, 21] and TDDFT [27] as well as other
+calculations [21, 24, 26] show a strong ellipticity for the
+emitted harmonics at certain alignment angles.
+Part of the explanation for the weak ellipticity is due
+to the ensemble angle average effect, which reduces the
+overall ellipticity, as observed before in N2 [27]. The ef-
+fect can be seen from the comparison of the harmonics
+ellipticity as a function of the alignment angle without (a)
+and with (b) average in Fig. 5. It is clearly seen that,
+in particular for the lower-order harmonics (below 15th
+harmonics), without averaging there is a strong elliptic-
+ity for certain alignment angles which disappears after
+alignment average is taken into account. In contrast, for
+the experimentally reported data in the range of 17th to
+23rd harmonics the averaging process does have a smaller
+effect only.
+In this latter range of harmonics from CO2 the main
+origin for the weak ellipticity is actually the role of mul-
+tielectron effects involving contributions from several or-
+bitals. In order to analyze these contributions, we com-
+pare in Fig. ?? the ellipticity of the harmonic response
+from the HOMO only (a) with those when adding subse-
+quently the contributions from the inner valence orbitals
+up to HOMO-5 (f). The comparison shows that the ellip-
+ticity of high-order harmonics from CO2 is influenced by
+the six valence orbitals considered. While the ellipticity
+of 17th to 23rd harmonics generated from the HOMO is
+rather large for certain alignments angles, the ellipticity
+FIG. 6:
+Ellipticity of high-order harmonics as a func-
+tion
+of
+alignment
+angle
+for
+CO2.
+Starting
+with
+the
+results
+from
+HOMO
+only
+(a):
+(1πg)4,
+contribu-
+tions
+from
+inner
+valence
+orbitals
+are
+added
+subse-
+quently
+in
+the
+other
+panels:
+(b)
+(3σu)2(1πg)4,
+(c)
+(1πu)4(3σu)2(1πg)4,
+(d)
+(4σg)2(1πu)4(3σu)2(1πg)4,
+(e)
+(2σu)2(4σg)2(1πu)4(3σu)2(1πg)4,
+(f)
+(3σg)2(2σu)2(4σg)2
+(1πu)4(3σu)2(1πg)4. Laser parameters as in Fig. 4.
+gradually gets weaker as more contributions are added.
+In contrast, for harmonics around the cutoff there re-
+mains a strong ellipticity at some alignment angles.
+The ellipticity of higher-order harmonics at certain
+alignment angles from the HOMO (3πg) can be under-
+stood based on the two-center interference effect, similar
+as in the case of H+
+2 and H2 above. The importance of
+such orbital structure effect for the harmonic generation
+from the HOMO of CO2 has been pointed out before
+[24]. The strong contributions from the inner valence or-
+bitals originate on a variety of effects. Both, HOMO-1
+(2σu) and HOMO-2 (1πu) have a different orbital symme-
+try than the HOMO of CO2. Therefore, ionization and,
+hence, harmonic generation, from HOMO is suppressed
+due to destructive interference at alignment angles of 0◦
+and 90◦ while it is at maximum around 45◦ [45]. In con-
+trast, the ionization rate is largest at 0◦ for HOMO-1 and
+90◦ for HOMO-2.
+Consequently, high-order harmonic
+generation from these two orbitals contributes strongly
+close to alignment angles at which the signal from the
+HOMO is weakest, despite the fact that the ionization
+
+30
+0.9
+25
+0.8
+0.7
+Harmonic order
+20
+0.6
+0.5
+15
+0.4
+10
+0.3
+0.2
+5
+0.1
+0
+20
+40
+60
+80
+Angle(Degrees25
+0.9
+20
+0.8
+0.7 
+Harmonic order
+15
+0.6
+0.5
+0.4
+10
+0.3
+0.2 
+5
+0.1 
+0
+20
+40
+60
+80
+Angle (Degrees)25
+25
+(a)
+(d)
+Harmonic order
+20
+0.8
+Harmonicorder
+20
+0.8
+15
+0.6
+15
+0.6
+10
+0.4
+10
+0.4
+0.2
+0.2
+0
+20
+40
+60
+80
+0
+20
+40
+60
+80
+Angle[degrees]
+Angle[degrees]
+25
+25
+(e)
+(b)
+0.8
+20
+0.8
+Harmonic
+15
+0.6
+15
+0.6
+10
+0.4
+10
+0.4
+0.2
+0.2
+0
+0
+20
+40
+60
+80
+0
+20
+40
+60
+80
+Angledegrees
+Angle [degrees]
+25
+25
+(c)
+(f)
+Harmonic order
+20
+0.8
+order
+20
+0.8
+15
+0.6
+Harmonic
+15
+0.6
+10
+0.4
+10
+0.4
+0.2
+0.2
+0
+20
+40
+60
+80
+0
+20
+40
+60
+80
+Angle[degrees
+Angle[degrees6
+FIG. 7: Rotational averaging assuming distribution 1.The rest
+of the notation and parameters as in fig.6.
+potential for the inner valence orbitals is smaller than
+that of the HOMO.
+As for the other inner valence orbitals, that have an
+even higher ionization potential, we have found that these
+are either strongly coupled to one of the higher lying
+states or among each other by the driving field. In the
+case of the HOMO-3 state (2σg), the projection onto the
+HOMO-1 state is shown in Fig. 9(a). We observe a strong
+coupling driven by the field although the frequency is
+non-resonant.
+This explains the significant change in
+the ellipticity pattern upon inclusion of the HOMO-3
+state (Fig. ??(d)). Finally, HOMO-4 and HOMO-5 states
+slightly contribute to the 17th to 23rd harmonic genera-
+tion at the given parameters and, hence, to the ellipticity
+pattern, since these two orbitals are coupled with each
+other, leading to a population transfer of about 40% (see
+Fig. 9(b)).
+To summarize, our results obtained within the time-
+dependent density functional theory indicate that high-
+order harmonic generation from CO2 is influenced by
+multielectron effects with contributions from a significant
+number of inner-valence orbitals, besides the contribution
+from the HOMO. The harmonic emission from these or-
+bitals is strongest at different alignment angles due to
+interference effects arising from the specific orbital struc-
+tures and there is a strong laser driven coupling between
+certain orbitals. As a result, the overall ellipticity of the
+FIG. 8: Rotational averaging assuming disributions 2. The
+rest of the notation and parameters as in fig.6.
+higher-order harmonics is rather small, except for the
+cutoff harmonics. The partial alignment and the related
+averaging of the results for different orientation angles
+further diminishes the ellipticity.
+Acknowledgments
+This work was supported by the U.S. National Science
+Foundation (Grants Nos. Grant No. PHY-1734006 and
+Grant No. PHY-2110628). This work utilized the Sum-
+mit supercomputer, which was supported by the U.S. Na-
+tional Science Foundation and the University of Colorado
+Boulder.
+[1] A. McPherson, G. Gibson, H. Jara, T.S. Luk, I.A. McIn-
+tyre, K. Boyer and C.K. Rhodes, J. Opt. Soc. Am. B 4,
+595 (1987).
+[2] M. Ferray, A. L’Huillier, X.F. Li, L.A. Lompre, G. Main-
+
+25
+25
+a
+(d)
+Harmonic order
+20
+0.8
+Harmonicorder
+20
+0.8
+5
+0.6
+15
+0.6
+10
+0.4
+10
+0.4
+0.2
+5
+0.2
+0
+0
+20
+40
+60
+80
+0
+20
+40
+60
+80
+Angle[degrees]
+Angle[degrees
+25
+25
+(b)
+e
+0.8
+Harmonic order
+20
+0.8
+15
+0.6
+15
+0.6
+10
+0.4
+10
+0.4
+0.2
+0.2
+0
+20
+40
+60
+80
+0
+20
+40
+60
+80
+Angle [degrees]
+Angle[degrees
+25
+25
+(c)
+(f)
+0.8
+Harmonicorder
+20
+0.8
+Harmonic
+15
+0.6
+15
+0.6
+10
+0.4
+10
+0.4
+5
+0.2
+5
+0.2
+0
+20
+40
+60
+80
+0
+20
+40
+60
+80
+Angle[degrees
+Angle[degrees25
+25
+a
+(d)
+Harmonic order
+20
+0.8
+20
+0.8
+Harmonic orde
+15
+0.6
+15
+0.6
+10
+0.4
+10
+0.4
+0.2
+5
+0.2
+0
+0
+20
+40
+60
+80
+0
+20
+40
+60
+80
+Angle[degrees]
+Angle[degrees]
+25
+25
+(b)
+le
+0.8
+20
+0.8
+15
+0.6
+15
+0.6
+10
+0.4
+10
+0.4
+0.2
+0.2
+0
+0
+20
+40
+60
+80
+0
+20
+40
+60
+80
+Angle [degrees]
+Angle[degrees
+25
+25
+(c)
+(f)
+0.8
+e 20
+0.8
+0.6
+Harmonic
+15
+0.6
+10
+0.4
+10
+0.4
+5
+0.2
+0.2
+0
+20
+40
+60
+80
+0
+20
+40
+60
+80
+Angle[degrees
+Angle[degrees]7
+FIG. 9:
+Projection of coupled inner valence orbitals (a)
+HOMO-3 (4σg) to HOMO-1 (3πu) (a) and (b) HOMO-5 (3σg)
+to HOMO-4 (2πu) (b) for an alignment angle of 20o. Laser
+parameters as in Fig. 4.
+fray and C. Manus, J. Phys. B 21, L31 (1988).
+[3] T. Popmintchev, M.-C. Chen, D. Popmintchev, P. Arpin,
+S. Brown, S. Alisauskas, G. Andriukaitis, T. Balciu-
+nas, O.D. M¨ucke, A. Pugzlys, A. Baltuska, B. Shim,
+S.E. Schrauth, A. Gaeta, C. Hernandez-Garcia, L. Plaja,
+A. Becker, A. Jaron-Becker, M.M. Murnane and H.C.
+Kapteyn H C, Science 91, 1287 (2012).
+[4] M. Hentschel, R. Kienberger, C. Spielmann, G.A. Reider,
+N. Milosevic, T. Brabec, P. Corkum, U. Heinzmann, M.
+Drescher and F. Krausz F, Nature 414, 509 (2001).
+[5] P.M. Paul, E.S. Toma, P. Breger, G. Mullot, F. Auge,
+Ph. Balcou, H.G. Muller and P. Agostini, Science 292,
+1689 (2001).
+[6] M. Lein, N. Hay, R. Velotta, J.P. Marangos and P.L.
+Knight, Phys. Rev. Lett. 88, 183903 (2002).
+[7] J. Itatani, J. Levesque, D. Zeidler, H. Niikura, H. Pepin,
+J.C. Kieffer, P.B. Corkum and D.M. Villeneuve, Nature
+432, 867 (2004).
+[8] H.J. W¨orner, H. Niikura, J.B. Bertrand, P.B. Corkum
+and D.M. Villeneuve, Phys. Rev. Lett. 102, 103901
+(2009).
+[9] S. Haessler, J. Caillat, W. Boutu, C. Giovanetti-Teixeira,
+T. Ruchon, T. Auguste, Z. Diveki, P. Breger, A. Maquet,
+B. Carre, R. Taieb and P. Salieres, Nat. Phys. 6, 200
+(2010).
+[10] C. Vozzi, M. Negro, F. Calegari, G. Sansone, M. Nisoli,
+S. De Silvestri and S. Stagira, Nat. Phys. 7, 822 (2011).
+[11] S.B. Schoun, R. Chirla, J. Wheeler, C. Roedig, P. Agos-
+tini, L.F. DiMauro, K.J. Schafer and M.B. Gaarde, Phys.
+Rev. Lett. 112, 153001 (2014).
+[12] S. Baker, J.S. Robinson, C.A. Haworth, H. Teng, R.A.
+Smith, C.C. Chirila, M. Lein, J.W.G. Tisch and J.P.
+Marangos, Science 312, 424 (2006).
+[13] W. Li, X. Zhou, R. Lock, S. Patchkovskii, A. Stolow, H.C.
+Kapteyn and M.M. Murnane, Science 322, 1207 (2008).
+[14] C.I. Blaga, J. Xu, A.D. DiChiara, E. Sistrunk, K. Zhang,
+P. Agostini, T.A. Miller, L.F. DiMauro and C.D. Lin,
+Nature 483, 194 (2012).
+[15] M. Miller, A. Jaron-Becker and A. Becker, Phys Rev A
+93 013406 (2016).
+[16] H.J. W¨orner,
+J.B. Bertrand,
+D.V. Kartashov,
+P.B.
+Corkum and D.M. Villeneuve, Nature 466, 604 (2010).
+[17] K.J. Schafer, B. Yang, L.F. DiMauro and K.C. Kulander,
+Phys. Rev. Lett. 70, 1599 (1993).
+[18] P.B. Corkum, Phys. Rev. Lett. 71, 1994 (1993).
+[19] M. Lewenstein, Ph. Balcou, M.Yu. Ivanov, A. L’Huillier
+and P.B. Corkum, Phys. Rev. A 49, 2117 (1994).
+[20] X. Zhao, R. Lock, N. Wagner, W. Li, H.C. Kapteyn and
+M.M. Murnane, Phys. Rev. Lett. 102, 073902 (2009).
+[21] Y. Mairesse, J. Higuet, N. Dudovich, D. Shafir, B. Fabre,
+E. Mevel, E. Constant, S. Patchkovskii, Z. Walters,
+M.Yu. Ivanov and O. Smirnova, Phys. Rev. Lett. 104,
+213601 (2010).
+[22] A. Etches, C.B. Madsen and L.B. Madsen, Phys. Rev. A
+81, 013409 (2010).
+[23] S. Ramakrishna, P.A.J. Sherratt, A.D. Dutoi and T. Sei-
+deman, Phys. Rev. A 81, 021802(R) (2010).
+[24] A.-T. Le, R.R. Lucchese and C.D. Lin, Phys. Rev. A 82,
+023814 (2010).
+[25] E.V. van der Zwan and M. Lein Phys. Rev. A 82, 033405
+(2010).
+[26] S.-K. Son, D.A. Telnov and S.I. Chu, Phys. Rev. A 82,
+043829 (2010).
+[27] Y. Xia and A. Jaron-Becker, Opt. Lett. 39, 1461 (2014).
+[28] G.N. Gibson, R.R. Freeman and T.J. McIllrath, Phys.
+Rev. Lett. 67, 1230 (1991).
+[29] A. Talebpour, S. Petit and S.L. Chin, Opt. Commun.
+,171 285 (1999).
+[30] A. Becker, A.D. Bandrauk and S.L. Chin, Chem. Phys.
+343, 345 (2001).
+[31] A. Jaron-Becker, A. Becker and F.H.M. Faisal, Phys.
+Rev. A 69, 023410 (2004).
+[32] B. McFarland, J.P. Farrell, P.H. Bucksbaum and M.
+G¨uhr, Science 322, 1232 (2008).
+[33] A.T. Le, R.R. Lucchese and C.D. Lin, J. Phys. B: At.
+Mol. Opt. Phys. 42, 211001 (2009).
+[34] O. Smirnova, Y. Mairesse, S. Patchkovskii, N. Dudovich,
+D. Villeneuve, P.B. Corkum and M.Yu. Ivanov, Nature
+460, 972 (2009).
+[35] Z. Diveki, A. Camper, S. Haessler, T. Auguste, T. Ru-
+chon, B. Carre, P. Salieres, R. Guichard, J. Caillat, A.
+Maquet and R. Taieb, New J. Phys. 14, 023062 (2012).
+[36] Y. Xia and A. Jaron-Becker, Opt. Express 24, 4689
+(2016).
+[37] D.L.
+Smith
+and
+G.N.
+Gibson,
+Phys.
+Rev.
+A
+97,
+021401(R) (2018).
+[38] E. Runge and E.K.U. Gross, Phys. Rev. Lett. 52, 997
+(1984).
+[39] E.S. Gustavo and N.S. Viktor, in Theory and Appli-
+cations of Computational Chemistry (Elsevier, 2005) p
+669ff.
+[40] R. van Leeuwen and E.J. Baerends, Phys. Rev. A 49,
+2421 (1994).
+[41] X. Chu and S.I. Chu, Phys. Rev. A 63, 023411 (2001).
+[42] X. Chu and S.I. Chu, Phys. Rev. A 64, 063404 (2001).
+[43] M.A.L. Marques, A. Castro, G. F. Bertsch, and A. Rubio,
+
+1.0
+(a)
+0.8
+Projection
+0.6
+0.4
+0.2
+0.0
+0
+5
+10
+15
+20
+25
+30
+35
+Time [fs]
+1.0
+(b)
+0.8
+Projection
+0.6
+0.4
+0.2
+0.0
+0
+5
+10
+15
+20
+25
+30
+35
+Time [fs]8
+Comput. Phys. Commun. 151, 60 (2003).
+[44] A. Castro, H. Appel, M. Oliveira, C.A. Rozzi, X. An-
+drade, F. Lorenzen, M.A.L. Marques, E.K.U. Gross, and
+A. Rubio, Phys. Stat. Sol. B 243, 2465 (2006).
+[45] A.-T. Le, R.R. Lucchese, M.T. Lee and C.D. Lin, Phys.
+Rev. Lett. 102, 203001 (2009).
+

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+Dynamical Signatures of Liouvillian Flat Band
+Yu-Guo Liu1 and Shu Chen1, 2, 3, ∗
+1Beijing National Laboratory for Condensed Matter Physics,
+Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
+2School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
+3Yangtze River Delta Physics Research Center, Liyang, Jiangsu 213300, China
+(Dated: January 16, 2023)
+Although flat-band structures have attracted intensive studies in condensed matter and optical physics due to
+their eigenstates exhibiting huge degeneracy and allowing for the localization of wave packet, it is not clear how
+the flat band of Liouvillian influences the relaxation dynamics of open quantum systems. To this end, we study
+the dynamical signatures of Liouvillian flat band in the scheme of Lindblad master equation. Considering a chain
+model with gain and loss, we demonstrate three kinds of band dispersion of Liouvillian: flat bland, dispersionless
+only in the real part and imaginary part, and capture their dynamical signatures: when the rapidity spectrum of
+Liouvillian is flat, the particle numbers in different sites relax to its steady state value with the same decay rate;
+when the real or imaginary part of rapidity spectrum is dispersionless, the relaxation behaviors have oscillating
+or forked characteristics. We also unveil that the Liouvillian flat band can lead to dynamical localization, which
+is characterized by the halt of propagation of a local perturbation on the steady state.
+Introduction.— Band structure of a Hamiltonian plays an
+important role in understanding the motion of particles in pe-
+riodic crystals. Usually, special band structures may give rise
+to exotic quantum phenomena, for example, low-energy ex-
+citations of electrons on a linear dispersive band in graphene
+behave like massless Dirac fermions [1, 2]. Another instance
+is the flat band (FB) in which all electrons carry the same en-
+ergy regardless of their momentum. Due to the dispersionless
+band structure, particles in FB have arbitrarily large effective
+mass, so they will be localized in real space. Especially, in
+strongly correlated systems, heavy degeneracy and zero ki-
+netic energy in FB can increase density of electronic states
+and highlight Coulomb interaction, leading to rich many-body
+phenomena [3–5].
+In open quantum systems, dynamics of density matrix ρ is
+described by Lindblad master equation (LME) under Born-
+Markov approximation [6–8]:
+dρ
+dt = L(ρ) := −i[H, ρ] +
+�
+µ
+�
+LµρL†
+µ − 1
+2{L†
+µLµ, ρ}
+�
+,
+(1)
+where L is called the Liouvillian superoperator, H is the
+Hamiltonian of system, and Lµ are Lindblad operators which
+reflect the coupling between system and environment. The
+Planck constant ℏ is set to unity throughout this Letter. There
+have been several methods developed to obtain the spectrum
+of L, especially for quadratic systems [9–15]. In Ref.[15],
+a route for realizing dispersionless bands is proposed based
+on the underlying mechanism with the emergence of a dis-
+sipationless dark space. Generally speaking, the short-time
+dynamics is related to the Liouvillian eigenvalues with large
+modulus of the real part, whereas the long-time relaxation to
+the smallest modulus beyond zero (the so called Liouvillian
+gap) [16–22]. However, how the structure of Liouvillian, es-
+pecially the Liouvillian flat band (LFB), influences dynamics
+is still a subtle and unexplored question.
+In this Letter, we focus on the dynamics of open quan-
+tum systems with LFB. In comparison with the real spectrum
+of Hamiltonian system, the Liouvillian spectrum is complex,
+and thus the corresponding rapidity spectrum can exhibit more
+rich structures with dispersionless band in both imaginary and
+real part or either of them. To make our study concrete, we
+shall first apply a geometrically intuitive method to construct
+lattice with correlated gain and loss, which supports LFB, and
+explore the generality of dynamical signatures associated with
+the structure of Liouvillian spectrum. We show that the ra-
+pidity spectra from Liouvillian and damping-matrix spectra
+of correlation functions have the same dispersion characteris-
+tics, which lead to different signatures of damping dynamics
+of local particle number distribution: oscillating, forked, syn-
+chronous damping are related to the band dispersionless only
+in imaginary part, real part and in both parts, respectively. Fur-
+thermore, we exactly solve the model and show that the LFB
+can induce dynamical localization, which is characterized by
+the halt of the propagation of a local perturbation on the non-
+equilibrium steady state (NESS).
+Formalism.— The density matrix ρ and Liouvillian super-
+operator L in Eq. (1) can be formally expressed as
+ρ =
+�
+IJ
+ρIJ|I⟩a⟨J|a, L(ρ) =
+�
+i j
+Fi(a, a†) ρ Fj(a, a†), (2)
+where a is the set of fermionic annihilation operators i.e. a =
+(a1, a2, · · · ), Fi(a, a†) is a function with variables among a
+and a†, I = (I1, I2, · · · ), J = (J1, J2, · · · ) and
+|I⟩a⟨J|a = (a†
+1)I1(a†
+2)I2 · · · (a†
+L)IL|0⟩a⟨0|a(aL)JL · · · (a1)J1, (3)
+where |0⟩a is the vacuum state for all a−fermions. For the
+convenience of analysis and calculation, we map fermionic
+LME into a new representation referred to as C by following
+the method in Ref. [10]:
+ρ → | ρ⟩C =
+�
+IJ
+ρIJ(a†
+1)I1 · · · (a†
+L)IL(c†
+1 ˆP)J1 · · · (c†
+L ˆP)JL |0⟩,
+(4a)
+L → ˆLC =
+�
+i j
+Fi(a, a†) F T
+j ( ˆPc, c† ˆP),
+(4b)
+arXiv:2301.05378v1  [cond-mat.other]  13 Jan 2023
+
+2
+where c = (c1, c2, · · · ) is the set of annihilation operators
+of c−fermions, which is a one-to-one mapping from a, T
+means matrix transpose, and |0⟩ is the vacuum state of both
+a− and c−fermions.
+ˆP is the parity operator defined by
+ˆP = exp
+�
+iπ �
+j(a†
+jaj + c†
+jcj)
+�
+, which is introduced to ensure
+fermionic anticommutation relations between a−fermions and
+c−fermions. Full mapping process is shown in the Supple-
+mental Material [25].
+Model.— We consider a Liouvillian in a periodic chain:
+L(ρ) = −i[H, ρ] + (1 − w)DL(ρ) + (1 + w)DR(ρ),
+(5)
+where H = �
+l J(a†
+l+1al + h.c.), w ∈ [−1, 1], and
+DL(ρ) =
+�
+l
+�
+2AlρA†
+l − A†
+l Alρ − ρA†
+l Al
+�
+,
+DR(ρ) =
+�
+l
+�
+2A†
+l ρAl − AlA†
+l ρ − ρAlA†
+l
+�
+,
+(6)
+where Al = √γ1a†
+l + √γ2al+1. The operators Al and A†
+l tie the
+gain and loss of neighboring sites together, which could be re-
+alized by optical superlattice with Bose-Einstein condensate
+reservoir [23]. The role of w ∈ [−1, 1] is analogous to the sta-
+tistical distribution from temperature [24]. Mapping Eq. (5)
+into the representation C, we get a ladder model consisting of
+a−fermion chain and c−fermion chain (see the Supplemental
+Material [25]). The L is mapped to ˆL = ˆH + (1 − w) ˆDL + (1 +
+w) ˆDR, where ˆH = �
+l
+�
+−iJ(a†
+l+1al + h.c.) + iJ(c†
+l+1cl + h.c.)
+�
+.
+ˆDL and ˆDR are illustrated in Fig. 1 (a) and (b), which have
+leftward and rightward hoppings, respectively, along two di-
+agonals of every plaquette in the ladder. The cross-stitch-type
+hopping is crucial for generating FB because it can form a
+destructive-interference structure, which consists with our ex-
+perience in the FB ladder models [34–37].
+In momentum space,
+ˆL can be expressed in BdG
+form
+as
+ˆL
+=
+0.5 ˆLk=0 + �π−
+k=0+ ˆLk,
+where
+ˆLk
+=
+(a†
+k c†
+k a−k c−k) Lk (ak ck a†
+−k c†
+−k)T − 4γ and γ = γ1 + γ2.
+Due to parity conservation in ˆL, the operator ˆP can be substi-
+tuted by a constant P which equals 1(−1) when ˆL acts on the
+state with even (odd) fermions. Then we have
+Lk = −i2J cos kσz ⊗ σz − 4 √γ1γ2 cos kPσz ⊗ σx −
+2γPσy ⊗ σy + 2w
+�
+(γ2 − γ1)σz ⊗ I + 2 √γ1γ2 sin kσy ⊗ σz
++i(γ2 − γ1)Pσx ⊗ σy + i2 √γ1γ2 sin kPI ⊗ σx
+�
+,
+(7)
+where I and σi are identity and Pauli matrices.
+ˆLk can
+be diagonalized as ˆLk
+= λ−(k)
+�
+ζ
+′
+1(k)ζ1(k) + ζ
+′
+4(k)ζ4(k)
+�
++
+λ+(k)
+�
+ζ
+′
+2(k)ζ2(k) + ζ
+′
+3(k)ζ3(k)
+�
+, where ζ
+′
+i(k) and ζj(k
+′) ful-
+fill anticommutation relations: {ζ
+′
+i(k), ζj(k
+′)} = δi jδkk′ and
+{ζ
+′
+i(k), ζ
+′
+j(k
+′)} = {ζi(k), ζj(k
+′)} = 0 [9]. The λ±(k) is called
+rapidity spectrum given by λ±(k) = −2γ ± 2mk for both odd
+and even parity [38], where
+mk =
+�����
+�
+(4γ1γ2 − J2) cos2 k,
+J2 ≤ 4γ1γ2,
+i
+�
+(J2 − 4γ1γ2) cos2 k,
+J2 > 4γ1γ2.
+(8)
+FIG. 1. ˆDL, ˆDR and ˆL = ˆH + ˆDL + ˆDR are sketched by (a), (b) and
+(c), where the color ovals, straight lines (with or without arrow) and
+wavy lines represent onsite loss, particle hopping and pair production
+and annihilation. The bule, red and orange ovals are corresponding
+to terms (γ1 − γ2)ˆna/c, l − γ1, (γ2 − γ1)ˆna/c, l − γ2 and constant loss
+−γ.
+ˆna/c, l is the particle number operator of a− or c−fermion on
+the site l. Horizontal black wavy lines represent ± √γ1γ2 ˆP(alal+1 +
+h.c.) or ± √γ1γ2 ˆP(clcl+1 +h.c.). The black arrows indicate directional
+hoppings with strength −2 √γ1γ2 ˆP. The bule, red and orange vertical
+wavy lines are corresponding to 2γ1 ˆPa†
+l c†
+l + 2γ2 ˆPclal, 2γ2 ˆPa†
+l c†
+l +
+2γ1 ˆPclal and 2γ ˆP(a†
+l c†
+l + clal). (d) shows the (c) in even parity and
+under flat band condition, where J = 2 √γ1γ2 = 1. The dashed wavy
+lines indicate the pairing terms have no effect on single particle- or
+hole- excitation on its steady state.
+The λ±(k) is independent with w and we show it in Fig. 2.
+When J2 = 4 √γ1γ2, λ is a FB of k. When J2 < 4 √γ1γ2
+(J2 > 4 √γ1γ2), λ is dispersionless in its imaginary (real) part.
+Especially, in Fig. 2 (c) and (f) the spectrum is pure real, which
+indicates Lk possessing a pseudo-Hermiticity [39–41], while
+in Fig. 2 (a) and (d) the complex spectrum shows the break-
+ing of pseudo-Hermiticity. Since the Liouvillian spectrum is
+obtained by sum of different number of λ±(k), it inherits the
+characteristics of rapidity spectrum, as shown in Fig. 2 (g)∼(i).
+When J2 = 4 √γ1γ2, Liouvillian spectrum consists of some
+highly degenerate discrete points (Fig. 2 (h)), corresponding
+to different occupations of the FB of rapidity spectrum, so we
+call this kind of Liouvillian spectrum as the LFB.
+Two-operator correlation functions.— By making Fourier
+transform, Eq. (5) becomes
+L(ρ) =
+π
+�
+k=−π
+�
+−i2J cos k[ˆnk, ρ]+(1−w)DL
+k(ρ)+(1+w)DR
+k (ρ)
+�
+,
+(9)
+where DL
+k(ρ) = 2BkρB†
+k − {B†
+kBk, ρ}, DR
+k (ρ) = 2B†
+kρBk −
+{BkB†
+k, ρ} and Bk = √γ1eika†
+k + √γ2a−k.
+We define two-
+operator correlation functions: Gk1, k2 = Tr(a†
+k1ak2ρ), Dk1, k2 =
+Tr(ak1ak2ρ), and D∗
+k1, k2 = Tr(a†
+k2a†
+k1ρ). In terms of the correla-
+tion function vector Ψk1k2 = (Gk1,k2,G−k2,−k1, Dk2,−k1, D∗
+k1,−k2)T,
+the dynamical evolution is governed by the following closed
+
+(a)
+(b)
+Y102.
+Y12P
+a
+a
+a
+C
+C
+C
+h121
+(c)
+(d)
+iJ
+2J
+2
+2
+a
+a
+a
+a
+a
+a
+c
+c
+C
+c
+C
+iJ
+iJ
+2
+23
+(b)
+(a)
+(d)
+(e)
+(c)
+(f)
+(g)
+(h)
+(i)
+FIG. 2. (a)∼(c) the real part of rapidity spectra λ±(k). (d)∼(f) the
+imaginary part of λ±(k). (g)∼(i) the Liouvillian spectra obtained by
+exactly diagonalizing 6-site lattice with w = 0, J = 1 and γ1 = 0.25
+for all subfigures. γ2 = 0.5 in (a), (d) and (g). γ2 = 1 in (b), (e) and
+(h). γ2 = 1.5 in (c), (f) and (i).
+equation:
+d
+dtΨk1k2 = Xk1k2Ψk1k2 + Vk1k2,
+(10)
+where
+Xk1k2 = −4γI ⊗ I + i2J cos k1σz ⊗ σz − i2J cos k2I ⊗ σz
++ 4 √γ1γ2 cos k1σx ⊗ σz − 4 √γ1γ2 cos k2σy ⊗ σy (11)
+and Vk1k2
+=
+δk1,k2
+�
+2γ + 2w(γ2 − γ1), 2γ + 2w(γ2 −
+γ1), i4w √γ1γ2 sin k1, −i4w √γ1γ2 sin k1
+�T.
+The
+damping
+matrix
+Xk1k2
+has
+four
+eigenstates
+which
+fulfill
+the
+equation
+Xk1k2|Γ±±
+k1k2⟩
+=
+Γ±±
+k1k2|Γ±±
+k1k2⟩
+with
+the
+eigenvalues
+given
+by
+Γ±±
+k1k2
+=
+−4γ ±
+2
+�
+4γ1γ2 − J2 �
+(| cos k1| ± | cos k2|sgn(4γ1γ2 − J2))2,
+where
+sgn(x) is a sign function. Γ also has a transition from the
+complex to the real by decreasing J due to the PT −symmetry
+of Xk1k2.
+In the Supplemental Material [25] we show that
+Xk1k2 has higher symmetry than ˆLk, which makes Xk1k2 have a
+similar band structure as ˆLk. In Fig. 3, we see that Γ±±
+k1k2 fully
+inherits the dispersion characteristics of real and imaginary
+part from the rapidity spectra in Fig. 2.
+(b)
+(a)
+(d)
+(e)
+(c)
+(f)
+FIG. 3. (a)∼(c) the real part of Γ±±
+k1k2. (d)∼(f) the imaginary part of
+Γ±±
+k1k2. J = 1 and γ1 = 0.25 are for all subfigures. γ2 = 0.5 in (a) and
+(d). γ2 = 1 in (b) and (e). γ2 = 1.5 in (c) and (f).
+Flat-band damping dynamics.— Damping dynamics dis-
+plays the converging processes from initial state to NESS [45].
+Here, we show that the “flat band” in real or imaginary
+or both parts will effectively influence the damping behav-
+iors in real space.
+We concentrate on the vector Ψl1l2 =
+(Gl1,l2,Gl2,l1, Dl2,l1, D∗
+l1,l2)T consisting of real-space correlation
+functions:
+Gl1, l2 = Tr(a†
+l1al2ρ), Dl1, l2 = Tr(al1al2ρ), D∗
+l1, l2 = Tr(a†
+l2a†
+l1ρ).
+Introduce the deviating expectation of operator ˆO as �
+O(t) =
+⟨ ˆO⟩(t) − ⟨ ˆO⟩S to describe the deviation from steady state ex-
+pectation value ⟨ ˆO⟩S
+= ⟨ ˆO⟩(∞).
+From Eq. (10), we get
+d
+dt �Ψk1k2 = Xk1k2�Ψk1k2.
+Making Fourier transformation, we
+have �Ψl1l2(t) = �
+k1k2 ei(−k1l1+k2l2)�Ψk1k2(t).
+Decomposing ar-
+bitrary initial state �Ψk1k2(0) by the eigenstates of Xk1k2 i.e.
+�Ψk1k2(0) = �
+αβ Cαβ
+k1k2|Γαβ
+k1k2⟩, where α and β take ±, then we
+have
+�Ψl1l2(t) =
+�
+k,µ
+eik·˜rCµ
+ket Γµ
+k |Γµ
+k⟩,
+(12)
+where k = (k1, k2), ˜r = (−l1, l2) and µ = (α, β). For non-zero
+Liouvillian gap, the system exponentially decays to NESS
+with time, so we can define instantaneous decay rate K(t) of
+the j component of �Ψl1l2(t) as
+K j
+l1l2 = d
+dt log
+�
+|�Ψj
+l1l2(t)|
+�
+.
+(13)
+Below we unveil how K(t) is affected by the dispersion of
+Γµ
+k through Fig. 4, in which the damping behaviors of local
+deviating particle number �nl = �
+Gll from the initial state with a
+single excitation on site 1 are shown:
+(i) When FB appears, Γµ
+k becomes a constant, denoted
+by Γ0. Then we have �Ψl1l2(t) = eΓ0t �
+k,µ eik·˜rCµ
+k|Γµ
+k⟩ and
+K j
+l1l2(t) = Re(Γ0), which means for arbitrary initial state dif-
+ferent two-operator correlation functions will synchronously
+relax to their steady state expectation values with the same
+decay rate, as demonstrated in Fig. 4 (b) and (e), where dif-
+ferent curves of log(˜nl) as a function with γt have the same
+constant slope, i.e. K1
+ll = 4γ for all l.
+(ii) When Γµ
+k is only dispersionless in its real part, we set
+Γµ
+k = −x0 ��� iyµ(k), where x0 and yµ(k) are real. Then we
+have �Ψl1l2(t) = e−x0t �
+k,µ Cµ
+keik·˜re−iyµ(k)t|Γµ
+k⟩ and
+K j
+l1l2 = −x0 + d
+dt log
+��������
+���
+�
+k,µ
+Cµ
+k|Γµ
+k⟩ jei�
+k·˜r−yµ(k)t����
+�������� .
+(14)
+The right side of Eq. (14) contains sum of a series of plane
+waves, which leads to K j
+l1l2(t) oscillating around x0, as shown
+in Fig. 4 (d). The oscillating slopes lead to continuously inter-
+secting curves in Fig. 4 (a).
+(iii) When Γµ
+k is only dispersionless in its imaginary part,
+we set Γµ
+k = −(xc + δxµ(k)) − iy0, where xc and δxµ(k)
+are the central value and the offset function of Re(Γµ
+k),
+
+-1.5
+-2-2
+-2.5
+-3k(元)k(元)J2
+>
+412J2
+ = 412J2
+412Im(入±ReReReIm2
+-4Re(X±5
+0
+5
+-20
+-10
+05
+5
+-30
+-15
+05
+5
+-40
+-20
+02
+0
+2
+0
+0.5
+10.5
+0
+-0.5
+0
+0.5
+10.5
+0
+-0.5
+0
+0.5k(元)2
+0
+-2
+1
+0
+0
+1
+-11
+0
+1
+1
+0
+0
+-1 -1J2
+412k1(元)k2(元)k1(元)k1(元)k1(元)k2(元)k1(元)k1(元)k2(元)1
+0
+1
+1
+0
+0k2(元)k2(元)k2(元)-2
+-3
+-4
+1
+1
+0
+0
+.1.4
+-5
+6
+1
+1
+0
+0
+1-5
+.10
+1
+1
+0
+0
+-1
+-1HH
+Re(T)
+k1k2HH
+k1k2J2
+>
+412J2
+ = 4124
+and y0 is the imaginary part.
+Then we have �Ψl1l2(t) =
+e−(xc+iy0)t �
+k,µ Cµ
+keik·˜re−δxµ(k)t|Γµ
+k⟩ and
+K j
+l1l2 = −xc + d
+dt log
+��������
+���
+�
+k,µ
+Cµ
+k|Γµ
+k⟩ jeik·˜re−δxµ(k)t���
+�������� .
+(15)
+Since δxµ(k) is real, the relaxation process does not display
+oscillating decay rates (see Fig. 4 (f)). This induces the forked
+damping curves typically as shown in Fig. 4 (c).
+(b)
+(a)
+(c)
+(d)
+(e)
+(f)
+FIG. 4. The damping of particle number at different sites. The lattice
+has 15 sites under the periodic boundary condition. Initial state is a
+single excitation on the first site from vacuum. The time evolutions of
+log(|˜nl|) are shown in (a), (b), (c), and their derivatives K1
+ll are shown
+in (d), (e) and (f). The blue, red and orange lines are corresponding
+to l = 1, l = 2 and l = 3, respectively. In (a) and (d), γ2 is set as 0.5.
+In (b) and (e), γ2 = 1. In (c) and (f), γ2 = 1.5. Others parameters are
+the same in all subfigures with J = 1, γ1 = 0.25 and w = 0.25. The
+black dashed line represents a constant decay rate as ˜nl ∝ e−4γt.
+The above damping dynamics is directly related to disper-
+sion of damping-matrix spectra. The damping-matrix spec-
+tra reflect the decay of correlation functions, however, the Li-
+ovillian spectra reflect the decay of the whole system. We
+prove that the damping-matrix spectra are included in Liouvil-
+lian spectra in the Supplemental Material [25]. Therefore, for
+more general models with closed evolution equations of two-
+operator correlation functions, the dispersionless Liouvillian
+bands will lead to dispersionless damping-matrix spectra, and
+then give rise to the same dynamical signatures as shown in
+our model.
+Localized
+normal
+master
+modes
+and
+dynamic
+localization.— In isolated system, FBs lead to localized
+eigenstates by destructive interference.
+Now, we exactly
+solve our model (see the Supplemental Material [25]) to show
+that the LFB can induce dynamic localization by localized
+normal master modes (LNMMs), which suppress propagation
+of local perturbation on NESS.
+Usually, the odd parity part of ˆL has no effect on the ex-
+pectation value of observation in pure fermionic system [25].
+Therefore, we focus on the balanced model (w = 0) with even
+parity (P = 1), whose Liouvillian is illustrated in Fig. 1 (c).
+By solving the equation ζi(k)|Ω⟩ = 0 for i = 1 ∼ 4, we get the
+steady state |Ω⟩ as
+|Ω⟩ = 1
+N
+π
+�
+k=−π
+(1 + a†
+kc†
+−k)|0⟩ = 1
+N
+L
+�
+l=1
+(1 + a†
+l c†
+l )|0⟩,
+(16)
+where N = 2L and this state is independent with γ1 and γ2.
+At the FB point with J = 2 √γ1γ2, the exceptional degeneracy
+occurs in the non-Hermitian matrix Lk of Eq. (7) with four
+eigenstates coalescing into two. Then ˆLk is reduced to ˆLk =
+−2γ
+�
+ζ
+′
+A(k)ζA(k) + ζ
+′
+B(k)ζB(k)
+�
+, where
+ζ
+′
+A(k) = −a†
+k + c−k,
+ζA(k) = 1
+2(−ak + ick + ia†
+−k + c†
+−k),
+ζ
+′
+B(k) = ak + c†
+−k,
+ζB(k) = 1
+2(a†
+k − ic†
+k + ia−k + c−k).
+(17)
+Making Fourier transformation, we get
+ζ
+′
+A(l) =
+�
+k
+e−iklζ
+′
+A(k) = −a†
+l +cl, ζ
+′
+B(l) =
+�
+k
+eiklζ
+′
+B(k) = al+c†
+l ,
+(18)
+which create local eigenstate ζ
+′
+A,B(l)|Ω⟩ of ˆL with eigenvalue
+−2γ and are coined as LNMMs.
+We can also understand LNMMs intuitively from the per-
+spective of destructive interference. Writing the real-space Li-
+ouvillian with w = 0, J = 2 √γ1γ2 = 1 as ˆL = �
+l(ˆhl + ˆfl −2γ),
+where the hopping term hl is defined as ˆhl = −i(a†
+l+1al +
+h.c.) + i(c†
+l+1cl + h.c.) − (a†
+l+1cl + c†
+l+1al + h.c) and the pair-
+ing term ˆfl is defined as fl = 2γ(a†
+l c†
+l + clal), we can check
+that ˆflal|Ω⟩ =
+ˆflcl|Ω⟩ =
+ˆfla†
+l |Ω⟩ =
+ˆflc†
+l |Ω⟩ = 0. This im-
+plies that the pairing terms do not affect a single particle or
+hole excited on the NESS. Therefore, for these states only
+hopping terms make sense. We schematically plot this re-
+duced ladder in Fig. 1 (d). It is easy to find another LNMM
+as ζ
+′
+C(l) = a†
+l − ic†
+l from the view of destructive interference,
+which forbids the state ζ
+′
+C(l)|Ω transferring to other sites. We
+can also check that ˆL ζ
+′
+C(l)|Ω⟩ = −2γ ζ
+′
+C(l)|Ω⟩.
+The LNMMs contain decay information of quantum
+jumps. To see it clearly, we map the C−representation state
+ζ
+′
+A(l)ζ
+′
+B(l)|Ω⟩, for example, back to density-matrix representa-
+tion:
+ζ
+′
+A(l)ζ
+′
+B(l)|Ω⟩ → −a†
+l alρs + ρsala†
+l + alρsa†
+l − a†
+l ρsal,
+(19)
+where ρs is the density matrix of NESS. The terms alρsa†
+l
+and a†
+l ρsal are exactly corresponding to local quantum jumps
+on NESS. Eq. (19) implies that the local perturbation on
+NESS from quantum jumps will relax to NESS without ex-
+panding its territory. To see it clearly, we simulate the evo-
+lution from an initial state described by the density matrix
+ρ0 = a†
+1ρsa1/Tr(a†
+1ρsa1), which is created by a quantum jump
+on the first site of NESS. In Fig 5, we demonstrate the time
+evolution of particle numbers of the first three sites in a lattice
+with 15 sites. In the initial time, a jump occurs on the first
+site of NESS, increasing only the particle number on the first
+site n1 to 1 with others sites keeping their steady state value
+
+2
+-6
+0
+2-2
+-4
+-6
+0
+1
+2K=
+-4ttK=
+-4K=
+-4tttK
+1
+1log
+(1i
+nt
+1)-2
+-6
+0
+1
+20
+n1
+n2
+n3
+-5
+-10
+0
+1
+20
+n1
+n2
+n3
+-5
+-10
+0
+1
+20
+n1
+n2
+n3
+~
+-5
+-10
+0
+1
+2J2
+>
+412J2
+ = 412J2
+412t5
+(a)
+(b)
+(c)
+FIG. 5. The time evolution of particle number on the first site n1
+shown in (a), second site n2 in (b) and third site n3 in (c). Initial
+state is a†
+1ρsa1/Tr(a†
+1ρsa1) corresponding to a quantum jump on the
+first site of steady state. The periodic lattice has 15 sites with w = 0,
+J = 1, γ1 = 0.25 in all subfigures. The black dotted, red solid, and
+blue dashed lines are corresponding to γ2 = 0.5, γ2 = 1 and γ2 = 1.5,
+respectively.
+0.5. The red solid line, black dotted line and blue dashed line
+are corresponding to the situation with J2 = 4γ1γ2 (LFB),
+J2 > 4γ1γ2, and J2 < 4γ1γ2, respectively. We can see that
+when J2 � 4γ1γ2, the perturbation can spread from n1 to n3.
+However, for the case with LFB, the perturbation excitation
+decays locally without going through to n2 and n3, indicating
+the occurrence of dynamical localization.
+Final remarks.— (i) We use a geometrically intuitive
+method to construct flat band models in open system and
+demonstrate that the dispersion of Liouvillian band can ef-
+fectively affect the damping dynamics of local particle num-
+ber, intermediated by damping matrix of correlation function
+vector. When the Liouvillian flat band appears, the particle
+number in different sites will relax to their stable values syn-
+chronously. When only the real or imaginary part of rapidity
+spectrum is dispersionless, the damping behaviors show the
+oscillating or forked characteristic.
+(ii) We show flat-band Liouvillian can induce dynamical
+localization on NESS by the localized normal master modes,
+which halt the propagation of perturbation from other sites to
+the target sites.
+(iii) Our model does not exhibit non-Hermitian skin ef-
+fect [42, 43], which was uncovered to cause many abnormal
+phenomena such as boundary sensitivity [44], chiral and he-
+lical damping [45, 46] and slowing down of relaxation pro-
+cesses [20]. The interplay between Liouvillian flat band and
+non-Hermitian skin effect is an interesting topic for future
+studies.
+Acknowledgments.— We thank X. L. Wang, Z. Y. Zheng
+and C. X. Guo for helpful discussions.The work is supported
+by National Key National Key Research and Development
+Program of China (Grant No.2021YFA1402104), the NSFC
+under Grants No.
+12174436 and No.
+T2121001, and the
+Strategic Priority Research Program of Chinese Academy of
+Sciences under Grant No.XDB33000000.
+∗ [email protected]
+[1] S.-Q. Shen, Topological insulators, Vol. 174 (Springer,2012).
+[2] A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov,
+and A. K. Geim, The electronic properties of graphene, Rev.
+Mod. Phys. 81, 109 (2009).
+[3] Z. Liu, F. Liu, and Y.-S. Wu, Exotic electronic states in the world
+of flat bands: From theory to material, Chinese Physics B 23,
+077308 (2014).
+[4] Congjun Wu, Doron Bergman, Leon Balents, and S. Das Sarma,
+Flat bands and wigner crystallization in the honeycomb optical
+lattice, Phys. Rev. Lett. 99, 070401 (2007).
+[5] N. Regnault and B. Andrei Bernevig, Fractional Chern Insulator,
+Phys. Rev. X 1, 021014 (2011).
+[6] Lindblad, G. On the generators of quantum dynamical semi-
+groups, Commun.Math. Phys. 48, 119-130 (1976).
+[7] C. W. Gardiner and M. J. Collett, Input and output in damped
+quantum systems: Quantum stochastic differential equations and
+the master equation, Phys. Rev. A 31, 3761 (1985).
+[8] D. Manzano, A short introduction to the Lindblad master equa-
+tion, AIP Advances 10, 025106 (2020).
+[9] T. Prosen, Third quantization: a general method to solve mas-
+ter equations for quadratic open Fermi systems, New Journal of
+Physics 10, 043026 (2008).
+[10] C. Guo and D. Poletti, Solutions for bosonic and fermionic
+dissipative quadratic open systems, Phys. Rev. A 95, 052107
+(2017).
+[11] N. Shibata and H. Katsura, Dissipative spin chain as a non-
+Hermitian Kitaev ladder, Phys. Rev. B 99, 174303 (2019).
+[12] B. Horstmann, J. I. Cirac, and G. Giedke, Noise-driven dynam-
+ics and phase transitions in fermionic systems, Phys. Rev. A 87,
+012108 (2013).
+[13] Y. Zhang and T. Barthel, Criticality and Phase Classification for
+Quadratic Open Quantum Many-Body Systems, Phys. Rev. Lett.
+129, 120401 (2022).
+[14] T. Barthel and Y. Zhang, Solving quasi-free and quadratic Lind-
+blad master equations for open fermionic and bosonic systems,
+arXiv:2112.08344 [quant-ph] (2022).
+[15] S. Talkington and M. Claassen, Dissipation Induced Flat Bands,
+Phys. Rev. B 106, L161109 (2022).
+[16] Z. Cai and T. Barthel, Algebraic versus Exponential Decoher-
+ence in Dissipative Many-Particle Systems, Phys. Rev. Lett. 111,
+150403 (2013).
+[17] M. ˇZnidariˇc, Relaxation times of dissipative many-body quan-
+tum systems, Phys. Rev. E 92, 042143 (2015).
+[18] F. Minganti, A. Biella, N. Bartolo, and C. Ciuti, Spectral theory
+of Liouvillians for dissipative phase transitions, Phys. Rev. A 98,
+042118 (2018).
+[19] T. Mori and T. Shirai, Resolving a Discrepancy between Liou-
+villian Gap and Relaxation Time in Boundary-Dissipated Quan-
+tum Many-Body Systems, Phys. Rev. Lett. 125, 230604 (2020).
+[20] T. Haga, M. Nakagawa, R. Hamazaki, and M. Ueda, Liouvillian
+
+---J2 > 412
+J2 = 412
+J2 < 412
+0.50.51
+0.50.501
+0.5
+0
+0.5
+1.5
+2n
+1n
+2n
+3t6
+Skin Effect: Slowing Down of Relaxation Processes without Gap
+Closing, Phys. Rev. Lett. 127, 070402 (2021).
+[21] Y. Nakanishi and T. Sasamoto, PT phase transition in open
+quantum systems with Lindblad dynamics, Phys. Rev. A 105,
+022219 (2022).
+[22] Y.-N. Zhou, L. Mao, and H. Zhai, R´enyi entropy dynamics and
+Lindblad spectrum for open quantum systems, Phys. Rev. Re-
+search 3, 043060 (2021).
+[23] S. Diehl, E. Rico, M. Baranov, and P. Zoller, Topology by dissi-
+pation in atomic quantum wires, Nature Phys 7, 971-977 (2011).
+[24] T. L. Landi, D. Poletti, and G. Schaller, Non-equilibrium
+boundary driven quantum systems: models, methods and prop-
+erties, arXiv:2104.14350 [quant-ph] (2022).
+[25] See Supplemental Material for (i) Mapping of Lindblad master
+equation, (ii) Diagonalization, exceptional point and symmetry
+of the Liouvillian, (iii) Exactly solution and discussion on parity,
+(iv) Evolution equations of correlation functions and the symme-
+try of damping matrix, (v) Particle number distribution of steady
+state, (vi) The relationship between the damping-matrix spectra
+and the Liouvillian spectra. The supplemental materias include
+also references [26–33].
+[26] M.-D. Choi, Completely positive linear maps on complex ma-
+trices, Linear Algebra Applications 10, 285 (1975).
+[27] A. Jamiołkowski, Linear transformations which preserve trace
+and positive semidefiniteness of operators, Rep. Math. Phys. 3,
+275 (1972).
+[28] J. E. Tyson, Operator-Schmidt decompositions and the Fourier
+transform, with applications to the operator-Schmidt numbers of
+unitaries, J. Phys. A: Math. Gen. 36, 10101 (2003).
+[29] M. Zwolak and G. Vidal, Mixed-State Dynamics in One-
+Dimensional Quantum Lattice Systems: A Time-Dependent Su-
+peroperator Renormalization Algorithm, Phys. Rev. Lett. 93,
+207205 (2004).
+[30] K. Kawabata, K. Shiozaki, M. Ueda, and M. Sato, Symmetry
+and Topology in Non-Hermitian Physics, Phys. Rev. X 9, 041015
+(2019).
+[31] A. W. W. Ludwig, Topological phases:
+classification of
+topological insulators and superconductors of noninteracting
+fermions, and beyond, Physica Scripta T168, 014001 (2015).
+[32] C.-H. Liu, H. Jiang, and S. Chen, Topological classification of
+non-Hermitian systems with reflection symmetry, Phys. Rev. B
+99, 125103 (2019).
+[33] C.-H. Liu and S. Chen, Topological classification of defects in
+non-Hermitian systems, Phys. Rev. B 100, 144106 (2019).
+[34] M. Creutz, End States, Ladder Compounds, and Domain-Wall
+Fermions, Phys. Rev. Lett. 83, 2636 (1999).
+[35] W. Maimaiti, A. Andreanov, H. C. Park, O. Gendelman, and S.
+Flach, Compact localized states and flat-band generators in one
+dimension, Phys. Rev. B 95, 115135 (2017).
+[36] W. Maimaiti and A. Andreanov, Non-Hermitian flatband gener-
+ator in one dimension, Phys. Rev. B 104, 035115 (2021).
+[37] Y. Kuno, T. Orito, and I. Ichinose, Flat-band many-body local-
+ization and ergodicity breaking in the Creutz ladder, New Journal
+of Physics 22, 013032 (2020).
+[38] Although the coefficient of ˆLk=0 is 0.5 in the equation ˆL =
+0.5 ˆLk=0 + �π−
+k=0+ ˆLk, the rapidity for k = 0 can be still described
+by λ±(0) due to the degeneracy as ζ
+′
+i (k = 0) = ζ
+′
+5−i(k = 0) and
+ζi(k = 0) = ζ5−i(k = 0).
+[39] A. Mostafazadeh, Pseudo-Hermiticity versus PT-symmetry. II.
+A complete characterization of non-Hermitian Hamiltonians
+with a real spectrum, Journal of Mathematical Physics 43, 2814
+(2002).
+[40] A. Mostafazadeh, Pseudo-Hermiticity for a class of nondiag-
+onalizable Hamiltonians, Journal of Mathematical Physics 43,
+6343 (2002).
+[41] Y. Ashida, Z. Gong, and M. Ueda, Non-Hermitian physics, Ad-
+vances in Physics 69, 249 (2020).
+[42] S. Yao and Z. Wang, Edge States and Topological Invariants of
+Non-Hermitian Systems, Phys. Rev. Lett. 121, 086803 (2018).
+[43] F. K. Kunst, E. Edvardsson, J. C. Budich, and E. J. Bergholtz,
+Biorthogonal Bulk-Boundary Correspondence in Non-Hermitian
+Systems, Phys. Rev. Lett. 121, 026808 (2018).
+[44] C.-X. Guo, C.-H. Liu, X.-M. Zhao, Y. Liu, and S. Chen, Exact
+Solution of Non-Hermitian Systems with Generalized Bound-
+ary Conditions: Size-Dependent Boundary Effect and Fragility
+of the Skin Effect, Phys. Rev. Lett. 127, 116801 (2021).
+[45] F. Song, S. Yao, and Z. Wang, Non-Hermitian Skin Effect and
+Chiral Damping in Open Quantum Systems, Phys. Rev. Lett.
+123, 170401 (2019).
+[46] C.-H. Liu, K. Zhang, Z. Yang, and S. Chen, Helical damp-
+ing and dynamical critical skin effect in open quantum systems,
+Physical Review Research 2, 043167 (2020).
+
+7
+SUPPLEMENTAL MATERIAL: Dynamics Signatures of Liouvillian Flat Band
+S1. Mapping of Lindblad master equation
+FIG. S1. Mapping of Lindblad master equation.
+The Lindblad master equation, formalized density matrix ρ and Liouvillian superoperator L is shown in Eq. (1) and Eq. (2) in
+the main text. First we carry out the Choi-Jamiolkwski isomorphism [1–4] to map the fermionic LME into representation B as
+d
+dt| ρ⟩B = ˆLB| ρ⟩B,
+(S1)
+where | ρ⟩B is vectorized from ρ and ˆLB is mapped from L. Specifically, the mapping is
+ρ → | ρ⟩B =
+�
+IJ
+ρIJ|I⟩a ⊗ |J⟩b,
+(S2a)
+L → ˆLB =
+�
+i j
+Fi(a, a†) ⊗ F T
+j (b, b†),
+(S2b)
+where b = (b1, b2, · · · ) is the set of annihilation operators of b−fermions, which is one-to-one mapping from a, and T means
+matrix transpose. |I⟩a and |J⟩b are defined as
+|I⟩a = (a†
+1)I1(a†
+2)I2 · · · (a†
+L)IL|0⟩a,
+(S3a)
+|J⟩b = (b†
+1)J1(b†
+2)J2 · · · (b†
+L)JL|0⟩b,
+(S3b)
+where |0⟩a and |0⟩b are vacuum state of all a−fermions and b−fermions, respectively. In this representation, the expectation
+value of observable becomes
+⟨ ˆOa⟩ =B ⟨S0| ˆOa ⊗ Ib| ρ⟩B,
+(S4)
+where B⟨S0| is a special state defined as:
+B⟨S0| =
+�
+S
+⟨S|a ⊗ ⟨S|b =
+�
+S
+�
+⟨0|a(aL)S L · ·(a1)S 1 ⊗ ⟨0|b(bL)S L · ·(b1)S 1�
+,
+(S5)
+and Ib is a unit operator of all b−fermions. The element S i of S = (S 1, S 2, · · · ) can take 0 or 1, and �
+S requires a sum over all
+possible configurations of S. Let us prove Eq. (S4):
+⟨ ˆOa⟩ =
+�
+IJS
+ρIJ ⟨S|a ˆOa|I⟩a⟨S|bIb|J⟩b
+=
+�
+IJS
+ρIJ a⟨0|aS L
+L · · · aS 1
+1 ˆOa(a†
+1)I1 · · · (a†
+L)IL|0⟩a δSJ
+=
+�
+IJ
+ρIJ a⟨0|aJL
+L · · · aJ1
+1 ˆOa(a†
+1)I1 · · · (a†
+L)IL|0⟩a
+=
+�
+IJ
+ρIJ a⟨0| ˆOa|0⟩a = Tr( ˆOaρ).
+(S6)
+
+p=p [I)a<Jla
+d
+0= L(p)= F;(a,at)pF,(a,at)
+Tr(Oap
+IJ
+ii
+IJ
+[p>c
+278
+In representation B, operators satisfy the following relations:
+{ai, a†
+j} = {bi, b†
+j} = δi j,
+{a†
+i , a†
+j} = {ai, a j} = {b†
+i , b†
+j} = {bi, bj} = 0,
+(S7a)
+[a†
+i , b j] = [a†
+i , b†
+j] = [ai, bj] = [ai, b†
+j] = 0.
+(S7b)
+The commutation relations in Eq. (S7b) are from the direct product between a−fermions and b−fermions, which are unfavorable
+for further analysis. To enforce fermionic anticommutation relations over all operators, we define operators of c−fermions as
+c† = b† ˆP and c = ˆPb, where ˆP is a parity operator defined as
+ˆP := exp
+�
+iπ
+�
+l
+(a†
+l al + b†
+l bl)
+�
+= exp
+�
+iπ
+�
+l
+(a†
+l al + c†
+l cl)
+�
+.
+(S8)
+It is easy to check the fermionic anticommutation relations in a−fermions and c−fermions:
+{ci, c†
+j} = δi j,
+{c†
+i , c†
+j} = {ci, c j} = 0
+(S9a)
+{a†
+i , cj} = {a†
+i , c†
+j} = {ai, cj} = {ai, c†
+j} = 0
+(S9b)
+By c we can fully fermionize system from representation B to representation C. The mapping is
+| ρ⟩B → | ρ⟩C =
+�
+IJ
+ρIJ(a†
+1)I1 · · · (a†
+L)IL(c†
+1 ˆP)J1 · · · (c†
+L ˆP)JL |0⟩,
+(S10a)
+ˆLB → ˆLC =
+�
+ij
+Fi(a, a†) F T
+j ( ˆPc, c† ˆP).
+(S10b)
+The LME and the expectation value of observable in representation C are
+d
+dt| ρ⟩C = ˆLC| ρ⟩C,
+(S11a)
+⟨ ˆOa⟩ =C ⟨S0| ˆOa| ρ⟩C,
+(S11b)
+where C⟨S0| is defined as:
+C⟨S0| =
+�
+S
+⟨0| ( ˆPcL)S L · · · ( ˆPc1)S 1aS L
+L · · · aS 1
+1 .
+(S12)
+Combining the mappings in Eq. (S2) and Eq. (S10), we get the final mapping, i.e., Eq. (4) in the main text. The mapping
+process is schematically shown in Fig. S1.
+S2. Model in representation C: diagonalization, exceptional point and symmetry
+In this section we map our Liouvillian in Eq. (5) in the main text into representation C and get its BdG form in momentum
+space. Based on the BdG form, we show the exceptional point and symmetry of our Liouvillian.
+A. The derivation of ˆL
+Our Liouvillian L in Eq. (5) is mapped into ˆL by the mapping (4b) in the main text:
+L(·) = −i[H, ·] + (1 − w)DL(·) + (1 + w)DR(·) → ˆL = ˆH + (1 − w) ˆDL + (1 + w) ˆDR.
+(S13)
+Note that our matrix representation of creation and annihilation operator is real, thus we have aT = a†, cT = c†, ˆPT = ˆP. Then
+we get
+− i[H, ·] → ˆH = −iH(a, a†) + iHT( ˆPc, c† ˆP) = −iJ
+�
+l
+(a†
+l+1al + a†
+l al+1) + iJ
+�
+l
+(c†
+l cl+1 + c†
+l+1cl).
+(S14)
+
+9
+Note that our matrix representation of Al = √γ1a†
+l + √γ2al+1 is real, thus we have AT
+l = A†
+l . Then we get
+DL(·) → ˆDL =
+�
+l
+�
+2Al(a, a†)Al( ˆPc, c† ˆP) − A†
+l (a, a†)Al(a, a†) − A†
+l ( ˆPc, c† ˆP)Al( ˆPc, c† ˆP)
+�
+=
+�
+l
+�
+2( √γ1a†
+l + √γ2al+1)( √γ1c†
+l ˆP + √γ2 ˆPcl+1) − ( √γ1al + √γ2a†
+l+1)( √γ1a†
+l + √γ2al+1)
+− ( √γ1 ˆPcl + √γ2c†
+l+1 ˆP)( √γ1c†
+l ˆP + √γ2 ˆPcl+1)
+�
+=
+�
+l
+�
+− 2 √γ1γ2 ˆP(a†
+l cl+1 + c†
+l al+1) + 2γ1 ˆPa†
+l c†
+l + 2γ2 ˆPcl+1al+1 − √γ1γ2(alal+1 + a†
+l+1a†
+l )
++ √γ1γ2(clcl+1 + c†
+l+1c†
+l ) − γ2(a†
+l+1al+1 + c†
+l+1cl+1) − γ1(ala†
+l + clc†
+l )
+�
+,
+(S15)
+DR(·) → ˆDR =
+�
+l
+�
+2A†
+l (a, a†)A†
+l ( ˆPc, c† ˆP) − Al(a, a†)A†
+l (a, a†) − Al( ˆPc, c† ˆP)A†
+l ( ˆPc, c† ˆP)
+�
+=
+�
+l
+�
+2( √γ1al + √γ2a†
+l+1)( √γ1 ˆPcl + √γ2c†
+l+1 ˆP) − ( √γ1a†
+l + √γ2al+1)( √γ1al + √γ2a†
+l+1)
+− ( √γ1c†
+l ˆP + √γ2 ˆPcl+1)( √γ1 ˆPcl + √γ2c†
+l+1 ˆP)
+�
+=
+�
+l
+�
+− 2 √γ1γ2 ˆP(a†
+l+1cl + c†
+l+1al) + 2γ1 ˆPclal + 2γ2 ˆPa†
+l+1c†
+l+1 + √γ1γ2(alal+1 + a†
+l+1a†
+l )
+− √γ1γ2(clcl+1 + c†
+l+1c†
+l ) − γ2(al+1a†
+l+1 + cl+1c†
+l+1) − γ1(a†
+l al + c†
+l cl)
+�
+.
+(S16)
+Due to [ ˆP, ˆL] = 0, the state will keep its parity in the evolution governed by the Lindblad master equation. Therefore, ˆP can
+reduce to a constant P, which equals 1 in even parity channel and −1 in odd parity channel.
+By Fourier transformation
+a†
+l =
+π
+�
+k=−π
+e−ikla†
+k,
+al =
+π
+�
+k=−π
+eiklak,
+c†
+l =
+π
+�
+k=−π
+e−iklc†
+k,
+cl =
+π
+�
+k=−π
+eiklck,
+(S17)
+we get ˆL in BdG form as
+ˆL = 1
+2
+ˆLk=0 +
+π−
+�
+k=0+
+ˆLk,
+(S18)
+where
+ˆLk = (a†
+k c†
+k a−k c−k) Lk (ak ck a†
+−k c†
+−k)T − 4γ,
+(S19)
+and
+Lk = −i2J cos kσz ⊗ σz − 4 √γ1γ2 cos kPσz ⊗ σx − 2γPσy ⊗ σy + 2w
+�
++(γ2 − γ1)σz ⊗ I + 2 √γ1γ2 sin kσy ⊗ σz
++i(γ2 − γ1)Pσx ⊗ σy + i2 √γ1γ2 sin kPI ⊗ σx
+�
+.
+(S20)
+B. Diagonalization of ˆLk
+We make a similarity transformation for ˆLk by matrix W:
+ˆLk = (a†
+k c†
+k a−k c−k) W W−1 Lk W W−1 (ak ck a†
+−k c†
+−k)T − 4γ
+= (ζ
+′
+1(k) ζ
+′
+2(k) ζ3(k) ζ4(k)) Λ (ζ1(k) ζ2(k) ζ
+′
+3(k) ζ
+′
+4(k))T − 4γ
+= λ1(k)ζ
+′
+1(k)ζ1(k) + λ2(k)ζ
+′
+2(k)ζ2(k) + λ3(k)ζ3(k)ζ
+′
+3(k) + λ4(k)ζ4(k)ζ
+′
+4(k) − 4γ,
+(S21)
+where
+(a†
+k c†
+k a−k c−k) W = (ζ
+′
+1(k) ζ
+′
+2(k) ζ3(k) ζ4(k)),
+W−1 (ak ck a†
+−k c†
+−k)T = (ζ1(k) ζ2(k) ζ
+′
+3(k) ζ
+′
+4(k))T
+(S22)
+
+10
+and Λ is a diagonal matrix given by
+Λ = W−1 Lk W = diag(λ1(k), λ2(k), λ3(k), λ4(k)).
+(S23)
+We write W and W−1 as
+W = (⃗v1 ⃗v2 ⃗v3 ⃗v4),
+W−1 =
+��������������
+⃗u t
+1
+⃗u t
+2
+⃗u t
+3
+⃗u t
+4
+��������������
+,
+(S24)
+where the column vector ⃗vi and row vector ⃗u t
+j satisfy ⃗u t
+j · ⃗vi = δi j. Then we have
+ζ
+′
+1(k) = (a†
+k c†
+k a−k c−k) · ⃗v1,
+ζ
+′
+2(k) = (a†
+k c†
+k a−k c−k) · ⃗v2,
+ζ
+′
+3(k) = (ak ck a†
+−k c†
+−k) · ⃗u3,
+ζ
+′
+4(k) = (ak ck a†
+−k c†
+−k) · ⃗u4,
+ζ1(k) = (ak ck a†
+−k c†
+−k) · ⃗u1,
+ζ2(k) = (ak ck a†
+−k c†
+−k) · ⃗u2,
+ζ3(k) = (a†
+k c†
+k a−k c−k) · ⃗v3,
+ζ4(k) = (a†
+k c†
+k a−k c−k) · ⃗v4.
+(S25)
+ζ
+′
+i(k) and ζj(k) hold anticommutation relations:
+{ζ
+′
+i(k), ζj(k)} = δi j,
+{ζ
+′
+i(k), ζ
+′
+j(k)} = {ζi(k), ζj(k)} = 0
+(S26)
+Calculating the eigenvalues of Eq. (S20), we get the same values for both even and odd parity: λ1(k) = −2γ − 2mk, λ2(k) =
+−2γ + 2mk, λ3(k) = 2γ − 2mk and λ4(k) = 2γ + 2mk , where
+mk =
+�����
+�
+(4γ1γ2 − J2) cos2 k,
+4γ1γ2 ≥ J2
+i
+�
+(J2 − 4γ1γ2) cos2 k,
+4γ1γ2 < J2
+(S27)
+Then Lk can be diagonalized as
+ˆLk = λ−(k)
+�
+ζ
+′
+1(k)ζl(k) + ζ
+′
+4(k)ζ4(k)
+�
++ λ+(k)
+�
+ζ
+′
+2(k)ζ2(k) + ζ
+′
+3(k)ζ3(k)
+�
+,
+(S28)
+where
+λ±(k) = −2γ ± mk.
+(S29)
+C. Exceptional point
+When J2 = 4γ1γ2, the exceptional point of Lk emerges. To see it clearly, we show real and imaginary part of the rapidity
+λ±(k) in Fig. S2. When the flat band condition is satisfied (γ2 = 1), it occurs exceptional degeneracy between λ+ and λ−.
+(a)
+(b)
+FIG. S2. The real (a) and imaginary (b) part of λ±(k) as a function with k and γ2. Other parameters are taken as J = 1 and γ1 = 0.25
+
+2
+0
+-2
+0
+0
+0.5
+1
+2
+10
+-5
+0
+0
+0.5
+1
+2
+1Im(入±Re(入±)k(元)k(元)11
+D. The symmetry of Liouvillian
+Due to ˆLk = ˆL−k, we can write ˆL in Eq. (S18) as ˆL = 1
+2
+�π
+k=−π ˆLk. Therefore, we can study the symmetry of the Liouvillian
+from Lk with k ∈ (−π, π). It is easy to check that Lk in Eq. (S20) has time-reversal symmetry (TRS), particle-hole symmetry
+(PHS) and chiral symmetry (CS)[5–8]:
+TRS : T+ L∗
+k T −1
++
+= L−k
+=⇒ T+ = σz ⊗ σx; T+T ∗
++ = 1
+PHS : C− LT
+k C−1
+− = −L−k
+=⇒ C− = σx ⊗ I; C−C∗
+− = 1
+CS : Γ L†
+k Γ−1 = −Lk
+=⇒ Γ = σy ⊗ σx; Γ2 = 1.
+(S30)
+Due to that Lk has full real spectrum in the region J2 < 4γ1γ2, the mathematical theorem ensures the Liouvillian having pseudo-
+Hermiticity i.e. there exists a Hermitian matrix η that η L†
+kη−1 = Lk. Especially, when w = 0, the system will additionally have
+inversion symmetry (IS) and the pseudo-Hermiticity will be enhanced to the parity-time symmetry (PTS):
+IS : P Lk P−1 = L−k
+=⇒ P = σy ⊗ σy
+PTS : PT L∗
+k PT −1 = Lk
+=⇒ PT = σx ⊗ σz.
+(S31)
+S3. Exactly solution of the model when w = 0
+In this section, we exactly solve our model both in even and odd channels. We show steady state and all the excited states
+of the open system. In addition, we prove that the odd parity states have no contribution on observations with even fermionic
+operators. Last, we calculate the correlation functions of steady state and local-quantum-jump states beyond the steady state.
+A. All the eigenstates of ˆL
+First, we diagonalize ˆLk in even channel (P = 1). Then normal master modes are show in Eq. (S25). The vectors ⃗v and ⃗u can
+be solved as
+⃗v1 = 1
+2
+�
+− 1 − iJ cos k/mk, −2 √γ1γ2 cos k/mk, −2 √γ1γ2 cos k/mk, 1 + iJ cos k/mk
+�T
+⃗v2 = 1
+2
+�
+− 1 + iJ cos k/mk, 2 √γ1γ2 cos k/mk, 2 √γ1γ2 cos k/mk, 1 − iJ cos k/mk
+�T
+⃗v3 = 1
+2
+�
+1, −iJ + mk/ cos k
+2 √γ1γ2
+, iJ − mk/ cos k
+2 √γ1γ2
+, 1
+�T
+⃗v4 = 1
+2
+�
+1, −iJ − mk/ cos k
+2 √γ1γ2
+, iJ + mk/ cos k
+2 √γ1γ2
+, 1
+�T
+⃗u1 = 1
+2
+�
+− 1, iJ − mk/ cos k
+2 √γ1γ2
+, iJ − mk/ cos k
+2 √γ1γ2
+, 1
+�T
+⃗u2 = 1
+2
+�
+− 1, iJ + mk/ cos k
+2 √γ1γ2
+, iJ + mk/ cos k
+2 √γ1γ2
+, 1
+�T
+⃗u3 = 1
+2
+�
+1 + iJ cos k/mk, 2 √γ1γ2 cos k/mk, −2 √γ1γ2 cos k/mk, 1 + iJ cos k/mk
+�T
+⃗u4 = 1
+2
+�
+1 − iJ cos k/mk, −2 √γ1γ2 cos k/mk, 2 √γ1γ2 cos k/mk, 1 − iJ cos k/mk
+�T.
+(S32)
+We make an ansatz for steady state |Ω⟩ as
+|Ω⟩ =
+π
+�
+k=0
+(z1 + z2a†
+kc†
+−k)(z3 + z4a†
+−kc†
+k)|0⟩.
+(S33)
+Solving the steady state equations: ζi|Ω⟩ = 0 for i = 1 ∼ 4, we get z1 = z2 and z3 = z4. Therefore, the solution of steady state
+(the Eq. (16) in the main text) is given by
+|Ω⟩ = 1
+N
+π
+�
+k=−π
+(1 + a†
+kc†
+−k)|0⟩.
+(S34)
+
+12
+By using Tr(ρs) = 1, we get the normalization factor N as
+N =C ⟨S0|
+π
+�
+k=−π
+(1 + a†
+kc†
+−k)|0⟩ = 2L,
+(S35)
+where L is the length of the chain. The details of N = 2L is given in subsection D. In addition, we get steady state in real space
+given by
+|Ω⟩ = 1
+N exp
+�
+π
+�
+k=−π
+a†
+kc†
+−k
+�
+|0⟩ = 1
+N exp
+�
+L
+�
+l=1
+a†
+l c†
+l
+�
+|0⟩ = 1
+N
+L
+�
+l=1
+(1 + a†
+l c†
+l )|0⟩.
+(S36)
+Under the parity constraint, valid eigenstates in even parity channel are |Ω⟩, ζ
+′
+α1(ki)ζ
+′
+α2(kj)|Ω⟩, ζ
+′
+α1(ki)ζ
+′
+α2(kj)ζ
+′
+α3(km)ζ
+′
+α4(kn)|Ω⟩, · · ·
+Secondly, we diagonalize ˆLk in the odd channel (P = −1). The process of diagonalization is the same as it in the even channel,
+however, the eigenvectors ⃗v and ⃗u of odd channel are different from them in even channel. We mark the eigenvectors and normal
+master modes of the odd channel with ’∗’:
+ζ
+′
+1∗(k) = (a†
+k c†
+k a−k c−k) · ⃗v1∗,
+ζ
+′
+2∗(k) = (a†
+k c†
+k a−k c−k) · ⃗v2∗,
+ζ
+′
+3∗(k) = (ak ck a†
+−k c†
+−k) · ⃗u3∗,
+ζ
+′
+4∗(k) = (ak ck a†
+−k c†
+−k) · ⃗u4∗,
+ζ1∗(k) = (ak ck a†
+−k c†
+−k) · ⃗u1∗,
+ζ2∗(k) = (ak ck a†
+−k c†
+−k) · ⃗u2∗,
+ζ3∗(k) = (a†
+k c†
+k a−k c−k) · ⃗v3∗,
+ζ4∗(k) = (a†
+k c†
+k a−k c−k) · ⃗v4∗,
+(S37)
+where
+⃗v1∗ = 1
+2
+�
+1 + iJ cos k/mk, −2 √γ1γ2 cos k/mk, 2 √γ1γ2 cos k/mk, 1 + iJ cos k/mk
+�T
+⃗v2∗ = 1
+2
+�
+1 − iJ cos k/mk, 2 √γ1γ2 cos k/mk, −2 √γ1γ2 cos k/mk, 1 − iJ cos k/mk
+�T
+⃗v3∗ = 1
+2
+�
+− 1, −iJ + mk/ cos k
+2 √γ1γ2
+, −iJ + mk/ cos k
+2 √γ1γ2
+, 1
+�T
+⃗v4∗ = 1
+2
+�
+− 1, −iJ − mk/ cos k
+2 √γ1γ2
+, −iJ − mk/ cos k
+2 √γ1γ2
+, 1
+�T
+⃗u1∗ = 1
+2
+�
+1, iJ − mk/ cos k
+2 √γ1γ2
+, −iJ + mk/ cos k
+2 √γ1γ2
+, 1
+�T
+⃗u2∗ = 1
+2
+�
+1, iJ + mk/ cos k
+2 √γ1γ2
+, −iJ − mk/ cos k
+2 √γ1γ2
+, 1
+�T
+⃗u3∗ = 1
+2
+�
+− 1 − iJ cos k/mk, 2 √γ1γ2 cos k/mk, 2 √γ1γ2 cos k/mk, 1 + iJ cos k/mk
+�T
+⃗u4∗ = 1
+2
+�
+− 1 + iJ cos k/mk, −2 √γ1γ2 cos k/mk, −2 √γ1γ2 cos k/mk, 1 − iJ cos k/mk
+�T.
+(S38)
+Solving the equation, ζi∗|Ω∗⟩ = 0 for i = 1 ∼ 4, we get
+|Ω∗⟩ = 1
+N
+π
+�
+k=−π
+(1 − a†
+kc†
+−k)|0⟩ = 1
+N
+L
+�
+l=1
+(1 − a†
+l c†
+l )|0⟩.
+(S39)
+Note that |Ω∗⟩ is even parity ( ˆP|Ω∗⟩ = +1|Ω∗⟩). Therefore, the valid eigenstates in odd parity channel are the states with odd
+numbers of excitations on the |Ω∗⟩, i.e. ζ
+′
+α1∗(ki)|Ω∗⟩, ζ
+′
+α1∗(ki)ζ
+′
+α2∗(k j)ζ
+′
+α3∗(km)|Ω∗⟩, · · ·
+In summary, the full eigenstates of ˆL are
+Steady state:
+|Ω⟩
+Single excitation:
+ζ
+′
+α1∗(ki) |Ω∗⟩
+Double excitation:
+ζ
+′
+α1(ki)ζ
+′
+α2(k j) |Ω⟩
+Triple excitation:
+ζ
+′
+α1∗(ki)ζ
+′
+α2∗(kj)ζ
+′
+α3∗(km) |Ω∗⟩
+Quadruple excitation:
+ζ
+′
+α1(ki)ζ
+′
+α2(kj)ζ
+′
+α3(km)ζ
+′
+α4(kn) |Ω⟩
+· · ·
+(S40)
+
+13
+B. Flat band condition
+When the condition J2 = 4γ1γ2 is satisfied, Liouvillian flat band occurs. We have λ1 = λ2 = −2γ, λ3 = λ4 = 2γ and mk = 0,
+which leads to divergence of eigenvectors ⃗v1, ⃗v2, ⃗u3, ⃗u4, ⃗v1∗, ⃗v2∗, ⃗u3∗ and ⃗u4∗. This indicates the exceptional point of ˆL. However,
+we can eliminate divergence by summing of these eigenvectors. Setting J = 2 √γ1γ2, we can get the normal master modes in
+even parity
+ζ
+′
+A(k) = (a†
+k c†
+k a−k c−k) · (⃗v1 + ⃗v2) = −a†
+k + c−k
+ζA(k) = (ak ck a†
+−k c†
+−k) · (⃗u1 + ⃗u2)/2 = 1
+2(−ak + ick + ia†
+−k + c†
+−k)
+ζ
+′
+B(k) = ak ck a†
+−k c†
+−k) · (⃗u3 + ⃗u4) = ak + c†
+−k
+ζB(k) = (a†
+k c†
+k a−k c−k) · (⃗v3 + ⃗v4)/2 = 1
+2(a†
+k − ic†
+k + ia−k + c−k),
+(S41)
+and in odd parity
+ζ
+′
+A∗(k) = (a†
+k c†
+k a−k c−k) · (⃗v1∗ + ⃗v2∗) = a†
+k + c−k
+ζA∗(k) = (ak ck a†
+−k c†
+−k) · (⃗u1∗ + ⃗u2∗)/2 = 1
+2(ak + ick − ia†
+−k + c†
+−k)
+ζ
+′
+B∗(k) = ak ck a†
+−k c†
+−k) · (⃗u3∗ + ⃗u4∗) = −ak + c†
+−k
+ζB∗(k) = (a†
+k c†
+k a−k c−k) · (⃗v3∗ + ⃗v4∗)/2 = 1
+2(−a†
+k − ic†
+k − ia−k + c−k).
+(S42)
+C. Ineffectiveness of odd parity
+Given an arbitrary state | ρ⟩, it can be decomposed into even and odd eigenstate of ˆL:
+| ρ⟩ =
+� �
+i
+Ce
+i |i⟩e
+�
++
+� �
+j
+Co
+j | j⟩o
+�
+,
+(S43)
+where |i⟩e and | j⟩o represents even and odd parity state in Eq. (S40). The expectation value of observation ˆO is
+C⟨S0| ˆO| ρ⟩ =
+� �
+i
+Ce
+i C⟨S0| ˆO|i⟩e
+�
++
+� �
+j
+Co
+j C⟨S0| ˆO| j⟩o
+�
+.
+(S44)
+When ˆO has even fermionic operators, we have C⟨S0| ˆO| j⟩o = 0. When ˆO has odd fermionic operators, we have C⟨S0| ˆO|i⟩e = 0.
+Usually, in pure fermionic system, fermionic operators appear in pairs, so the odd parity part of ˆL does not influence the
+expectation value of observation.
+D. Correlation functions of steady state and quantum jump states
+Firstly, we show the details for the calculation of normalization factor N:
+N =C ⟨S0|
+L
+�
+l=1
+(1 + a†
+l c†
+l )|0⟩
+=
+�
+S
+⟨0|( ˆPcL)S L · · · ( ˆPc1)S 1aS L
+L · · · aS 1
+1 (1 + a†
+1c†
+1) · · · (1 + a†
+Lc†
+L)|0⟩
+=
+�
+S
+⟨0|( ˆPcLaL)S L · · · ( ˆPc1a1)S 1 (1 + a†
+1c†
+1) · · · (1 + a†
+Lc†
+L)|0⟩
+= ⟨0|(1 + ˆPcLaL) · · · (1 + ˆPc1a1) (1 + a†
+1c†
+1) · · · (1 + a†
+Lc†
+L)|0⟩
+= ⟨0|
+L
+�
+l=1
+�
+(1 + ˆPclal)(1 + a†
+l c†
+l )
+�
+|0⟩
+= 2L.
+(S45)
+
+14
+Secondly, we show the particle number distribution of the steady state ns
+j
+ns
+j =C ⟨S0|a†
+jaj|Ω⟩
+= 1
+N
+�
+S
+⟨0|( ˆPcL)S L · · · ( ˆPc1)S 1aS L
+L · · · aS 1
+1 a†
+ja j (1 + a†
+1c†
+1) · · · (1 + a†
+Lc†
+L)|0⟩
+= 2L−1
+N ⟨0|(1 + ˆPc jaj)a†
+jaj(1 + a†
+jc†
+j)|0⟩
+= 1
+2
+(S46)
+The other correlation functions of steady state can be calculated by the same method. The results are
+Gs
+j1,j2 = 0 (j1 � j2), Ds
+j1,j2 = 0, Ds∗
+j1, j2 = 0.
+(S47)
+Thirdly, we focus on a state from a quantum jump on the site l of the steady state. We denote this state as |φl⟩:
+|φl⟩ :=
+a†
+l ρsal
+Tr(a†
+l ρsal)
+=
+a†
+l c†
+l |Ω⟩
+C⟨S0|a†
+l c†
+l |Ω⟩
+.
+(S48)
+The particle number on site j of |φl⟩, denoted as nl
+j:
+nl
+j=l =C ⟨S0|a†
+l al|φl⟩
+=
+⟨0|(1 + ˆPclal) a†
+l al a†
+l c†
+l (1 + a†
+l c†
+l )|0⟩
+⟨0|(1 + ˆPclal)a†
+l c†
+l (1 + a†
+l c†
+l )|0⟩
+= 1.
+(S49)
+nl
+j�l =C ⟨S0|a†
+jaj|φl⟩
+=
+⟨0|(1 + ˆPclal)a†
+l c†
+l (1 + a†
+l c†
+l ) (1 + ˆPcjaj)a†
+jc†
+j(1 + a†
+jc†
+j)|0⟩
+⟨0|(1 + ˆPclal)a†
+l c†
+l (1 + a†
+l c†
+l ) (1 + ˆPcja j)(1 + a†
+jc†
+j)|0⟩
+= 1
+2.
+(S50)
+By the same way, we get the other correlation functions of |φl⟩. The results are
+Gl
+j1,j2 = 0 (j1 � j2), Dl
+j1,j2 = 0, Dl∗
+j1, j2 = 0.
+(S51)
+S4. Evolution equations of correlation functions
+In this section, we derive the evolution equations of two-operator correlation functions both in real space and momentum
+space and show the symmetry of damping matrix in momentum space.
+A. Evolution equations of correlation functions in real space
+The evolution equation of the expectation value of operator ˆO in the open system is
+d
+dtTr( ˆOρ(t)) = Tr( ˆO d
+dtρ) = Tr( ˆOL(ρ)).
+(S52)
+By considering the Liouvillian L in Eq. (5) in the main text, the equation becomes
+d
+dtTr( ˆOρ(t)) = −iTr( ˆO[H, ρ]) + (1 − w)Tr( ˆODL(ρ)) + (1 + w)Tr( ˆODR(ρ)).
+(S53)
+
+15
+Using the relation Tr(ABC) = Tr(CAB), we have
+Tr( ˆO[H, ρ]) = Tr([ ˆO, H] ρ) = J
+�
+l
+Tr([ ˆO, a†
+l+1al + a†
+l al+1] ρ),
+(S54)
+Tr( ˆODL(ρ)) =
+�
+l
+�
+Tr( ˆO2Al ρ A†
+l ) − Tr( ˆOA†
+l Al ρ) − Tr( ˆO ρ A†
+l Al)
+�
+=
+�
+l
+�
+Tr([A†
+l , ˆO]Al ρ) + Tr(A†
+l [ ˆO, Al] ρ)
+�
+,
+(S55)
+Tr( ˆODR(ρ)) =
+�
+l
+�
+Tr( ˆO2A†
+l ρ Al) − Tr( ˆOAlA†
+l ρ) − Tr( ˆO ρ AlA†
+l )
+�
+=
+�
+l
+�
+Tr([Al, ˆO]A†
+l ρ) + Tr(Al[ ˆO, A†
+l ] ρ)
+�
+.
+(S56)
+Substituting ˆO = a†
+l1al2, ˆO = al1al2 and ˆO = a†
+l2a†
+l1 into Eq.(S52) ∼ Eq.(S55), we get the evolution equations of Gl1,l2, Dl1,l2 and
+D∗
+l1,l2, respectively. Namely, the evolution equations of correlation functions in real space are
+d
+dtGl1,l2 = − 4γGl1,l2 + iJ(Gl1−1,l2 + Gl1+1,l2 − Gl1,l2−1 − Gl1,l2+1) + 2�γ + w(γ2 − γ1)� δl1,l2
++ √γ1γ2 (−Dl1−1,l2 − Dl1+1,l2 + Dl2,l1−1 + Dl2,l1+1) + √γ1γ2 (D∗
+l1,l2−1 + D∗
+l1,l2+1 − D∗
+l2−1,l1 − D∗
+l2+1,l1),
+(S57)
+d
+dt Dl1,l2 = + 2 √γ1γ2(−Gl1−1,l2 − Gl1+1,l2 + Gl2−1,l1 + Gl2+1,l1) + 2w √γ1γ2
+�δl1,l2−1 − δl2,l1−1
+�
+− 4γDl1,l2 − iJ/2 (Dl1−1,l2 + Dl1+1,l2 + Dl1,l2−1 + Dl1,l2+1) + iJ/2 (Dl2,l1−1 + Dl2,l1+1 + Dl2−1,l1 + Dl2+1,l1),
+(S58)
+d
+dt D∗
+l1,l2 = + 2 √γ1γ2(−Gl2,l1−1 − Gl2,l1+1 + Gl1,l2−1 + Gl1,l2+1) + 2w √γ1γ2
+�δl1,l2−1 − δl2,l1−1
+�
+− 4γD∗
+l1,l2 + iJ/2 (D∗
+l1−1,l2 + D∗
+l1+1,l2 + D∗
+l1,l2−1 + D∗
+l1,l2+1) − iJ/2 (D∗
+l2,l1−1 + D∗
+l2,l1+1 + D∗
+l2−1,l1 + D∗
+l2+1,l1).
+(S59)
+B. Evolution equations of correlation functions in momentum space
+The Liouvillian of our model in momentum space is shown in Eq. (9) in the main text. Substituting this equation into Eq. (S52),
+we have
+d
+dtTr( ˆOρ(t)) =
+π
+�
+k=−π
+�
+− i2J cos kTr( ˆO[ˆnk, ρ]) + (1 − w)Tr( ˆODL
+k(ρ)) + (1 + w)DR
+k (ρ)
+�
+=
+π
+�
+k=−π
+�
+− i2J cos kTr([ ˆO, ˆnk]ρ) + (1 − w)� Tr([B†
+k, ˆO]Bkρ) + Tr(B†
+k[ ˆO, Bk]ρ) �
++ (1 + w)� Tr([Bk, ˆO]B†
+kρ) + Tr(Bk[ ˆO, B†
+k]ρ) ��
+.
+(S60)
+Substituting ˆO = a†
+k1ak2, ˆO = a†
+−k2a−k1, ˆO = ak2a−k1 and ˆO = a†
+−k2a†
+k1 into Eq.(S60), we get the evolution equations of correlation
+functions Gk1,k2, G−k2,−k1, Dk2,−k1 and D∗
+k1,−k2:
+d
+dt
+���������������
+Gk1,k2
+G−k2,−k1
+Dk2,−k1
+D∗
+k1,−k2
+���������������
+= Xk1k2
+���������������
+Gk1,k2
+G−k2,−k1
+Dk2,−k1
+D∗
+k1,−k2
+���������������
++ Vk1k2,
+(S61)
+where
+Xk1k2 =
+��������������
+−4γ + i2J(cos k1 − cos k2)
+0
+4 √γ1γ2 cos k1
+4 √γ1γ2 cos k2
+0
+−4γ + i2J(cos k2 − cos k1)
+−4 √γ1γ2 cos k2
+−4 √γ1γ2 cos k1
+4 √γ1γ2 cos k1
+−4 √γ1γ2 cos k2
+−4γ − i2J(cos k1 + cos k2)
+0
+4 √γ1γ2 cos k2
+−4 √γ1γ2 cos k1
+0
+−4γ + i2J(cos k1 + cos k2)
+��������������
+(S62)
+
+16
+and
+Vk1k2 = δk1,k2
+��������������
+2γ + 2w(γ2 − γ1)
+2γ + 2w(γ2 − γ1)
+i4w √γ1γ2 sin k1
+−i4w √γ1γ2 sin k1
+��������������
+.
+(S63)
+Eq. (S61) ∼ Eq. (S63) are just the same equations as Eq. (10) and Eq. (11) in the main text.
+C. Symmetry of damping matrix
+Denoting k = (k1, k2), then we can check the damping matrix Xk has TRS, PHS, CS, IS and PTS:
+TRS : UT X∗
+k U−1
+T = X−k
+=⇒
+UT = σz ⊗ σx; UT U∗
+T = 1
+PHS : UC XT
+k U−1
+C = −X−k
+=⇒ UC = I ⊗ σx; UCU∗
+C = 1
+CS : UΓ X†
+k U−1
+Γ = −Xk
+=⇒
+UΓ = σz ⊗ I; U2
+Γ = 1
+IS : UP Xk U−1
+P = X−k
+=⇒
+UP = I ⊗ I
+PTS : UPT X∗
+k U−1
+PT = Xk
+=⇒ UPT = σz ⊗ σx.
+(S64)
+Compared with the symmetry of the Liouvillian in Eq.(S30), Xk has higher symmetry.
+S5. Particle number distribution of steady state
+As for steady state, the Eq. (S61) equals to 0, so we can get the correlation functions of steady state by
+�
+Gs
+k1,k2 Gs
+−k2,−k1 Ds
+k2,−k1 Ds∗
+k1,−k2
+�T = −X−1
+k1k2Vk1k2
+(S65)
+When k1 = k2 = k, we get particle number distribution of steady state in momentum space ns
+k:
+ns
+k = Gs
+kk = (1 − w)γ1 + (1 + w)γ2
+2γ
+−
+2Jwγ1γ2 cos2 k sin k
+γ3 + γ(J2 − 4γ1γ2) cos2 k.
+(S66)
+Due to the translation invariance of our system, the particle number distributes uniformly on each site. Therefore, particle
+number on site l in the thermodynamic limit can be calculated by
+ns
+l = 1
+L
+L
+�
+j=1
+ns
+j = 1
+L
+�
+k
+ns
+k = 1
+2π
+� π
+k=−π
+dk ns
+k = 1
+2 + w(γ2 − γ1)
+2γ
+.
+(S67)
+S6. The relationship between the damping-matrix spectra and the Liouvillian spectra
+In this section, we demonstrate that for a real physical process with closed evolution equations of correlation functions the
+damping-matrix spectra are the subset of the Liouvillian spectra.
+The general form of closed evolution equations of correlation functions is
+d
+dtΨ = X Ψ + V,
+(S68)
+where
+X
+is
+the
+damping
+matrix,
+Ψ
+is
+the
+vector
+of
+correlation
+functions,
+for
+example,
+Ψ
+is
+taken
+as
+(Gk1,k2,G−k2,−k1, Dk2,−k1, D∗
+k1,−k2)T in our model. The vector V induces the correlation function vector of steady state ΨS as
+ΨS = −X−1V. By deducting ΨS , we have
+d
+dt(Ψ(t) − ΨS ) = X (Ψ(t) − ΨS ).
+(S69)
+
+17
+If the correlation function vector ΨΓ is governed by the eigen equation of damping matrix, we have
+X (ΨΓ(t) − ΨS ) = Γ (ΨΓ(t) − ΨS ),
+(S70)
+where Γ is the eigenvalue of X. The equation in the initial time is
+X (ΨΓ(0) − ΨS ) = Γ (ΨΓ(0) − ΨS ),
+(S71)
+Then from Eq. (S69), we obtain
+ΨΓ(t) − ΨS = eΓt (ΨΓ(0) − ΨS ).
+(S72)
+If ΨΓ is in a real physical process, we will have
+ΨΓ(t) = C⟨S0| ˆΨ e ˆLC t| ρ(0)⟩C,
+(S73a)
+ΨS = C⟨S0| ˆΨ e ˆLC t|Ω⟩C = C⟨S0| ˆΨ |Ω⟩C,
+(S73b)
+where ˆLC is the Liouvillian of system in representation C, | ρ(0)⟩C is the initial state of system and |Ω⟩C is the steady state
+of system.
+ˆΨ is the vector of operators in terms of correlation function vector Ψ, for example, in our model ˆΨ equals to
+(a†
+k1ak2, a†
+−k2a−k1, a−k1ak2, a†
+−k2a†
+k1)T. Substituting Eq.(S73) into Eq.(S72), we obtain
+C⟨S0| ˆΨ e ˆLC t� | ρ(0)⟩C − |Ω⟩C
+� = C⟨S0| ˆΨ eΓ t� | ρ(0)⟩C − |Ω⟩C
+�.
+(S74)
+Comparing the two sides of the above equation, we have
+e ˆLC t� | ρ(0)⟩C − |Ω⟩C
+� = eΓ t� | ρ(0)⟩C − |Ω⟩C
+�,
+(S75)
+and thus the eigenvalue Γ of damping matrix X is also the eigenvalue of Liouvillian ˆLC.
+∗ [email protected]
+[1] M.-D. Choi, Completely positive linear maps on complex matrices, Linear Algebra Applications 10, 285 (1975).
+[2] A. Jamiołkowski, Linear transformations which preserve trace and positive semidefiniteness of operators, Rep. Math. Phys. 3, 275 (1972).
+[3] J. E. Tyson, Operator-Schmidt decompositions and the Fourier transform, with applications to the operator-Schmidt numbers of unitaries,
+J. Phys. A: Math. Gen. 36, 10101 (2003).
+[4] M. Zwolak and G. Vidal, Mixed-State Dynamics in One-Dimensional Quantum Lattice Systems: A Time-Dependent Superoperator Renor-
+malization Algorithm, Phys. Rev. Lett. 93, 207205 (2004).
+[5] K. Kawabata, K. Shiozaki, M. Ueda, and M. Sato, Symmetry and Topology in Non-Hermitian Physics, Phys. Rev. X 9, 041015 (2019).
+[6] A. W. W. Ludwig, Topological phases: classification of topological insulators and superconductors of noninteracting fermions, and beyond,
+Physica Scripta T168, 014001 (2015).
+[7] C.-H. Liu, H. Jiang, and S. Chen, Topological classification of non-Hermitian systems with reflection symmetry, Phys. Rev. B 99, 125103
+(2019).
+[8] C.-H. Liu and S. Chen, Topological classification of defects in non-Hermitian systems, Phys. Rev. B 100, 144106 (2019).
+

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+Ultrafast switching of persistent electron and hole currents in ring molecules
+Tennesse Joyce and Agnieszka Jaron
+JILA and Department of Physics, University of Colorado, Boulder, CO-80309, USA
+(Dated: January 3, 2023)
+A circularly polarized laser pulse can induce persistent intra-molecular currents by either exciting
+or ionizing molecules. These two cases are identified as electron currents and hole currents, respec-
+tively, and up to now they have been studied only separately. We report ab initio time-dependent
+density-functional theory (TDDFT) simulations of currents during resonance-enhanced two-photon
+ionization of benzene, which reveal for the first time that both electron and hole currents can be
+present simultaneously. By adjusting the intensity of the laser pulse, the balance between the two
+types of current can be controlled, and the overall sign of the current can be switched. We provide
+a physical explanation for the effect in terms of complex molecular orbitals which is consistent with
+the TDDFT simulations.
+It has long been understood that, in response to an ap-
+plied magnetic field, the delocalized electrons of an aro-
+matic molecule circulate in so-called aromatic ring cur-
+rent [1, 2]. This effect is important in nuclear magnetic
+resonance spectroscopy, where the internal magnetic field
+generated by the ring current is responsible for diamag-
+netic shielding [3]. In 2006, it was proposed that ring
+currents in molecules could also be induced by ultra-
+short laser pulses with circular or elliptical polarization
+[4, 5]. The basic mechanism is that angular momentum
+carried by light is transfered to electrons in a molecule.
+Due to conservation of angular momentum, the current
+persists after the pulse has ended—even without an ex-
+ternal magnetic field.
+Various experiments on atomic
+targets have confirmed the existence of the effect [6, 7],
+although no direct observational data is available in the
+case of molecules. Recent interest in photoinduced ring
+currents is motivated by the rapid technological advances
+in polarization control of high-harmonic radiation made
+in the last few years [8–10], which may enable experimen-
+tal study of these phenomena in the near future [11].
+There are several major advantages of photoinduced
+ring currents compared to those induced by static mag-
+netic fields. First, the current is expected to be orders of
+magnitude stronger, and so is the induced magnetic field
+[12]. Second, they enable femtosecond (or even attosec-
+ond) time-resolved studies of aromaticity and magnetism
+[13, 14]. Lastly, they establish the possibility for coherent
+control of ring currents [15], which may have applications
+for controlling chemical reactions or the operation of ad-
+vanced opto-electronic devices.
+In this Letter we predict a novel effect which causes
+the dominant charge carrier of the ring current to transi-
+tion from electrons to holes as the peak laser intensity in-
+creases past around 1012 W/cm2. We illustrate the effect
+with a series of ab initio time-dependent density func-
+tional theory (TDDFT) simulations of benzene (C6H6),
+which is the prototypical aromatic molecule. Lastly, we
+demonstrate that the effect is not accounted for in the
+commonly used few level model of ring currents, due to
+the fact that it neglects ionization. This calls into ques-
+tion the results of several previous studies (e.g. [4, 5, 15])
+where it was assumed that the few level model is accurate
+for laser intensities on the order of 1012 W/cm2.
+We begin by introducing the distinction between elec-
+tron and hole current: when an electron is promoted to
+an orbital with nonzero angular momentum, this creates
+an electron current; when an electron is removed (e.g.,
+ionized) from an orbital with nonzero angular momen-
+tum, this creates a hole current. So far, hole currents
+have mostly been studied in the context of strong field
+ionization of atoms by circularly polarized laser pulses,
+and it was recently confirmed experimentally that a hole
+can be created with a specific angular momentum relative
+to the laser polarization [16–19]. Electron currents on the
+other hand do not involve ionization, only excitation.
+However, in the interaction of atoms and molecules
+with strong laser fields, excitation and ionization are of-
+ten closely related and occur together. A typical example
+is resonance-enhanced multiphoton ionization (REMPI)
+[20, 21], a two-step ionization process wherein an atom
+or molecule is first excited to an intermediate state (that
+must be resonant with some multiple of the laser fre-
+quency) and then subsequently ionized.
+Now consider
+REMPI in a system where the intermediate excited state
+corresponds to an electron current, and the final ionized
+state corresponds to a hole current (we will show that
+benzene is such a system). The balance between excita-
+tion and ionization (and therefore electron and hole cur-
+rent) will depend on the laser intensity because the pro-
+cesses involve different numbers of photons (and therefore
+scale with different powers of intensity). In particular at
+low intensities we expect electron current to dominate
+(excitation), and at high intensities we expect hole cur-
+rent to dominate (ionization).
+Our main theoretical method is TDDFT, as imple-
+mented by Octopus [22–24], which provides a fully non-
+perturbative description of the light-matter interaction.
+As a reference point to compare against the full TDDFT
+simulations, we also consider the few level model of ring
+currents (e.g. [5]). We discuss the implementations of
+both models in [25]. Because the few level model does
+not include ionization, we expect the two models to di-
+verge at high enough laser intensities.
+The laser pulse in our simulations is described in the
+arXiv:2301.00380v1  [physics.chem-ph]  1 Jan 2023
+
+2
+FIG. 1.
+(a) Visualization of the current density based on the component passing through a plane bisecting the molecule as
+shown (averagea over all possible orientations of that plane [see Eq. (2)]) (b) Cross sections of the current density taken at
+the end of the laser pulse (t = 200 a.u.) for several different simulations. At low laser intensity the co-rotating current (red)
+dominates, while at high intensity the counter-rotating current (blue) dominates. Note: Each plot is scaled individually relative
+to the maximum absolute value within that plot. The nuclei lie in the plane z = 0 with the carbon ring at x = ±2.63 a.u. and
+the hydrogen ring at x = ±4.69 a.u..
+dipole approximation by the following electric field,
+E(t) =
+�
+E sin2 (πt/T) Re
+�
+ˆϵeiω(t−T/2)�
+,
+0 < t < T,
+0,
+otherwise,
+(1)
+with central frequency ω = 6.76 eV (183 nm), dura-
+tion T = 16π/ω = 202 a.u. = 4.9 fs, circular polar-
+ization ˆϵ = (ˆx + iˆy)/
+√
+2 (with the molecule in the xy-
+plane), and a variable peak amplitude E.
+The central
+frequency was chosen to be resonant with the doubly
+degenerate E1u state (as computed with linear response
+TDDFT [25]), which is predominantly associated with
+the HOMO-LUMO transition (HOMO = Highest Occu-
+pied Molecular Orbital; LUMO = Lowest Unoccupied
+Molecular Orbital).
+Because the computed ionization
+threshold is 9.0 eV < 2ω, this laser pulse is designed
+to drive 1+1 REMPI where one photon is enough to
+promote electron to the excited state and one additional
+photon to ionize.
+After interacting with the laser pulse (t > T), the ben-
+zene molecule is in a superposition of the A1g ground
+state and the E1u excited state and also, to an extent,
+ionized. This causes oscillations in the charge and current
+densities ρ(r, t) and J(r, t), respectively, with period 612
+as (corresponding to the energy difference between the
+ground state and excited states), which are an example
+of attosecond charge migration [26].
+In order to visualize the current we isolate the sta-
+tionary component of the current density, by computing
+an angle averaged cross section defined by the following
+integral (in cylindrical coordinates ρ, z, φ),
+J(x, z, t) = 1
+2π
+� 2π
+0
+ˆφ · J(|x|, z, φ)dφ.
+(2)
+The geometric interpretation of this integral is given in
+Fig.
+1.
+The angle averaging procedure for the few
+level model causes that the fast-oscillating component
+effectively vanishes. Within the few-level model, the fast-
+oscillating component of the current density is zeroed out
+by this averaging procedure because of its parity.
+It has
+similar effect on TDDFT results, and therefore J(x, z, y)
+has only a very gradual time dependence for t > T. The
+same is true for TDDFT. These integrated current densi-
+ties are plotted in Fig. 1b. At low intensities the current
+is a combination of a strong co-rotating current (red) and
+a weak counter-rotating current (blue), while at high in-
+tensities the counter-rotating current dominates. As we
+explain below (see Fig. 4), the reversal is a signature of
+the transition from electron to hole current regime.
+The oscillatory component of the charge motion is best
+visualized by plotting the charge displacement,
+∆ρ(r, t) = ρ(r, t) − ρ(r, 0),
+(3)
+shown in Fig. 2. The cloud of displaced charge circulates
+around the molecule with the expected period of 612 as,
+and this continues even after the pulse ends.
+Overall,
+both the magnitude and shape of the charge displace-
+ment are remarkably similar between the two models,
+however there are some subtle differences.
+First, long
+after the laser pulse the two models gradually become
+desynchronized. Second, in TDDFT there appears to be
+a rearrangement of charge in the plane of the molecule,
+whereas the few level model only predicts the dynamics
+above and below the plane.
+
+(a)
+(b)
+3.8 × 1011 W/cm2
+Few-level Model
+4.0
+2.0
+('n
+-0.0
+N
+-2.0
+-4.0
+5 × 1012 W/cm²
+1013 W/cm²
+4.0
+2.0
+-0.0
+-2.0
+-4.0
+-6.0
+-3.0
+0.0
+3.0
+-6.0
+-3.0
+0.0
+3.0
+6.0
+x (a.u.)
+x (a.u.)3
+FIG. 2.
+Snapshots of the charge displacement induced by a circularly-polarized laser pulse with peak intensity 5×1012 W/cm2
+taken around the peak of the laser pulse (first three columns t ≈ 100 a.u.) and after the laser pulse (last three columns t ≈ 400
+a.u.). Light areas indicate excess electrons while dark areas indicate fewer electrons, as compared to the ground state charge
+density before the laser pulse. We compare the results between the two theoretical models, TDDFT (top row) and the few
+level model (bottom row).
+FIG. 3.
+Comparison of full TDDFT simulations (solid blue
+line) to the few level model (orange dashed line). For peak
+intensities, when ionization (dotted green line) becomes non-
+negligible, the two models begin to disagree.
+The smooth
+lines have been interpolated between the calculated intensities
+using the method described in [25].
+Another important observation about the density dif-
+ference is that the dark areas are generally larger than
+the light areas. In the TDDFT results one reason for this
+is ionization, with the ionization probability given by
+P ionize = −
+�
+∆ρ(r, 2T)d3r,
+(4)
+where the integral ranges over the simulation box. Unex-
+pectedly, the few level model also appears to have dark
+areas larger than light areas even though it does not
+include ionization, and in fact the charge displacement
+must integrate to zero in that model. The reason for this
+is that the E1u is of mixed character, part of which in-
+volves excitation to LUMO + 3 [25].
+Note: The excess
+of darker areas in the TDDFT model is a combination of
+both ionization and excitation to LUMO +3 orbital.
+The intensity dependence of the dynamics is illustrated
+in Fig. 3. using the current. Note: this current is directly
+proportional to z-component of the magnetic moment as
+well as z-component of electronic angular momentum),
+Since the domain of integration is the simulation box,
+ionized electrons are not included.
+For this reason we
+plot Lz(2T) so that the ionizing wavepacket has enough
+time to leave the box. Whereas in the few level model
+the magnetic moment increases monotonically with the
+laser intensity (up to about 1013 W/cm2, after which
+the system Rabi oscillates back to the ground state), in
+TDDFT the current starts to decrease already around
+1012 W/cm2, and reverses sign for even higher intensities.
+We also plot the ionization probability (defined in Eq. 4),
+and conclude that the reversal occurs precisely when the
+ionization probability becomes non-negligible.
+The implications of the transition from electron to hole
+current on the charge dynamics, and the underlying phys-
+ical mechanism responsible for that transition, can be
+understood in more detail describe in more detail using
+complex molecular orbitals, as illustrated schematically
+in Fig. 4. These orbitals represent a change of basis from
+the usual real-valued Kohn-Sham orbitals ψn(r) (defined
+in [25]),
+ψHOMO
+±
+(r) = [ψ14(r) ± iψ15(r)] /
+√
+2,
+(5)
+ψLUMO
+±
+(r) = [ψ16(r) ± iψ17(r)] /
+√
+2.
+(6)
+The advantage of using complex orbitals is that they are
+eigenfunctions of the 6-fold symmetry operator (rotation
+
+t = 105 a.u.
+t = 110 a.u.
+t = 390 a.u.
+t = 395 a.u.
+t = 100 a.u.
+t = 400 a.u.
+TDDFT
+Few
+Level
+Model0.8 
+Lz (TDDFT)
+Lz (Few level)
+0.6 
+lonizationprobability(TDDFT)
+or probability
+0.4
+0.2
+(a.u.)
+0.0
+0.2
+1011
+1012
+1013
+Peak intensity (W/cm2)4
+FIG. 4.
+Schematic illustrating the complex molecular or-
+bitals and the physical mechanism for the transition from
+electron to hole current. Color indicates the complex phase.
+about the molecular axis by 60◦),
+exp
+�
+− iπ
+3¯h
+ˆLz
+¯h
+�
+ψHOMO
+±
+(r) = exp
+�
+∓iπ
+3
+�
+ψHOMO
+±
+(r),(7)
+exp
+�
+−iπ
+3
+ˆLz
+¯h
+�
+ψLUMO
+±
+(r) = exp
+�
+∓2iπ
+3
+�
+ψLUMO
+±
+(r).(8)
+The complex orbitals have magnetic quantum numbers
+m defined modulo 6: ψHOMO
+±
+have m = ±1 and ψLUMO
+±
+have m = ±2. Just as for atomic orbitals, the sign of
+m indicates the direction the electron circulates around
+the molecule, and the magnitude indicates more-or-less
+the angular speed. We have chosen our conventions such
+that m > 0 electrons are co-rotating with the laser field,
+and m < 0 electrons are counter-rotating.
+Using the notation of complex orbitals, Fig. 4 illus-
+trates how in the ground state, both ψHOMO
+±
+are dou-
+bly occupied, and consequently there is zero net cur-
+rent. When the benzene molecule is exposed to a cir-
+cularly polarized laser pulse, the usual selection rule
+∆m = 1 applies (here we assume the laser is polar-
+ized in the molecular plane, see [25] for the more gen-
+eral case), so that the only dipole-allowed transition is
+ψHOMO
++
+to ψLUMO
++
+, which is the dominant component of
+the E1u excited state. The electron excited to LUMO
+contributes a strong co-rotating current (m = +2), but
+the imbalance of electrons in the HOMO contributes a
+weaker counter-rotating current (m = −1). This can al-
+ternatively be interpreted as a positively charged hole
+occupying ψHOMO
++
+producing a co-rotating hole current
+(rather than a counter-rotating electron current). This
+is precisely what we see in the top row of Fig. 1b, two
+components to the current with opposite sign (red and
+blue).
+In order to explain the reversal of the current at higher
+intensity (bottom row of Fig. 1b), we simply recognize
+that the electron previously excited to ψLUMO
++
+can ab-
+sorb a second photon from the same laser pulse, ionizing,
+and leaving behind only the hole current. The balance
+between the one-photon excitation and the two-photon
+ionization processes can be controlled by varying the laser
+intensity, because the first process scales with I while the
+second process scales with I2 (with I ∝ E2 the laser in-
+tensity). Furthermore, it is now apparent that the sign
+reversal can be interpreted as a change in the dominant
+charge carrier from electrons to holes.
+In conclusion, we have shown that both electron and
+hole currents are present during resonance-enhanced two-
+photon ionization of benzene, and the balance between
+the two current regimes can be controlled by varying the
+peak laser intensity. We have proposed a simple expla-
+nation for the effect in terms of molecular orbitals, which
+is consistent with the results of full TDDFT simulations.
+Variants of complex orbital model should apply to a wide
+variety of molecules other than benzene, meaning that
+the structure of the complex molecular orbitals can be
+used to predict the interplay between electron and hole
+currents during REMPI. In order to measure this effect
+in experiment, several pump-probe schemes have been
+proposed that are sensitive to the magnitude and direc-
+tion of the ring current [7, 11]. In [25], we demonstrate
+that the reversal is independent of the orientation of the
+molecule, which greatly simplifies any potential exper-
+iment.
+Finally, our results suggest that the few level
+model typically used to study photoinduced ring currents
+may be insufficient even for moderate laser intensities
+around 1012 W/cm2. A more ab initio nonperturbative
+theory such as TDDFT, as used in present paper, is more
+appropriate for this regime.
+This work was supported by the NSF Grant No. PHY-
+1734006 and Grant No. PHY-2110628. This work uti-
+lized resources from the University of Colorado Boulder
+Research Computing Group, which is supported by the
+National Science Foundation.
+[1] J. A. N. F. Gomes and R. B. Mallion, Chemical Reviews
+101, 1349 (2001).
+[2] T. M. Krygowski, H. Szatylowicz, O. A. Stasyuk, J. Do-
+minikowska,
+and M. Palusiak, Chemical Reviews 114,
+6383 (2014).
+[3] T. Heine, C. Corminboeuf, and G. Seifert, Chemical Re-
+views 105, 3889 (2005).
+[4] I. Barth and J. Manz, Angewandte Chemie International
+
+2元
+m=-2
+m = +2
+LUMO
+Phase
+7元
+0
+m=-1
+m=+1
+HOMO5
+Edition 45, 2962 (2006).
+[5] I. Barth, J. Manz, Y. Shigeta, and K. Yagi, Journal of
+the American Chemical Society 128, 7043 (2006).
+[6] M. Wollenhaupt, M. Krug, J. K¨ohler, T. Bayer, C. Sarpe-
+Tudoran,
+and T. Baumert, Applied Physics B 95, 245
+(2009).
+[7] S. Eckart, M. Kunitski, M. Richter, A. Hartung, J. Rist,
+F. Trinter, K. Fehre, N. Schlott, K. Henrichs, L. P. H.
+Schmidt, T. Jahnke, M. Sch¨offler, K. Liu, I. Barth,
+J. Kaushal, F. Morales, M. Ivanov, O. Smirnova,
+and
+R. D¨orner, Nature Physics 14, 701 (2018).
+[8] A. Fleischer, O. Kfir, T. Diskin, P. Sidorenko, and O. Co-
+hen, Nature Photonics 8, 543 (2014).
+[9] D. D. Hickstein, F. J. Dollar, P. Grychtol, J. L. Ellis,
+R. Knut, C. Hern´andez-Garc´ıa, D. Zusin, C. Gentry,
+J. M. Shaw, T. Fan, K. M. Dorney, A. Becker, A. Jaro´n-
+Becker, H. C. Kapteyn, M. M. Murnane,
+and C. G.
+Durfee, Nature Photonics 9, 743 (2015).
+[10] P.-C. Huang, C. Hern´andez-Garc´ıa, J.-T. Huang, P.-Y.
+Huang, C.-H. Lu, L. Rego, D. D. Hickstein, J. L. Ellis,
+A. Jaron-Becker, A. Becker, S.-D. Yang, C. G. Durfee,
+L. Plaja, H. C. Kapteyn, M. M. Murnane, A. H. Kung,
+and M.-C. Chen, Nature Photonics 12, 349 (2018).
+[11] O. Neufeld and O. Cohen, Physical Review Letters 123,
+103202 (2019).
+[12] K.-J. Yuan and A. D. Bandrauk, Physical Review A 88,
+013417 (2013).
+[13] I. S. Ulusoy and M. Nest, Journal of the American Chem-
+ical Society 133, 20230 (2011).
+[14] K.-J. Yuan, C.-C. Shu, D. Dong,
+and A. D. Ban-
+drauk, The Journal of Physical Chemistry Letters 8, 2229
+(2017).
+[15] H. Mineo and Y. Fujimura, The Journal of Chemical
+Physics 147, 224301 (2017).
+[16] I. Barth and O. Smirnova, Physical Review A 84, 063415
+(2011).
+[17] T. Herath, L. Yan, S. K. Lee, and W. Li, Physical Review
+Letters 109, 043004 (2012).
+[18] X. Zhu, P. Lan, K. Liu, Y. Li, X. Liu, Q. Zhang, I. Barth,
+and P. Lu, Optics Express 24, 4196 (2016).
+[19] K. Liu and I. Barth, Physical Review A 94, 043402
+(2016).
+[20] V. S. Antonov, I. N. Knyazev, V. S. Letokhov, V. M.
+Matiuk, V. G. Movshev,
+and V. K. Potapov, Optics
+Letters 3, 37 (1978).
+[21] W. Dietz, H. J. Neusser, U. Boesl, E. W. Schlag,
+and
+S. H. Lin, Chemical Physics 66, 105 (1982).
+[22] M. A. L. Marques, A. Castro, G. F. Bertsch, and A. Ru-
+bio, Computer Physics Communications 151, 60 (2003).
+[23] A. Castro, H. Appel, M. Oliveira, C. A. Rozzi, X. An-
+drade, F. Lorenzen, M. a. L. Marques, E. K. U. Gross,
+and A. Rubio, physica status solidi (b) 243, 2465 (2006).
+[24] X. Andrade, D. Strubbe, U. D. Giovannini, A. H. Larsen,
+M. J. T. Oliveira,
+J. Alberdi-Rodriguez,
+A. Varas,
+I. Theophilou, N. Helbig, M. J. Verstraete, L. Stella,
+F. Nogueira, A. Aspuru-Guzik, A. Castro, M. A. L.
+Marques,
+and A. Rubio, Physical Chemistry Chemical
+Physics 17, 31371 (2015).
+[25] See Supplemental Material for details on the TDDFT
+and few-level models, orientation dependence, and the
+method for interpolating over intensity.
+[26] H. J. W¨orner, C. A. Arrell, N. Banerji, A. Cannizzo,
+M. Chergui, A. K. Das, P. Hamm, U. Keller, P. M. Kraus,
+E. Liberatore, P. Lopez-Tarifa, M. Lucchini, M. Meuwly,
+C. Milne, J.-E. Moser, U. Rothlisberger, G. Smolentsev,
+J. Teuscher, J. A. van Bokhoven, and O. Wenger, Struc-
+tural Dynamics 4, 061508 (2017).
+

VdAyT4oBgHgl3EQfhfjt/content/tmp_files/load_file.txt

@@ -0,0 +1,447 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf,len=446
+page_content='Ultrafast switching of persistent electron and hole currents in ring molecules  Tennesse Joyce and Agnieszka Jaron  JILA and Department of Physics, University of Colorado, Boulder, CO-80309, USA  (Dated: January 3, 2023)  A circularly polarized laser pulse can induce persistent intra-molecular currents by either exciting  or ionizing molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' These two cases are identified as electron currents and hole currents, respec-  tively, and up to now they have been studied only separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' We report ab initio time-dependent  density-functional theory (TDDFT) simulations of currents during resonance-enhanced two-photon  ionization of benzene, which reveal for the first time that both electron and hole currents can be  present simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' By adjusting the intensity of the laser pulse, the balance between the two  types of current can be controlled, and the overall sign of the current can be switched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' We provide  a physical explanation for the effect in terms of complex molecular orbitals which is consistent with  the TDDFT simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  It has long been understood that, in response to an ap-  plied magnetic field, the delocalized electrons of an aro-  matic molecule circulate in so-called aromatic ring cur-  rent [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' This effect is important in nuclear magnetic  resonance spectroscopy, where the internal magnetic field  generated by the ring current is responsible for diamag-  netic shielding [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' In 2006, it was proposed that ring  currents in molecules could also be induced by ultra-  short laser pulses with circular or elliptical polarization  [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' The basic mechanism is that angular momentum  carried by light is transfered to electrons in a molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  Due to conservation of angular momentum, the current  persists after the pulse has ended—even without an ex-  ternal magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  Various experiments on atomic  targets have confirmed the existence of the effect [6, 7],  although no direct observational data is available in the  case of molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Recent interest in photoinduced ring  currents is motivated by the rapid technological advances  in polarization control of high-harmonic radiation made  in the last few years [8–10], which may enable experimen-  tal study of these phenomena in the near future [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  There are several major advantages of photoinduced  ring currents compared to those induced by static mag-  netic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' First, the current is expected to be orders of  magnitude stronger, and so is the induced magnetic field  [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Second, they enable femtosecond (or even attosec-  ond) time-resolved studies of aromaticity and magnetism  [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Lastly, they establish the possibility for coherent  control of ring currents [15], which may have applications  for controlling chemical reactions or the operation of ad-  vanced opto-electronic devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  In this Letter we predict a novel effect which causes  the dominant charge carrier of the ring current to transi-  tion from electrons to holes as the peak laser intensity in-  creases past around 1012 W/cm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' We illustrate the effect  with a series of ab initio time-dependent density func-  tional theory (TDDFT) simulations of benzene (C6H6),  which is the prototypical aromatic molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Lastly, we  demonstrate that the effect is not accounted for in the  commonly used few level model of ring currents, due to  the fact that it neglects ionization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' This calls into ques-  tion the results of several previous studies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' [4, 5, 15])  where it was assumed that the few level model is accurate  for laser intensities on the order of 1012 W/cm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  We begin by introducing the distinction between elec-  tron and hole current: when an electron is promoted to  an orbital with nonzero angular momentum, this creates  an electron current;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' when an electron is removed (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=',  ionized) from an orbital with nonzero angular momen-  tum, this creates a hole current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' So far, hole currents  have mostly been studied in the context of strong field  ionization of atoms by circularly polarized laser pulses,  and it was recently confirmed experimentally that a hole  can be created with a specific angular momentum relative  to the laser polarization [16–19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Electron currents on the  other hand do not involve ionization, only excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  However, in the interaction of atoms and molecules  with strong laser fields, excitation and ionization are of-  ten closely related and occur together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' A typical example  is resonance-enhanced multiphoton ionization (REMPI)  [20, 21], a two-step ionization process wherein an atom  or molecule is first excited to an intermediate state (that  must be resonant with some multiple of the laser fre-  quency) and then subsequently ionized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  Now consider  REMPI in a system where the intermediate excited state  corresponds to an electron current, and the final ionized  state corresponds to a hole current (we will show that  benzene is such a system).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' The balance between excita-  tion and ionization (and therefore electron and hole cur-  rent) will depend on the laser intensity because the pro-  cesses involve different numbers of photons (and therefore  scale with different powers of intensity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' In particular at  low intensities we expect electron current to dominate  (excitation), and at high intensities we expect hole cur-  rent to dominate (ionization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  Our main theoretical method is TDDFT, as imple-  mented by Octopus [22–24], which provides a fully non-  perturbative description of the light-matter interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  As a reference point to compare against the full TDDFT  simulations, we also consider the few level model of ring  currents (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' [5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' We discuss the implementations of  both models in [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Because the few level model does  not include ionization, we expect the two models to di-  verge at high enough laser intensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  The laser pulse in our simulations is described in the  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='00380v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='chem-ph]  1 Jan 2023  2  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  (a) Visualization of the current density based on the component passing through a plane bisecting the molecule as  shown (averagea over all possible orientations of that plane [see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' (2)]) (b) Cross sections of the current density taken at  the end of the laser pulse (t = 200 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=') for several different simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' At low laser intensity the co-rotating current (red)  dominates, while at high intensity the counter-rotating current (blue) dominates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Note: Each plot is scaled individually relative  to the maximum absolute value within that plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' The nuclei lie in the plane z = 0 with the carbon ring at x = ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='63 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' and  the hydrogen ring at x = ±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='69 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='.  dipole approximation by the following electric field,  E(t) =  �  E sin2 (πt/T) Re  �  ˆϵeiω(t−T/2)�  ,  0 < t < T,  0,  otherwise,  (1)  with central frequency ω = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='76 eV (183 nm), dura-  tion T = 16π/ω = 202 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='9 fs, circular polar-  ization ˆϵ = (ˆx + iˆy)/  √  2 (with the molecule in the xy-  plane), and a variable peak amplitude E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  The central  frequency was chosen to be resonant with the doubly  degenerate E1u state (as computed with linear response  TDDFT [25]), which is predominantly associated with  the HOMO-LUMO transition (HOMO = Highest Occu-  pied Molecular Orbital;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' LUMO = Lowest Unoccupied  Molecular Orbital).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  Because the computed ionization  threshold is 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='0 eV < 2ω, this laser pulse is designed  to drive 1+1 REMPI where one photon is enough to  promote electron to the excited state and one additional  photon to ionize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  After interacting with the laser pulse (t > T), the ben-  zene molecule is in a superposition of the A1g ground  state and the E1u excited state and also, to an extent,  ionized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' This causes oscillations in the charge and current  densities ρ(r, t) and J(r, t), respectively, with period 612  as (corresponding to the energy difference between the  ground state and excited states), which are an example  of attosecond charge migration [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  In order to visualize the current we isolate the sta-  tionary component of the current density, by computing  an angle averaged cross section defined by the following  integral (in cylindrical coordinates ρ, z, φ),  J(x, z, t) = 1  2π  � 2π  0  ˆφ · J(|x|, z, φ)dφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  (2)  The geometric interpretation of this integral is given in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  The angle averaging procedure for the few  level model causes that the fast-oscillating component  effectively vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Within the few-level model, the fast-  oscillating component of the current density is zeroed out  by this averaging procedure because of its parity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  It has  similar effect on TDDFT results, and therefore J(x, z, y)  has only a very gradual time dependence for t > T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' The  same is true for TDDFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' These integrated current densi-  ties are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' At low intensities the current  is a combination of a strong co-rotating current (red) and  a weak counter-rotating current (blue), while at high in-  tensities the counter-rotating current dominates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' As we  explain below (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' 4), the reversal is a signature of  the transition from electron to hole current regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  The oscillatory component of the charge motion is best  visualized by plotting the charge displacement,  ∆ρ(r, t) = ρ(r, t) − ρ(r, 0),  (3)  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' The cloud of displaced charge circulates  around the molecule with the expected period of 612 as,  and this continues even after the pulse ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  Overall,  both the magnitude and shape of the charge displace-  ment are remarkably similar between the two models,  however there are some subtle differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  First, long  after the laser pulse the two models gradually become  desynchronized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Second, in TDDFT there appears to be  a rearrangement of charge in the plane of the molecule,  whereas the few level model only predicts the dynamics  above and below the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  (a)  (b)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='8 × 1011 W/cm2  Few-level Model  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content="0  ('n  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='0  N  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='0  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='0  5 × 1012 W/cm²  1013 W/cm²  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='0  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='0  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='0  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='0  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='0  x (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=')  x (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' )3  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  Snapshots of the charge displacement induced by a circularly-polarized laser pulse with peak intensity 5×1012 W/cm2  taken around the peak of the laser pulse (first three columns t ≈ 100 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=') and after the laser pulse (last three columns t ≈ 400  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Light areas indicate excess electrons while dark areas indicate fewer electrons, as compared to the ground state charge  density before the laser pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' We compare the results between the two theoretical models, TDDFT (top row) and the few  level model (bottom row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  Comparison of full TDDFT simulations (solid blue  line) to the few level model (orange dashed line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' For peak  intensities, when ionization (dotted green line) becomes non-  negligible, the two models begin to disagree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  The smooth  lines have been interpolated between the calculated intensities  using the method described in [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  Another important observation about the density dif-  ference is that the dark areas are generally larger than  the light areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' In the TDDFT results one reason for this  is ionization, with the ionization probability given by  P ionize = −  �  ∆ρ(r, 2T)d3r,  (4)  where the integral ranges over the simulation box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Unex-  pectedly, the few level model also appears to have dark  areas larger than light areas even though it does not  include ionization, and in fact the charge displacement  must integrate to zero in that model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' The reason for this  is that the E1u is of mixed character, part of which in-  volves excitation to LUMO + 3 [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  Note: The excess  of darker areas in the TDDFT model is a combination of  both ionization and excitation to LUMO +3 orbital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  The intensity dependence of the dynamics is illustrated  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' using the current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Note: this current is directly  proportional to z-component of the magnetic moment as  well as z-component of electronic angular momentum),  Since the domain of integration is the simulation box,  ionized electrons are not included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  For this reason we  plot Lz(2T) so that the ionizing wavepacket has enough  time to leave the box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Whereas in the few level model  the magnetic moment increases monotonically with the  laser intensity (up to about 1013 W/cm2, after which  the system Rabi oscillates back to the ground state), in  TDDFT the current starts to decrease already around  1012 W/cm2, and reverses sign for even higher intensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  We also plot the ionization probability (defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' 4),  and conclude that the reversal occurs precisely when the  ionization probability becomes non-negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  The implications of the transition from electron to hole  current on the charge dynamics, and the underlying phys-  ical mechanism responsible for that transition, can be  understood in more detail describe in more detail using  complex molecular orbitals, as illustrated schematically  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' These orbitals represent a change of basis from  the usual real-valued Kohn-Sham orbitals ψn(r) (defined  in [25]),  ψHOMO  ±  (r) = [ψ14(r) ± iψ15(r)] /  √  2,  (5)  ψLUMO  ±  (r) = [ψ16(r) ± iψ17(r)] /  √  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  (6)  The advantage of using complex orbitals is that they are  eigenfunctions of the 6-fold symmetry operator (rotation  t = 105 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  t = 110 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  t = 390 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  t = 395 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  t = 100 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  t = 400 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  TDDFT  Few  Level  Model0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='8  Lz (TDDFT)  Lz (Few level)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='6  lonizationprobability(TDDFT)  or probability  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='2  (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=')  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='2  1011  1012  1013  Peak intensity (W/cm2)4  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  Schematic illustrating the complex molecular or-  bitals and the physical mechanism for the transition from  electron to hole current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Color indicates the complex phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  about the molecular axis by 60◦),  exp  �  − iπ  3¯h  ˆLz  ¯h  �  ψHOMO  ±  (r) = exp  �  ∓iπ  3  �  ψHOMO  ±  (r),(7)  exp  �  −iπ  3  ˆLz  ¯h  �  ψLUMO  ±  (r) = exp  �  ∓2iπ  3  �  ψLUMO  ±  (r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' (8)  The complex orbitals have magnetic quantum numbers  m defined modulo 6: ψHOMO  ±  have m = ±1 and ψLUMO  ±  have m = ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Just as for atomic orbitals, the sign of  m indicates the direction the electron circulates around  the molecule, and the magnitude indicates more-or-less  the angular speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' We have chosen our conventions such  that m > 0 electrons are co-rotating with the laser field,  and m < 0 electrons are counter-rotating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  Using the notation of complex orbitals, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' 4 illus-  trates how in the ground state, both ψHOMO  ±  are dou-  bly occupied, and consequently there is zero net cur-  rent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' When the benzene molecule is exposed to a cir-  cularly polarized laser pulse, the usual selection rule  ∆m = 1 applies (here we assume the laser is polar-  ized in the molecular plane, see [25] for the more gen-  eral case), so that the only dipole-allowed transition is  ψHOMO  +  to ψLUMO  +  , which is the dominant component of  the E1u excited state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' The electron excited to LUMO  contributes a strong co-rotating current (m = +2), but  the imbalance of electrons in the HOMO contributes a  weaker counter-rotating current (m = −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' This can al-  ternatively be interpreted as a positively charged hole  occupying ψHOMO  +  producing a co-rotating hole current  (rather than a counter-rotating electron current).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' This  is precisely what we see in the top row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' 1b, two  components to the current with opposite sign (red and  blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  In order to explain the reversal of the current at higher  intensity (bottom row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' 1b), we simply recognize  that the electron previously excited to ψLUMO  +  can ab-  sorb a second photon from the same laser pulse, ionizing,  and leaving behind only the hole current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' The balance  between the one-photon excitation and the two-photon  ionization processes can be controlled by varying the laser  intensity, because the first process scales with I while the  second process scales with I2 (with I ∝ E2 the laser in-  tensity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Furthermore, it is now apparent that the sign  reversal can be interpreted as a change in the dominant  charge carrier from electrons to holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  In conclusion, we have shown that both electron and  hole currents are present during resonance-enhanced two-  photon ionization of benzene, and the balance between  the two current regimes can be controlled by varying the  peak laser intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' We have proposed a simple expla-  nation for the effect in terms of molecular orbitals, which  is consistent with the results of full TDDFT simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  Variants of complex orbital model should apply to a wide  variety of molecules other than benzene, meaning that  the structure of the complex molecular orbitals can be  used to predict the interplay between electron and hole  currents during REMPI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' In order to measure this effect  in experiment, several pump-probe schemes have been  proposed that are sensitive to the magnitude and direc-  tion of the ring current [7, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' In [25], we demonstrate  that the reversal is independent of the orientation of the  molecule, which greatly simplifies any potential exper-  iment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  Finally, our results suggest that the few level  model typically used to study photoinduced ring currents  may be insufficient even for moderate laser intensities  around 1012 W/cm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' A more ab initio nonperturbative  theory such as TDDFT, as used in present paper, is more  appropriate for this regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  This work was supported by the NSF Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' PHY-  1734006 and Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' PHY-2110628.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' This work uti-  lized resources from the University of Colorado Boulder  Research Computing Group, which is supported by the  National Science Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Gomes and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Mallion, Chemical Reviews  101, 1349 (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  [2] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Krygowski, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Szatylowicz, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Stasyuk, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Do-  minikowska,  and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Palusiak, Chemical Reviews 114,  6383 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  [3] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Heine, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Corminboeuf, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Seifert, Chemical Re-  views 105, 3889 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  [4] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Barth and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Manz, Angewandte Chemie International  2元  m=-2  m = +2  LUMO  Phase  7元  0  m=-1  m=+1  HOMO5  Edition 45, 2962 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  [5] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Barth, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Manz, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Shigeta, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Yagi, Journal of  the American Chemical Society 128, 7043 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  [6] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Wollenhaupt, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Krug, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' K¨ohler, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Bayer, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Sarpe-  Tudoran,  and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Baumert, Applied Physics B 95, 245  (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  [7] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Eckart, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Kunitski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Richter, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Hartung, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Rist,  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Trinter, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Fehre, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Schlott, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Henrichs, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  Schmidt, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Jahnke, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Sch¨offler, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Liu, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Barth,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Kaushal, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Morales, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Ivanov, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Smirnova,  and  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' D¨orner, Nature Physics 14, 701 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  [8] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Fleischer, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Kfir, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Diskin, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Sidorenko, and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Co-  hen, Nature Photonics 8, 543 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  [9] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Hickstein, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Dollar, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Grychtol, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Ellis,  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Knut, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Hern´andez-Garc´ıa, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Zusin, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Gentry,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Shaw, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Fan, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Dorney, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Becker, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Jaro´n-  Becker, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Kapteyn, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Murnane,  and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  Durfee, Nature Photonics 9, 743 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  [10] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Huang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Hern´andez-Garc´ıa, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Huang, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  Huang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Lu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Rego, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Hickstein, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Ellis,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Jaron-Becker, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Becker, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Yang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Durfee,  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Plaja, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Kapteyn, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Murnane, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Kung,  and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Chen, Nature Photonics 12, 349 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  [11] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Neufeld and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Cohen, Physical Review Letters 123,  103202 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  [12] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Yuan and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Bandrauk, Physical Review A 88,  013417 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  [13] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Ulusoy and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Nest, Journal of the American Chem-  ical Society 133, 20230 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  [14] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Yuan, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Shu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Dong,  and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Ban-  drauk, The Journal of Physical Chemistry Letters 8, 2229  (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  [15] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Mineo and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Fujimura, The Journal of Chemical  Physics 147, 224301 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  [16] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Barth and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Smirnova, Physical Review A 84, 063415  (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  [17] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Herath, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Yan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Lee, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Li, Physical Review  Letters 109, 043004 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  [18] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Zhu, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Lan, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Liu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Li, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Liu, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Zhang, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Barth,  and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Lu, Optics Express 24, 4196 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  [19] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Liu and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Barth, Physical Review A 94, 043402  (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  [20] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Antonov, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Knyazev, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Letokhov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  Matiuk, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Movshev,  and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Potapov, Optics  Letters 3, 37 (1978).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  [21] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Dietz, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Neusser, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Boesl, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Schlag,  and  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Lin, Chemical Physics 66, 105 (1982).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  [22] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Marques, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Castro, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Bertsch, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Ru-  bio, Computer Physics Communications 151, 60 (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  [23] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Castro, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Appel, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Oliveira, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Rozzi, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' An-  drade, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Lorenzen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Marques, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Gross,  and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Rubio, physica status solidi (b) 243, 2465 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  [24] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Andrade, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Strubbe, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Giovannini, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Larsen,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Oliveira,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Alberdi-Rodriguez,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Varas,  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Theophilou, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Helbig, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Verstraete, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Stella,  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Nogueira, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Aspuru-Guzik, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Castro, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  Marques,  and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Rubio, Physical Chemistry Chemical  Physics 17, 31371 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  [25] See Supplemental Material for details on the TDDFT  and few-level models, orientation dependence, and the  method for interpolating over intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='  [26] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' W¨orner, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Arrell, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Banerji, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Cannizzo,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Chergui, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Das, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Hamm, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Keller, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Kraus,  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Liberatore, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Lopez-Tarifa, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Lucchini, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Meuwly,  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Milne, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content='-E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Moser, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Rothlisberger, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Smolentsev,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Teuscher, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' van Bokhoven, and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}
+page_content=' Wenger, Struc-  tural Dynamics 4, 061508 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAyT4oBgHgl3EQfhfjt/content/2301.00380v1.pdf'}

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+SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT
+EPIDEMIC MODELS
+I: THE REPLACEMENT NUMBER DYNAMICS
+FLORIAN NILL
+31-DEC-2022
+Abstract. As shown recently by the author, constant population SI(R)S models map to
+Hethcote’s classic endemic model originally proposed in 1973. This unifies a whole class
+of models with up to 10 parameters being all isomorphic to a simple 2-parameter master
+model for endemic bifurcation. In this work this procedure is extended to a 14-parameter
+SSISS Model, including social behavior parameters, a (diminished) susceptibility of the
+R-compartment and unbalanced constant per capita birth and death rates, thus covering
+many prominent models in the literature. Under mild conditions, in the dynamics for
+fractional variables in this model all vital parameters become redundant at the cost of
+possibly negative incidence rates. There is a symmetry group GS acting on parameter
+space A, such that systems with GS-equivalent parameters are isomorphic and map to the
+same normalized system. Using (Xrep, I) as canonical coordinates, Xrep the replacement
+number, normalization reduces to parameter space A/GS with 5 parameters only. This
+approach reveals unexpected relations between various models in the literature. Part two
+of this work will analyze equilibria, stability and backward bifurcation and part three
+will further reduce the number of essential parameters from 5 to 3.
+Contents
+1.
+Introduction
+2
+2.
+The SSISS model
+6
+2.1.
+Constant population
+8
+2.2.
+Time varying population
+9
+2.3.
+Classifying parameter space
+10
+2.4.
+Examples from the literature
+12
+2.5.
+Absence of periodic solutions
+14
+3.
+Normalization
+15
+3.1.
+Phase space
+15
+3.2.
+Canonical coordinates
+16
+3.3.
+Main results
+17
+3.4.
+Examples revisited
+22
+4.
+Summary and outlook
+23
+Appendix A.
+Normalizing linear vital dynamics
+24
+Appendix B.
+Scaling the SI(R)S model
+24
+Appendix C.
+The case α1 = α2 = 0
+27
+References
+27
+E-mail address: [email protected].
+2020 Mathematics Subject Classification. 34C23, 34C26, 37C25, 92D30.
+Key words and phrases. SIRS model, SSISS model, normalization, symmetry, stability, endemic bifur-
+cation, backward bifurcation.
+The author is retired physicist, Dr.rer.nat.habil., formerly senior research fellow at Inst. theor. Physik,
+Freie Universität Berlin.
+1
+arXiv:2301.00159v1  [q-bio.PE]  31 Dec 2022
+
+2
+SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT EPIDEMIC MODELS I
+1. Introduction
+Building mathematical models to describe phenomena in natural sciences one typically
+encounters dynamical variables and external parameters. Within the model values for
+external parameters are considered to be given from outside, like fundamental natural
+constants (speed of light c, Planck’s constant ℏ), parameters describing material or bi-
+ological properties (spring constant κ, birth rate δ, recovery rate γ) or social behavior
+(contact rate β). Naturally, reducing the number of essential parameters is always a goal
+to detect redundancies within parameter space and to simplify computations by unload-
+ing formulas. In the simplest case a pure dimensional scale parameter may without loss
+be put equal to one by choosing dimensional units appropriately. For example, putting
+c = 1 amounts to measuring spatial distances by light running times and masses in units
+of energies, putting ℏ = 1 amounts to measuring energies by angular frequencies and
+putting γ = 1 amounts to measuring time in units of the recovery time in an epidemic
+model.
+More generally a normalization program consists of finding appropriate coordinate
+transformations in variable+parameter space such that the transformed system only de-
+pends on a maximally reduced subset of transformed parameters. Examples are1
+Harmonic oscillator
+Predator-prey model
+˙u
+=
+v
+˙u
+=
+−uv + c1u
+˙v
+=
+−u
+˙v
+=
+uv − v
+Classic SIR model
+Classic endemic model
+˙u
+=
+−uv
+˙u
+=
+−uv − c1u + c2
+˙v
+=
+uv − v
+˙v
+=
+uv − v
+(1.1)
+Following this strategy the 6-parameter SI(R)S model (≡ combined SIRS/SIS model)
+with standard incidence, constant vaccination and immunity waning rates and a balanced
+birth and death rate has recently been shown by the author (Nill 2022) to admit a nor-
+malized version looking like the classic endemic model above2.
+In this work (including two follow ups to be denoted as parts II and III (Nill n.d.[b],[c]))
+this method is extended to the case where immunity after recovery (or vaccination) is
+incomplete right from the onset and where also compartment dependent constant per
+capita birth and death rates lead to a time varying population size N.
+In this way
+one is naturally lead to replacing the SI(R)S model by a SSISS model, where in place
+of the usual S, I and R compartments we have two susceptible compartments S1 and
+S2 and one infectious compartment I. Infection transmission from I to S2 is diminished
+as compared to transmission to S1. There is a vaccination flow from S1 to S2 and an
+immunity waning flow from S2 to S1. The model could also be interpreted by considering
+1The variables in these examples are:
+- Harmonic oscillator: u = q, v = p/
+√
+mk, where q, p, κ, m are coordinate, momentum, spring constant
+and particle mass and where the oscillation period is normalized to T = 2π by putting m/k = 1.
+- Predator-prey model: (u, v) denote appropriately rescaled prey and predator populations, respectively,
+and the predator mortality rate is normalized to one.
+- SIR model: u = r0S, v = r0I, where r0 is the basic reproduction number, (S, I) are susceptible and
+infectious fractions of the population and where the recovery rate is normalized to γ = 1.
+- Endemic model:
+(u, v, r0, γ) as above, c1 = δ/(γ + δ) and c2 = r0c1, where δ is the balanced
+birth/mortality rate and where now time scale is normalized to γ + δ = 1.
+2Aapart from allowing also values u ∈ R and an enlarged parameter range (c1, c2) ∈ R+ × R ∪ {0, 0}.
+
+SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT EPIDEMIC MODELS I
+3
+S2 as the “lock-down” fraction and S1 as the “freedom fraction”. In this picture flows from
+S1 to S2 and vice-versa are described by an I-linear (respectively (N −I)-linear) flow with
+rate parameters θi, i = 1, 2, modeling social behavior in reaction to published prevalence
+data.
+Combining both interpretations it turns out to be convenient to start with an
+abstract version of a SSISS model staying completely symmetric under interchanging S1
+and S2, see Fig. 1.
+The present part I provides a normalization prescription reducing the number of inde-
+pendent parameters in this model from initially fourteen to essentially five (four in the
+SI(R)S model sub-case). Based on this approach, part II will give a complete review on
+equilibria and stability in the master SSISS model, thereby also recovering an exceptional
+scenario which had been overlooked in the literature so far. In part III the scaling sym-
+metry for SI(R)S models mentioned above will be generalized to the full SSISS model,
+thereby reducing the number of parameters again by two. So, the total reduction from
+fourteen to three reveals a great hidden redundancy in parameter space. It also provides
+a unifying view on results in the literature concerning equilibrium states, endemic bifur-
+cation and stability properties for all kinds of sub-classes of this model. Put differently,
+in the presence of a common normalized version presenting basically repeated arguments
+for various subsets of non-vanishing parameters becomes obsolete.
+Relating this work to the literature, let me focus on deterministic SIR-type 3-compartment
+dynamical systems, which conveniently may be classified according to
+A) constant vs. time-varying total population size N,
+B) infection transmission only from I to S vs. also from I to R (in which case it makes
+sense to rename S ≡ S1 and R ≡ S2).
+Also, I will restrict this survey to models with standard bi-linear incidence flows βiSiI/N,
+such that the vector field ˙Y = V(Y), Y = (S1, S2, I), is homogeneous of first order. This
+applies to diseases where the number of effective contacts per capita is independent of N.
+ad A) Endemic models with constant population have first been constructed by adding
+a non-zero balanced birth and death rate to the classic SIR model of (Kermack and
+McKendrick 1927). As shown by (Hethcote 1974) (see also (Hethcote 1976, 1989)), in
+this way already the simplest model without vaccination and loss of immunity shows
+a bifurcation from a stable disease-free equilibrium point (DFE) to a stable endemic
+scenario when raising the basic reproduction number R0 above one. Nowadays this is
+considered as Hethcote’s classic endemic model. Including linear vaccination and/or loss
+of immunity terms and optionally also considering recovery without immunity one ends up
+with various types of constant population SI(R)S models without changing this picture,
+see for example (Batistela et al. 2021; Chauhan, Misra, and Dhar 2014; Korobeinikov and
+Wake 2002; O’Regan et al. 2010). As remarked above (and reviewed in more detail in
+Appendix B), the true reason lies in the fact that constant population SI(R)S models with
+up to 10 parameters all map to the same normalized 2-parameter version of the classic
+endemic model as given in Eq. (1.1).
+Models with variable population are mostly studied under the assumption of a constant
+(i.e. N-independent) birth flow. Heuristically this may be justified by assuming that
+N varies slowly on characteristic epidemic time scales. But truly speaking, as already
+pointed out by (Mena-Lorca and Hethcote 1992), this Ansatz rather models a constant
+immigration scenario. So in this work I will follow the more natural proposal of modeling
+vital dynamics by possibly department dependent constant per capita birth and death
+
+4
+SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT EPIDEMIC MODELS I
+rates. Note that, unless fine tuning parameters, this implies that either N(t) → ∞ or
+N(t) → 0 as t → ∞. So in this type of models one always analyzes the dynamics of
+fractional variables Si := Si/N, I := I/N, which is well known to be independent of N(t).
+Apparently, this stream of models has been initiated by (Busenberg and Driessche 1990,
+1991; Derrick and Driessche 1993). (Razvan 2001) has studied a SIRS model in this sense
+with infection transmission also from outside and a SIS-version with varying population
+size has been analyzed by (J. Li and Ma 2002). For generalizations to SEIR models see
+e.g. (Greenhalgh 1997; M. Y. Li et al. 1999; G. Lu and Z. Lu 2018; Sun and Hsieh 2010).
+ad B) A different approach to modeling partial and/or waning immunity consists of
+introducing a diminished incidence flow with rate βR ≡ β2 > 0 directly from R ≡ S2
+to I. This has presumably first been proposed in the so-called SIRI model of (Derrick
+and Driessche 1993), see above. In addition, the authors also introduced a time varying
+population size N(t) and an excess mortality ∆µI in compartment I to this model. In turn,
+they didn’t use linear vaccination nor immunity waning terms. In this way they identified
+a range of parameters in the domain R0 < 1, for which besides the locally asymptotically
+stable disease free equilibrium there also coexist two endemic equilibria, one being a
+saddle and the other one also being locally asymptotically stable. Later (Hadeler and
+Castillo-Chavez 1995) found the same phenomenon in their combined SIS/SIRS core group
+model with linear vaccination, constant population and also two incidence rates βi for
+S → I and R → I. Meanwhile it is well known that models with infection incidents
+from several compartments may show a so-called backward bifurcation from the disease-
+free to an endemic scenario (Hadeler and Driessche 1997). This means that two locally
+asymptotically stable equilibrium states may coexist for some range below threshold,
+causing also hysteresis effects upon varying parameters. Apparently, a varying population
+size is not needed for this. In (Kribs-Zaleta and Velasco-Hernandez 2000) the authors have
+improved and extended these results by adding also a linear immunity waning rate to the
+model of (Hadeler and Driessche 1997).
+One may also distinguish vaccinated and recovered people into separate compartments.
+This leads to 4-compartment models, where similar results have been obtained by, e.g.
+(J. Arino, Mccluskey, and Driessche 2003; Yang, Sun, and Julien Arino 2010).
+Backward bifurcation has lately also been observed in SEIRS-type models for Covid-
+19 by considering two distinguished susceptible compartments.
+In (Nadim and Chat-
+topadhyay 2020) the less susceptible compartment had been interpreted as an incomplete
+lockdown and in (Diagne et al. 2021) as an incomplete vaccination efficacy.
+More recently, in (Avram, Adenane, Basnarkov, et al. 2021; Avram, Adenane, Bianchin,
+et al. 2022) the authors have given a thorough stability analysis of an eight parameter
+SIRS-type model by adding a varying population size to the model of (Kribs-Zaleta and
+Velasco-Hernandez 2000) (apparently without being aware of that paper).
+Closing this overview I should also remark that backward bifurcation is also observed
+when considering I-dependent contact or recovery rates to model reactive behavior or
+infection treatment. However the list of papers on this topic over the last 20 years becomes
+too huge to be quoted at this place.
+This paper extends the normalization algorithm for constant population SI(R)S models to
+models as above, i.e. with time varying population size and/or a non-zero incidence rate
+βR ≡ β2 from R ≡ S2 to I. As a starting observation, there is an ambiguity in deriving the
+dynamics ˙y = F(y) for fractional variables y = (S1, S2, I), see Appendix A. This allows
+choosing the vector field F such that all vital dynamics parameters become redundant,
+
+SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT EPIDEMIC MODELS I
+5
+provided the birth-minus-death rates νi = δi −µi in S1 and S2 coincide, ν1 = ν2 = ν. This
+redundancy already reduces the number of parameters in the master SSISS model from
+fourteen to eight. More than that, F depends on the incidence rates βi only as a function
+of ˜βi = βi + νI − ν, where νI = δI − µI is the birth-minus-death rate in I. Assuming for
+simplicity compartment independent birth rates gives ˜βi = βi − ∆µI, where ∆µI denotes
+the excess mortality in I. In this way models with variable population, ∆µI > 0, and
+absence of a incidence rate from R, β2 = 0, look like models with constant population,
+∆µI = 0, and a negative incidence rate β2 = ˜β2 < 0. Conversely, models with positive
+incidence rates βi > 0 and excess mortality ∆µI < min{β1, β2} behave like models with
+constant population size and incidence rates βi = ˜βi > 0. So, the above classification
+schemes A) and B) become blurred and, instead, it is more expedient to view all models
+as if they had constant population size and two distinguished and possibly also negative
+incidence rates ˜βi ∈ R.
+In this way most of the above bench marking 3-compartment models (if necessary after
+imposing the constraint ν1 = ν2) become comparable as sub-cases of the master SISS
+model, with tilde parameters swallowing all birth and death rates and possibly with
+negative incidence rates ˜βi ∈ R. As an example, the models of (Hadeler and Castillo-
+Chavez 1995) and (Kribs-Zaleta and Velasco-Hernandez 2000) become isomorphic and
+they completely cover the sub-case µ1 = µ2 and 0 < min{˜β1, ˜β2} in (Avram, Adenane,
+Bianchin, et al. 2022). Also, apart from an irrelevant boundary case, the complementary
+sub-case µ1 = µ2 and 0 > min{˜β1, ˜β2} in (Avram, Adenane, Bianchin, et al. 2022) is
+covered by the model of (J. Li and Ma 2002). So, applying the normalization procedure
+of this paper, all results in Section 5 and 6 of (Avram, Adenane, Bianchin, et al. 2022)
+already follow from the previous literature. A more detailed list of unexpected relations
+between the above models is given in Section 2.4.
+The plan of this paper is as follows. In Sections 2.1 and 2.2 we pass to fractional com-
+partment variables, Si = Si/N and I = I/N, and prove redundancy of all vital dynamics
+parameters at the cost of possibly negative incidence rates ˜βi. For convenience, time scale
+is also normalized by putting the total expected waiting time in compartment I equal to
+one. In this way the number of essential parameters is already reduced from fourteen to
+seven. Thus, denoting A the space of essential parameters, we have dim A = 7.
+Section 2.3 classifies various useful subsets in parameter space like Aphys ⊂ A, guaran-
+teeing forward invariance of the physical triangle
+Tphys := {(S1, S2, I) ∈ R3
+≥0 | S1 + S2 + I = 1},
+and Abio ⊂ Aphys, guaranteeing an epidemiological interpretation of parameters by re-
+quiring in particular θ1 ≥ 0 ≥ θ2.
+Section 2.4 identifies eight examples from the above list of models as sub-cases of the
+master SSISS model. In this way we obtain various relations between these models as
+indicated above, which apparently have not been recognized before.
+In Section 2.5 we adapt methods from (Busenberg and Driessche 1990) to prove ab-
+sence of periodic solutions for all parameters non-negative, except βi. The extension to
+parameters a ∈ Abio (requiring θ2 ≤ 0) heavily relies on the symmetry results in Section
+3 and will be proven in Section 3.3.
+Section 3 starts from the observation, that the time-normalized equation of motion for
+I takes the generic form ˙I = (Xrep − 1)I, where Xrep = β1S1 + β2S2 is the replacement
+
+6
+SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT EPIDEMIC MODELS I
+number (Hethcote 2000), i.e.
+the expected number of secondary cases produced by a
+typical infectious individual during its time of infectiousness (nowadays mostly called
+effective reproduction number).
+A coordinate free formulation of the model naturally
+leads to taking (Xrep, I) as independent canonical coordinates3 in the physical triangle
+Tphys. In this way, we arrive at formulating the SSISS model as a dynamical system in
+(Xrep, I)-space, called the replacement number (RN) dynamics (Section 3.2).
+˙Xrep = f(Xrep, I),
+˙I = (Xrep − 1)I.
+(1.2)
+Since f(Xrep, I) turns out to be a 5-parameter quadratic polynomial with no term ∼ X2
+rep,
+the number of free parameters is now reduced from seven to five.
+The main results of this paper are derived in Section 3.3. Denoting D the new parameter
+set, dim D = 5, the above approach yields a surjective submersion A ∋ a �→ x(a) ∈ D.
+Moreover, A becomes a principal fibre bundle with respect to a group right action ◁ :
+A × GS → A such that x(a ◁ g) = x(a) and D ∼= A/GS. Here GS ⊂ GL+(R2) is the
+group acting on (S1, S2) ∈ R2 and leaving S1 + S2 invariant. Eq. (1.2) implies that SSISS
+dynamical systems at parameter values a, a′ ∈ A are isomorphic whenever a and a′ are
+GS-equivalent, i.e. x(a) = x(a′) or equivalently a′ = a ◁ g for some g ∈ GS. In this way
+we also get
+-
+Absence of periodic solutions also for parameters a ∈ Abio,
+-
+Conditions under which the social behavior parameters θi can be “gauged to zero”, i.e.
+there exists g ∈ GS such that a ◁ g ∈ Aθ=0.
+Section 3.4 revisits the examples from the literature within the new formalism and Sec-
+tion 4 gives a summary and outlook to parts II and III of this work. Finally, Appen-
+dix A provides a normalization prescription for the dynamics of fractional variables in
+n-compartment models with linear (i.e. constant per capita) birth and death rates, Ap-
+pendix B reviews the scaling symmetry in SI(R)S models introduced in (Nill 2022) and
+Appendix C discusses a boundary case in parameter space.
+Acknowledgement I would like to thank Florin Avram for encouraging interest and
+useful discussions.
+2. The SSISS model
+This Section starts with proposing an abstract completely symmetrized SSISS model
+consisting of three compartments, S1, S2 and I, with total population N = S1 + S2 + I.
+Members of I are infectious, members of S1 are highly susceptible (socially active or not
+immune) and members of S2 are less susceptible (partly immune or reducing contacts).
+The flow diagram between compartments is depicted in Fig. 1.
+The parameters in this model may be given the following interpretations
+3Here “canonical” is not meant in the sense of Hamiltonian systems.
+
+SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT EPIDEMIC MODELS I
+7
+Figure 1. Completely symmetric flow diagram of the SSISS model.
+All pa-
+rameters are nonnegative except θ2 ∈ [−α2, 0]. Also q1 + q2 = 1, γ1 + γ2 > 0
+and β1 > β2. Generalizing to compartment dependent birth rates amounts to
+replacing δN by δ1S1 + δ2S2 + δII.
+α1
+:
+Vaccination rate of susceptibles moving from S1 → S2 (assuming
+θ1 = θ2 = 0, see below).
+α2
+:
+Immunity waning rate inducing a flow from S2 → S1 (assuming
+θ2 = 0, see below).
+βi
+:
+Number of effective contacts per unit time of a susceptible from Si.
+γi
+:
+Recovery rate from I → Si.
+θ1
+:
+Willingness to get vaccinated (alternatively to reduce contacts)
+given the actual prevalence I/N.
+In reality only one of the two
+parameters α1 and θ1 should be chosen non-zero.
+θ2
+:
+Epidemiologically one should restrict to θ2 = 0 or (θ2 = −α2 < 0
+and α1 = 0). In this latter case the meaning of the S2-compartment
+is “contact reducing” and α2 = −θ2 parametrizes the readiness to
+increase contacts proportional to 1 − I/N.
+µi
+:
+Mortality rate in Si.
+µI
+:
+Mortality rate in I. One could also consider vertical transmission,
+in which case µI would be the mortality rate diminished by the rate
+of infected newborns.
+∆µI
+:
+Mortality excess ∆µI = µI − µ in case µ1 = µ2 = µ, which will be
+assumed most of the time.
+δ
+:
+Rate of not infected newborns. Generalizing to compartment de-
+pendent birth rates amounts to replacing δN = δ1S1 + δ2S2 + δII.
+qi
+:
+Split ratio of newborns between S1 and S2, q1 + q2 = 1. In the
+reduced-immunity interpretation q2 would be the portion of vacci-
+nated newborns.
+
+8
+SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT EPIDEMIC MODELS I
+So in total this model counts 15 independent parameters (12 if we require constant total
+population, δi = µi, δI = µI). Epidemiologically all parameters except 0 ≥ θ2 ≥ −α2
+are assumed non-negative and also β2 < β1. A more technical classification of admissible
+parameter ranges will be given below. Here is a list of prominent examples in the literature
+-
+Hethcotes classic 3-parameter endemic model (Hethcote 1974, 1976, 1989) by putting
+δ = µi = µI > 0, q1 = 1, β1 > 0, γ2 > 0 and all other parameters vanishing.
+-
+The 7-parameter SIRS model with time varying population size in (Busenberg and
+Driessche 1990), adding to Hethcote’s model an immunity waning rate α2 and allowing
+different (constant per capita) mortality and birth rates.
+-
+The 6-parameter SIRI model of (Derrick and Driessche 1993), replacing the immunity
+waning rate α2 in (Busenberg and Driessche 1990) by the incidence rate β2 > 0 and
+also requiring µ1 = µ2.
+-
+An extended 10-parameter constant population SI(R)S (i.e. mixed SIRS/SIS) model
+with constant and I-linear vaccination rates α1, θ1, an immunity waning rate α2 and
+two recovery flows I ← Si. Hence δi = µi, δI = µI and θ2 = β2 = 04.
+-
+The 6-parameter isolated core system in (Hadeler and Castillo-Chavez 1995), with
+two incidence and recovery rates, βi, γi > 0, a vaccination term α1 > 0 and a constant
+population with balanced birth and death rates, δ = µi = µI > 0 and q1 = 1.
+-
+The 7-parameter vaccination models of (Kribs-Zaleta and Velasco-Hernandez 2000)
+adding an immunity waning rate α2 > 0 to the model of (Hadeler and Castillo-Chavez
+1995). As we will see in Eq. (2.24) below, due to a redundancy of parameters the two
+models actually stay isomorphic.
+-
+The 8-parameter SIS-model with vaccination and varying population size of (J. Li and
+Ma 2002) keeping only θi = γ2 = β2 = 0 and assuming µ1 = µ2 = µ.5 As we will see
+in (2.25), after a parameter transformation this model becomes isomorphic to the case
+where only θi = 0 and β2 ≤ 0.
+-
+The 8-parameter SIRS-type model analyzed recently by (Avram, Adenane, Bianchin,
+et al. 2022), keeping only γ1 = θ1 = θ2 = q2 = 0 and all other parameters positive.
+The authors allow a varying population size by first discussing the general case of all
+mortality rates being different and then concentrate on µ1 = µ2 ̸= δ and ∆µI > 0.
+Their paper is closest to the present work and in fact initiated it.
+In a “zeroth normalization” step I will now show that passing to fractional variables and
+requiring δ1 − µ1 = δ2 − µ2 all vital dynamic parameters in the SSISS model become
+redundant6. In this way the number of essential parameters reduces from 14 to 8. The
+price to pay in the non-constant population case is possibly getting negative incidence
+rates βi.
+2.1. Constant population. To get a constant population N the birth rates have to obey
+δi = µi and δI = µI, or more generally
+δ = (µ1S1 + µ2S2 + µII)/N .
+(2.1)
+In case µ1 = µ2 = µ this would read δ = µ+I∆µI. Heuristically this should be understood
+as an approximation for ∆µI/µ ≪ 1. Under this assumption, denoting fractions of the
+4Here I have chosen enlarge the conventional setting for SI(R)S models by also allowing θ1 > 0.
+5Actually the authors let µ be a function of N, which however disappears when passing to fractional
+variables.
+6Redundancy of constant per capita birth and death rates may in fact be shown under quite general
+assumptions in n-compartment models, see Appendix A.
+
+SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT EPIDEMIC MODELS I
+9
+total population by Si = Si/N and I = I/N and introducing the notations
+˜α1
+:=
+α1 + q2µ1 ,
+˜γ1
+:=
+γ1 + q1µI ,
+˜α2
+:=
+α2 + q1µ2 ,
+˜γ2
+:=
+γ2 + q2µI ,
+(2.2)
+S =
+�
+S1
+S2
+�
+,
+D(β) =
+�
+β1
+0
+0
+β2
+�
+,
+E(α) =
+�
+α1
+−α2
+−α1
+α2
+�
+,
+˜γ =
+�
+˜γ1
+˜γ2
+�
+(2.3)
+the dynamical system described by the flow diagram Fig. 1 becomes
+˙S
+=
+− [E( ˜α) + IE(θ) + ID(β)] S + I ˜γ ,
+(2.4)
+˙I
+=
+˜γ(Xrep − 1)I ,
+˜γ = ˜γ1 + ˜γ2
+(2.5)
+Xrep
+:=
+(β1S1 + β2S2)/˜γ .
+(2.6)
+Note that ˜γ−1 ≡ (γ1 + γ2 + µI)−1 is the expected waiting time in I and hence Xrep is the
+replacement number (Hethcote 2000), i.e. the expected number of secondary cases pro-
+duced by a typical infectious individual during its time of infectiousness. In conventional
+SI(R)S models, i.e. for β2 = θ2 = 0, the replacement number in the limit S1 = 1 would
+become the basic reproduction number r0 = β1/γ. This is why nowadays the replacement
+number is mostly called effective reproduction number. Later we will also have the notion
+of a reduced reproduction number R0 as the value of Xrep at the disease-free equilibrium.
+To avoid misunderstandings, I prefer to keep the various notions of “reproduction num-
+bers” for parameters, whereas the replacement number Xrep is considered as a dynamical
+variable.
+Now obviously, by (2.2), all vital dynamics parameters become redundant and may be
+absorbed by redefining αi and γi. Note that this observation is independent of the choice
+of βi and θi, i.e. it already holds in a combined SI(R)S model.
+2.2. Time varying population. To derive the equations of motion in case of a time vary-
+ing population keep compartment dependent per capita birth and death rates δi, δI, µi, µI
+constant and put Y = (S1, S2, I), y = N−1Y and
+ν ≡ (ν1, ν2, νI) := (δ1 − µ1, δ2 − µ2, δI − µI).
+Then ˙y = ˙Y/N − y ˙N/N and ˙N/N = ⟨ν | y⟩. Using S1 + S2 + I = 1 we may rewrite
+S1 ˙N/N = S1[ν1 + (ν2 − ν1)S2 + (νI − ν1)I]
+S2 ˙N/N = S2[ν2 + (ν1 − ν2)S1 + (νI − ν2)I]
+I ˙N/N = I[νI + (ν1 − νI)S1 + (ν2 − νI)S2].
+So now introduce
+˜α1
+:=
+α1 + q2δ1 ,
+˜α2
+:=
+α2 + q1δ2 ,
+˜γ1
+:=
+γ1 + q1δI ,
+˜γ2
+:=
+γ2 + q2δI ,
+˜β1
+:=
+β1 + νI − ν1 ,
+˜β2
+:=
+β2 + νI − ν2 .
+(2.7)
+With the same notation as in Eq. (2.3) and e(ν) :=
+�
+ν1 − ν2
+ν2 − ν1
+�
+we then get
+˙S
+=
+−
+�
+E( ˜α) + IE(θ) + ID( ˜β)
+�
+S + I ˜γ + S1S2e(ν) ,
+(2.8)
+˙I
+=
+˜γ(Xrep − 1)I ,
+(2.9)
+Xrep
+:=
+(˜β1S1 + ˜β2S2)/˜γ ,
+˜γ := ˜γ1 + ˜γ2.
+(2.10)
+
+10
+SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT EPIDEMIC MODELS I
+So, imposing the condition ν1 = ν2 =: ν and putting ∆νI := ν−νI we get e(ν) = 0 and the
+equations of motion look exactly as in the case of constant population (2.4)-(2.6). Again
+all vital dynamics parameters become redundant and may be absorbed by redefining βi,
+αi and γi. The difference this time is that ˜βi = βi − ∆νI may become negative! Thus we
+arrive at
+Proposition 2.1. Assume ν1 = ν2.
+i)
+If ∆νI ≤ min{β1, β2} the SSISS model with variable population maps to the model with
+constant population.
+ii) If ∆νI > min{β1, β2} it maps to the model with min{β1, β2} = 0 and variable popula-
+tion with �
+∆νI = ∆νI − min{β1, β2}.
+iii) If ∆νI = β2 < β1 and θ2 = 0 it becomes the extended SI(R)S model with θ1 ≥ 0 and
+two recovery flows I → S1 and I → S2.
+Remark 2.2. Note that under the usual assumptions δi = δI = δ and µ1 = µ2 = µ, ∆νI
+coincides with the excess mortality in the infectious compartment, ∆νI = µI −µ = ∆µI.
+Remark 2.3. The observation that on the level of fractional variables in both scenarios
+(constant vs. variable population, the latter provided ν1 = ν2) all vital dynamics param-
+eters are redundant seems to be new7. Essential for this is allowing all four parameters
+(αi, γi) being positive and βi possibly being negative. The introduction of parameters θi
+is not needed to assure this. Redundancy of constant per capita birth and death rates
+may in fact be shown under quite general assumptions in n-compartment models, see
+Appendix A.
+2.3. Classifying parameter space. In this subsection assume ν1 = ν2. Then the re-
+formulation in terms of possibly negative incidence rates ˜βi leads to a new classification
+scheme identifying seven sectors in this model. For θi = 0 these are labeled by the signa-
+tures of ˜β1 + ˜β2 and ˜β1 ˜β2 (in case of a compartment independent birth rate δ equivalently
+by the size of the excess mortality ∆µI), see Table 1. For θi ̸= 0 this classification will be
+refined in Section 3, Table 3.
+To simplify notation, in what follows let me drop the tilde above parameters. The case
+β1 = β2 will be ignored, since in this case putting S = S1 + S2 one easily checks that
+(S, I) obeys the dynamics of a SIS model, which can immediately be solved by separation
+of variables. Also, due to the permutation symmetry 1 ↔ 2, there is no loss assuming
+β1 > β2. Next, choosing time scale to be measured in units of γ−1, we may without
+loss also put γ = 1. Thus, assume γi ∈ [0, 1] and γ1 + γ2 = 1. So, having started from
+fourteen, essentially we are now left with seven free parameters (think of all greek symbols
+of dimension [time]−1 being divided by γ).
+To further classify the space of admissible parameters some formalism will be needed. Put
+C := {(αi, γi, θi) ∈ R6 | α1 + α2 > 0 ∧ γ1 + γ2 = 1}
+(2.11)
+C+ := C ∩ {(αi, γi) ∈ R4
+≥0}
+(2.12)
+Csplit := C ∩ {θ1 ≥ 0 ≥ θ2}
+(2.13)
+Cphys := C+ ∩ {θi + αi ≥ 0 , i = 1, 2}
+(2.14)
+Cbio := Csplit ∩ Cphys
+(2.15)
+7As communicated privately this had also been realized recently in a talk by Florin Avram.
+
+SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT EPIDEMIC MODELS I
+11
+Table 1. Seven sectors in the SSISS-model at θi = 0 and for compartment
+independent birth rate δ. By Corollary 2.9 Sector I is isomorphic to the models
+of (Hadeler and Castillo-Chavez 1995; Kribs-Zaleta and Velasco-Hernandez 2000)
+and Sectors III-VII are largely covered by (J. Li and Ma 2002). Sector II is a
+mixed SI(R)S model with two recovery flows I → R and I → S.
+Sector
+sign(˜β1 + ˜β2)
+sign(˜β1 ˜β2)
+Interval [˜β1, ˜β2]
+Excess mortality ∆µI
+I
++
++
+0 < ˜β2 < ˜β1
+∆µI < β2
+II (SIRS)
++
+0
+0 = ˜β2 < ˜β1
+∆µI = β2
+III
++
+−
+0 < −˜β2 < ˜β1
+β2 < ∆µI < (β1 + β2)/2
+IV
+0
+−
+0 < −˜β2 = ˜β1
+∆µI = (β1 + β2)/2
+V
+−
+−
+˜β2 < −˜β1 < 0
+(β1 + β2)/2 < ∆µI < β1
+VI
+−
+0
+˜β2 < ˜β1 = 0
+β1 = ∆µI
+VII
+−
++
+˜β2 < ˜β1 < 0
+β1 < ∆µI
+Note that for θi = 0 we have C+ = Cphys = Cbio. Denoting
+B := {β = (β1, β2) ∈ R2 | β2 < β1}.
+(2.16)
+the full parameter sets are then given by A := C × B or Ax := Cx × B, respectively. I will
+also use obvious notations like Aθ=0 := A ∩ {θi = 0} and Aα≥0 := A ∩ {αi ≥ 0}.
+Remark 2.4. In the definition of C in (2.11) the border case α1 = α2 = 0 (i.e. absence of
+constant vaccination and waning immunity rates) has been excluded, see Appendix C for
+a short discussion. For the body of this paper I will stick with the assumption α1+α2 > 0.
+Next, it is easy to check, that for a ∈ Aphys the physical triangle
+Tphys := {(S1, S2, I) ∈ R3
+≥0 | S1 + S2 + I = 1}
+(2.17)
+stays forward invariant under the dynamics (2.8)-(2.9), i.e. on Tphys we have I = 0 ⇒ ˙I =
+0 and Si = 0 ⇒ ˙Si ≥ 0. Note that θi + αi ≥ 0 in (2.14) is sufficient but not necessary to
+assure this.
+Lemma 2.5. In the SSISS model (2.8)-(2.9) the physical triangle stays forward invariant
+for all parameters (αi, βi, γi, θi) ∈ Aphys, also including the border case α1 = α2 = 0.
+□
+We are now ready to state a main result of this paper. Assuming ν1 = ν2 the normaliza-
+tion procedure to be introduced in Section 3 will further reduce the number of essential
+parameters from seven to five. This means, SSISS models fall into isomorphy classes map-
+ping to the same normalized system. It turns out, that these isomorphy classes coincide
+with orbits under a parameter symmetry group GS acting simultaneously on phase P
+and parameter space A, such that parameters for the normalized system are naturally
+identified as elements of A/GS.
+Theorem 2.6. For y = (S1, S2, I)T ∈ R3 and parameter values a = (α, β, γ, θ) ∈ A
+denote ˙y = Fa(y) the dynamical system (2.8)-(2.9) with vector field Fa : R3 → R3. Let
+GS ⊂ GL+(R2) be the subgroup acting on S ∈ R2 from the left and leaving S1 + S2
+invariant.
+
+12
+SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT EPIDEMIC MODELS I
+i)
+Then there exists a free right action ◁ : A × GS → A such that A becomes a principal
+GS-bundle and
+Fa ◦ Tg = Tg ◦ Fa◁ g ,
+Tg :=
+�
+� g
+0
+0
+0
+0
+1
+�
+� ,
+∀(a, g) ∈ A × GS.
+(2.18)
+ii) Put j := ( 0 1
+1 0 ) and for g ∈ GS denote ¯g := jgj ∈ GS. Viewing α, γ, θ ∈ R2 as column
+vectors and β ∈ B as a row vector and writing a ◁ g = a′ = (α′, β′, γ′, θ′) we have
+α′ = ¯g−1α,
+θ′ = ¯g−1θ + ϑ
+γ′ = g−1γ,
+ϑ =
+1
+β′
+1 − β′
+2
+�
+−(β1 − β′
+1)(β2 − β′
+1)
+(β1 − β′
+2)(β2 − β′
+2)
+�
+β′ = βg
+iii) The GS-right action B × GS ∋ (β, g) �→ βg ∈ B is free and transitive and A ∼=
+A/GS × B as trivial principal fiber bundles.
+iv) Put S′ = g−1S. Then ⟨β|S⟩ = ⟨β′|S′⟩ ≡ Xrep and therefore ˙Xrep = fa(Xrep, I) where
+fa = fa◁ g is GS-invariant, i.e. it only depends on A/GS.
+v) If θ1 ≥ θ2 or θ1θ2 > 08, then there exists g ∈ GS such that a′ := a ◁ g ∈ Aθ=0, i.e. the
+parameters θi may be “gauged to zero”. If in this case a ∈ Abio then also a′ ∈ Abio.
+Remark 2.7. As we will see, although the linear transformation Tg preserves the condition
+S1 + S2 + I = 1, it does not necessarily leave R3
+≥0 (and hence Tphys) invariant.
+Remark 2.8. Since dim GS = 2 we have dim A/GS = dim A − 2.
+So, using (Xrep, I)
+as independent coordinates in Tphys, the number of essential parameters of the SSISS
+dynamical system reduces from seven to five.
+Parts i)-iv) of Theorem 2.6 will be proven in Corollary 3.7 and Lemma 3.8 and part v) in
+Lemma 3.18. Before coming to this let me close this Section
+-
+in Subsection 2.4 with shortly revisiting some bench-marking models in the literature
+within the present framework,
+-
+in Subsection 2.5 with proving absence of periodic solutions by optimizing the methods
+of (Busenberg and Driessche 1990).
+2.4. Examples from the literature. For simplicity, in this subsection let me assume a
+compartment independent birth rate δ. Formulating the dynamics for fractional variables
+y = (S1, S2, I) there always remains an ambiguity by adding a vectorfield vanishing on
+Tphys. In Eqs. (2.8)-(2.9) the vector field F ≡ Fa has the special form
+F(y) = My + Γ(y ⊗ y),
+⟨1|M = ⟨1|Γ = 0,
+(2.19)
+where M ∈ R3×3, 1 = (1, 1, 1) and Γ ∈ Hom (R3 ⊗ R3, R3). As is shown in Appendix A,
+n-compartment models with at most quadratic terms and population size varying only
+due to constant per capita birth and death rates may always be normalized in this way.
+Using different conventions bears the risk of overlooking redundancies in parameter space.
+Moreover, it also makes it tedious to pin down the differences between (or equivalence of)
+various models in the literature. Table 2 shows how the examples quoted at the beginning
+of this Section9 compare with each other when mapped to the present set of parameters.
+8Actually these conditions are sufficient but not necessary. For a weaker condition see Section 3.3.
+9Heth = (Hethcote 1974, 1976, 1989); SIRI = (Derrick and Driessche 1993); BuDr = (Busenberg
+and Driessche 1990); SI(R)S = 10-parameter mixed SIRS/SIS model with constant population size and
+
+SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT EPIDEMIC MODELS I
+13
+Table 2. Mapping models in the literature9 expressed in non-normalized vari-
+ables (S1, S2, I) to the present choice of parameters. The column # counts the
+number of free parameters in the original models.
+After passing to fractional
+variables (S1, S2, I) and tilde parameters, Eq. (2.2) or Eq. (2.7), and resetting
+time scale to ˜γ = 1, the column #eff counts the number of effectively independent
+parameters as determined in Eqs. (2.20)-(2.26).
+α1
+α2
+β1
+β2
+γ1
+γ2
+δ
+µ1
+µ2
+µI
+q1
+q2
+#
+#eff
+Heth
+0
+0
+✓
+0
+0
+✓
+δ = µ1 = µ2 = µI
+1
+0
+3
+2
+SIRI1
+0
+0
+✓
+✓
+0
+✓
+✓
+µ1 = µ2
+✓
+1
+0
+6
+3
+SIRI2
+0
+0
+✓
+✓
+✓
+0
+✓
+µ1 = µ2
+✓
+0
+1
+6
+3
+BuDr
+0
+✓
+✓
+0
+0
+✓
+✓
+✓
+✓
+✓
+1
+0
+7
+5
+SI(R)S
+✓
+✓
+✓
+0
+✓
+✓
+δ = µ1 = µ2 = µI
+✓
+✓
+7
+4
+HaCa
+✓
+0
+✓
+✓
+✓
+✓
+δ = µ1 = µ2 = µI
+1
+0
+6
+5
+KZVH
+✓
+✓
+✓
+✓
+✓
+✓
+δ = µ1 = µ2 = µI
+1
+0
+7
+5
+LM
+✓
+✓
+✓
+0
+✓
+0
+✓
+µi = f(N)
+✓
+✓
+✓
+8
+5
+AABH1
+✓
+✓
+✓
+✓
+0
+✓
+✓
+µ1 = µ2
+10
+✓
+1
+0
+8
+5
+AABH2
+✓
+✓
+✓
+✓
+✓
+0
+✓
+µ1 = µ2
+10
+✓
+0
+1
+8
+5
+Applying the transformations (2.2) or (2.7), respectively, maps the above 11-parameter
+set to the redundancy-free 6-parameter set (˜αi, ˜βi, ˜γi). After resetting time scale to ˜γ ≡
+˜γ1 + ˜γ2 = 1 the classification of the above models looks as follows:
+AHeth = Abio ∩ Aθ=0 ∩ {˜α1 = 0 ∧ ˜γ2 > 0 ∧ ˜γ1 = ˜α2 ∧ ˜β2 = 0}
+(2.20)
+ASIRIi = Abio ∩ Aθ=0 ∩ {˜αi = 0 ∧ ˜γj > 0 ∧ ˜γi = ˜αj, j ̸= i}
+(2.21)
+ABuDr = Abio ∩ Aθ=0 ∩ {˜α1 = 0 ∧ ˜γ2 > 0 ∧ ˜β2 < 0}11
+(2.22)
+ASIRS = Abio ∩ Aθ2=0 ∩ {˜β2 = 0}
+(2.23)
+AKZVH = Abio ∩ Aθ=0 ∩ {˜β2 > 0} = AHaCa
+(2.24)
+ALM = Abio ∩ Aθ=0 ∩ {˜β2 < 0 ∧ ˜γ1 > 0}
+(2.25)
+AAABHi = Abio ∩ Aθ=0 ∩ {˜γj > 0, j ̸= i}
+(2.26)
+The dimensions of these parameter spaces are displayed in the last column of Table 211.
+To verify Eqs. (2.20)-(2.26) the following explanations should suffice.
+•
+The SIRI model of (Derrick and Driessche 1993) with varying population requires
+αi = γ1 = 0. Since for βR > βS the mapping to the SISS model permutes 1 ↔ 2 (i.e.
+maps R → S1 and S → S2), if βR < βS we get ˜α1 = 0, ˜α2 = ˜γ1 = δ and ˜γ2 = γ2 > 0,
+and if βR > βS we get ˜α2 = 0, ˜α1 = ˜γ2 = δ and ˜γ1 = γ1 > 0.
+•
+The SIRS model of (Busenberg and Driessche 1990) differs from SIRI by allowing
+α2 > 0 and µ1 < µ2, but in turn it requires βS > βR = 0. Thus, we have ˜α1 = 0
+θ2 = β2 = 0; HaCa = core system in (Hadeler and Castillo-Chavez 1995); KZVH = (Kribs-Zaleta and
+Velasco-Hernandez 2000); LM = (J. Li and Ma 2002); AABH = (Avram, Adenane, Bianchin, et al. 2022).
+BuDr and AABH come in two versions, the subscript 1 refers to βS > βR and 2 to βS < βR.
+10 The bulk of results in Section 5 and 6 of (Avram, Adenane, Bianchin, et al. 2022) assumes µ1 = µ2.
+11To be comparable Eq. (2.22) refers to the sub-case µ1 = µ2 in (Busenberg and Driessche 1990), so
+dim ABuDr = 4. Allowing also an excess mortality µ2 − µ1 > 0 gives #eff = 5 in Table 2.
+
+14
+SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT EPIDEMIC MODELS I
+and ˜γ1 = δ as in SIRI1, but ˜α2 = α2 + δ becomes independent. If, for comparison, we
+restrict to µ1 = µ2 = µ then β2 = 0 implies ˜β2 = −∆µI ≤ 0.
+•
+If q1 > 0 then one of the three parameters (γ1, α2, δ) always becomes redundant.
+So the models of (Hadeler and Castillo-Chavez 1995) and (Kribs-Zaleta and Velasco-
+Hernandez 2000) are isomorphic, in spite of the latter containing the additional im-
+munity waning rate α2. Also, they both satisfy ˜β2 = β2 > 0.
+•
+Putting q2 = 1 in the SIS-type model of (J. Li and Ma 2002) the mapping (α1, α2, γ1, δ) �→
+(˜αi, ˜γi) is bijective. Also, the authors have defined µi = f(N) and µI = f(N) + ∆µI.
+Hence, the only restrictions in this model are ˜β2 = −∆µI < 0 and ˜γ1 > 0.
+In summary we get the following conclusions, which apparently have not yet been realized
+in the literature.
+Corollary 2.9. Assume µ1 = µ2 =: µ and put ∆µI := µI − µ.
+i)
+For β1 > β2 = ∆µI the SIRI model of (Derrick and Driessche 1993) is isomorphic to
+Hethcote’s classic endemic model.
+Moreover, restricting to ˜γ1 > 0 and β2 ̸= ∆µI we have
+ii) The SIRS-type model of (Busenberg and Driessche 1990) reduces to a sub-case of the
+SIS-type model of (J. Li and Ma 2002), which in turn covers Sectors III-VII of the
+SSISS model at θi = 0.
+iii) The models of (Hadeler and Castillo-Chavez 1995) and (Kribs-Zaleta and Velasco-
+Hernandez 2000) are isomorphic and cover Sector I of the SSISS model at θi = 0.
+iv) The models of (J. Li and Ma 2002) and (Hadeler and Castillo-Chavez 1995; Kribs-
+Zaleta and Velasco-Hernandez 2000) only differ by the sign of ˜β2.
+v) Their disjoint union covers the SIRI model of (Derrick and Driessche 1993) and co-
+incides with the model of (Avram, Adenane, Bianchin, et al. 2022).
+An equivalent formulation of Corollary 2.9 based on normalized parameters and vari-
+ables is given in Corollary 3.19 in Section 3.4.
+2.5. Absence of periodic solutions. In this subsection I will specify parameter ranges
+guaranteeing absence of periodic solutions by optimizing methods from (Busenberg and
+Driessche 1990) (see also (Busenberg and Driessche 1991; Derrick and Driessche 1993)) for
+the present situation, including θi ̸= 0. To start with, the Busenberg-Driessche version of
+the classical Bendixson–Dulac Theorem may be given the following alternative formulation
+Lemma 2.10. (Busenberg and Driessche 1990) Let F : R3 → R3 be smooth in a neigh-
+borhood of Tphys and assume Tphys forward invariant under the flow of ˙y = F(y). Assume
+there exists a smooth scalar function u(y) defined in a neighborhood of Tphys such that
+Ψ(y) := ∇ · (uF)(y) − (y · ∇)(u
+�
+i
+Fi)(y) ≤ 0 ,
+∀y ∈ Tphys
+(2.27)
+and Ψ(y) < 0 for some y ∈ Tphys. Then in Tphys \ ∂Tphys there exist no periodic solutions,
+homoclinic loops or oriented phase polygons of the dynamical system ˙y = F(y).
+Proof. Put 1 := (1, 1, 1) and g := y × uF. Then g · F = 0 and ⟨1 | ∇ × g⟩|Tphys = Ψ|Tphys,
+where the second identity easily follows from ⟨1 | F⟩|Tphys = 0. Now the claim follows by
+Stoke’s Theorem as in the proof of Theorem 4.1 of (Busenberg and Driessche 1990).
+□
+Remark 2.11. In Lemma A.1 in Appendix A it is shown that for models with constant
+per capita birth and death rates one may always replace F by ˜F obeying F|Tphys = ˜F|Tphys
+
+SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT EPIDEMIC MODELS I
+15
+and ⟨1 | ˜F⟩ = 0 also outside Tphys. So in this case the second term in (2.27) vanishes and
+the condition ∇(u˜F) ≤ 0 looks like in the classical Bendixson-Dulac theorem.
+As in (Busenberg and Driessche 1990) putting y = (S1, S2, I) and u = 1/(S1S2I) we now
+apply this to the dynamical system Eqs. (2.8)-(2.10). We have uF(y) = uMy + uf(y)
+where
+M =
+�
+�
+−˜α1
+˜α2
+˜γ1
+˜α1
+−˜α2
+˜γ2
+0
+0
+−1
+�
+� ,
+(uf)(y) =
+�
+�
+−( ˜β1 + θ1)/S2 + θ2/S1 + (ν1 − ν2)/I
+−( ˜β2 + θ2)/S1 + θ1/S2 + (ν2 − ν1)/I
+˜β1/S2 + ˜β2/S1
+�
+� .
+(2.28)
+Here the time scale normalization ˜γ1 + ˜γ2 = 1 is understood.
+Theorem 2.12. Under the following conditions there exist no periodic solutions, homo-
+clinic loops or oriented phase polygons of the SSISS system (2.8)-(2.10) in Tphys.
+i) (˜αi, ˜γi, θi) ∈ R6
+≥0.
+ii) (˜αi, ˜γi, θi) ∈ Cbio and ν1 = ν2.
+Proof. First note that ˜γ1 + ˜γ2 = 1 implies that the boundary lines {S1 = 0} and {S2 = 0}
+cannot both be forward invariant. Hence, ∂Tphys cannot be a phase polygon. Next, the
+second term in (2.27) vanishes, because we have ⟨1 | F⟩ = 0 also outside of Tphys. We are
+left to compute ∇·(u(y)My) = − �
+i̸=j Mi,jyj/yi < 0 and ∇·f = −θ2/S2
+1 −θ1/S2
+2. Part i)
+follows by Lemma 2.5 and Lemma 2.10. The proof of part ii) relies on the normalization
+formalism of Section 3 and follows from Corollary 3.17.
+□
+Remark 2.13. Note that Theorem 2.12ii) doesn’t follow directly from Theorem 2.6, because
+there the equivalence transformation Tg need not preserve Tphys, see also Remark 2.7.
+Remark 2.14. Usually in the literature on models with constant per capita birth and death
+rates the vector field F appears in the form F = FM + f, where FM = My − ⟨1 | My⟩y,
+the second term being nonzero. This makes computations more involved but still yields
+ΨM|Tphys ≡ ∇ · (uFM)|Tphys − (y · ∇)⟨1 | uFM⟩|Tphys = − �
+i̸=j Mi,jyj/yi, see Eq. (3.8)
+in (Derrick and Driessche 1993). The fact that M may be chosen to satisfy ⟨1|M = 0
+(Lemma A.1 in Appendix A, see also remark 2.11) is rarely noticed in the literature.
+3. Normalization
+3.1. Phase space. From now on we drop again the tilde above parameters and also
+require ν1 = ν2. To proceed one has to choose suitable coordinates (X, Y ) on a phase space
+P ⊃ Tphys. Let’s first do some linear algebra. Put V = R2 and consider S ≡ |S⟩ =
+� S1
+S2
+�
+,
+α ≡ |α⟩ = ( α1
+α2 ), γ ≡ |γ⟩ = ( γ1
+γ2 ), θ ≡ |θ⟩ =
+� θ1
+θ2
+�
+as elements of V (“ket-” or “column-”
+vectors). Denote
+e ≡ ⟨e| := (1, 1) ,
+β ≡ ⟨β| := (β1, β2)
+(3.1)
+as a basis in the dual space V ∗ (“bra-” or “row-” vectors). Putting L(β, θ) := D(β)+E(θ)
+we then have
+⟨e|E(α) = 0,
+⟨e|L(β, θ) = ⟨β|,
+⟨e | γ⟩ = 1
+(3.2)
+where ⟨· | ·⟩ denotes the dual pairing V ∗ ⊗ V → R. Generalizing this setting, pick (e, β)
+any oriented12 basis in V ∗ and γ ∈ V satisfying ⟨e | γ⟩ = 1. Denote E ⊂ End V the right
+12The requirement of being oriented (with respect to a given orientation in V ) is a coordinate free
+version of the condition β2 < β1.
+
+16
+SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT EPIDEMIC MODELS I
+ideal anihilated by ⟨e| and L := {L ∈ End V | ⟨e|L = ⟨β|}. On V × R = R3 consider the
+dynamical system
+˙S = − [E + IL] S + Iγ,
+S ∈ V, E ∈ E, L ∈ L,
+(3.3)
+˙I = (X − 1)I,
+I ∈ R, X := ⟨β | S⟩.
+(3.4)
+Fixing e and varying (β, E, L, γ) under the above constraints defines a 7-parameter dy-
+namical system which in fact provides a coordinate free reformulation of the SSISS model
+(2.4). Note that the conditions imply ⟨e | ˙S⟩ + ˙I ≡ ˙S1 + ˙S2 + ˙I = 0, so the dynamics
+(3.3)-(3.4) leaves the cosets {⟨e | S⟩ + I = const.} ⊂ R3 invariant. Since I = 0 implies
+˙I = 0 also the half spaces {I ∈ R±} as well as the plane {I = 0} stay invariant.
+Definition 3.1. The dynamical system (3.3)-(3.4) on phase space P = {(S, I) ∈ V ×R≥0 |
+⟨e | S⟩ + I = 1} with parameter space A = C × B is called the extended SSISS model.
+Remark 3.2. The extension to negative values of variables Si and parameters a is needed
+to construct the symmetry operation of GS in Theorem 2.6.
+3.2. Canonical coordinates. Putting I := 1 − ⟨e | S⟩ and using S as independent
+coordinates on P Eq. (3.4) becomes redundant and we end up with a two-dimensional
+system. However, based on the coordinate free formulation (3.3)-(3.4), there is another
+natural set of canonical coordinates for this system. Put
+X := ⟨β | S⟩,
+Y := ⟨e | S⟩,
+(3.5)
+or equivalently choose the basis dual to (3.1) in V
+e⊥ ≡ |e⊥⟩ :=
+1
+β1 − β2
+�
+1
+−1
+�
+,
+β⊥ ≡ |β⊥⟩ :=
+1
+β1 − β2
+�
+−β2
+β1
+�
+(3.6)
+Hence we have X ≡ Xrep, Y ≡ S1 + S2 and
+S = Xe⊥ + Y β⊥.
+(3.7)
+Lemma 3.3. In canonical coordinates the extended SSISS model becomes
+˙X
+=
+(−aX + b) + (−cX + d)I − ϵI2 ,
+(3.8)
+˙Y
+=
+(1 − X)I = − ˙I ,
+(3.9)
+where I = 1 − Y and where the new parameters are given by
+a := α1 + α2
+(3.10)
+b := α2β1 + α1β2
+(3.11)
+c := β1 + β2 + θ1 + θ2
+(3.12)
+d := γ1β1 + γ2β2 − b + ϵ
+(3.13)
+ϵ := β1β2 + β1θ2 + β2θ1 .
+(3.14)
+Proof. In canonical coordinates the matrices E(α) and L(β, θ) := D(β) + E(θ) take the
+normal form
+E(α) =
+�
+a
+−b
+0
+0
+�
+,
+L(β, θ) =
+�
+c
+−ϵ
+1
+0
+�
+.
+(3.15)
+Using |γ⟩ = (β1γ1 +β2γ2)|e⊥⟩+|β⊥⟩ the claim follows by straightforward calculation.
+□
+
+SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT EPIDEMIC MODELS I
+17
+The canonical form of the SSISS dynamical system (3.8)-(3.9) will also be called the RN-
+dynamical system (RN = replacement number). Beware that unless β2 ≥ 0 the “would-be”
+replacement number X may take negative values even for Si ≥ 0. In fact, in canonical
+coordinates the physical triangle takes the form
+Tphys(β) = {(X, Y ) ∈ R × [0, 1] | β2Y ≤ X ≤ β1Y }
+= {(X, I) ∈ R × [0, 1] | β2(1 − I) ≤ X ≤ β1(1 − I)}.
+(3.16)
+So in (X, I)-space Tphys is given by the corners T< = (β2, 0), T> = (β1, 0) and T∧ = (0, 1).
+To stay with epidemiological conventions, from now on I will use X ≡ Xrep and I ≡ 1−Y
+as independent variables, in terms of which phase space is now given by
+P = {(X, I) ∈ R × R≥0}.
+Also note that in canonical coordinates the dynamics is reduced from seven to five pa-
+rameters, i.e. the system no longer depends on β. So, the role of β is reduced to fixing
+the image of physical triangles Tphys in canonical coordinates. Equivalently this means
+that fixing x = (a, b, c, d, ϵ) and varying β ∈ B we get an equivalence class of isomorphic
+dynamical systems, albeit physical triangles are not mapped onto each other under these
+isomorphisms.
+Proposition 3.4. For a, a′ ∈ A, a = (α, β, γ, θ) and a′ = (α′, β′, γ′, θ′), assume x(a) =
+x(a′). Following Eq. (3.7) put
+S := Xe⊥(β) + (1 − I)β⊥ ,
+S′ := Xe⊥(β′) + (1 − I)β′⊥ .
+(3.17)
+Then S1 + S2 = S′
+1 + S′
+2 = 1 − I and S = gS′ where g ∈ GL+(R2) is uniquely defined by
+g = |β⊥⟩⟨e| + |e⊥(β)⟩⟨β′| =
+1
+β1 − β2
+�
+β′
+1 − β2
+β′
+2 − β2
+β1 − β′
+1
+β1 − β′
+2 ,
+�
+(3.18)
+implying det g = (β′
+1 − β′
+2)/(β1 − β2) > 0. Moreover, (S, I) satisfies the SSISS dynamics
+(3.3)-(3.4) at parameter values a iff (S′, I) satisfies it at parameter values a′.
+Proof. Eq. (3.17) implies ⟨e|S⟩ = ⟨e|S′⟩ = 1−I and ⟨β|S⟩ = ⟨β′|S′⟩ = X. Hence, g must
+satisfy ⟨e|g = ⟨e| and ⟨β|g = ⟨β′| with unique solution (3.18).
+□
+Remark 3.5. Apparently we have g ∈ GS := {g ∈ GL+(R2) | ⟨e|g = ⟨e|} and by Eq.
+(3.18) β �→ βg defines a transitive and free right action of GS on B13. In Corollary 3.7
+below this action will be transported to a free GS-action on A, thus proving parts i)-iv)
+of Theorem 2.6.
+3.3. Main results. In this subsection we study the constraints on the new parameters
+x := (a, b, c, d, ϵ) and admissible ranges of β - or equivalently Tphys(β) - for given values
+of x, which will finally lead to a proof of Theorems 2.6 and 2.12. Recalling A ≡ C × B
+denote
+φ : A ∋ a �→ (x(a), β) ∈ D × B,
+D := R+ × R4
+(3.19)
+where x(a) is given by (3.10)-(3.14). The proof of the following Lemma is by straight
+forward calculation and hence omitted.
+13Note that dim GS = 2.
+The parametrization of g in (3.18) is redundant by invariance under
+(β1, β2) �→ (β1 + λ, β2 + λ) and (β1, β2) �→ (χβ1, χβ2), (λ, χ) ∈ R × R+.
+
+18
+SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT EPIDEMIC MODELS I
+Lemma 3.6. The map φ : A → D × B provides a diffeomorphism with φ−1 given by
+αi = b − aβi
+βj − βi
+,
+γi = d + b − ϵ − βj
+βi − βj
+,
+θi = β2
+i − cβi + ϵ
+βj − βi
+,
+j ̸= i
+(3.20)
+□
+Corollary 3.7. Consider D×B as a trivial principal GS-bundle with fiber B and GS right
+action (x, β) ◁ g := (x, βg), see Remark 3.5. Defining a ◁ g := φ−1(x(a), βg) we get an
+isomorphic GS-bundle structure on A. Putting y := (S1, S2, I) and writing the dynamical
+system (3.3)-(3.4) with parameters a ∈ A as ˙y = Fa(y), Proposition 3.4 becomes
+Fa ◦ Tg = Tg ◦ Fa◁ g ,
+Tg := g ⊕ id ,
+g ∈ GS.
+This proves parts i), iii) and iv) of Theorem 2.6.
+□
+The remaining transformation rules in part ii) of Theorem 2.6 now boil down to an exercise
+in linear algebra.
+Lemma 3.8. Let D(β) and E(α) be given as in Eq. (2.3) and ϑ(β, β′) as in part ii) of
+Theorem 2.6. Then for all g ∈ GS, α ∈ R2 and β′ = βg ∈ B
+E(¯gα)g = gE(α),
+D(β)g = g [D(β′) + E(ϑ(β, β′))]
+Applying these identities to the dynamical system (3.3)-(3.4) proves Theorem 2.6ii).
+□
+Remark 3.9. Beware that the transformation matrix g preserves S1+S2 but not necessarily
+R2
+≥0.
+Also, if a ∈ Aphys (or Abio) and x(a) = x(a′) then it depends on β′ whether
+a′ ∈ Aphys (or Abio), see Proposition 3.15 below. Hence, the above equivalencies may
+produce scenarios where a ∈ Aphys and a′ = a ◁ g ̸∈ Aphys and T−1
+g Tphys ̸∈ R3
+≥0 but still
+T−1
+g Tphys is forward invariant under the flow of Fa′.
+Next, on D define the functions
+R0(x) := b/a
+≡ α2β1 + α1β2
+α1 + α2
+,
+(3.21)
+R1(x) := d + b − ϵ ≡ γ1β1 + γ2β2 .
+(3.22)
+Obviously we may also use x ≡ (a, R0, R1, c, ϵ) ∈ R+ × R4 as independent parameters in
+D. Moreover we clearly have
+φ(A+) = {(x, β) ∈ D × B | β2 ≤ Ri ≤ β1 , i = 1, 2} ,
+(3.23)
+i.e. on A+ the functions Ri may be interpreted as two kinds of mean values of β1 and β2.
+Again beware that for β2 < 0 we may have Ri < 0 even on A+. To explain the meaning
+of R0 note that for a > 0 the value of the replacement number X at the disease-free
+equilibrium (DFE) of the RN-dynamical system (3.8)-(3.9) is precisely given by X∗
+0 = R0.
+Following results of (Driessche and Watmough 2002) this leads to
+Definition 3.10. R0 is called the reduced reproduction number.
+Remark 3.11. As has been shown by (Driessche and Watmough 2002, 2008), in models
+with just one infectious compartment the more general notion of R0 as the spectral
+radius of the next generation matrix ((Diekmann, Heesterbeek, and Metz 1990), see also
+(Diekmann and Heesterbeek 2000)) reduces to the above definition. Denoting the values
+
+SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT EPIDEMIC MODELS I
+19
+of Si at the DFE by S∗
+i we have R0 = β1S∗
+1 + β2S∗
+2, which is the usual formula, see e.g.
+(Kribs-Zaleta and Velasco-Hernandez 2000) or (Avram, Adenane, Bianchin, et al. 2022).
+Remark 3.12. Mostly in the literature R0 is called the basic reproduction number. But in
+case β2 = 0 this terminology is already occupied by r0 := β1/γ as the expected number
+of secondary cases produced by a typical infectious individual in a totally susceptible
+population. So to avoid confusion I prefer to call R0 the reduced reproduction number.
+Next put Dx := πD(φ(Ax)), x = phys or x = bio, where πD : D × B → D denotes
+the canonical projection. We look for suitable coordinates describing Dx and then derive
+additional bounds on β to describe φ(Ax). Consider the following functions on D.
+A±(x) := 1
+2
+�
+a + c ±
+�
+(a + c)2 − 4(b + ϵ)
+�
+,
+(3.24)
+B±(x) := 1
+2
+�
+c ±
+√
+c2 − 4ϵ
+�
+.
+(3.25)
+Then by (3.15) and the trace-det formula A± and B± provide the eigenvalues of E + L
+and L, respectively. The meaning of these eigenvalues becomes clear by looking at (3.20)
+β1 = A+ ⇔ α1 + θ1 = 0
+β1 = B+ ⇔ θ1 = 0
+(3.26)
+β2 = A− ⇔ α2 + θ2 = 0
+β2 = B− ⇔ θ2 = 0
+(3.27)
+βi = R0 ⇔ αi = 0
+βi = R1 ⇔ γj = 0, j ̸= i
+(3.28)
+More generally from (3.20) we get
+θi = (βi − B−)(βi − B+)
+βj − βi
+,
+j ̸= i ,
+(3.29)
+αi + θi = (βi − A−)(βi − A+)
+βj − βi
+,
+j ̸= i .
+(3.30)
+Hence A± will serve to fix the constraints on (x, β) ∈ φ(Aphys) and B± (B ≡ “bio”) to fix
+constraints on (x, β) ∈ φ(Abio). First we gather some trivial identities.
+c = B+ + B− = A+ + A− − a ,
+ϵ = B+B− = A+A− − b ,
+(3.31)
+a = A+ + A− − B+ − B− ,
+aR0 ≡ b = A+A− − B+B−
+(3.32)
+From these one immediately computes
+a(A± − R0) = (A± − B+)(A± − B−) = A2
+± − cA± + ϵ
+(3.33)
+a(R0 − B±) = (B± − A+)(B± − A−) = B2
+± − (a + c)B± + (b + ϵ)
+(3.34)
+Now let’s introduce the notation
+DA := D ∩ {A± ∈ R}
+(3.35)
+DB := D ∩ {B± ∈ R ∧ B− < B+}
+(3.36)
+DAB := DA ∩ DB ∩ {B− ≤ A− ≤ B+ ≤ A+}.
+(3.37)
+Lemma 3.13. The following identities hold
+DAB = {x ∈ DA | A− ≤ R0 ≤ A+ ∧ c2 ̸= 4ϵ} = {x ∈ DB | B− ≤ R0 ≤ B+}
+(3.38)
+Hence in DAB we always have the additional bound
+B− ≤ A− ≤ R0 ≤ B+ ≤ A+ .
+(3.39)
+
+20
+SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT EPIDEMIC MODELS I
+Proof. By Eqs. (3.33) and (3.34) on DAB we always have A− ≤ R0 ≤ A+ and B− ≤
+R0 ≤ B+.
+Conversely, if A± ∈ R, c2 ̸= ϵ and A− ≤ R0 ≤ A+ then (3.33) implies
+A2
+− − cA− + ϵ ≤ 0 ≤ A2
++ − cA+ + ϵ and therefore c2 > 4ϵ. Hence B− < B+ ∈ R and again
+by (3.33) B− ≤ A− ≤ B+ ≤ A+. The second identity follows analogously.
+□
+Lemma 3.14. Denoting iB : DB ∋ x �→ (x, B+, B−) ∈ DB × B the following identities
+hold
+φ(Asplit) = {(x, β) ∈ DB × B | B− ≤ β2 < β1 ≤ B+} ,
+(3.40)
+φ(Aθ+α≥0) = {(x, β) ∈ DA × B | β2 ≤ A− ≤ β1 ≤ A+} .
+(3.41)
+φ(Aθ=0) = iB(DB)
+(3.42)
+φ(Aθ=0 ∩ Aα≥0) = iB(DAB)
+(3.43)
+Proof. We have B± ∈ R iff there exists β ∈ R such that β2 − cβ + ϵ ≤ 0. Hence, by
+(3.20), if θ1 ≥ 0 and θ2 ≤ 0 then B± ∈ R and B− ≤ β2 < β1 ≤ B+, proving the “⊂”-part
+in (3.40). The opposite direction follows from (3.29). Similarly, A± ∈ R iff there exists
+β ∈ R such that β2 − (a + c)β + b + ϵ ≤ 0. Hence, by (3.20), if α1 + θ1 ≥ 0 then A± ∈ R
+and A− ≤ β1 ≤ A+. If in addition α2 + θ2 ≥ 0 then also β2 ≤ A−, proving the “⊂”-part
+in (3.41). The opposite direction follows from (3.30). Eq. (3.42) follows since in Aθ=0 we
+have β1 = B+ and β2 = B−. If in addition αi ≥ 0 then (3.30) implies Eq. (3.43).
+□
+We are now in the position to summarize the constraints describing φ(Aphys) and φ(Abio).
+Proposition 3.15. For Ax = Cx × B as defined in (2.14) - (2.15) we have
+φ(Aphys) ≡ φ(A+) ∩ φ(Aθ+α≥0)
+= (DA × B) ∩ {β2 ≤ {A−, R0, R1} ≤ β1 ≤ A+} ,
+(3.44)
+Dphys = DA ∩ {R0,1 ≤ A+} ,
+(3.45)
+φ(Abio) ≡ φ(Asplit) ∩ φ(Aphys)
+= (DAB × B) ∩ {B− ≤ β2 ≤ A− ≤ R0 ≤ β1 ≤ B+} ∩ {R1 ∈ [β2, β1]} ,
+(3.46)
+Dbio = DAB ∩ {R1 ∈ [B−, B+]} ⊂ Dphys .
+(3.47)
+Proof. This is a summary of Eq. (3.23) and Lemmas 3.13 - 3.14.
+□
+Proposition 3.15 motivates the following notation and definition
+Definition 3.16. For x ∈ Dbio put
+βmax
+2
+(x) := min{A−, R1},
+βmin
+1
+(x) := max{R0, R1}.
+(3.48)
+Then β ∈ B is called bio-compatible with x if B− ≤ β2 ≤ βmax
+2
+and βmin
+1
+≤ β1 ≤ B+,
+equivalently if φ−1(x, β) ∈ Abio.
+Similarly, β is called compatible if β2 ≤ βmax
+2
+and
+βmin
+1
+≤ β1 ≤ A+, equivalently if φ−1(x, β) ∈ Aphys. A physical triangle Tphys(β) is called
+(bio)-compatible, if β is (bio)-compatible.
+Hence, (bio-)compatible physical triangles are always forward invariant under the RN-
+dynamics (3.8)-(3.9) and the smallest one is just Tphys(βmin
+1
+, βmax
+2
+). The following Corollary
+also proves part ii) of Theorem 2.12.
+
+SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT EPIDEMIC MODELS I
+21
+Corollary 3.17. Let x ∈ Dbio and let β ∈ B be compatible with x. Then there exist
+no periodic solutions, homoclinic loops or oriented phase polygons of the RN-dynamical
+system (3.8)-(3.9) in Tphys(β).
+Proof. Let Z ⊂ Tphys(β) be a solution cycle (image of a periodic solution, a homoclinic
+loop or an oriented phase polygon). As argued in the proof of Theorem 2.12, we must
+have Z ̸= ∂Tphys(β′) for all β′ ∈ B. Hence, by forward invariance, Z must lie inside the
+smallest compatible triangle, Z ⊂ Tphys(βmin
+1
+, βmax
+2
+) ⊂ Tphys(B+, B−). But, by Proposition
+3.15 and Eq.
+(3.43), φ−1(x, B+, B−) ∈ Abio ∩ Aθ=0 and we get a contradiction with
+Theorem 2.12i).
+□
+Finally, to prove Theorem 2.6v), note that Lemma 3.14 and Proposition 3.15 in particular
+imply (use that GS acts transitively on B)
+Aθ=0 ◁ GS = Asplit ◁ GS = φ−1(DB × B)
+(3.49)
+Aθ+α≥0 ◁ GS = φ−1(DA × B)
+(3.50)
+(Aθ=0 ∩ Aα≥0) ◁ GS = φ−1(DAB × B)
+(3.51)
+Aphys ◁ GS = φ−1(Dphys × B)
+(3.52)
+Abio ◁ GS = φ−1(Dbio × B)
+(3.53)
+Aθ=0 ∩ Abio ◁ GS ⊂ Abio
+(3.54)
+where the last equation follows from (Dbio × B) ∩ iB(DB) = iB(Dbio) ⊂ φ(Abio). Part v)
+of Theorem 2.6 now follows from Eqs. (3.49), (3.54) and Lemma 3.18 below.
+Lemma 3.18. Put AB := φ−1(DB × B) = Aθ=0 ◁ GS, then
+AB ⊃ A ∩ {θ1 ≥ θ2 ∨ θ1θ2 > 0} ⊃ Asplit ⊃ Abio.
+Proof. The second and third inclusions are obvious from the definitions (2.13) and (2.15)
+and the first inclusion follows from DB = D ∩ {c2 > 4ϵ} and
+c2 − 4ϵ = (β1 − β2)2 + (θ1 + θ2)2 + 2(β1 − β2)(θ1 − θ2)
+= (β1 − β2 + θ1 − θ2)2 + 4θ1θ2.
+□
+Table 3. Sector classification in Abio generalizing Table 1.
+Sector
+c = B− + B+
+ϵ = B−B+
+Interval [B−, B+]
+I
++
++
+0 < B− < B+
+II (SIRS)
++
+0
+0 = B− < B+
+III
++
+−
+0 < −B− < B+
+IV
+0
+−
+0 < −B− = B+
+V
+−
+−
+B− < −B+ < 0
+VI
+−
+0
+B− < B+ = 0
+VII
+−
++
+B− < B+ < 0
+
+22
+SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT EPIDEMIC MODELS I
+Let me close by mentioning that the parametrizations (3.31) can now be used to generalize
+the Sector classification of Table 1 from the special case Aθ=0 to all of Abio (more generally
+to AB := φ−1(DB × B) ⊃ Abio) as shown in Table 3.
+3.4. Examples revisited. For completeness let us revisit the examples in Section 2.4
+within the present setting. Eqs. (2.20)-(2.26) translate into14
+DHeth = Dbio ∩ {R0 = B+ ∧ a < 1 ∧ d = B− = 0}
+(3.55)
+DSIRI1,2 = Dbio ∩ {R0 = B± ∧ a < 1 ∧ d = B∓(B± + 1 − a)}
+(3.56)
+DBuDr = Dbio ∩ {R1 < R0 = B+ ∧ B− < 0}15
+(3.57)
+DSIRS = Dbio ∩ {B− = 0}
+(3.58)
+DLM = Dbio ∩ {B− < min{0, R1}}
+(3.59)
+DKZVH = Dbio ∩ {B− > 0} = DHaCa
+(3.60)
+DAABH1 = Dbio ∩ {B− ≤ R1 < B+}15
+(3.61)
+DAABH2 = Dbio ∩ {B− < R1 ≤ B+}15
+(3.62)
+Note that all models except SI(R)S already satisfy θi = 0 whence ˜β1 = B+, ˜β2 = B−
+by Eqs. (3.26)-(3.28). In the SI(R)S model we have instead 0 = ˜β2 = B− < ˜β1 ≤ B+.
+Corollary 2.9 may now be reformulated as follows
+Corollary 3.19. Referring to the sub-cases µ1 = µ2 in (Avram, Adenane, Bianchin, et al.
+2022; Busenberg and Driessche 1990) and putting DAABH := DAABH1 ∪ DAABH2 we have
+DHeth = DSIRI1 ∩ {B− = 0}
+(3.63)
+= DSIRS ∩ {a < 1 ∧ R0 = c ∧ d = 0}
+(3.64)
+DLM ⊃ DBuDr ∩ {B− ̸= R1}
+(3.65)
+DLM = DAABH2 ∩ {B− < 0}
+(3.66)
+DKZVH = DAABH ∩ {B− > 0}
+(3.67)
+Finally, we are now in the position to generalize the scaling symmetry for SI(R)S models
+of (Nill 2022) to the present setting. First note that having started from the 10-parameter
+extended SI(R)S model we now have arrived at dim DSIRS = 4. Also, dim DHeth = 2 with
+independent parameters a ∈ (0, 1) and c = R0 = B+ > 0. In particular, if x ∈ DHeth
+then putting (u, v) := (X, cI) the RN-dynamical system (3.8)-(3.9) reduces to the classic
+endemic model in Eq. (1.1). In a second normalization step the number of parameters in
+the SI(R)S case may now be reduced again by two. In this way, for c > d,16 the normalized
+SI(R)S model also looks like the classic endemic model
+˙u = −uv − c1u + c2 ,
+˙v = uv − v ,
+(3.68)
+the difference being that coming from DHeth we have c1 = a ∈ (0, 1) and c2 = aR0 ≥ 0,
+whereas coming from DSIRS gives (c1, c2) ∈ R+ × R17. However, since endemic bifurcation
+14Heth = (Hethcote 1974, 1976, 1989); SIRI = (Derrick and Driessche 1993); BuDr = (Busenberg
+and Driessche 1990); SIRS = 10-parameter mixed SIRS/SIS model with constant population size and
+θ2 = β2 = 0; HaCa = core system in (Hadeler and Castillo-Chavez 1995); KZVH = (Kribs-Zaleta and
+Velasco-Hernandez 2000); LM = (J. Li and Ma 2002); AABH = (Avram, Adenane, Bianchin, et al. 2022).
+BuDr and AABH come in two versions, the subscript 1 refers to βS > βR and 2 to βS < βR.
+15Referring to the sub-case µ1 = µ2 in these models, see Footnotes 10 and 11.
+16Note that in DSIRS we have c = B+ ≥ R1 − aR0 = d where equality implies R0 = 0 and R1 = B+.
+17In case a = 0 we would get c1 = c2 = 0.
+
+SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT EPIDEMIC MODELS I
+23
+in the model (3.68) occurs at R0 = c2/c1 = 1, extending this model to the SI(R)S case
+by including also values c2 < 0 and c1 ≥ 1 doesn’t change its characteristic behavior.
+In particular, various proofs in the literature on variants of constant population SI(R)S
+models with standard incidence become obsolete, it’s all contained in Hethcote’s work.
+Eq. (3.68) is proven in Appendix B. In principle, the proof relies on the same structure
+as in Theorem 2.6, with the symmetry group GS acting on A replaced by a dilatation
+group Gdil = R2
++ acting on D. Since these dilatations may blow up physical triangles to
+arbitrary size, we also get the following
+Lemma 3.20. For x ∈ DSIRS the forward flow of the RN-dynamical system (3.8)-(3.9)
+stays bounded for all initial conditions (X0, I0) ∈ R × R≥0.
+This result may be used to prove, that SI(R)S models as above are always Hamiltonian
+(Nill n.d.[a]). Lemma 3.20 is also proven in Appendix B.
+4. Summary and outlook
+In summary we have seen, that in canonical coordinates the 14-parameter SSISS model,
+constraint by ν1 = ν2, effectively depends on at most five parameters x = (a, b, c, d, ϵ). De-
+pending on natural model restrictions like “phys” or “bio” these parameters obey various
+relations which can be encoded by further reparametrizations like x = (a, R0, R1, B+, B−),
+see Eqs. (3.21), (3.22), (3.31) and Proposition 3.15. The incidence rates βi have disap-
+peared from the equations of motion. Their role is reduced to fixing physical triangles
+Tphys(β) in (X, I)-space, see Eq.
+(3.16).
+If x ∈ Dbio, then for all compatible values
+β = (β1, β2) the triangles Tphys(β) stay forward invariant under the RN-dynamics (3.8)-
+(3.9). Independence of β also means that SSISS models at parameter values φ−1(x, β)
+for fixed x ∈ D and varying β ∈ B are all isomorphic to each other18 (Proposition 3.4).
+The isomorphisms are provided by a parameter symmetry group GS ⊂ GL+(R2) acting
+simultaneously on phase space P and parameter space A (Theorem 2.6i-iv). If x ∈ DB
+then a representative in A of the equivalence class x may always be chosen by putting
+β1 = B+ and β2 = B− and hence θi = 0 (Theorem 2.6v). In combination with methods
+from (Busenberg and Driessche 1990) this also leads to a proof of absence of periodic
+solutions for all a ∈ Abio (Theorem 2.12).
+In part III of this work it will be shown, that the model also admits an additional scaling
+symmetry leading to a second normalization step, similar as described for the SI(R)S
+model in Appendix B, see also (Nill 2022). In this way the number of essential parameters
+will further reduce from five to three (respectively two in Sectors II and VI).
+Part II of this work will reanalyze equilibrium points and their stability properties in
+all Sectors of Abio, thereby recovering and extending the results of (Avram, Adenane,
+Bianchin, et al. 2022; Hadeler and Castillo-Chavez 1995; Kribs-Zaleta and Velasco-Hernandez
+2000; J. Li and Ma 2002), which had been obtained for θi = 0 and some more parameter
+restrictions, see Table 2 and Corollary 2.9/3.19. This approach will differ from previ-
+ous papers by relying on the normalization formalism and sector classification of the
+present work. In this way the search for endemic equilibria (X∗, I∗) simplifies consider-
+ably, since always X∗ = 1. So one is left with analyzing roots of the quadratic equation
+h(I∗) := ˙X(X∗ = 1, I∗) = 0. This will also uncover an exceptional scenario in Sectors
+III-V, which apparently has been overlooked in the literature so far.
+18By Remark 3.9, physical triangles are not mapped onto each other under these isomorphisms.
+
+24
+SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT EPIDEMIC MODELS I
+Appendix A. Normalizing linear vital dynamics
+This Appendix gives a normalization prescription for the dynamics of fractional vari-
+ables in an n-compartment model with linear vital dynamics. Let the vectorfield V :
+Rn → Rn be homogeneous of degree one and assume there exists ν = (ν1, · · · , νn) such
+that ⟨1|V(Y)⟩ ≡ �
+i Vi(Y) = ⟨ν|Y⟩ for all Y ∈ Rn, where 1 := (1, · · · , 1).
+Call
+N(Y) := ⟨1|Y⟩ the total population and y := N−1Y the fractional compartment vari-
+ables, then the dynamical system ˙Y = V(Y) implies
+˙y = V(y) − ⟨ν | y⟩y =: F(y).
+Denote S := {y ∈ Rn | ⟨1|y⟩ = 1}, then clearly ⟨1|F⟩|S = 0. The aim is to substitute F
+by ˜F such that F|S = ˜F|S and ⟨1|˜F⟩ = 0 holds as an identity on all of Rn. The following
+Lemma holds by straight forward calculation.
+Lemma A.1. Put Λijk := (δij − δik)(νk − νj) and Λi(y) := �
+j,k Λijkyjyk.
+i)
+For all y ∈ Rn and i = 1, · · · , n we have
+1
+2Λi(y) =
+�
+k
+(νk − νi)yiyk ≡ yi⟨ν|y⟩ − νiyi⟨1|y⟩.
+(A.1)
+ii) Put
+˜F := V − diag(ν) − 1
+2Λ.
+(A.2)
+Then F|S = ˜F|S and ⟨1|˜F⟩ = 0 as an identity on Rn.
+By this method we also get conditions guaranteeing that constant per capita birth and
+death rates become redundant as in Eq. (2.7).
+Lemma A.2. Let V(Y) be of the form
+Vi(Y) =
+�
+j
+MijYj + 1
+2
+�
+j,k
+ΓijkYjYk/N +
+�
+j
+LijYj
+where without loss Γijk = Γikj and where �
+i Mij = �
+i Γijk = 0. Hence, all vital dynamics
+parameters are encoded in (Lij) and νj := �
+i Lij satisfies ⟨1|V = ⟨ν|. If in this case
+Lij ̸= νiδij ⇒ Mij ̸= 0 and νj ̸= νk ⇒ (Γjjk ̸= 0 ∧ Γkkj ̸= 0), then for the dynamics of
+fractional variables all parameters Lij are redundant.
+Proof. Applying (A.2) we have ˜Fi(y) = �
+j ˜
+Mijyj+ 1
+2 ˜Γijkyjyk, where ˜
+Mij = Mij+Lij−νiδij
+and ˜Γijk = Γijk − Λijk. The claim follows since Λijk = Λikj, Λjjk = −Λkkj and Λijk = 0 if
+νj = νk or if j ̸= i ̸= k, which also yields �
+i Λijk = 0 .
+□
+Appendix B. Scaling the SI(R)S model
+In this appendix we extend the dilatation symmetry as proposed for a 6-parameter
+SI(R)S model in (Nill 2022) to the 10-parameter extended SI(R)S model as classified in
+this paper. Denote Sector II in DB by DII := DB ∩ {B− = 0} and DSIRS := DII ∩ Dbio.
+Recall that in DII we have c = B+ > 0 and in DSIRS we have 0 ≤ Ri ≤ B+ and hence
+d − c = R1 − aR0 − B+ ≤ 0, where equality implies R0 = 0 and R1 = B+. Hence the
+following Lemma in particular includes Lemma 3.20.
+
+SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT EPIDEMIC MODELS I
+25
+Lemma B.1. Consider the RN-dynamical system (3.8) - (3.9) on phase space P ≡ R×R≥0
+for parameter values x = (a, b, c, d, ϵ = 0) ∈ DII ∩ {d ≤ c ∧ d = c ⇒ R0 < 1} ⊃ DSIRS.
+Let T ⊂ P be a rectangular triangle with corners T◁ = (X◁, 0), T▷ = (X▷, 0) and T△ =
+(X◁, I△), where X◁ < X▷. Call T compatible with x if
+I△ = (X▷ − X◁)/c
+X◁ ≤ min{R0, d/c}
+R0 − X◁ ≤ I△ min{c, (c − d)/a}
+i)
+Then every x-compatible triangle T is forward invariant.
+ii) The forward flow for arbitrary initial conditions (X0, I0) ∈ P stays bounded.
+Proof. To prove part i), the upper bounds on X◁ imply ˙X > 0 on the line {X = X◁}. We
+are left to show ˙X + c ˙I ≤ 0 on the hypotenuse X(I) = X◁ + c(I△ − I), 0 ≤ I ≤ I△.
+˙X + c ˙I = a(R0 − X(I)) + (d − c)I
+= a(R0 − X◁ − c(I△ − I)) + (d − c)I
+≤ I△ min{ac, c − d} − ac(I△ − I) + (d − c)I
+≤ 0
+Part ii) follows since for d < c we may always choose X◁ < X0 and X▷ large enough,
+such T is x-compatible and (X0, I0) ∈ T . For d = c and R0 < 1 x-compatibility requires
+X◁ = R0. If in this case X0 < R0 glue the rectangle R = [X0, R0]×[0, I△] to the left of T .
+Then (X0, I0) ∈ R ∪ T for X▷ large enough and R ∪ T is forward invariant, since ˙I < 0
+and ˙X > 0 for (X, I) ∈ R.
+□
+Given x ∈ DII ∩ {d ≤ c ∧ d = c ⇒ R0 < 1} as above and T compatible with x
+we now show that the RN-dynamical system (3.8) - (3.9) may always be rescaled to an
+isomorphic system with parameters x′ ∈ DSIRS such that T maps to the physical triangle
+Tphys(B′
++, 0) of the SI(R)S system. Following (Nill 2022) the dilatation symmetry group
+Gdil ≡ GX × GI ≡ R2
++ is defined by rescaling (X, I) variables according to
+X(ξ,λ)(t) − 1 := ξ(X(ξt) − 1),
+I(ξ,λ)(t) := λI(ξt),
+(ξ, λ) ∈ R2
++
+The following Lemma is easily verified by straightforward calculation.
+Lemma B.2. Let the group action ▷ : Gdil × D ∋ (ξ, λ, x) �→ (ξ, λ) ▷ x ∈ D be given by
+(ξ, λ) ▷ (a, R0 − 1, c, d − c, ϵ) := (ξa, ξ(R0 − 1), ξc/λ, ξ2(d − c)/λ, ξ2ϵ/λ2)
+(B.1)
+and for x ∈ D let fx(X, I) denote the vector field of the system (3.8) - (3.9). Then
+( ˙X, ˙I) = fx(X, I) ⇐⇒ ( ˙X(ξ,λ), ˙I(ξ,λ)) = fx′(X(ξ,λ), I(ξ,λ)),
+x′ = (ξ, λ) ▷ x.
+□
+Note that this action leaves all Sectors in DB invariant, but in general not Dbio ⊂ DB. We
+now determine Gdil ▷ DSIRS, thereby also providing an alternative proof of Lemma B.1i).
+Proposition B.3.
+i)
+Let T be compatible with x ∈ DII ∩ {d ≤ c ∧ d = c ⇒ R0 < 1} in the sense of Lemma
+B.1.
+Then there exists a unique dilatation transformation (ξ, λ) ∈ Gdil such that
+x′ := (ξ, λ) ▷ x ∈ Dbio and such that the rescaled triangle satisfies T(ξ,λ) = Tphys(B′
++, 0).
+ii) Gdil ▷ DSIRS = DII ∩ {d ≤ c ∧ d = c ⇒ R0 < 1}.
+
+26
+SYMMETRIES AND NORMALIZATION IN 3-COMPARTMENT EPIDEMIC MODELS I
+Proof. To prove part i) denote transformed quantities by a prime.
+The requirements
+T′
+◁ = (0, 0) and T′
+△ = (0, 1) fix ξ = (1 − X◁)−1 and λ = I−1
+△ .
+Hence X▷ maps to
+ξcI△ = c′ = B′
++ and therefore T(ξ,λ) = Tphys(B′
++, 0). To show 0 ≤ R′
+i ≤ B′
++ use R′
+0 =
+ξ(R0 − 1) + 1 = ξ(R0 − X◁) and therefore
+0 ≤ R′
+0 ≤ ξ
+λ min{c, (c − d)/a} = min{c′, (c′ − d′)/a′} ≤ B′
++
+By the above we also have R′
+1 = a′R′
+0 + d′ ≤ c′ = B′
++ and we are left to show R′
+1 ≥ 0.
+Sufficient is d′ ≥ 0 which follows from 1−d′/c′ = ξ(1−d/c) ≤ ξ(1−X◁) = 1. This proves
+part i) and therefore also the “⊃”-direction of part ii). To prove the “⊂”-direction use that
+the action of Gdil on D preserves the sign of d − c and in case d = c we have R0 = 0 and
+therefore R′
+0 = ξ(R0 − 1) + 1 = 1 − ξ < 1.
+□
+As in (Nill 2022), the above dilatation symmetry leads to a second normalization step
+for the SIRS-Sector, thus further reducing its number of essential parameters from four
+to two. Equivalently this means, that equivalence classes of Gdil-isomorphic systems with
+parameters in Gdil ▷ DSIRS are naturally parametrized by KSIRS := (Gdil ▷ DSIRS)/Gdil. A
+convenient realization of the normalized system on phase space P = {(q, p) ∈ R × R≥0} is
+given by putting
+q(t) := 1
+a(X(t/a) − 1) ,
+p(t) := c
+aI(t/a)
+(B.2)
+In terms of these variables the RN-dynamical system (3.8) - (3.9) becomes
+˙q = −q(p + 1) + κ0 − κ1p ,
+˙p = qp ,
+(B.3)
+where the new Gdil-invariant parameters are given by
+κ0 := R0 − 1
+a
+,
+κ1 := c − d
+ac
+.
+(B.4)
+The only remaining constraint on the reduced parameter space says
+KSIRS = {(κ0, κ1) ∈ R × R≥0 | κ1 = 0 ⇒ κ0 < 0} .
+(B.5)
+Thus, after normalization the whole SIRS Sector just looks like Hethcote’s classic endemic
+model except for a somewhat less restricted parameter space. In fact, by Eq. (3.55),
+DHeth ⊂ DSIRS is already two-dimensional with independent parameters a ∈ (0, 1) and
+c = R0 = B+ > 0. These map injectively to KSIRS via κ0 = (c − 1)/a and κ1 = 1/a,
+whence
+DHeth ∼= KHeth = KSIRS ∩ {κ1 > 1 ∧ κ0 + κ1 > 0}
+(B.6)
+The normalization convention in Eq. (3.68) is obtained under the restriction c > d or
+equivalently κ1 > 0. In this case one may alternatively use
+u(t) − 1 :=
+c
+c − d(X(ct/(c − d)) − 1) = 1
+κ1
+q(t/κ1) ,
+(B.7)
+v(t) :=
+c2
+c − dI(ct/(c − d)) = 1
+κ1
+p(t/κ1) .
+(B.8)
+In terms of these variables we recover the normalization convention (1.1), (3.68)
+˙u = −uv − c1u + c2 ,
+˙v = uv − v ,
+(B.9)
+where c1 = 1/κ1 and c2 = 1/κ1 + κ0/κ2
+1, which is also the version given in (Nill 2022).
+In part III of this work the above normalization step will be generalized to all Sectors of
+Dbio. In this way the equation for ˙q in (B.3) gets an additional term −κ2p2, and so our
+
+REFERENCES
+27
+initial 14-parameter19 SSISS model boils down to a much simpler 3-parameter dynamical
+system.
+Appendix C. The case α1 = α2 = 0
+This Appendix shortly discusses the border case α1 = α2 = 020. In this case define
+parameter spaces C0
+x as in Eqs. (2.11)-(2.15) with αi = 0 and A0
+x := C0
+x ×B. In particular,
+in A0
+bio we have θ1 ≥ 0, θ2 = 0, γi ≥ 0 and γ1 + γ2 = 1. Lemma 3.3 still holds with
+a = b = 0 and d = R1 + ϵ, i.e. the replacement number dynamics becomes
+˙X = (d − cX)I − ϵI2 ,
+˙I = (X − 1)I .
+(C.1)
+In this case R0 is undefined and there is a continuum of disease free equilibria at I = 0,
+which are locally stable for X < 1 and unstable for X > 1. Proposition 3.4 remains
+unchanged provided a = a′ ∈ A0. Putting D0 = {(c, d, ϵ) ∈ R3} Lemma 3.6 still holds
+with A replaced by A0 and D replaced by D0. Moreover, in A0
+bio we get A+ = B+ = β1+θ1,
+A− = B− = β2, c = β1+β2+θ1, ϵ = β2(β1+θ1) and putting D0
+A = D0
+B = D0
+AB := D0∩{c2 >
+4ϵ} Proposition 3.15 becomes
+φ(A0
+phys) = (D0
+B × B) ∩ {β2 ≤ {B−, R1} ≤ β1 ≤ B+} ,
+(C.2)
+D0
+phys = D0
+B ∩ {R1 ≤ B+} ,
+(C.3)
+φ(A0
+bio) = (D0
+B × B) ∩ {B− = β2 ≤ R1 ≤ β1 ≤ B+} ,
+(C.4)
+D0
+bio = D0
+B ∩ {B− ≤ R1 ≤ B+} ⊂ D0
+phys .
+(C.5)
+So, for x ∈ D0
+phys physical triangles Tphys(β1, β2) are forward invariant provided (β1, β2)
+satisfy the bounds C.2. Finally, Eq. (3.42) becomes φ−1(iB(D0
+B)) = φ(Aθ=0 ∩ Aα=0) and
+Theorem 2.12, Theorem 2.6 and Corollary 3.17 stay valid also for α = 0.
+References
+Arino, J., C.C. Mccluskey, and P. van den Driessche (2003). “Global results for an epidemic
+model with vaccination that exhibits backwad bifurcation.” In: SIAM J. Appl. Math.
+64, pp. 260–276. doi: 10.1137/S0036139902413829.
+Avram, F., R. Adenane, L. Basnarkov, et al. (Dec. 2021). “On matrix-SIR Arino models
+with linear birth rate, loss of immunity, disease and vaccination fatalities, and their
+approximations.” In: arXiv preprint. url: http://arxiv.org/abs/2112.03436.
+Avram, F., R. Adenane, G. Bianchin, et al. (2022). “Stability analysis of an eight parameter
+SIR- type model including loss of immunity, and disease and vaccination fatalities.” In:
+Mathematics 10.3, p. 402. doi: 10.3390/math10030402.
+Batistela, C.M. et al. (2021). “Vaccination and social distance to prevent Covid-19.” In:
+IFAC PapersOnLine 54-15, pp. 151–156.
+Busenberg, S. N. and P. van den Driessche (1990). “Analysis of a disease transmission
+model in a population with varying size.” In: J. Math. Biol. 28, pp. 257–270.
+Busenberg, S. N. and P. van den Driessche (1991). “Nonexistence of periodic solutions
+for a class of epidemiological models.” In: Biology, Epidemiology, and Ecology. Ed. by
+S.N. Busenberg and M. Martelli. Vol. 92. Lecture Notes in Biomath. Berlin Heidelberg
+New York: Springer, pp. 70–79.
+19i.e. constraint by ν1 = ν2.
+20Here, for simplicity of notation, the tilde is still omitted.
+So beware that truly this appendix
+addresses the cases αi = νi = νI = 0 (constant population (2.2)) or αi = δi = 0 and µ1 = µ2 (time
+varying population (2.7)).
+
+28
+REFERENCES
+Chauhan, S., O.P. Misra, and J. Dhar (2014). “Stability Analysis of Sir Model with Vac-
+cination.” In: American J. Comp. Appl. Math. 2014.4(1), pp. 17–23. doi: 10.5923/j.
+ajcam.20140401.03.
+Derrick, W.R. and P. van den Driessche (1993). “A disease transmission model in a non-
+constant population.” In: J Math Biol 31.5, pp. 495–512. doi: 10.1007/BF00173889.
+Diagne, M.L. et al. (2021). “A Mathematical Model of COVID-19 with Vaccination and
+Treatment.” In: Computational and Mathematical Methods in Medicine 2021, p. 1250129.
+doi: 10.1155/2021/1250129.
+Diekmann, O. and J.A.P. Heesterbeek (2000). Mathematical epidemiology of in-fectious
+diseases. Wiley series in mathematical and computational biology. West Sussex, Eng-
+land: John Wiley & Sons.
+Diekmann, O., J.A.P. Heesterbeek, and J.A.J. Metz (1990). “On the definition and the
+computation of the basic reproduction ratio R0 in models for infectious diseases in
+heterogeneous populations.” In: J. Math. Biol. 28, p. 365.
+Driessche, P. van den and J. Watmough (2002). “Reproduction numbers and sub-threshold
+endemic equilibria for compartmental models of disease transmission.” In: Math.Biosci.
+180, pp. 29–48.
+Driessche, P. van den and J. Watmough (2008). “Further notes on the basic reproduction
+number.” In: Mathematical Epidemiology. Ed. by F. Bauer, P. van den Driessche, and
+J. Wu. Vol. 1945. Lecture Notes in Mathematics, pp. 159–178. isbn: 978-3-540-78910-9.
+doi: 10.1007/978-3-540-78911-6.
+Greenhalgh, D. (1997). “Hopf bifurcation in epidemic models with a latent period and
+nonpermanent immunity.” In: Mathematical and Computer Modelling 25.2, pp. 85–107.
+Hadeler, K. P. and C. Castillo-Chavez (1995). “A Core Group Model for Disease Trans-
+mission.” In: Math.Biosci. 128, pp. 41–55. url: https://www.researchgate.net/
+publication/216242267.
+Hadeler, K. P. and P. van den Driessche (1997). “Backward Bifurcation in epidemic Con-
+trol.” In: Math.Biosci. 146, pp. 15–35.
+Hethcote, H.W. (1974). “Asymptotic behavior and stability in epidemic models.” In: Math-
+ematical Problems in Biology. Victoria Conference 1973. Ed. by P. van den Driessche.
+Vol. 2. Lecture Notes in Biomathematics. Springer Verlag, pp. 83–92. doi: 10.1007/
+978-3-642-45455-4_10.
+Hethcote, H.W. (1976). “Qualitative analysis for communicable disease models.” In: Math.
+Biosci. 28, pp. 335–356.
+Hethcote, H.W. (1989). “Three basic epidemiological models.” In: Applied Mathematical
+Ecology. Ed. by L. Gross, T.G. Hallam, and S.A. Levin. Vol. 18. Biomathematics. Berlin
+Heidelberg New York: Springer Verlag, pp. 119–144.
+Hethcote, H.W. (2000). “The Mathematics of Infectious Diseases.” In: SIAM Rev. 42,
+p. 599.
+Kermack, W. O. and A. G. McKendrick (1927). “Contribution to the mathematical theory
+of epidemics, part I.” In: Proc. Roy. Soc. Lond A 115, pp. 700–721.
+Korobeinikov, A. and G.C. Wake (2002). “Lyapunov Functions and Global Stability for
+SIR, SIRS, and SIS Epidemiological Models.” In: Appl. Math. Lett. 15, pp. 955–960.
+Kribs-Zaleta, C.M. and J.X. Velasco-Hernandez (2000). “A simple vaccination model with
+multiple endemic states.” In: Mathematical Biosciences 164, pp. 183–201. doi: 10.
+1007/s00285-021-01629-8.
+
+REFERENCES
+29
+Li, Jianquan and Zhien Ma (2002). “Qualitative analyses of SIS epidemic model with vac-
+cination and varying total population size.” In: Mathematical and Computer Modelling
+35, pp. 1235–1243.
+Li, Michael Y et al. (1999). “Global dynamics of a SEIR model with varying total popu-
+lation size.” In: Mathematical biosciences 160.2, pp. 191–21325.
+Lu, Guichen and Zhengyi Lu (2018). “Global asymptotic stability for the seirs models
+with varying total population size.” In: Mathematical biosciences 296, pp. 17–25.
+Mena-Lorca, J. and H.W. Hethcote (1992). “Dynamic models of infectious diseases as
+regulators of population sizes.” In: J. Math. Biol 30, pp. 693–716.
+Nadim, S.S. and J. Chattopadhyay (2020). “Occurrence of backward bifurcation and pre-
+diction of disease transmission with imperfect lockdown: A case study on COVID-19.”
+In: Chaos, Solitons and Fractals 140, p. 110163.
+Nill, Florian (Mar. 13, 2022). “Endemic oscillations for SARS-COV-2 Omicron - A SIRS
+model analysis.” In: doi: https://doi.org/10.48550/arXiv.2211.09005. eprint:
+arXiv:2211.09005[q-bio.PE].
+Nill, Florian (n.d.[a]). “Hamiltonian structure in epidemic SIRS models.” paper to be
+written up.
+Nill, Florian (n.d.[b]). “Symmetries and normalization in 3-compartment epidemic models.
+II: Equilibria and stability.” paper to be written up.
+Nill, Florian (n.d.[c]). “Symmetries and normalization in 3-compartment epidemic models.
+III: The dilatation symmetry.” paper to be written up.
+O’Regan, S.M. et al. (2010). “Lyapunov functions for SIR and SIRS epidemic models.”
+In: Applied Mathematics Letters 23, pp. 446–448. doi: 10.1016/j.aml.2009.11.014.
+Razvan, M.R. (2001). “Multiple Equilibria for an SIRS Epidemiological System.” In: arXiv
+preprint arXiv:math/0101051v1. doi: 10.48550/arXiv.math/0101051.
+Sun, Chengjun and Ying-Hen Hsieh (2010). “Global analysis of an SEIR model with vary-
+ing population size and vaccination.” In: Applied Mathematical Modelling 34, pp. 2685–
+2697.
+Yang, Wei, Chengjun Sun, and Julien Arino (2010). “Global analysis for a general epidemi-
+ological model with vaccination and varying population.” In: Journal of Mathematical
+Analysis and Applications 372.1, pp. 208–223.
+

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+arXiv:2301.00180v1  [math.DG]  31 Dec 2022
+A blow-up formula for stationary quaternionic maps ∗†
+Jiayu Li‡
+Chaona Zhu§
+Abstract
+Let (M, Jα, α = 1, 2, 3) and (N, J α, α = 1, 2, 3) be Hyperk¨ahler manifolds. Suppose
+that uk is a sequence of stationary quaternionic maps and converges weakly to u in
+H1,2(M, N), we derive a blow-up formula for limk→∞ d(u∗
+kJ α), for α = 1, 2, 3, in the
+weak sense. As a corollary, we show that the maps constructed by Chen-Li [CL2] and by
+Foscolo [F] can not be tangent maps (c.f [LT], Theorem 3.1) of a stationary quaternionic
+map satisfing d(u∗J α) = 0.
+1
+Introduction and the main result
+A hyperk¨ahler manifold is a Riemannian manifold (M, g) with three parallel com-
+plex structures {J1, J2, J3} compatible with the metric g such that (J1)2 = (J2)2 =
+(J3)2 = J1J2J3 = −id. The simplest hyperk¨ahler manifold is the Euclidean space
+R4m. It is well-known that the only compact hyperk¨ahler manifolds of dimension 4 are
+K3 surfaces and complex tori. Let (M, g, Jα, α = 1, 2, 3) and (N, h, J α, α = 1, 2, 3) be
+hyperk¨ahler manifolds. Let ωα(·, ·) = g(·, Jα·) and Ωα(·, ·) = h(·, J α·), (α = 1, 2, 3)
+be the K¨ahler forms on M and N respectively. A smooth map u : M → N is called a
+quaternionic map (triholomorphic map) if
+AαβJ β ◦ du ◦ Jα = du
+(1)
+where Aαβ denote the entries of a matrix A in SO(3). For simplicity, we choose
+Aαβ = δαβ.
+The quaternionic maps (triholomorphic maps) between Hyperk¨ahler manifolds
+has been studied by many aothors (cf. [BT], [Ch], [CL1, [CL2], [FKS], [W]). Quater-
+nionic maps automatically minimize the energy functional in their homotopy classes
+(cf. [Ch], [CL1] and [FKS]) and hence they are harmonic. It can be verified that
+holomorphic and anti-holomorphic maps with respect to some complex structures on
+M and N are quaternionic maps. However, Chen-Li constructed quaternionic maps
+which are not holomorphic with respect to any complex structures on M and N (cf.
+[CL1]).
+∗This work is supported by NSF grant 11721101.
+†MSC (2000): 53C26, 53C43, 58E12, 58E20. Keywords: Stationary harmonic maps, quaternionic maps,
+blow-up formula.
+‡[email protected]
+§[email protected]
+1
+
+Definition 1.1 A map u from M to N is called a stationary quaternionic map if it
+is a stationary harmonic map and it is a quaternionic map outside its singular set.
+It is clear that (c.f. [BT]), if u satisfies (1) almost everywhere, and
+d(u∗J α) = 0,
+for α = 1, 2, 3,
+(2)
+then u is a stationary quaternionic map.
+Chen-Li ([CL2]) proved that, if there is a harmonic sphere φ : S2 → N which
+satisfies
+dφ JS2 = −
+3
+�
+k=1
+akJ k dφ,
+(3)
+where ⃗a = (a1, a2, a3) : S2 → S2, and
+�
+S2 xi|∇φ|2dσ = 0, i = 1, 2, 3, (x1, x2, x3) ∈ S2,
+(4)
+then
+u(x, x4) = φ( x
+|x|) for any x ∈ R3\{0}
+is a stationary quaternionic map with the x4-axis as its singular set.
+Chen-Li ([CL2]) showed that there does exist a complete noncompact hyperk¨ahler
+manifold, into which there is a harmonic S2 which satisfies (3) and (4). Recently,
+Foscolo [F] showed that there exists a compact K3 surface with the above property.
+However, the map u constructed by Chen-Li or by Foscolo does not satisfy (2). Now
+the question is whether the maps constructed by Chen-Li or by Foscolo could be a
+tangent map of a stationary quaternionic map with identity (2), if not the singular
+set of a stationary quaternionic map with identity (2) might be of codimensional 4
+(Remark 1.2 in [BT]).
+Suppose that uk is a sequence of stationary quaternionic maps with bounded
+energies E(uk) ≤ Λ. The blow-up set of uk can be defined as
+Σ = ∩r>0{x ∈ M| lim inf
+k→∞ r2−m
+�
+Br(x)
+| ▽ uk|2dy ≥ ǫ0}.
+We can always assume that uk ⇀ u weakly in W 1,2(M, N) and that
+| ▽ uk|2dx ⇀ | ▽ u|2dx + ν
+in the sense of measure as k → ∞. Here ν is a nonnegative Radon measure on M
+with support in Σ. It is known that Σ is a Hm−2-rectifiable set, and we may write
+ν = θ(x)Hm−2⌊Σ. It is clear that strongly convergence in H1,2(M, N) preserves the
+identity (2). In this paper we mainly prove the following blow-up formula for weakly
+convergence sequence of stationary quaternionic maps.
+2
+
+Theorem 1.2 Let uk be a sequence of stationary quaternionic map with E(uk) ≤ Λ.
+Assume that uk → u weakly in H1(M, N). Then there exist
+(a1, a2, a3) ∈ R3 with
+�3
+α=1(aα)2 = 1 such that, for any smooth (m − 3)-form η with compact support in
+M,
+lim
+k→∞
+3
+�
+α=1
+aα
+�
+M
+dη ∧ u∗
+kJ α =
+3
+�
+α=1
+aα
+�
+M
+dη ∧ u∗J α +
+�
+Σ
+θdη|Σ
+(5)
+and for any (b1, b2, b3) ⊥ (a1, a2, a3), there holds
+lim
+k→∞
+3
+�
+α=1
+bα
+�
+M
+dη ∧ u∗
+kJ α =
+3
+�
+α=1
+bα
+�
+M
+dη ∧ u∗J α.
+As a corollary of the theorem, the maps constructed by Chen-Li [CL2] and by Fos-
+colo [F] can not be tangent maps (c.f [LT], Theorem 3.1) of a stationary quaternionic
+map satisfing d(u∗J α) = 0.
+2
+The proof of the blow-up formula
+If u is a strong limit of a sequence of stationary quaternionic maps in H1,2(M, N),
+then it’s easy to see that u satisfies (2). If u is just a weak limit, i.e. there exists a
+sequence of stationary quaternionic maps uk satisfying uk → u weakly in H1,2(M, N)
+and |∇uk|2dV → |∇u|2dV +θHm−2|Σ in the sense of measure, we prove in this section
+a formula for the blow-up set θHm−2|Σ and the limiting map u.
+Without loss of generality, we may assume that m = 4. Because Σ is a Hm−2-
+rectifiable set, so we may assume that Σ = ∪∞
+i=0Σi, Σi ∩Σi′ = φ if i ̸= i′, Hm−2(Σ0) =
+0, Σi ⊂ Ni and Ni (i = 1, 2, · · ·) is an (m − 2)-dimensional embedded C1 submanifold
+of M. It is important that (see p. 61 in [Si]) TxΣ = TxNi for Hm−2-a.e. x ∈ Σi.
+It is known that ν = θ(x)Hm−2⌊Σ, where θ(x) is upper semi-continuous with
+ǫ0 ≤ θ(x) ≤ C(l1) for Hm−2-a.e. x ∈ Σ, C(l1) is a positive constant depending only
+on M and l1 (cf. [Lin], Lemma 1.6). Since Hm−2(Σ) < +∞, for any 1. > 0, there
+exist Σ1. ⊂ Σ and i0 such that Hm−2(Σ1. ) < 1., Σc
+1. = Σ\Σ1. = ∪i0
+i=1Σ1.
+i where Σ1.
+i ⊂ Σi
+(i = 1, · · ·, i0) is a bounded closed set. We choose a covering {Brn|n = 1, 2, · · ·} of
+Σ1. such that �
+n rm−2
+n
+< C1.. Here and in the sequel, C always denotes a uniform
+constant depending only on M and N.
+Suppose that (x1, ..., x4) is a local normal coordinate system in Bǫ(Σδ
+i), and that
+(x3, x4) is the corresponding coordinate system in Σi, and the matrix expressions of
+the complex structures are given by (6), (7) and (8).
+J1 =
+
+
+
+
+0
+0
+0
+−1
+0
+0
+1
+0
+0
+−1
+0
+0
+1
+0
+0
+0
+
+
+
+ ,
+A1βJ β =
+
+
+
+
+J1
+·
+·
+J1
+
+
+
+
+(6)
+3
+
+J2 =
+
+
+
+
+0
+−1
+0
+0
+1
+0
+0
+0
+0
+0
+0
+1
+0
+0
+−1
+0
+
+
+
+ ,
+A2βJ β =
+
+
+
+
+J2
+·
+·
+J2
+
+
+
+
+(7)
+J3 =
+
+
+
+
+0
+0
+1
+0
+0
+0
+0
+1
+−1
+0
+0
+0
+0
+−1
+0
+0
+
+
+
+ ,
+A3βJ β =
+
+
+
+
+J3
+·
+·
+J3
+
+
+
+
+(8)
+where AαβJ β are 4n×4n-matrices, Aαβ are the entries of a matrix A in SO(3). Then
+the quaternionic equation is
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+u1
+1 + u2
+2 + u3
+3 + u4
+4
+=
+0
+u2
+1 − u1
+2 + u4
+3 − u3
+4
+=
+0
+u3
+1 − u1
+3 − u4
+2 + u2
+4
+=
+0
+u4
+1 − u1
+4 − u2
+3 + u3
+2
+=
+0
+u5
+1 + u6
+2 + u7
+3 + u8
+4
+=
+0
+u6
+1 − u5
+2 + u8
+3 − u7
+4
+=
+0
+u7
+1 − u5
+3 − u8
+2 + u6
+4
+=
+0
+u8
+1 − u5
+4 − u6
+3 + u7
+2
+=
+0
+· · ·
+u4n−3
+1
++ u4n−2
+2
++ u4n−1
+3
++ u4n
+4
+=
+0
+u4n−2
+1
+− u4n−3
+2
++ u4n
+3 − u4n−1
+4
+=
+0
+u4n−1
+1
+− u4n−3
+3
+− u4n
+2 + u4n−2
+4
+=
+0
+u4n
+1 − u4n−3
+4
+− u4n−2
+3
++ u4n−1
+2
+=
+0.
+(9)
+Theorem 2.1 For any smooth (m − 3)-form η with compact support in M, we have
+lim
+k→∞
+3
+�
+α=1
+Aαβ
+�
+M
+dη ∧ u∗
+kJ β =
+3
+�
+α=1
+Aαβ
+�
+M
+dη ∧ u∗J β +
+�
+Σ
+θdη|Σ
+and
+lim
+k→∞ A1β
+�
+M
+dη ∧ u∗
+kJ β = A1β
+�
+M
+dη ∧ u∗J β,
+lim
+k→∞ A3β
+�
+M
+dη ∧ u∗
+kJ β = A3β
+�
+M
+dη ∧ u∗J β,
+Proof. Assume that η = �
+I ηIdxI. We have
+lim
+k→∞
+�
+M
+dη ∧ u∗
+k(AαβJ β) =
+�
+M
+dη ∧ u∗(AαβJ β)
++
+lim
+δ→0 lim
+ǫ→0 lim
+k→∞
+�
+Bǫ(∪i0
+i=1Σδ
+i )
+dη ∧ u∗
+k(AαβJ β)
++
+lim
+δ→0 lim
+ǫ→0 lim
+k→∞
+�
+∪nBrn\Bǫ(∪i0
+i=1Σδ
+i )
+dη ∧ u∗
+k(AαβJ β).
+(10)
+4
+
+It’s easy to see that
+lim
+δ→0 lim
+ǫ→0 lim
+k→∞
+�
+∪nBrn
+dη ∧ u∗
+k(J β) = 0
+(11)
+By Lemma 2.2 in [LT], we get
+lim
+δ→0 lim
+ǫ→0 lim
+k→∞
+�
+Bǫ(Σδ
+i )
+dη ∧ u∗
+k(AαβJ β)
+=
+lim
+δ→0 lim
+ǫ→0 lim
+k→∞
+�
+Bǫ(Σδ
+i )
+2∂ηI
+∂xl
+∂uσ
+k
+∂x1 (AαβJ β)σγ
+∂uγ
+k
+∂x2 dxl ∧ dxI ∧ dx1 ∧ dx2
+(12)
+Substituting (9) to (12) and applying Lemma 2.2 in [LT], we have
+lim
+δ→0 lim
+ǫ→0 lim
+k→∞
+�
+Bǫ(Σδ
+i )
+dη ∧ u∗
+k(A1βJ β) = lim
+δ→0 lim
+ǫ→0 lim
+k→∞
+�
+Bǫ(Σδ
+i )
+dη ∧ u∗
+k(A3βJ β) = 0
+and
+lim
+δ→0 lim
+ǫ→0 lim
+k→∞
+�
+Bǫ(Σδ
+i )
+dη ∧ u∗
+k(A2βJ β) = lim
+δ→0 lim
+ǫ→0 lim
+k→∞
+�
+Bǫ(Σδ
+i )
+|∇uk|2dη ∧ dx1 ∧ dx2
+=
+lim
+δ→0 lim
+ǫ→0(
+�
+Bǫ(Σδ
+i )
+|∇u|2dη ∧ dx1 ∧ dx2 +
+�
+Bǫ(Σδ
+i )∩Σ
+θdη|Σ) =
+�
+Σi
+θdη|Σ
+(13)
+Then the proof of the theorem is completed.
+Q.E.D.
+Remark 2.2 From this theorem, we see that if uk satisfies (2), the weak limit u still
+satisfies (2) if and only if θ = constant.
+As a corollary, we can derive that θ(x) is locally constant. Precisely,
+Corollary 2.3 Under the assumption of Theorem 1.2, and assume that there is an
+open ball Bm ⊂ M \ Singu with Hm−2(Σ ∩ Bm) > 0. We have θ(x) is constant on
+Σ ∩ Bm.
+Proof.
+In (5), we choose cutoff function η such that suppη ⊂ Bm.
+Since Bm ⊂
+M \ Singu, we have u is smooth on Bm. Then du∗J β = 0 on Bm for β = 1, 2, 3. In
+view of (5), we conclude that θ is constant on Σ ∩ Bm.
+Q.E.D.
+Let φ : S2 → N be a nonconstant smooth map satisfying (3) and (4). Set
+u(x, x4) = φ( x
+|x|) for any x ∈ R3\{0} x4 ∈ Rm−3
+(14)
+as Chen-Li ([CL2]) did. Then we have
+5
+
+Proposition 2.4 For any smooth (m − 3)-form η with compact support in Rm, we
+have
+�
+Rm dη ∧ u∗J α = −Eα
+T (φ)
+�
+Rm−3 η(0, x4),
+(15)
+where
+ET(φ) =
+�
+S2⟨Jα
+S2, u∗J α⟩dσ.
+Proof. We choose a spherical coordinate system (r, ϕ, θ) in R3, because u is smooth
+for any r > 0, we have
+�
+Rm dη ∧ u∗J α
+=
+�
+Rm−3
+� ∞
+0
+∂ηI
+∂r dr ∧ dxI
+�
+S2 φ∗J α
+=
+−
+�
+Rm−3 η(0, x4)
+�
+S2 φ∗J α
+=
+−Eα
+T (φ)
+�
+Rm−3 η(0, x4)
+Q.E.D.
+By Theorem 2.1 and Proposition 2.4, we have the following corollary.
+Corollary 2.5 The map u defined in (14) can not be a tangent map (c.f [LT], The-
+orem 3.1) of a stationary quaternionic map with the property (2) at a singular point.
+Proof. Suppose that u is defined as in (14). If it is a tangent map, then we have
+by Theorem 2.1,
+3
+�
+α=1
+Aαβ
+�
+M
+dη ∧ u∗J β +
+�
+Σ
+θdη|Σ = 0.
+By Proposition 2.4, we obtain
+3
+�
+α=1
+AαβEβ
+T(φ)
+�
+Rm−3 η(0, x4) =
+�
+Σ
+θdη|Σ.
+Since u is stationary, by the blow-up formula of Li-Tian [LT], we have Σ is station-
+ary. Using the constancy theorem (Theorem 41.1 in [Si]), it follows that the density
+function θ is constant in every connected component of Σ, which implies that φ is
+homotopy to a constant map. We therefore get a contradiction.
+Q.E.D.
+REFERENCES
+[BT] C. Bellettini and G. Tian, Compactness results for triholomorphic maps, J. Eur. Math. Soc., 2(2019),
+1271-1317.
+6
+
+[Ch] J. Chen, Complex anti-self-dual connections on product of Calabi-Yau surfaces and triholomorphic
+curves, Commun. Math. Phys. 201(1999), 201-247.
+[CL1] J. Chen and J. Li, Quaternionic maps between Hyperk¨ahler manifolds, J. Diff. Geom. 55(2000), no.
+2, 355-384.
+[CL2] J. Chen and J. Li, Quarternionic maps and minimal surfaces, Ann. Sc. Norm. Super. Pisa Cl. Sci.
+4 (2005), no. 3, 375-388.
+[FKS] J.M. Figuroa-O’Farrill, C. K¨ohl and B. Spence, Supersymmetric Yang-Mills, octonionic instantons
+and triholomorphic curves, Nucl. Phys. B 521 (1998) no. 3, 419-443.
+[F] L. Foscolo, ALF gravitational instantons and collapsing Ricci-flat metrics on the K3 surface, J. Diff.
+Geom., 112(2019), 79-120.
+[LT] J. Li, and G. Tian, A blow-up formula for stationary harmonic maps, IMRN, 14(1998), 735-755.
+[Lin] F.-H. Lin, Gradient estimates and blow-up analysis for stationary harmonic maps I, Ann. of Math.
+149(1999), 785-829.
+[Si] L. Simon, Lectures on Geometric Measure Theory, Proc. Center Math. Anal. 3(1983), Australian
+National Univ. Press.
+[W] C. Wang, Energy quantization for triholomorphic maps, Calc. Var. PDE 18(2003), 145-158.
+7
+

b9AyT4oBgHgl3EQfXPer/content/tmp_files/load_file.txt

@@ -0,0 +1,337 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf,len=336
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='00180v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='DG]  31 Dec 2022  A blow-up formula for stationary quaternionic maps ∗†  Jiayu Li‡  Chaona Zhu§  Abstract  Let (M, Jα, α = 1, 2, 3) and (N, J α, α = 1, 2, 3) be Hyperk¨ahler manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' Suppose  that uk is a sequence of stationary quaternionic maps and converges weakly to u in  H1,2(M, N), we derive a blow-up formula for limk→∞ d(u∗  kJ α), for α = 1, 2, 3, in the  weak sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' As a corollary, we show that the maps constructed by Chen-Li [CL2] and by  Foscolo [F] can not be tangent maps (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='f [LT], Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='1) of a stationary quaternionic  map satisfing d(u∗J α) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='  1  Introduction and the main result  A hyperk¨ahler manifold is a Riemannian manifold (M, g) with three parallel com-  plex structures {J1, J2, J3} compatible with the metric g such that (J1)2 = (J2)2 =  (J3)2 = J1J2J3 = −id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' The simplest hyperk¨ahler manifold is the Euclidean space  R4m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' It is well-known that the only compact hyperk¨ahler manifolds of dimension 4 are  K3 surfaces and complex tori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' Let (M, g, Jα, α = 1, 2, 3) and (N, h, J α, α = 1, 2, 3) be  hyperk¨ahler manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' Let ωα(·, ·) = g(·, Jα·) and Ωα(·, ·) = h(·, J α·), (α = 1, 2, 3)  be the K¨ahler forms on M and N respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' A smooth map u : M → N is called a  quaternionic map (triholomorphic map) if  AαβJ β ◦ du ◦ Jα = du  (1)  where Aαβ denote the entries of a matrix A in SO(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' For simplicity, we choose  Aαβ = δαβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='  The quaternionic maps (triholomorphic maps) between Hyperk¨ahler manifolds  has been studied by many aothors (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' [BT], [Ch], [CL1, [CL2], [FKS], [W]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' Quater-  nionic maps automatically minimize the energy functional in their homotopy classes  (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' [Ch], [CL1] and [FKS]) and hence they are harmonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' It can be verified that  holomorphic and anti-holomorphic maps with respect to some complex structures on  M and N are quaternionic maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' However, Chen-Li constructed quaternionic maps  which are not holomorphic with respect to any complex structures on M and N (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='  [CL1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='  ∗This work is supported by NSF grant 11721101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='  †MSC (2000): 53C26, 53C43, 58E12, 58E20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' Keywords: Stationary harmonic maps, quaternionic maps,  blow-up formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='  ‡jiayuli@ustc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='cn  §zcn1991@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='ustc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='cn  1  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='1 A map u from M to N is called a stationary quaternionic map if it  is a stationary harmonic map and it is a quaternionic map outside its singular set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='  It is clear that (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' [BT]), if u satisfies (1) almost everywhere, and  d(u∗J α) = 0,  for α = 1, 2, 3,  (2)  then u is a stationary quaternionic map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='  Chen-Li ([CL2]) proved that, if there is a harmonic sphere φ : S2 → N which  satisfies  dφ JS2 = −  3  �  k=1  akJ k dφ,  (3)  where ⃗a = (a1, a2, a3) : S2 → S2, and  �  S2 xi|∇φ|2dσ = 0, i = 1, 2, 3, (x1, x2, x3) ∈ S2,  (4)  then  u(x, x4) = φ( x  |x|) for any x ∈ R3\\{0}  is a stationary quaternionic map with the x4-axis as its singular set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='  Chen-Li ([CL2]) showed that there does exist a complete noncompact hyperk¨ahler  manifold, into which there is a harmonic S2 which satisfies (3) and (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' Recently,  Foscolo [F] showed that there exists a compact K3 surface with the above property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='  However, the map u constructed by Chen-Li or by Foscolo does not satisfy (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' Now  the question is whether the maps constructed by Chen-Li or by Foscolo could be a  tangent map of a stationary quaternionic map with identity (2), if not the singular  set of a stationary quaternionic map with identity (2) might be of codimensional 4  (Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='2 in [BT]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='  Suppose that uk is a sequence of stationary quaternionic maps with bounded  energies E(uk) ≤ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' The blow-up set of uk can be defined as  Σ = ∩r>0{x ∈ M| lim inf  k→∞ r2−m  �  Br(x)  | ▽ uk|2dy ≥ ǫ0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='  We can always assume that uk ⇀ u weakly in W 1,2(M, N) and that  | ▽ uk|2dx ⇀ | ▽ u|2dx + ν  in the sense of measure as k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' Here ν is a nonnegative Radon measure on M  with support in Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' It is known that Σ is a Hm−2-rectifiable set, and we may write  ν = θ(x)Hm−2⌊Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' It is clear that strongly convergence in H1,2(M, N) preserves the  identity (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' In this paper we mainly prove the following blow-up formula for weakly  convergence sequence of stationary quaternionic maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='  2  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='2 Let uk be a sequence of stationary quaternionic map with E(uk) ≤ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='  Assume that uk → u weakly in H1(M, N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' Then there exist  (a1, a2, a3) ∈ R3 with  �3  α=1(aα)2 = 1 such that, for any smooth (m − 3)-form η with compact support in  M,  lim  k→∞  3  �  α=1  aα  �  M  dη ∧ u∗  kJ α =  3  �  α=1  aα  �  M  dη ∧ u∗J α +  �  Σ  θdη|Σ  (5)  and for any (b1, b2, b3) ⊥ (a1, a2, a3), there holds  lim  k→∞  3  �  α=1  bα  �  M  dη ∧ u∗  kJ α =  3  �  α=1  bα  �  M  dη ∧ u∗J α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='  As a corollary of the theorem, the maps constructed by Chen-Li [CL2] and by Fos-  colo [F] can not be tangent maps (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='f [LT], Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='1) of a stationary quaternionic  map satisfing d(u∗J α) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='  2  The proof of the blow-up formula  If u is a strong limit of a sequence of stationary quaternionic maps in H1,2(M, N),  then it’s easy to see that u satisfies (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' If u is just a weak limit, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' there exists a  sequence of stationary quaternionic maps uk satisfying uk → u weakly in H1,2(M, N)  and |∇uk|2dV → |∇u|2dV +θHm−2|Σ in the sense of measure, we prove in this section  a formula for the blow-up set θHm−2|Σ and the limiting map u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='  Without loss of generality, we may assume that m = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' Because Σ is a Hm−2-  rectifiable set, so we may assume that Σ = ∪∞  i=0Σi, Σi ∩Σi′ = φ if i ̸= i′, Hm−2(Σ0) =  0, Σi ⊂ Ni and Ni (i = 1, 2, · · ·) is an (m − 2)-dimensional embedded C1 submanifold  of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' It is important that (see p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' 61 in [Si]) TxΣ = TxNi for Hm−2-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' x ∈ Σi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='  It is known that ν = θ(x)Hm−2⌊Σ, where θ(x) is upper semi-continuous with  ǫ0 ≤ θ(x) ≤ C(l1) for Hm−2-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' x ∈ Σ, C(l1) is a positive constant depending only  on M and l1 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' [Lin], Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' Since Hm−2(Σ) < +∞, for any 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' > 0, there  exist Σ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' ⊂ Σ and i0 such that Hm−2(Σ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' ) < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=', Σc  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' = Σ\\Σ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' = ∪i0  i=1Σ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='  i where Σ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='  i ⊂ Σi  (i = 1, · · ·, i0) is a bounded closed set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' We choose a covering {Brn|n = 1, 2, · · ·} of  Σ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' such that �  n rm−2  n  < C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='. Here and in the sequel, C always denotes a uniform  constant depending only on M and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='  Suppose that (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=', x4) is a local normal coordinate system in Bǫ(Σδ  i), and that  (x3, x4) is the corresponding coordinate system in Σi, and the matrix expressions of  the complex structures are given by (6), (7) and (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='  J1 =  \uf8eb  \uf8ec  \uf8ec  \uf8ed  0  0  0  −1  0  0  1  0  0  −1  0  0  1  0  0  0  \uf8f6  \uf8f7  \uf8f7  \uf8f8 ,  A1βJ β =  \uf8eb  \uf8ec  \uf8ec  \uf8ed  J1 J1  \uf8f6  \uf8f7  \uf8f7  \uf8f8  (6)  3  J2 =  \uf8eb  \uf8ec  \uf8ec  \uf8ed  0  −1  0  0  1  0  0  0  0  0  0  1  0  0  −1  0  \uf8f6  \uf8f7  \uf8f7  \uf8f8 ,  A2βJ β =  \uf8eb  \uf8ec  \uf8ec  \uf8ed  J2 J2  \uf8f6  \uf8f7  \uf8f7  \uf8f8  (7)  J3 =  \uf8eb  \uf8ec  \uf8ec  \uf8ed  0  0  1  0  0  0  0  1  −1  0  0  0  0  −1  0  0  \uf8f6  \uf8f7  \uf8f7  \uf8f8 ,  A3βJ β =  \uf8eb  \uf8ec  \uf8ec  \uf8ed  J3 J3  \uf8f6  \uf8f7  \uf8f7  \uf8f8  (8)  where AαβJ β are 4n×4n-matrices, Aαβ are the entries of a matrix A in SO(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' Then  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='the quaternionic equation is  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='\uf8f1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
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+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
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+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='+ u4n−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='  (9)  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='1 For any smooth (m − 3)-form η with compact support in M, we have  lim  k→∞  3  �  α=1  Aαβ  �  M  dη ∧ u∗  kJ β =  3  �  α=1  Aαβ  �  M  dη ∧ u∗J β +  �  Σ  θdη|Σ  and  lim  k→∞ A1β  �  M  dη ∧ u∗  kJ β = A1β  �  M  dη ∧ u∗J β,  lim  k→∞ A3β  �  M  dη ∧ u∗  kJ β = A3β  �  M  dη ∧ u∗J β,  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' Assume that η = �  I ηIdxI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' We have  lim  k→∞  �  M  dη ∧ u∗  k(AαβJ β) =  �  M  dη ∧ u∗(AαβJ β)  +  lim  δ→0 lim  ǫ→0 lim  k→∞  �  Bǫ(∪i0  i=1Σδ  i )  dη ∧ u∗  k(AαβJ β)  +  lim  δ→0 lim  ǫ→0 lim  k→∞  �  ∪nBrn\\Bǫ(∪i0  i=1Σδ  i )  dη ∧ u∗  k(AαβJ β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='  (10)  4  It’s easy to see that  lim  δ→0 lim  ǫ→0 lim  k→∞  �  ∪nBrn  dη ∧ u∗  k(J β) = 0  (11)  By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='2 in [LT], we get  lim  δ→0 lim  ǫ→0 lim  k→∞  �  Bǫ(Σδ  i )  dη ∧ u∗  k(AαβJ β)  =  lim  δ→0 lim  ǫ→0 lim  k→∞  �  Bǫ(Σδ  i )  2∂ηI  ∂xl  ∂uσ  k  ∂x1 (AαβJ β)σγ  ∂uγ  k  ∂x2 dxl ∧ dxI ∧ dx1 ∧ dx2  (12)  Substituting (9) to (12) and applying Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='2 in [LT], we have  lim  δ→0 lim  ǫ→0 lim  k→∞  �  Bǫ(Σδ  i )  dη ∧ u∗  k(A1βJ β) = lim  δ→0 lim  ǫ→0 lim  k→∞  �  Bǫ(Σδ  i )  dη ∧ u∗  k(A3βJ β) = 0  and  lim  δ→0 lim  ǫ→0 lim  k→∞  �  Bǫ(Σδ  i )  dη ∧ u∗  k(A2βJ β) = lim  δ→0 lim  ǫ→0 lim  k→∞  �  Bǫ(Σδ  i )  |∇uk|2dη ∧ dx1 ∧ dx2  =  lim  δ→0 lim  ǫ→0(  �  Bǫ(Σδ  i )  |∇u|2dη ∧ dx1 ∧ dx2 +  �  Bǫ(Σδ  i )∩Σ  θdη|Σ) =  �  Σi  θdη|Σ  (13)  Then the proof of the theorem is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='  Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='2 From this theorem, we see that if uk satisfies (2), the weak limit u still  satisfies (2) if and only if θ = constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='  As a corollary, we can derive that θ(x) is locally constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' Precisely,  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='3 Under the assumption of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='2, and assume that there is an  open ball Bm ⊂ M \\ Singu with Hm−2(Σ ∩ Bm) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' We have θ(x) is constant on  Σ ∩ Bm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='  In (5), we choose cutoff function η such that suppη ⊂ Bm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='  Since Bm ⊂  M \\ Singu, we have u is smooth on Bm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' Then du∗J β = 0 on Bm for β = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' In  view of (5), we conclude that θ is constant on Σ ∩ Bm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='  Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='  Let φ : S2 → N be a nonconstant smooth map satisfying (3) and (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' Set  u(x, x4) = φ( x  |x|) for any x ∈ R3\\{0} x4 ∈ Rm−3  (14)  as Chen-Li ([CL2]) did.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' Then we have  5  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='4 For any smooth (m − 3)-form η with compact support in Rm, we  have  �  Rm dη ∧ u∗J α = −Eα  T (φ)  �  Rm−3 η(0, x4),  (15)  where  ET(φ) =  �  S2⟨Jα  S2, u∗J α⟩dσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' We choose a spherical coordinate system (r, ϕ, θ) in R3, because u is smooth  for any r > 0, we have  �  Rm dη ∧ u∗J α  =  �  Rm−3  � ∞  0  ∂ηI  ∂r dr ∧ dxI  �  S2 φ∗J α  =  −  �  Rm−3 η(0, x4)  �  S2 φ∗J α  =  −Eα  T (φ)  �  Rm−3 η(0, x4)  Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='  By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='1 and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='4, we have the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='5 The map u defined in (14) can not be a tangent map (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='f [LT], The-  orem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='1) of a stationary quaternionic map with the property (2) at a singular point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' Suppose that u is defined as in (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' If it is a tangent map, then we have  by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='1,  3  �  α=1  Aαβ  �  M  dη ∧ u∗J β +  �  Σ  θdη|Σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='  By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='4, we obtain  3  �  α=1  AαβEβ  T(φ)  �  Rm−3 η(0, x4) =  �  Σ  θdη|Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='  Since u is stationary, by the blow-up formula of Li-Tian [LT], we have Σ is station-  ary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' Using the constancy theorem (Theorem 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='1 in [Si]), it follows that the density  function θ is constant in every connected component of Σ, which implies that φ is  homotopy to a constant map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' We therefore get a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='  Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='  REFERENCES  [BT] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' Bellettini and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' Tian, Compactness results for triholomorphic maps, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=', 2(2019),  1271-1317.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='  6  [Ch] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' Chen, Complex anti-self-dual connections on product of Calabi-Yau surfaces and triholomorphic  curves, Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' 201(1999), 201-247.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='  [CL1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' Chen and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' Li, Quaternionic maps between Hyperk¨ahler manifolds, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' Diff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' 55(2000), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='  2, 355-384.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='  [CL2] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' Chen and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' Li, Quarternionic maps and minimal surfaces, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' Super.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' Pisa Cl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='  4 (2005), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' 3, 375-388.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='  [FKS] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' Figuroa-O’Farrill, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' K¨ohl and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' Spence, Supersymmetric Yang-Mills, octonionic instantons  and triholomorphic curves, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' B 521 (1998) no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' 3, 419-443.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='  [F] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' Foscolo, ALF gravitational instantons and collapsing Ricci-flat metrics on the K3 surface, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' Diff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='  Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=', 112(2019), 79-120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='  [LT] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' Li, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' Tian, A blow-up formula for stationary harmonic maps, IMRN, 14(1998), 735-755.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='  [Lin] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' Lin, Gradient estimates and blow-up analysis for stationary harmonic maps I, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' of Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='  149(1999), 785-829.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='  [Si] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' Simon, Lectures on Geometric Measure Theory, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' Center Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' 3(1983), Australian  National Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='  [W] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' Wang, Energy quantization for triholomorphic maps, Calc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' Var.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content=' PDE 18(2003), 145-158.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}
+page_content='  7' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfXPer/content/2301.00180v1.pdf'}

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+1 
+ 
+Many-body Hybrid Excitons with strong molecular orientation 
+dependence 
+in 
+Organic-Inorganic 
+van 
+der 
+Waals 
+Heterostructures  
+Shaohua Fu,1,2,4 # Jianwei Ding3 #, Haifeng Lv,5 Shuangyan Liu,1 Kun Zhao1, Zhiying 
+Bai1, Dawei He,1 Rui Wang,6 Jimin Zhao,4 Xiaojun Wu,5 Dongsheng Tang,2 * Xiaohui 
+Qiu,3 * Yongsheng Wang1, Xiaoxian Zhang,1 * 
+1Key Laboratory of Luminescence and Optical Information, Ministry of Education, 
+Institute of Optoelectronic Technology, Beijing Jiaotong University, Beijing 100044, 
+China  
+2Synergetic Innovation Center for Quantum Effects an Application, Key Laboratory of 
+Low-dimensional Quantum Structures and Quantum Control of Ministry of Education, 
+School of Physics and Electronics, Hunan Normal University, Changsha 410081, China 
+3CAS Key Laboratory of Standardization and Measurement for Nanotechnology, CAS 
+Center for Excellence in Nanoscience, National Center for Nanoscience and 
+Technology, Beijing 100190, P. R. China. 
+4Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, 
+Chinese Academy of Sciences, Beijing 100190, China 
+5Hefei National Laboratory for Physical Sciences at the Microscale, CAS Key 
+Laboratory of Materials for Energy Conversion, Synergetic Innovation of Quantum 
+Information & Quantum Technology, School of Chemistry and Materials Sciences, and 
+CAS Center for Excellence in Nanoscience, University of Science and Technology of 
+China, Hefei, Anhui 230026, P.R. China 
+
+2 
+ 
+6Beijing Information technology college, Beijing 100015, P. R. China 
+#These authors contributed equally 
+∗e-mail: [email protected]; [email protected]; 
+[email protected]  
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+3 
+ 
+Abstract 
+The coherent many-body interaction at the organic-inorganic interface can give rise to 
+intriguing hybrid excitons that combine the advantages of the Wannier-Mott and 
+Frenkel excitons simultaneously. Unlike the 2D inorganic heterostructures that suffer 
+from moment mismatch, the hybrid excitons formed at the organic-inorganic interface 
+have a momentum-direct nature, which have yet to be explored. Here, we report hybrid 
+excitons at the copper phthalocyanine/molybdenum diselenide (CuPc/MoSe2) interface 
+with 
+strong 
+molecular 
+orientation 
+dependence 
+using 
+low-temperature 
+photoluminescence spectroscopy. The new emission peaks observed in the 
+CuPc/MoSe2 heterostructure indicate the formation of interfacial hybrid excitons. The 
+density functional theory (DFT) calculation confirms the strong hybridization between 
+the lowest unoccupied molecular orbital (LUMO) of CuPc and the conduction band 
+minimum (CBM) of MoSe2, suggesting that the hybrid excitons consist of electrons 
+extended in both layers and holes confined in individual layers. The temperature- 
+dependent measurements show that the hybrid excitons can gain the signatures of the 
+Frenkel excitons of CuPc and the Wannier-Mott excitons of MoSe2 simultaneously. The 
+out-of-plane molecular orientation is used to tailor the interfacial hybrid exciton states. 
+Our results reveal the hybrid excitons at the CuPc/MoSe2 interface with tunability by 
+molecular orientation, which suggests that the emerging organic-inorganic 
+heterostructure can be a promising platform for many-body exciton physics. 
+ 
+ 
+
+4 
+ 
+Introduction 
+Hybrid excitons are many-body exciton states that originate from the hybridization 
+of electronic states at interfaces1,2, which have been realized in distinct systems, 
+including quantum dots coupled to a Fermi sea1,3, coupled quantum-dot molecules4,5, 
+and emerging van der Waals heterostructures2,6-8, displaying great potential in kondo 
+physics1,8, quantum optics4,7, and strongly correlated electronic physics2. Transition 
+metal dichalcogenides (TMDs) have become a promising building block for hybrid 
+excitons due to their strong light-matter interaction9,10 and rich exciton physics11-14, 
+such as valley polarized excitons15-17, long-lived interlayer excitons18 and moiré 
+excitons19-21. However, in TMD heterostructures, the momentum-mismatch problem 
+severely restricts the formation of prominent hybrid excitons, which is moment-direct 
+only at a small twist angle, requiring a precise control of the interlayer angle alignment 
+during fabrication2,6,7.  
+Unlike the momentum-mismatch issue encountered in inorganic heterostructures, 
+the hybrid excitons formed at the organic-inorganic heterostructures have a momentum-
+direct nature22, which could simplify the fabrication process and maintain the novel 
+exciton physics at the same time. In addition, theoretical calculations have predicted 
+that the hybrid excitons at the organic-inorganic interfaces can gain the signature of the 
+Wannier-Mott excitons of inorganics and the Frenkel excitons of organics 
+simultaneously23,24. Nevertheless, the coupling at the organic-inorganic interfaces is 
+generally weak25, and an ordered structure of the organic materials is required to 
+achieve strong electronic coupling24. TMDs can be an ideal building brick for realizing 
+
+5 
+ 
+hybrid excitons at the organic-inorganic interfaces, because they not only contain rich 
+exciton physics but can also serve as a suitable template for the growth of well-ordered 
+organic films through the van der Waals epitaxial method26-31. Moreover, the short-
+range interactions, such as ultrafast charge transfer31-34 and interfacial spin orbital 
+coupling35, have been observed at organic-TMD interfaces, which suggest that the 
+coherent superposition of the electron wavefunctions and the formation of hybrid 
+excitons are possible at such interfaces. However, experimental evidence for hybrid 
+Wannier-Mott-Frenkel excitons at organic-TMD interfaces remains to be explored.  
+   It is well-known that the interfacial hybridization strength depends sensitively on 
+the interlayer twist angle and stacking configuration in inorganic heterostructures6,7. 
+Similarly, the molecular orientation at the organic-inorganic interface can be used to 
+tune the interfacial distance, i.e., the hybridization strength, so the interfacial hybrid 
+exciton behavior could be effectively tailored, which is beneficial for designing the 
+organic-inorganic interface functionality.  
+In this article, we report the formation of hybrid excitons at the CuPc/MoSe2 
+interface and their further modulation by molecular orientation. The new emission 
+peaks observed in photoluminescence spectroscopy and the further DFT calculations 
+confirm the emergence of new hybrid excitons. The temperature-dependent 
+measurements reveal that the hybrid excitons combine the signature of both Wannier-
+Mott and Frenkel exciton species. The out-of-plane molecular orientation is also 
+applied to tailor the interfacial hybrid excitons. Our results suggest that the organic-
+inorganic heterostructure is a promising platform to explore many-body exciton physics. 
+
+6 
+ 
+Results  
+ 
+Fig. 1 | Sample configuration and basic optical characterization. a Schematic illustration of the 
+different molecular orientations (face-on and edge-on) at CuPc/MoSe2 heterostructure interface. b 
+AFM topographic image of a typical CuPc/MoSe2 heterostructure on Si/SiO2 substrate. c PL spectra 
+of MoSe2 and CuPc/MoSe2 heterostructure at 298 K. d Time-resolved PL spectra of MoSe2 and 
+CuPc/MoSe2 heterostructure at 298 K. The solid curves are the fitted results.  
+Sample configuration and basic optical characterization 
+Figure 1a shows a schematic of the CuPc/MoSe2 heterostructure configuration. 
+We consider two different molecular orientations, i.e., face-on and edge-on, at the 
+CuPc/MoSe2 interface, which will sensitively influence the interfacial coupling strength. 
+The CuPc/MoSe2 heterostructure is prepared by directly evaporating CuPc molecules 
+on top of a monolayer MoSe2 surface in vacuum (see details in methods). A film 
+thickness of ~5 nm is determined by AFM (Fig. 1b and Supplementary Fig. 1c). The 
+
+a
+b
+10 nm
+edge-on
+Cu
+CuPc
+N
+face-on
+CuPc
+CuPclMoSe
+C
+H
+Mo
+Se
+MoSe2
+600nm
+0
+c
+d
+×20
+PL intensity (a.u.)
+-MoSe,
+Normalized PL
+- CuPc/MoSe2
+0.1
+1.4
+1.5
+1.6
+1.7
+0
+1
+2
+Photon energy (eV)
+Delay time (ns)7 
+ 
+optimal molecular orientation at interface of the as-grown sample is the face-on 
+orientation, which has been reported in similar systems34,36 and revealed by our 
+theoretical calculation. The edge-on orientation is introduced by using the CuPc single 
+crystal later. At the face-on orientation, the planar conjugated structure of the CuPc 
+molecule37 and the atomic flat surface of monolayer MoSe210 without dangling bonds 
+facilitate interfacial coupling between them (Fig. 1a). The photoluminescence (PL) 
+spectra of MoSe2 and CuPc/MoSe2 heterostructure acquired at room temperature are 
+shown in Fig. 1c and Supplementary Fig.1d. The MoSe2 exhibits a pronounced PL peak 
+located at ~1.58 eV from the A excitonic transition38. In contrast, a remarkable redshift 
+of ~20 meV in the PL peak energy and a strong quenching in the PL peak intensity are 
+observed in CuPc/MoSe2 heterostructure. The pure CuPc film shows no detectable PL 
+signal (Supplementary Fig. 2a) due to the weak absorption at approximately 514 nm34 
+and its strong intersystem crossing39. Control experiments are further performed to 
+examine the underlying possibilities of the observed phenomena. By changing the 
+thickness of CuPc thin film, it is found that the PL quenching ratio remains almost 
+unchanged (Supplementary Fig. 2c), indicating that absorption of the CuPc film is not 
+the main reason and the phenomena may stem from interfacial interaction. We also 
+adopt another heterostructure configuration by dry transferring MoSe2 on top of the 
+CuPc film (Supplementary Fig.3) and observe similar phenomena, which indicates that 
+the dielectric environment change has negligible influence here40,41. Therefore, the 
+observed phenomena should originate from the intrinsic interfacial coupling between 
+CuPc and MoSe2. Time-resolved PL measurements are performed to compare the PL 
+
+8 
+ 
+lifetimes of MoSe2 and CuPc/MoSe2 heterostructure (Fig. 1c). The PL decays of both 
+MoSe2 and CuPc/MoSe2 heterostructure can be well fitted with a biexponential function, 
+thus, two processes can be derived from both decay curves. For MoSe2, the fast decay 
+constituent has a lifetime of ~85 ps, which is consistent with the lifetime of the A 
+exciton42. The slow decay component has a lifetime of ~1495 ps and is likely to 
+originate from defect-bound excitons43,44. The PL decay of heterostructure exhibits 
+much shorter lifetime than the A exciton of MoSe2, which is not consistent with the 
+behavior of interfacial charge transfer exciton because it usually has a longer lifetime 
+due to the spatial indirect nature31,45.  
+ 
+Fig. 2 | Hybrid excitons in CuPc/MoSe2 heterostructure. a PL spectra of the MoSe2 and 
+CuPc/MoSe2 heterostructure at 78 K. b The PL peak intensity of MoSe2 and CuPc/MoSe2 
+heterostructure as a function of the excitation power. c PL spectra of the MoSe2 and CuPc/MoSe2 
+
+a
+b
+1.0
+x
+hX2
+X
+MoSe2
+105
+T
+78 K
+78 K
+★
+hx
+~p1.13
+CuPc/MoSe2
+intensity (a.u.)
+hx,
+PL
+104
+hX
+P1.13
+-P1.19
+103
+P1.26
+Normal
+PLi
+102
+P1.09
+hX?
+T
+hX4
+101
+~P1.26
+0.0
+1.50
+1.65
+1.80
+1.95
+1
+10
+100
+Photon energy (eV)
+Excitation power(μW)
+C 1.0
+d
+hX2
+T
+4 K
+MoSe2
+10
+000
+Ix/lT
+hX,
+Peak intensity ratio
+CuPc/MoSe2
+PL
+Normalized I
+Q00
+00.0000
+0.5
+Ihx1/lhx2
+X
+hX3
+00
+hX4
+0.1
+0.0
+1.50
+1.65
+1.80
+1.95
+0
+50
+100
+150
+200
+Photon energy (eV)
+Temperature (K)9 
+ 
+heterostructure at 4 K. d The PL peak intensity ratio of exciton versus trion in MoSe2 (IX/IT) and hx1 
+versus hx2 in the CuPc/MoSe2 heterostructure (IhX1/IhX2) as a function of temperature. 
+Emergence of interfacial hybrid excitons  
+Low-temperature PL spectra under 514 nm excitation are obtained at 78 K to 
+further reveal the possible mechanism. We still observe no detectable PL signal in the 
+pure CuPc thin film (Supplementary Fig. 2b). As displayed in Fig. 2a, two emission 
+peaks located at ~1.648 eV and 1.618 eV are observed in the PL spectrum of MoSe2, 
+which can be ascribed to the emission from the A exciton (X) and trion (T) of MoSe238. 
+A striking contrast is observed in the PL spectrum of heterostructure, with four new 
+emission peaks located at ~1.630 eV, ~1.606 eV, ~1.727 eV and ~1.848 eV emerging, 
+which are labeled hX1, hX2, hX3, and hX4, respectively. The hX1 and hX2 show a clear 
+redshift compared with the A exciton of MoSe2, and the hX4 peak displays an obvious 
+redshift with respect to the B exciton of MoSe2 (Supplementary Fig. 4). The hX3 peak 
+is a totally new PL peak that are not observed in pure MoSe2 and CuPc films. Charge 
+transfer exciton31 or dark exciton46 of MoSe2 is also excluded because it has a much 
+higher energy (~79 meV) than the A exciton of MoSe2. In addition, the PL peaks of 
+heterostructure show clear broadening compared with those of MoSe2. We ascribe the 
+observed PL peak redshift and broadening to the signature of interfacial hybridization 
+as reported in similar MoSe2/WS2 heterostructure6. Power-dependent PL spectra are 
+further obtained to examine the origin of the new peaks in heterostructure (Fig. 2b and 
+Supplementary Fig. 5). It is obvious that the peak intensity is enhanced with increasing 
+power. The relationship between excitation power and PL intensity can be expressed 
+as47 𝐼 ∝ 𝑃𝛼, in which 𝐼 represents the PL intensity and 𝑃 represents the excitation 
+
+10 
+ 
+power. The intensities of X and T peaks in MoSe2 show a linear relationship with 
+excitation power with a slope of ~1.13, which indicates recombination from excitons48. 
+Interestingly, all the new peaks in heterostructure also show the linear relationship with 
+similar slopes, which suggests similar exciton behavior with no biexciton49 or defect 
+effect50. Since we have observed new PL peaks with exciton behavior and the signature 
+of hybridization, it is possible that new hybrid excitons are formed in CuPc/MoSe2 
+heterostructure due to interfacial hybridization. 
+We further perform PL measurements at 4 K to examine the influence of interfacial 
+hybridization. As illustrated in Fig. 2c, the PL spectrum of MoSe2 at 4 K is dominated 
+by trion rather than A exciton due to the enhanced trion localization, in great contrast 
+to that at 78 K42. On the contrary, the PL spectrum of heterostructure is still dominated 
+by hX1 and hX2, similar to that at 78 K. Figure 2d displays the evolution of X/T from 4 
+K to 200 K, it is clear that the ratio of X/T has increased from 0.09 to 10 when the 
+temperature is increased from 4 K to 200 K due to the increased thermal perturbance to 
+trion formation. Nevertheless, the radio of hX1/hX2 shows a weak temperature 
+dependence from 4 K to 200 K, differing from the pure exciton and trion behavior in 
+MoSe2, which indicates that the interfacial hybridization effect has changed the exciton 
+behavior in heterostructure. In addition, the PL spectrum of heterostructure displays a 
+highly asymmetric line-shape with prominent low energy tail (Fig. 2a, c), which can be 
+ascribed to an energy shakeup process during the recombination of hybrid excitons 8. 
+ 
+
+11 
+ 
+Fig. 3 | Theoretical calculation of electronic structure in CuPc/MoSe2 heterostructure. a Top 
+-view and side-view for the optimized structure of CuPc/MoSe2 heterostructure. b Calculated band 
+structure of CuPc/MoSe2 heterostructure. c Conduction bands (CB), CB+1, CB+2 and CB+3 in the 
+energy range of 0.72 to 0.80 eV, which correspond to the blue square in b. d Projected charge 
+density for bands in b. ΔE (23 meV) is defined as the energy difference between CB+1 and CB+3, 
+which is mostly contributed by CuPc and MoSe2, respectively. e Schematic illustration of the 
+formation of interfacial hybrid excitons due to the hybridization between LUMO of CuPc and CBM 
+of MoSe2. 
+The above results indicate that the interfacial hybridization effect could lead to the 
+formation of hybrid excitons and change the exciton behavior at the CuPc/MoSe2 
+interface. First-principles calculations are further performed to confirm this. The 
+heterostructure is built by adsorbing a CuPc molecule on a 5×3√3 supercell of MoSe2 
+and the optimal molecular orientation is the face-on orientation (Fig. 3a). The calculated 
+electronic structure of CuPc/MoSe2 heterostructure with face-on orientation is shown 
+
+a
+Top-view
+Side-view
+e
+CuPc
+3.36A
+hX
+CuPc
+h2o
+MoSe2
+MoSe2
+b
+1.0
+C 0.80
+CB
+(eV)
+CB+1
+0.5
+出
+AE
+W-0.5
+CB+2
+CB+3
+CB
+CB+1
+CB+2
+CB+3
+-1.0
+0.72
+x
+Y12 
+ 
+in Fig. 3b. We could recognize two nearly flat bands near 0.5 and -0.5 eV, which 
+correspond to the singly occupied and unoccupied molecular orbitals (SOMO and 
+SUMO) of the CuPc molecule. Then, we concentrate on the bands in the energy range 
+of 0.72 to 0.80 eV (Fig. 3c), which are denoted as conduction band CB, CB+1, CB+2 
+and CB+3. As shown in Fig. 3d, the projected charge density shows that CB and CB+1 
+are mostly contributed by CuPc, and CB+3 is mostly contributed by MoSe2. Notably, 
+CB+2 is contributed both by CuPc and MoSe2, which could be regarded as the 
+hybridization between the LUMO of CuPc and the conduction band of MoSe2. The 
+energy difference between CB+3 and CB+1 in the same spin channel is approximately 
+23 meV, leading to strong hybridization between CuPc and MoSe2 at the face-on 
+orientation, which can explain the observed new hybrid excitons at CuPc/MoSe2 
+interface. The calculation also reveals that the formed hybrid excitons consist of 
+electrons extended in both layers and holes confined in individual layers (Fig. 3e), 
+which can be used to achieve novel quantum control at organic-inorganic interfaces7. 
+ 
+ 
+
+13 
+ 
+ 
+Fig. 4 | Temperature dependence of the hybrid excitons. a Two-dimensional PL spectrum of 
+CuPc/MoSe2 heterostructure as a function of temperature. b PL spectra of CuPc/MoSe2 
+heterostructure in the energy range of 1.4 - 1.95 eV at the temperatures of 78 K, 98 K, 138 K, 178 
+K, and 208 K, respectively. c PL spectra of CuPc/MoSe2 heterostructure in the energy range of 1.68-
+1.95 eV at the temperatures of 78 K, 98 K, 138 K, 178 K, and 208 K, respectively. d The peak 
+energy of hX1 (black), hX2 (red), and hX3 (purple) as a function of temperature. 
+Temperature-dependent behavior of interfacial hybrid excitons  
+The temperature-dependent behavior of the observed hybrid excitons is carefully 
+examined from 78 K to 298 K. For CuPc /MoSe2 heterostructure (Fig. 4a), we clearly 
+observed a remarkable increase in the whole PL intensity when cooling from room 
+temperature (298 K) to low temperature (78 K), which can be explained by the 
+suppression of nonradiative recombination51. When the temperature is higher than 178 
+K, hX2 becomes undistinguishable and hX1 dominates the PL spectra (Fig. 4b). For 
+MoSe2, the trion peak disappears at 98 K and the A exciton becomes dominant 
+
+a
+78
+b
+hX
+C
+6
+hX,
+hX4
+hX3
+2
+208 K
+hX3
+208 K
+max
+98
+hX4
+1
+3
+hX3hX4
+118
+0
+0
+hX
+148
+178K
+178 K
+2
+4
+hX4
+1
+188
+hX3 hX4
+2
+(×103)
+0
+0
+228
+138K
+hx
+138 K
+min
+intensity
+intensity
+4
+278
+2
+hX
+1.4
+1.6
+1.8
+2.0
+hX3 hX4
+2
+Photon energy (eV)
+0
+d 1.75
+μ20
+xyaxy
+PL
+0
+98K
+hX.
+98 K
+14
+1.70
+10
+hX
+7
+4
+(eV)
+hX3
+hX
+hX4
+hX
+0
+hX2/
+0
+40
+hX,
+78 K
+78 K
+18
+1.60
+...
+20
+hX3
+hX3 hX4
+9
+0
+1.55
+0
+100
+150
+200
+250
+300
+1.4
+1.6
+1.8
+1.71
+1.80
+1.89
+Temperature (K)
+Photon energy (eV)
+Photonenergy (eV)14 
+ 
+(Supplementary Fig. 6). To our surprise, the peak energy of the hybrid excitons shows 
+different temperature dependence (Fig. 4a, d). We first focus on the hX1, hX2, and hX4 
+peaks, of which the peak energy shows obvious redshift with increasing temperature 
+(Fig. 4b, d) due to the increased electron-phonon interactions51, similar to the 
+temperature-dependent behavior of exciton and trion in MoSe2 (Supplementary Fig. 6). 
+By fitting the peak energy with the standard semiconductor bandgap model52:𝐸𝑔(0) =
+𝐸𝑔(𝑇) − 𝑆ℏ𝜔 [𝑐𝑜𝑡ℎ (
+ℏ𝜔
+2𝑘𝐵𝑇) −1] , where 𝐸𝑔  represents the bandgap, ℏ����  represents 
+the phonon energy, 𝑆  represents the electron-phonon coupling strength, and 𝑇 
+represents the temperature, we obtain a similar phonon energy for the CuPc/MoSe2 
+heterostructure and MoSe2 (Supplementary Fig. 7a, and Supplementary Table 1), 
+suggesting that these hybrid excitons are also influenced by the phonons of MoSe2. 
+The hX3 peak located at ~1.72 eV is more unique among the four hybrid excitons. 
+The peak energy shows a rather weak temperature dependence (Fig 4c, d), which is 
+totally different from the other three peaks. Such weak temperature-dependent behavior 
+of excitons has been observed in organic molecules, in which the effects of thermal 
+expansion and exciton-phonon coupling almost cancel out53, indicating that the hX3 
+peak displays the signature of Frenkel excitons in CuPc. However, this peak cannot be 
+simply assigned to the emission of the CuPc molecules since no PL signals of CuPc 
+film were observed at 78 K (Supplementary Fig. 2b). Furthermore, we can even observe 
+this peak in the heterostructure region when we dry transferred monolayer MoSe2 on 
+top of the CuPc thin film immediately without any further treatment (Supplementary 
+Fig. 8), which can exclude the influence of the CuPc film morphology. This also 
+
+15 
+ 
+indicates that the coupling at the CuPc/MoSe2 interface is very robust and the hybrid 
+excitons can be formed immediately once they are in contact without any further 
+treatment. On the other hand, it also combines the character of the Wannier-Mott 
+exciton in MoSe2. For example, the peak intensity of hX3 shows similar temperature-
+dependent behavior with the A exciton of MoSe2 (Supplementary Fig. 7b), which 
+suggests that it gains a large oscillator strength from MoSe2 and shows a detectable PL 
+signal compared with the pure CuPc film. Therefore, the peak energy of hX3 shows the 
+signature of the Frenkel excitons in the organic CuPc film and its emission properties 
+display the character of Wannier-Mott excitons in the inorganic MoSe2 monolayer, 
+which unambiguously reveal the formation of hybrid Frenkel-Wannier-Mott excitons 
+at the CuPc/MoSe2 interface.  
+ 
+
+a
+b
+hX,
+100°C
+?200°C
+X
+100°℃
+200°C
+anneal
+hX2
+MOSe
+CL
+UPC
+MoSe,/cuPc
+(crvstal
+crystal)
+Normalized PL
+Edge-on
+MoSe,/CuPccrystal region
+C
+hX3
+hX4
+hx
+mixed Face-on
+hX3
+3mixedFace-on
+and Edge-on
+andEdge-on
+hX
+hX
+hx.
+hx4
+MoSe, region
+Face-on
+Face-on
+hX3
+hX4
+1.5
+1.6
+1.7
+1.8
+1.91.5
+1.6
+1.7
+1.8
+1.9
+1.7
+1.8
+1.9
+Photon energy (ev)
+Photon energy (ev)
+Photonenergy (eV)
+d
+e
+HS(film)
+Face-on
+('n'e)
+CuPc
+hybridization
+MoSe,
+Raman intensity
+?
+h
+Hybrid Exciton
+HybridExciton
+CuPc
+200°℃
+Mixed
+Face-on
+(h)
+Intralayer Exciton
+Interlayer
+HS(crystal)Edge-on
+MoSe,region
+100℃
+@h
+200240
+1380
+1610
+100°C
+200°C
+Ramanshift(cm-1)
+Annealingtemperature16 
+ 
+Fig. 5 | Molecular orientation-dependent hybrid excitons. a The PL spectra in the MoSe2 region 
+and MoSe2/CuPc crystal region for the same MoSe2/CuPc crystal heterostructure after annealing at 
+100°C and 200°C. After annealing at 200°C, the CuPc crystal partially decomposes and the CuPc 
+molecules can migrate on MoSe2, which lead to the formation hybrid excitons in both regions. b 
+AFM topographic image of the MoSe2/CuPc crystal heterostructure after annealing at 100℃ and 
+200℃. c Enlarged view of the image in the gray dotted box in a. d Raman spectra of the MoSe2 
+region in the MoSe2/CuPc crystal heterostructure sample after annealing at 100°C and 200°C. e 
+Schematic illustration of the relationship between molecular orientation (face-on, edge-on) of CuPc 
+and interlayer hybridization. 
+Tailoring the hybrid exciton using molecular orientation 
+The molecular orientation is introduced as a new degree of freedom to modulate the 
+interfacial hybridization strength, which will further tailor the interfacial hybrid 
+excitons. In general, the CuPc molecule tends to adopt a face-on orientation on the 
+MoSe2 surface that allows efficient interfacial hybridization, as revealed by our 
+theoretical calculation. In contrast, the edge-on orientation will experience insufficient 
+interfacial hybridization due to the larger interfacial distance. To demonstrate that the 
+molecular orientation can be used to tailor the interfacial hybrid exciton states, we 
+carefully prepared MoSe2/CuPc film heterostructure and MoSe2/CuPc crystal 
+heterostructure simultaneously by the dry transfer method. Since the CuPc molecule 
+stacks randomly in the CuPc film, it easily adopts a face-on orientation on the MoSe2 
+surface. However, because the CuPc molecule shows a herringbone stacking in the 
+crystal54, it can only adopt an edge-on orientation on MoSe2 surface before crystal 
+decomposition. Because the monolayer MoSe2 partially covers the CuPc crystal, we 
+could compare the measurements from the MoSe2 region and MoSe2/CuPc crystal 
+
+17 
+ 
+region in the same sample (Supplementary Fig. 9a). After annealing simultaneously at 
+100°C, the PL spectra of MoSe2/CuPc film heterostructure and MoSe2/CuPc crystal 
+heterostructure display great contrast as expected. The PL of crystal heterostructure 
+shows similar spectral features and slight quenching compared with monolayer MoSe2 
+(Fig. 5a and Supplementary Fig. 9b), indicating a weak interfacial hybridization 
+strength. However, the PL of film heterostructure presents obvious quenching and the 
+formation of hybrid excitons (Supplementary Fig. 9b, c). The influence of the 
+morphology of CuPc can be excluded since the surface of CuPc crystal is flatter than 
+that of the CuPc film (Supplementary Fig. 10). Therefore, the above results suggest that 
+the molecular orientation can be used to tune the interlayer hybridization strength and 
+further tailor the interfacial hybrid excitons.  
+To confirm this deduction, the MoSe2/CuPc crystal heterostructure is further 
+annealed at 200oC to decompose the CuPc crystal. When the CuPc crystal decomposes, 
+the CuPc molecules can easily adopt a face-on orientation on the MoSe2 surface, thus, 
+the interfacial hybrid excitons should also be observed. After annealing at 200°C, the 
+PL spectra display obvious quenching in both the MoSe2/CuPc crystal region and 
+MoSe2 region (Supplementary Fig. 9d), and clearly shows the formation of hybrid 
+excitons (Fig. 5a, c), which indicates that the molecular orientation is changed from 
+edge-on to face-on after CuPc crystal decomposition. The decomposition of the CuPc 
+crystal is confirmed by AFM, as shown in Fig. 5b. It is obvious that the CuPc crystal 
+partially decomposes after annealing at 200°C, as evidenced by the change in the AFM 
+height profile. Note that we can even observe hybrid excitons in the MoSe2 region 
+
+18 
+ 
+because the CuPc molecules can migrate on MoSe2 surface after the decomposition of 
+CuPc crystal. The Raman spectra further confirms this, as the Raman peaks of both 
+MoSe2 and CuPc molecules appear in the MoSe2 region after annealing at 200°C (Fig. 
+5d). These results unambiguously show that we can successfully tailor the interfacial 
+hybrid excitons by changing the molecular orientation (Fig. 5e). The theoretical 
+calculation also supports our results. As shown in the electronic structure of 
+heterostructure at the edge-on orientation (Supplementary Fig. 11), we observe no 
+obvious interfacial hybridization, which coincides with the observed phenomena in 
+MoSe2/CuPc crystal heterostructure. 
+Discussion 
+  We have demonstrated the formation of interfacial hybrid excitons in CuPc/MoSe2 
+heterostructure due to the hybridization between CuPc and MoSe2. The observed 
+phenomenon is unusual as the coupling at the organic-inorganic interface is generally 
+weak24. The first principles calculations rationalize the results as the LUMO of CuPc 
+strongly hybridized with the CBM of MoSe2, which leads to the emergence of new 
+eigenstates. The new hybrid excitons consist of electrons delocalized in both layers and 
+holes confined in individual layer, enabling simultaneous large optical and electrical 
+dipoles7. The temperature-dependent behavior suggests that the hybrid excitons 
+simultaneously gain the signature of the Wannier-Mott excitons in MoSe2 and the 
+Frenkel excitons in CuPc. The excellent agreement between the theoretical and 
+experimental results not only validates the observed strong coupling phenomenon, but 
+also provides a basis for manipulating hybrid excitons at the organic-inorganic interface. 
+
+19 
+ 
+For instance, the large electrical dipole in the out-of-plane direction can be used to 
+achieve novel electrical control of the hybrid excitons55,56. Our result is also of great 
+importance for realizing tunable interlayer hybridization strength by changing the 
+molecular orientation, which can be used to tailor the exciton states at the organic-
+inorganic interface. In conclusion, we report the formation of interfacial hybrid excitons 
+with strong molecular orientation dependence that originate from the hybridization 
+between CuPc and MoSe2, which is meaningful for many-body exciton physics at the 
+organic-inorganic interface. 
+ 
+Methods 
+Sample preparation 
+(1) Construction of the CuPc film /MoSe2 heterostructure 
+Monolayer MoSe2 was mechanically exfoliated on a SiO2/Si substrate from bulk 
+crystals and further annealed in vacuum at 200 °C to remove surface contaminants. The 
+thickness of MoSe2 was confirmed by optical contrast, atomic force microscopy (AFM), 
+and Raman measurements (Supplementary Fig. 1). To construct the CuPc (film)/MoSe2 
+heterostructure, CuPc thin film was directly deposited on top of monolayer MoSe2 using 
+thermal evaporation in vacuum (home-built evaporator). The heating current was 
+maintained at 5 amperes and the average evaporation speed was 0.25 nm/min.  
+(2) Construction of the MoSe2/CuPc film heterostructure 
+The CuPc film was firstly thermally evaporated on a SiO2/Si substrate using the same 
+conditions as (1). Then, monolayer MoSe2 was mechanically exfoliated on the PDMS 
+
+20 
+ 
+substrate from bulk crystals, and further transferred on top of the CuPc film using the 
+dry transfer method. 
+(3) Construction of the MoSe2/CuPc crystal heterostructure 
+The single crystals of CuPc were grown by the physical vapor deposition (PVT) method 
+in a quartz tube with a hot zone temperature of 400°C. To construct the MoSe2/CuPc 
+(crystal) heterostructure, monolayer MoSe2 was mechanically exfoliated on a PDMS 
+substrate, and further transferred on top of a CuPc single crystal using the dry transfer 
+method. 
+Low temperature PL Measurements.  
+The measurements at 78 K were conducted in a temperature-controlled cryostat 
+(THMS600, Linkam) with a diffraction-limited excitation beam diameter of 1µm.The 
+signal was collected using a 50X long-working distance objective and detected on a 
+commercial Renishaw inVia spectrometer. The excitation power was selected to be 
+below 200 µW to avoid heating damage to the sample. The measurements at 4 K were 
+conducted in a temperature-controlled cryostat (Montana Instruments) with an 
+excitation beam diameter of 1µm. The signal was collected using a 100X objective and 
+detected on a commercial Ocean Optics spectrometer.  
+Theoretical calculation. First-principles calculations were carried out based on the 
+density functional theory (DFT) framework by utilizing the Vienna Ab initio Simulation 
+Package (VASP) 5.4.4 package57,58. Pseudopotentials were used to describe the 
+electron-ion interactions within the PAW approach and generalized gradient 
+approximations (GGA) of Perdew-Burke-Ernzerhof (PBE) were adopted for the 
+
+21 
+ 
+exchange-correlation potential59-61. To better describe the interlayer van der Waals 
+(vdW) interactions, we adopt optB88-vdW corrections for the optimization of 
+structures62. The electron wave functions are expanded on a plane-wave basis set with 
+an energy cutoff of 520 eV. The atomic coordinates of all structures were allowed to 
+relax until the forces acting on the ions were less than 0.01 eV Å-1. The convergence 
+criterion for the electronic self-consistent cycle is fixed at 1×10-5 eV. The integrations 
+in the reduced Brillouin zone are performed on a 3×3×1 Monkhorst-Pack special k-
+points for optimization and self-consistent calculations63,64. A vacuum slab above 15 Å 
+was used in all calculations to avoid interlayer interactions. The CuPc/MoSe2 
+heterostructure is modeled by adsorbing one CuPc molecule on a 5×3√3 supercell of 
+MoSe2, which can be written as Mo30Se60C32N8H16Cu. The lattice parameters of the 
+CuPc/MoSe2 heterostructure were calculated to be a = 16.62 Å, b = 17.27 Å, and 
+α=β=γ=90°. The interlayer distance between CuPc and MoSe2 substrate is 
+approximately 3.36 Å.× 
+ 
+Data availability  
+The data that support the findings of this study are available from the corresponding 
+authors upon reasonable request. 
+ 
+References 
+1. 
+Kleemans, N.A.J.M., et al. Many-body exciton states in self-assembled quantum dots coupled 
+to a Fermi sea. Nat. Phys. 6, 534-538 (2010). 
+2. 
+Shimazaki, Y., Schwartz, I., Watanabe, K., Taniguchi, T., Kroner, M. & Imamoglu, A. Strongly 
+correlated electrons and hybrid excitons in a moire heterostructure. Nature 580, 472-477 (2020). 
+3. 
+Dalgarno, P.A., et al. Optically induced hybridization of a quantum dot state with a filled 
+
+22 
+ 
+continuum. Phys. Rev. Lett. 100, 176801 (2008). 
+4. 
+Stinaff, E.A., et al. Optical signatures of coupled quantum dots. Science 311, 636-639 (2006). 
+5. 
+Krenner, H.J., et al. Optically probing spin and charge interactions in a tunable artificial 
+molecule. Phys. Rev. Lett. 97, 076403 (2006). 
+6. 
+Alexeev, E.M., et al. Resonantly hybridized excitons in moire superlattices in van der Waals 
+heterostructures. Nature 567, 81-86 (2019). 
+7. 
+Hsu, W.T., et al. Tailoring excitonic states of van der Waals bilayers through stacking 
+configuration, band alignment, and valley spin. Sci .Adv. 5, eaax7407 (2019). 
+8. 
+Brotons-Gisbert, M., et al. Coulomb blockade in an atomically thin quantum dot coupled to a 
+tunable Fermi reservoir. Nat. Nanotechnol. 14, 442-446 (2019). 
+9. 
+Mak, K.F., Lee, C., Hone, J., Shan, J. & Heinz, T.F. Atomically thin MoS2: a new direct-gap 
+semiconductor. Phys. Rev. Lett. 105, 136805 (2010). 
+10. 
+Ugeda, M.M., et al. Giant bandgap renormalization and excitonic effects in a monolayer 
+transition metal dichalcogenide semiconductor. Nat. Mater. 13, 1091-1095 (2014). 
+11. 
+Wang, G., et al. Colloquium: Excitons in atomically thin transition metal dichalcogenides. Rev. 
+Mod. Phys. 90 (2): 021001 (2018). 
+12. 
+Regan, E.C., et al. Emerging exciton physics in transition metal dichalcogenide heterobilayers. 
+Nat. Rev. Mater. 1-18 (2022). 
+13. 
+Huang, D., Choi, J., Shih, C.K. & Li, X. Excitons in semiconductor moire superlattices. Nat. 
+Nanotechnol. 17, 227-238 (2022). 
+14. 
+Wilson, N.P., Yao, W., Shan, J. & Xu, X. Excitons and emergent quantum phenomena in stacked 
+2D semiconductors. Nature 599, 383-392 (2021). 
+15. 
+Xiao, D., Liu, G.B., Feng, W., Xu, X. & Yao, W. Coupled spin and valley physics in monolayers 
+of MoS2 and other group-VI dichalcogenides. Phys. Rev. Lett. 108, 196802 (2012). 
+16. 
+Rivera, P., et al. Valley-polarized exciton dynamics in a 2D semiconductor heterostructure. 
+Science 351, 688-691 (2016). 
+17. 
+Rivera, P., Yu, H., Seyler, K.L., Wilson, N.P., Yao, W. & Xu X. Interlayer valley excitons in 
+heterobilayers of transition metal dichalcogenides. Nat. Nanotechnol. 13, 1004-1015 (2018). 
+18. 
+Rivera, P., et al. Observation of long-lived interlayer excitons in monolayer MoSe2-WSe2 
+heterostructures. Nat. Commun. 6, 6242 (2015). 
+19. 
+Jin, C., et al. Observation of moire excitons in WSe2/WS2 heterostructure superlattices. Nature 
+567, 76-80 (2019). 
+20. 
+Seyler, K.L., et al. Signatures of moiré-trapped valley excitons in MoSe2/WSe2 heterobilayers. 
+Nature 567, 66-70 (2019). 
+21. 
+Tran, K., et al. Evidence for moiré excitons in van der Waals heterostructures. Nature, 567, 71-
+75 (2019). 
+22. 
+Ulman, K. & Quek, S.Y. Organic-2D material heterostructures: a promising platform for exciton 
+Condensation and multiplication. Nano Lett. 21, 8888-8894 (2021). 
+23. 
+Agranovich, V.M., Basko, D.M., Rocca, G.C.L. & Bassani, F. Excitons and optical 
+nonlinearities in hybrid organic-inorganic nanostructures. J. Phys.: Condens. Matter 10, 9369-
+9400 (1998). 
+24. 
+Agranovich, V.M., Gartstein, Y.N. & Litinskaya, M. Hybrid resonant organic-inorganic 
+nanostructures for optoelectronic applications. Chem. Rev. 111, 5179-5214 (2011). 
+25. 
+Blumstengel, S., Sadofev, S., Xu, C., Puls, J. & Henneberger, F. Converting Wannier into 
+
+23 
+ 
+Frenkel excitons in an inorganic/organic hybrid semiconductor nanostructure. Phys. Rev. Lett. 
+97, 237401 (2006). 
+26. 
+Gobbi, M., Orgiu, E. & Samori, P. When 2D materials meet molecules: opportunities and 
+challenges of hybrid organic/inorganic van der Waals Heterostructures. Adv. Mater. 30, 
+e1706103 (2018). 
+27. 
+Huang, Y.L., Zheng, Y.J., Song, Z., Chi, D., Wee, A.T.S. & Quek, S.Y. The organic-2D transition 
+metal dichalcogenide heterointerface. Chem. Soc. Rev. 47, 3241-3264 (2018). 
+28. 
+Cho, K., Pak, J., Chung, S. & Lee, T. Recent advances in interface engineering of transition-
+metal dichalcogenides with organic molecules and polymers. ACS Nano 13, 9713-9734 (2019). 
+29. 
+He, D., et al. Two-dimensional quasi-freestanding molecular crystals for high-performance 
+organic field-effect transistors. Nat. Commun. 5, 5162 (2014). 
+30. 
+Wang, S., et al. A MoS2 /PTCDA hybrid heterojunction synapse with efficient photoelectric dual 
+modulation and versatility. Adv. Mater. 31, e1806227 (2019). 
+31. 
+Ding, J., Fu, S., Hu, K., Zhang, G., Liu, M., Zhang, X., Wang, R. & Qiu, X. Efficient hot electron 
+capture in CuPc/MoSe2 heterostructure assisted by intersystem crossinng. Nano Lett. 22, 8463-
+8469 (2022). 
+32. 
+Homan, S.B., Sangwan, V.K., Balla, I., Bergeron, H., Weiss, E.A. & Hersam, M.C. Ultrafast 
+exciton dissociation and long-lived charge separation in a photovoltaic pentacene-MoS2 van der 
+Waals heterojunction. Nano Lett. 17, 164-169 (2017). 
+33. 
+Zhong, C., Sangwan, V.K., Wang, C., Bergeron, H., Hersam, M.C. & Weiss, E.A. Mechanisms 
+of ultrafast charge separation in a PTB7/monolayer MoS2 van der Waals Heterojunction. J. Phys. 
+Chem. Lett. 9, 2484-2491 (2018). 
+34. 
+Padgaonkar, S., et al. Molecular-orientation-dependent interfacial charge transfer in 
+phthalocyanine/MoS2 mixed-dimensional heterojunctions. J. Phys. Chem. C 123, 13337-13343 
+(2019). 
+35. 
+Kafle, T.R., Kattel, B., Lane, S.D., Wang, T., Zhao, H. & Chan, W.L. Charge transfer exciton 
+and spin flipping at organic-transition-metal dichalcogenide interfaces. ACS Nano 11, 10184-
+10192 (2017). 
+36. 
+Choi, J., Zhang, H. & Choi, J.H. Modulating optoelectronic properties of two-dimensional 
+transition metal dichalcogenide semiconductors by photoinduced charge transfer. ACS Nano 10, 
+1671-1680 (2016). 
+37. 
+Aristov, V.Y., et al. Electronic structure of the organic semiconductor copper phthalocyanine: 
+experiment and theory. J. Chem. Phys. 128, 034703 (2008). 
+38. 
+Ross, J.S., et al. Electrical control of neutral and charged excitons in a monolayer semiconductor. 
+Nat. Commun. 4, 1474 (2013). 
+39. 
+Caplins, B.W., Mullenbach, T.K., Holmes, R.J. & Blank, D.A. Femtosecond to nanosecond 
+excited state dynamics of vapor deposited copper phthalocyanine thin films. Phys. Chem. Chem. 
+Phys. 18, 11454-11459 (2016). 
+40. 
+Shi, N. & Ramprasad, R. Dielectric properties of Cu-phthalocyanine systems from first 
+principles. Appl. Phys. Lett. 89, 102904 (2006). 
+41. 
+Robertson, J. High dielectric constant oxides. Eur.Phys.J-Appl. Phys. 28, 265-291 (2004). 
+42. 
+Godde, T., et al. Exciton and trion dynamics in atomically thin MoSe2 and WSe2: effect of 
+localization. Phys. Rev. B 94, 165301 (2016). 
+43. 
+Anghel, S., Passmann, F., Ruppert, C., Bristow, A.D. & Betz, M. Coupled exciton-trion spin 
+
+24 
+ 
+dynamics in a MoSe2 monolayer. 2D Mater. 5, 045024 (2018). 
+44. 
+Ruppert, C., Chernikov, A., Hill, H.M., Rigosi, A.F. & Heinz, T.F. The role of electronic and 
+phononic excitation in the optical response of monolayer WS2 after ultrafast excitation. Nano 
+Lett. 17, 644-651 (2017). 
+45. 
+Zhu, T., et al. Highly mobile charge-transfer excitons in two-dimensional WS2/tetracene 
+heterostructures. Sci. Adv. 4, eaao3104 (2018). 
+46. 
+Robert, C., et al. Measurement of the spin-forbidden dark excitons in MoS2 and MoSe2 
+monolayers. Nat. Commun. 11, 4037 (2020). 
+47. 
+Schmidt, T., Lischka, K. & Zulehner, W. Excitation-power dependence of the near-band-edge 
+photoluminescence of semiconductors. Phys. Rev. B Condens. Matter 45, 8989-8994 (1992). 
+48. 
+Li, Z., et al. Revealing the biexciton and trion-exciton complexes in BN encapsulated WSe2. 
+Nat. Commun. 9, 3719 (2018). 
+49. 
+Kylänpää, I. & Komsa, H.P. Binding energies of exciton complexes in transition metal 
+dichalcogenide monolayers and effect of dielectric environment. Phys. Rev. B 92, 205418 (2015). 
+50. 
+Tongay, S., et al. Defects activated photoluminescence in two-dimensional semiconductors: 
+interplay between bound, charged, and free excitons. Sci. Rep. 3, 2657 (2013). 
+51. 
+Li, D., et al. Large‐area 2D/3D MoS2–MoO2 heterostructures with thermally stable exciton and 
+intriguing electrical transport behaviors. Adv. Electron. Mater. 3, 1600335 (2017). 
+52. 
+O’Donnell, K.P. & Chen, X. Temperature dependence of semiconductor band gaps. Appl. Phys. 
+Lett. 58, 2924-2926 (1991). 
+53. 
+Alvertis, A.M., et al. Impact of exciton delocalization on exciton-vibration interactions in 
+organic semiconductors. Phys. Rev. B 102, 081122 (2020). 
+54. 
+Zou, T., et al. Controllable molecular packing motif and overlap type in organic nanomaterials 
+for advanced optical properties. Crystals 8, 22 (2018). 
+55. 
+Peimyoo, N., et al. Electrical tuning of optically active interlayer excitons in bilayer MoS2. Nat. 
+Nanotechnol. 16, 888-893 (2021). 
+56. 
+Leisgang, N., et al. Giant Stark splitting of an exciton in bilayer MoS2. Nat. Nanotechnol. 15, 
+901-907 (2020). 
+57. 
+Kresse, G. & Furthmüller, J. Efficient iterative schemes for ab initio total-energy calculations 
+using a plane-wave basis set. Phys. Rev. B 54, 11169-11186 (1996). 
+58. 
+Kresse, G., & Hafner, J. Ab initio molecular dynamics for liquid metals. Phys. Rev. B 47, 558-
+561 (1993). 
+59. 
+Perdew, J.P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made simple. Phys. 
+Rev. Lett. 77, 3865-3868 (1996). 
+60. 
+Blöchl, P.E. Projector augmented-wave method. Phys. Rev. B 50, 17953-17979 (1994). 
+61. 
+Kresse, G., & Joubert, D. From ultrasoft pseudopotentials to the projector augmented-wave 
+method. Phys. Rev. B 59, 1758-1775 (1999). 
+62. 
+Grimme, S., Antony, J., Ehrlich, S. & Krieg, H. A consistent and accurate ab initio 
+parametrization of density functional dispersion correction (DFT-D) for the 94 elements H-Pu. 
+J. Chem. Phys. 132, 154104 (2010). 
+63. 
+Monkhorst, H.J. & Pack, J.D. Special points for Brillouin-zone integrations. Phys Rev B 13, 
+5188-5192 (1976). 
+64. 
+Methfessel, M. & Paxton, A.T. High-precision sampling for Brillouin-zone integration in metals. 
+Phys. Rev. B 40, 3616-3621 (1989). 
+
+25 
+ 
+ 
+Acknowledgements 
+This work was supported by the National Nature Science Foundation of China (Grant 
+Nos. 11974088, 12074116, 21790353, 61875236, 61975007), the National Key 
+Research and Development Program of China (Grant Nos.2016YFA0202302, 
+2017YFA0205000, 2021YFA1400201), the Strategic Priority Research Program of 
+CAS (Grant No. XDB30000000), the CAS Project for Young Scientists in Basic 
+Research (Grant No. YSBR-059). 
+Author contributions 
+X.Z., X.Q. and D.T. conceived the idea; S.F. and J.D. prepared the samples and 
+conducted all the optical measurements and the corresponding data analysis; X.W. and 
+H.L. performed the DFT calculations; This manuscript was prepared primarily by X.Z., 
+S.F. and J.D., and all authors contributed to discussing and commenting on the paper. 
+Competing interests 
+The authors declare no competing interests 
+Additional information 
+Correspondence and requests for materials should be addressed to Xiaoxian Zhang. 
+

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+Regularized Optimal Mass Transport with Nonlinear Diffusion
+Kaiming Xu, Xinan Chen, Helene Benveniste, Allen Tannenbaum ∗†‡§
+January 10, 2023
+Abstract
+In this paper, we combine nonlinear diffusion with the regularized optimal mass
+transport (rOMT) model. As we will demonstrate, this new approach provides further
+insights into certain applications of fluid flow analysis in the brain. From the point
+of view of image processing, the anisotropic diffusion method, based on Perona-Malik,
+explicitly considers edge information. Applied to rOMT analysis of glymphatic trans-
+port based on dynamic contrast-enhanced magnetic resonance imaging data, this new
+framework appears to capture a larger advection-dominant volume.
+1
+Introduction
+The theory of optimal mass transport(OMT) was first proposed by Gaspard Monge in 1781
+and has since evolved into a unique scientific field which has had significant impact on
+research in many disciplines [22, 23]. Mass transport theory has been applied to diverse
+fields including physics, biology, economics and engineering. OMT defines a distance called
+the Wasserstein distance, and thus creates a natural geometry on the space of probability
+distributions.
+Our study is based on a fluid dynamics reformulation of OMT [1] which
+allows us to calculate the flow fields between two density distributions.
+Regularized optimal mass transport (rOMT), an extension of fluid dynamics reformulation
+of OMT, is a tool to study temporal flow fields as a physically inspired model of optical flow.
+It has the ability to capture the flow dynamics, handle noise and simulate diffusion [3, 5, 9].
+rOMT utilizes an advection-diffusion equation as its flow-driven partial different equation
+and is endpoint free. A source term may be added to rOMT in which case the total mass
+preservation condition can be circumvented. This line of research will be pursued in other
+work.
+Anisotropic diffusion, a major tool for image segmentation, edge detection and image de-
+noising, was first proposed by Perona and Malik [17]. Notably, instead of using a constant
+diffusion coefficient, Perona and Malik considered a nonnegative function (conductivity
+∗K. Xu is with the Department of Applied Mathematics & Statistics, Stony Brook University, NY; email:
+[email protected]
+†X. Chen is with the Department of Medical Physics, Memorial Sloan Kettering Cancer Center, NY
+‡H. Benveniste is with the Department of Anesthesiology, Yale School of Medicine, CT
+§A. Tannenbaum is with the Departments of Computer Science and Applied Mathematics & Statistics,
+Stony Brook University, NY; email: [email protected]
+1
+arXiv:2301.03428v1  [physics.flu-dyn]  3 Dec 2022
+
+coefficient) of the magnitude of the local density gradient; see equation (8). The authors
+suggested two possible conductivity coefficients (see (9) and (10)), wherein the diffusion will
+be very small near the edges, i.e. reflecting the fact that near edges images tend to have
+very large intensity gradients. In this work, we show that anisotropic diffusion enhances
+the interpretation of glymphatic dynamic contrast-enhanced magnetic resonance imaging
+(DCE-MRI) flow data and may be used in conjunction with the constant diffusion coefficient
+approach [3]. The anisotropic diffusion equation may be derived via the steepest descend
+method for solving an energy minimization problem [25].
+The glymphatic system is involved in transporting waste products from the brain to the
+meningeal lymphatic system which connects to the cervical lymph nodes [14]. The function-
+ing of the glymphatic and lymphatic systems decrease with age and have been implicated in
+the pathophysiology of a wide range of neurodegenerative diseases including cerebral amy-
+loid angiopathy [3, 24] and Alzheimer’s disease [4, 10, 13, 16]. We study glymphatic trans-
+port using a temporal series of DCE-MRI data acquired from the rodent brain [6, 11, 12].
+Since the data are acquired at discrete time points, our work is motivated by the need to
+find a dynamic physically based model of the transport. Several different versions of OMT
+[18] and rOMT [3, 5, 9] have been used to model the glymphatic flow.
+In the present work, we propose a new version of rOMT. Specifically, we replace the lin-
+ear diffusion in rOMT [3, 5, 9] with the Perona-Malik based anisotropic diffusion. Here,
+we argue that this gives us enhanced flexibility to study image-based flows inherent to
+glymphatic transport. Notably, many diffusion processes in fluids are better captured by
+nonlinear models, e.g., axisymmetric surface diffusion [2] and thin fluid films [7, 8]. We
+utilize Lagrangian coordinates for visualizing the glymphatic transport pathlines. Several
+properties of solute particle movement are computed along the pathlines such as speed and
+the P´eclet number. Here we compare various parameters of the anisotropic diffusion coef-
+ficient, and observe the impact of different values on several data metrics including P´eclet
+plots which can map diffusion dominated versus advection dominated regions of the brain.
+We briefly summarize the contents of the present paper. In Section 2, we review the theory
+of OMT, rOMT and nonlinear diffusion. Section 3 introduces the algorithm and numerical
+methods we employ for our current work. In Section 4, we explicate the application of the
+model to glymphatic DCE-MRI data and analyze the experimental results and we conclude
+our paper in Section 5.
+2
+Model
+2.1
+OMT
+In this section, we introduce OMT and its fluid dynamics formulation. All the technical
+details as well as a complete set of references may be found in [22, 23].
+The original
+formulation of OMT was given by Gaspard Monge and may be expressed as
+inf
+T {
+�
+Ω
+c(x, T(x))ρ0(x)dx | T#ρ0 = ρ1},
+(1)
+where c(x, y) is the cost function of moving the unit mass from x to y, ρ0 and ρ1 are two
+probability distributions in the domain Ω ⊆ Rd, T is the transport map, and T# is the
+2
+
+push-forward of T. This formulation assumes that ρ0 and ρ1 have the same total mass, i.e.
+�
+Ω ρ0(x)dx =
+�
+Ω ρ1(x)dx and then seeks for the optimal transport map T to minimize the
+total cost, the integral in equation (1), subject to the push-forward constraint.
+Later, Leonid Kantorovich formulated a relaxed version of OMT as follows:
+inf
+π∈Π(ρ0,ρ1)
+�
+Ω×Ω
+c(x, y)π(dx, dy),
+(2)
+where Π(ρ0, ρ1) denotes the set of all couplings (joint distributions) between the marginals
+ρ0 and ρ1. From here on, the cost function c will be taken as the square of the Euclidean
+distance c(x, y) = ∥x − y∥2.
+Benemou and Brenier [1] proved that for c(x, y) = ∥x − y∥2, the specific infimum of Monge-
+Kantorovich formulation is equal to the result in following fluid dynamics formulation for
+density/probability distributions with compact support:
+inf
+ρ,v
+� 1
+0
+�
+Ω
+ρ(t, x)|v(t, x)|2dxdt,
+(3)
+∂ρ
+∂t + ∇ · (ρv) = 0,
+(4)
+ρ(0, x) = ρ0(x),
+ρ(1, x) = ρ1(x),
+(5)
+where ρ : [0, 1]×Ω → R≥0 is the family of density/probability distributions defining geodesic
+path from ρ0 to ρ1, and v : [0, 1] × Ω → Rd is the velocity vector field.
+2.2
+rOMT
+The regularized OMT model (rOMT) [5, 9] adds two assumptions: 1. the image data we
+use are noisy observations and thus we do not want to make the final density we calculate
+coincide with the MR images; and 2. the flow is driven by an advection-diffusion equation.
+Based on these two assumptions, the rOMT formulation may be written as:
+inf
+ρ,v
+� 1
+0
+�
+Ω
+ρ(t, x)|v(t, x)|2dxdt + β
+�
+Ω
+(ρ(1, x) − ρ1(x))2dx,
+(6)
+∂ρ
+∂t + ∇ · (ρv) = ∇ · (σ0∇ρ),
+(7)
+ρ(0, x) = ρ0(x).
+In this formulation, the final marginal condition is removed and a penalty of the error
+between final density and ground truth is added in the objective function (6), where β is
+the penalty parameter. Equation (7) is an advection-diffusion equation with a constant σ0
+denoting the diffusion coefficient.
+2.3
+Nonlinear diffusion
+Instead of using linear diffusion in which σ0 is a constant, nonlinear diffusion seems to have
+certain advantages that we will now describe. Perona and Malik proposed an anisotropic
+3
+
+diffusion [17], which is a useful tool for image segmentation, edge detection and image
+denoising. The anisotropic diffusion equation is
+∂ρ
+∂t = ∇ · (σ(|∇ρ|)∇ρ),
+(8)
+where σ(·) is a nonnegative strictly decreasing function. If we consider a 3D problem, then
+|∇ρ| =
+�
+ρ2x + ρ2y + ρ2z. The proper diffusion should be large in smooth homogeneous areas
+and become smaller near edges, the places where |∇ρ| is large.
+Perona and Malik [17]
+suggested two versions of the diffusion (conductivity) coefficient:
+σ(x) = σ0
+1
+1 + ( x
+K )2 ,
+(9)
+σ(x) = σ0e−( x
+K )2.
+(10)
+Both are 0 when x approaches ∞ and attend upper bound σ0 while x = 0. K is a constant
+and controls the sensitivity to edges and can be tuned for different applications.
+Following [25], we may derive the anisotropic diffusion equation (8) via the steepest descent
+from an energy minimization problem. More precisely, considering the following minimiza-
+tion problem:
+min
+�
+Ω
+f(|∇ρ|)dΩ,
+(11)
+then the steepest descend equation may be computed to be
+∂ρ
+∂t = ∇ · (f′(|∇ρ| ∇ρ
+|∇ρ|)).
+(12)
+Obviously, (12) is identical to (8) if
+f′(x) = xσ(x).
+(13)
+For example, the corresponding f function of σ function (9) is
+f(x) = σ0K2
+2
+ln[1 + ( x
+K )2]
+(14)
+2.4
+rOMT with nonlinear diffusion
+In this section, we present our new rOMT formulation. We replace the diffusion in (7) by
+anisotropic diffusion in (8) and obtain the following formulation:
+inf
+ρ,v
+� 1
+0
+�
+Ω
+ρ(t, x)|v(t, x)|2dxdt + β
+�
+Ω
+(ρ(1, x) − ρ1(x))2dx,
+∂ρ
+∂t + ∇ · (ρv) = ∇ · (σ(|∇ρ|)∇ρ),
+(15)
+ρ(0, x) = ρ0(x).
+One may employ various versions of the σ function and in this work, we choose the function
+given in (9). Note that, there are two parameters σ0 and K which may be tuned based on
+the data we use.
+4
+
+Equation (15) may be written in conservation form as
+∂ρ
+∂t + ∇ · (ρ(v − σ(|∇ρ|)∇ log ρ)) = 0,
+and after defining an augmented velocity
+vaug = v − σ(|∇ρ|)∇ log ρ,
+we derive a simple conservation form of equation (15)
+∂ρ
+∂t + ∇ · (ρvaug) = 0.
+The Lagrangian representation X = X(x, t) of the optimal trajectory for this rOMT with
+nonlinear diffusion model is given by
+X(x, 0) = x,
+∂X(x, t)
+∂t
+= vaug
+opt (X(x, t), t),
+(16)
+where
+vaug
+opt = vopt − σ(|∇ρopt|)∇ log ρopt,
+(17)
+and vopt and ρopt denote the optimal solution of the rOMT with nonlinear diffusion model.
+In Section 4, we exhibit the pathlines in Figure 2 and Figure 3 derived from the Lagrangian
+coordinates (16).
+3
+Numerical scheme
+In this section, we focus on the numerical solution of the nonlinear diffusive rOMT model.
+The pipeline that comes from [5, 9] is based on the Gauss-Newton method:
+1. Give initial guess of v at each time and spatial point.
+2. Use v, ρ0 and the advection-diffusion equation (15) to calculate ρ at each subsequent
+time step.
+3. Calculate the objective function (6), which we will denote with Γ(v) as the discrete
+form.
+4. Calculate the gradient g(v) and the Hessian matrix H(v) of Γ(v) with respect to v.
+5. Solve the descent direction s by solving H(v)s = −g(v).
+6. Do line search to find l and update v by setting v = v + ls.
+7. Repeat step 2-6 until the results attain the final condition.
+Space is discretized into a cell-center grid of size nx × ny × nz with a total number of N
+cells, each with width ∆x, height ∆y and depth ∆z. Time is divided into m intervals of
+length ∆t with m + 1 time steps. Moreover, the superscript 0 corresponds to initial time
+t = 0, M corresponds to final time t = 1 and dt × m = 1. We use ρ = [(ρ0)T , . . . , (ρm)T ]T
+and v = [(v1)T , . . . , (vm)T ]T to represent temporal density and velocity, respectively. Note
+that the velocity vi describes the velocity field from (i − 1)th time step to ith time step.
+5
+
+3.1
+Advection-diffusion equation
+Here we describe the numerical scheme for equation (15).
+The discrete form of equation (15) between time tn and tn+1 is
+ρn+1 − ρn
+∆t
++ A(ρ, v) = D(ρ),
+(18)
+where A and D are discretizations of advective and diffusive terms, respectively. We will
+describe these in greater detail below. Following the work of Steklova and Haber [21], we
+split equation (18) into two parts,
+ρadv − ρn
+∆t
++ A(ρ, v) = 0,
+(19)
+ρn+1 − ρadv
+∆t
+= D(ρ),
+(20)
+where ρadv is an auxiliary variable. Simply by adding (19) and (20), we obtain the equation
+(18).
+So far we have not chosen the time step of ρ in the advective part A(ρ, v) and
+diffusive part D(ρ). We use a standard forward scheme, i.e. ρ = ρn in our implementation.
+Summarizing up to this point, to solve for the next time step density ρn+1, we first calculate
+ρadv by solving equation (19) and use ρadv and ρn to calculate ρn+1 following equation
+(20).
+For the advective part A(ρ, v), we utilize a particle-in-cell method which is also how Steklova
+and Haber[21] dealt with their advective part to solve equation (19):
+ρadv = S(v)ρ.
+(21)
+S(v) is the averaging matrix with respect to v.
+The basic idea of particle-in-cell method is moving density the ρi in the cell center to
+the target ρnew
+i
+according to its velocity vi and using its nearest neighbor cell centers to
+interpolate.
+The numerical techniques of solving equation (20) are based on hyperbolic conservation
+laws and the theory of viscosity solutions [15, 19, 20], and we explicitly write D in the next
+section.
+3.2
+Anisotropic diffusion
+From now on, we explore in 3-dimension (d = 3), following [19, 25] and discretize the
+anisotropic diffusion as follows:
+D(ρi,j,k) = ∆x
+−{σ[
+�
+(∆x
++ρi,j,k)2 + m2(∆y
++ρi,j,k, ∆y
+−ρi,j,k) + m2(∆z
++ρi,j,k, ∆z
+−ρi,j,k)]∆x
++ρi,j,k}
++ ∆y
+−{σ[
+�
+(∆y
++ρi,j,k)2 + m2(∆x
++ρi,j,k, ∆x
+−ρi,j,k) + m2(∆z
++ρi,j,k, ∆z
+−ρi,j,k)]∆y
++ρi,j,k}
++ ∆z
+−{σ[
+�
+(∆z
++ρi,j,k)2 + m2(∆x
++ρi,j,k, ∆x
+−ρi,j,k) + m2(∆y
++ρi,j,k, ∆y
+−ρi,j,k)]∆z
++ρi,j,k}.
+(22)
+6
+
+Here,
+∆x
+−ai,j,k = ai,j,k − ai−1,j,k
+∆x
+,
+∆x
++ai,j,k = ai+1,j,k − ai,j,k
+∆x
+,
+∆y
+−ai,j,k = ai,j,k − ai,j−1,k
+∆y
+,
+∆y
++ai,j,k = ai,j+1,k − ai,j,k
+∆y
+,
+∆z
+−ai,j,k = ai,j,k − ai,j,k−1
+∆z
+,
+∆z
++ai,j,k = ai,j,k+1 − ai,j,k
+∆z
+,
+m(a, b) = median(a, b, 0).
+We note that the solution of equation (18) may be written recursively:
+ρn+1 = S(vn)ρn + ∆tD(ρn).
+(23)
+3.3
+Objective function Γ(v)
+A straightforward way [5, 9] to discretize the objective function Γ(v) in (6) is
+hd ∗ ∆t ∗ ρT (Im ⊗ [IN|IN|IN])(v ⊙ v) + β|ρm − ρT |2.
+(24)
+Here hd = ∆x∗∆y∗∆z, ρ, v are column vectors, ⊗ is Kronecker product and ⊙ is Hadamard
+product.
+3.4
+Gradient, hessian and sensitivity
+In order to apply the Gauss-Newton minimization procedure such as described in Steklova
+and Haber [21], we need expressions for the gradient g(v) and the Hessian H(v). Taking
+the gradient of (24) with respect to v, we find
+g(v) = ∂Γ(v)
+∂v
+= hd ∗ ∆t ∗ [2ρT Mdiag(v) + (M(v ⊙ v))T J] + β(ρm − ρ1)T ∂ρm
+∂v ,
+(25)
+where M = Im ⊗ [IN|IN|IN], matrix J = (Jk
+j ). Here Jk
+j = ∂ρk
+∂vj , k = 1, . . . , m and j =
+0, . . . , m − 1.
+The Hessian matrix is
+H(v) = ∂g
+∂v = hd∗∆t∗[2ρT ∇(Mdiag(v))+2∇(ρ)Mdiag(v)+M(v⊙v)∇J +∇[M(v⊙v)]J]
++ β[(∂ρm
+∂v )T (∂ρm
+∂v ) + (ρm − ρ1)∂2ρm
+∂v2 ].
+(26)
+Numerically we approximate the Hessian by
+H(v) = 2hd ∗ ∆t ∗ ρT ∇(Mdiag(v)) + β(∂ρm
+∂v )T (∂ρm
+∂v )
+= 2hd ∗ ∆t ∗ diag(ρT M) + β(∂ρm
+∂v )T (∂ρm
+∂v ).
+(27)
+In the formulae for the gradient (25) and Hessian (27), we still need to know the sensitivity
+of the density ρ with respect to the velocity v. We recall equation (23)
+ρn+1 = S(vn)ρn + ∆tD(ρn).
+7
+
+From that, the sensitivity can be calculated as below:
+∂ρk
+∂vj =
+�
+�
+�
+S(vk−1) ∂ρk−1
+∂vj
++ ∆tD′(ρk−1) ∂ρk−1
+∂vj
+k ≥ j + 2
+∂
+∂vj (S(vj)ρj)
+k = j + 1
+0
+k ≤ j
+(28)
+4
+Experimental results
+In this section, we test our proposed methodology on 3D DCE-MRI data derived from
+[3]. In this dataset, rats were anesthetized, and a Gd-tagged tracer was injected into the
+cerebrospinal fluid (CSF). The rat underwent dynamic 3D MRI scanning every 5 minutes
+to collect a total 29 3D brain images with a voxel size of 100×106×100. Post-processing of
+the DCE-MRI data included head motion correction, intensity normalization, and voxel-by-
+voxel conversion to percentage of baseline signal. In our experiment, we chose a 12-month-
+old wild type rat for demonstrating the results.
+The new algorithm was run for data covering a 100-minute time period (60 minutes to
+160 minutes) which includes 23 frames, and we used every other image as inputs to reduce
+runtime, leaving 12 frames for the numerical experiment.
+We use In, n = 1, . . . , 12 to
+represent these frames. To derive the interpolations, we applied our model between each of
+two consecutive frames, i.e. Ik and Ik+1. To ensure continuity, (except for the first step),
+the initial density originates from the previous step. For example, if we are considering
+the problem between I2 and I3, and we will use the final density I′
+2 calculated between I1
+and I2 as the new initial density here and apply our model between I′
+2 and I3. One of the
+metrics that can measure the model accuracy is the error between the final density I′
+k and
+the ground truth Ik at each step.
+Here we are using σ function (9) with σ0 = 0.002. The choice of σ0 follows [3]. We tested
+rOMT on the 3D DCE-MRI data set with σ0 = 0.00002, 0.0002, 0.002, 0.02, 0.2. The speed
+maps in Figure 4 show a stable trend between σ0 = 0.00002 and σ0 = 0.002 and among
+these three σ0 (0.00002,0.0002 and 0.002), 0.002 has the minimal interpolation error (see
+Figure 5).
+We computed pathlines based on Lagrangian coordinates (16). We compared different K’s
+and the results are shown in Figures 1-3. Figure 1 shows the relative error
+e = |I′ − I|2
+|I|2
+on each frame with different K’s. The x-axis represents the indices of frames and the y-
+axis is the relative error. From Figure 1, we observe that rOMT with anisotropic diffusion
+has similar accuracy as the original rOMT model. Figure 2 compares the P´eclet number
+along pathlines in the right lateral view plane for different K’s. Further, Figure 3 shows
+the ventral surface of the brain. Red color represents larger P´eclet numbers (advection
+dominant) and blue represents smaller P´eclet numbers (diffusion dominant). As shown in
+Figure 2 and Figure 3, a smaller K value results in more advection dominated transport
+in ‘surface’ areas of the brain which corresponds to the CSF compartment. When we set
+K = ∞, then clearly σ(x) = σ0, since
+lim
+K→∞ σ0
+1
+1 + ( x
+K )2 = σ0.
+8
+
+Figure 1: Relative interpolation error plot for different parameter K’s.
+Original means
+constant diffusion coefficient, i.e. K = ∞.
+9
+
+relativeinterpolationerror
+0.25
+0.2
+0.15
+original
+0.1
+K=10
+K=100
+K=1000
+K=10000
+K=100000
+0.05
+K=1000000
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11Figure 2: Pathlines endowed with P´eclet Number shown in the lateral view plane. Parameter
+K = 10, 100, 1000, 10000, 100000, ∞. The maximal limit of color bar is 300. When K is
+small, the advective (red) pathline dominates in CSF rich areas.
+10
+
+K=10
+Pseudocolor
+Var: PathPoint
+300.0
+225.0
+150.0
+75.00
+0.000
+Max: 2.210e+13
+Min: 0.000K=100
+Pseudocolor
+Var: PathPoint
+300.0
+225.0
+150.0
+75.00
+0.000
+Max: 6.551e+13
+Min: 0.000K= 1000
+Pseudocolor
+Var: PathPoint
+300.0
+225.0
+150.0
+75.00
+0.000
+Max: 7.892e+13
+Min: 0.000K=10000
+Pseudocolor
+Var: PathPoint
+300.0
+225.0
+150.0
+75.00
+0.000
+Max:1.703e+14
+Min: 0.000K=100000
+Pseudocolor
+Var: PathPoint
+300.0
+225.0
+150.0
+75.00
+0.000
+Max:1.770e+14
+Min: 0.000K=Infinity
+Pseudocolor
+Var: PathPoint
+300.0
+225.0
+150.0
+75.00
+0.000
+Max:1.839e+13
+Min: 0.000Figure 3: P´eclet number endowed pathlines shown in ventral view plane. Parameter K =
+10, 100, 1000, 10000, 100000, ∞. The maximal limit of the color bar is 300.
+11
+
+K=1000
+Pseudocolor
+Var: PathPoint
+300.0
+225.0
+150.0
+75.00
+0.000
+Max:7.892e+13
+Min: 0.000K= 10000
+Pseudocolor
+Var: PathPoint
+300.0
+225.0
+150.0
+75.00
+0.000
+Max: 1.703e+1
+Min: 0.000K=100000
+Pseudocolor
+Var: PathPoint
+300.0
+225.0
+150.0
+75.00
+0.000
+Max:1.770e+14
+Min: 0.000K=Infinity
+Pseudocolor
+Var: PathPoint
+300.0
+225.0
+150.0
+75.00
+0.000
+Max:1.839e+13
+Min: 0.000K=10
+Pseudocolor
+Var: PathPoint
+300.0
+225.0
+150.0
+75.00
+0.000
+Max: 2.210e+13
+Min: 0.000K=100
+Pseudocolor
+Var: PathPoint
+300.0
+225.0
+150.0
+75.00
+0.000
+Max: 6.551e+13
+Min: 0.000Figure 4: Speed map for different σ0’s. The maximal limit of the color bar is 0.6. The first
+three speed maps exhibit a stable trend. The last two speed maps with higher values of
+diffusion dramatically (and erroneously) increase speed suggesting that σ0 is too large.
+Figure 5: Mean speed (blue line) and interpolation error (orange line) of different σ0’s. The
+interpolation error is the relative error between interpolated frames and data image of the
+last frame. The interpolation error reflects the closeness between interpolations from rOMT
+and the data image. Lower interpolation error means more accurate the rOMT is fitting
+the real data. This figure shows larger σ0 has better interpolation error but when σ0 goes
+to 0.2, the mean speed accelerates dramatically, which is unrealistic given previous data of
+the expected magnitude of solute transport in brain tissue.
+12
+
+。=0.00020.=0.002.=0.020.6
+0。=0.2
+0.5
+0.4
+0.3
+0.2
+0.10.25
+0.2
+meanspeed
+interpolation error
+0.15
+0.1
+0.05
+e
+0.00002
+0.0002
+0.002
+0.02
+0.2
+do5
+Discussion
+In this paper, we proposed a novel extension of the rOMT model. Specifically, we replaced
+the linear diffusion term in the advection-diffusion equation by a nonlinear diffusion term
+based on the Perona-Malik anisotropic diffusion approach. The updated model was tested
+on glymphatic DCE-MRI data comparing different parameter K’s in the conductivity co-
+efficient (σ) function and we observed that smaller K yields increased number of advective
+pathlines in CSF rich areas. More uniform advective solutes flow in the CSF compartment
+including at the level of the basal cisterns, ambient cistern and subarachnoid space above
+the cerebellum may be more biologically realistic.
+This paper only applied the model on glymphatic DCE-MRI data, but it can be generally
+applied to other types of biological imaging data.
+In the future, we plan to apply our
+approach to tumor vasculature imagery also derived from DCE-MRI, since the mass (tracer)
+is injected and may leak, we also plan to explore an unbalanced version of rOMT with
+nonlinear diffusion.
+Acknowledgments
+This research was funded in part by AFOSR grant FA9550-20-1-0029, NIH grant R01-
+AG048769, a grant from Breast Cancer Research Foundation BCRF-17-193, Army Research
+Office grant W911NF2210292, and a grant from the Cure Alzheimer’s Foundation.
+References
+[1] Jean-David Benamou and Yann Brenier. A computational fluid mechanics solution to
+the monge-kantorovich mass transfer problem. Numerische Mathematik, 84(3):375–393,
+2000.
+[2] Andrew J Bernoff, Andrea L Bertozzi, and Thomas P Witelski. Axisymmetric surface
+diffusion: dynamics and stability of self-similar pinchoff. Journal of statistical physics,
+93(3):725–776, 1998.
+[3] Xinan Chen, Xiaodan Liu, Sunil Koundal, Rena Elkin, Xiaoyue Zhu, Brittany Monte,
+Feng Xu, Feng Dai, Maysam Pedram, Hedok Lee, et al. Cerebral amyloid angiopathy
+is associated with glymphatic transport reduction and time-delayed solute drainage
+along the neck arteries. Nature Aging, 2(3):214–223, 2022.
+[4] Sandro Da Mesquita, Antoine Louveau, Andrea Vaccari, Igor Smirnov, R Chase Cor-
+nelison, Kathryn M Kingsmore, Christian Contarino, Suna Onengut-Gumuscu, Emily
+Farber, Daniel Raper, et al. Functional aspects of meningeal lymphatics in ageing and
+alzheimer’s disease. Nature, 560(7717):185–191, 2018.
+[5] Rena Elkin, Saad Nadeem, Eldad Haber, Klara Steklova, Hedok Lee, Helene Ben-
+veniste, and Allen Tannenbaum. Glymphvis: visualizing glymphatic transport path-
+ways using regularized optimal transport. In International Conference on Medical Im-
+age Computing and Computer-Assisted Intervention, pages 844–852. Springer, 2018.
+13
+
+[6] Jeffrey J Iliff, Hedok Lee, Mei Yu, Tian Feng, Jean Logan, Maiken Nedergaard, He-
+lene Benveniste, et al. Brain-wide pathway for waste clearance captured by contrast-
+enhanced mri. The Journal of clinical investigation, 123(3):1299–1309, 2013.
+[7] JR King. Emerging areas of mathematical modelling. Philosophical Transactions of the
+Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences,
+358(1765):3–19, 2000.
+[8] JR King. Two generalisations of the thin film equation. Mathematical and Computer
+modelling, 34(7-8):737–756, 2001.
+[9] Sunil Koundal, Rena Elkin, Saad Nadeem, Yuechuan Xue, Stefan Constantinou, Simon
+Sanggaard, Xiaodan Liu, Brittany Monte, Feng Xu, William Van Nostrand, et al.
+Optimal mass transport with lagrangian workflow reveals advective and diffusion driven
+solute transport in the glymphatic system. Scientific reports, 10(1):1–18, 2020.
+[10] Benjamin T Kress, Jeffrey J Iliff, Maosheng Xia, Minghuan Wang, Helen S Wei, Dou-
+glas Zeppenfeld, Lulu Xie, Hongyi Kang, Qiwu Xu, Jason A Liew, et al. Impairment
+of paravascular clearance pathways in the aging brain. Annals of neurology, 76(6):845–
+861, 2014.
+[11] Hedok Lee, Kristian Mortensen, Simon Sanggaard, Palle Koch, Hans Brunner, Bjørn
+Quistorff, Maiken Nedergaard, and Helene Benveniste. Quantitative gd-dota uptake
+from cerebrospinal fluid into rat brain using 3d vfa-spgr at 9.4 t. Magnetic resonance
+in medicine, 79(3):1568–1578, 2018.
+[12] Hedok Lee, Lulu Xie, Mei Yu, Hongyi Kang, Tian Feng, Rashid Deane, Jean Logan,
+Maiken Nedergaard, and Helene Benveniste.
+The effect of body posture on brain
+glymphatic transport. Journal of Neuroscience, 35(31):11034–11044, 2015.
+[13] Qiaoli Ma, Benjamin V Ineichen, Michael Detmar, and Steven T Proulx. Outflow of
+cerebrospinal fluid is predominantly through lymphatic vessels and is reduced in aged
+mice. Nature communications, 8(1):1–13, 2017.
+[14] Maiken Nedergaard. Garbage truck of the brain. Science, 340(6140):1529–1530, 2013.
+[15] Stanley Osher and James A Sethian. Fronts propagating with curvature-dependent
+speed: Algorithms based on hamilton-jacobi formulations. Journal of computational
+physics, 79(1):12–49, 1988.
+[16] Weiguo Peng, Thiyagarajan M Achariyar, Baoman Li, Yonghong Liao, Humberto
+Mestre, Emi Hitomi, Sean Regan, Tristan Kasper, Sisi Peng, Fengfei Ding, et al.
+Suppression of glymphatic fluid transport in a mouse model of alzheimer’s disease.
+Neurobiology of disease, 93:215–225, 2016.
+[17] Pietro Perona and Jitendra Malik. Scale-space and edge detection using anisotropic
+diffusion. IEEE Transactions on pattern analysis and machine intelligence, 12(7):629–
+639, 1990.
+[18] Vadim Ratner, Yi Gao, Hedok Lee, Rena Elkin, Maiken Nedergaard, Helene Ben-
+veniste, and Allen Tannenbaum. Cerebrospinal and interstitial fluid transport via the
+glymphatic pathway modeled by optimal mass transport. Neuroimage, 152:530–537,
+2017.
+14
+
+[19] Leonid I Rudin, Stanley Osher, and Emad Fatemi. Nonlinear total variation based
+noise removal algorithms. Physica D: nonlinear phenomena, 60(1-4):259–268, 1992.
+[20] James A Sethian. A review of recent numerical algorithms for hypersurfaces moving
+with curvature dependent speed. J. Differential Geometry, 31(31):131–161, 1989.
+[21] Klara Steklova and Eldad Haber. Joint hydrogeophysical inversion: state estimation
+for seawater intrusion models in 3d. Computational Geosciences, 21(1):75–94, 2017.
+[22] C´edric Villani. Optimal transport: old and new, volume 338. Springer, 2009.
+[23] C´edric Villani. Topics in optimal transportation, volume 58. American Mathematical
+Soc., 2021.
+[24] Jiajie Xu, Ya Su, Jiayu Fu, Xiaoxiao Wang, Benedictor Alexander Nguchu, Bensheng
+Qiu, Qiang Dong, and Xin Cheng. Glymphatic dysfunction correlates with severity of
+small vessel disease and cognitive impairment in cerebral amyloid angiopathy. European
+Journal of Neurology, 29(10):2895–2904, 2022.
+[25] Yu-Li You, Wenyuan Xu, Allen Tannenbaum, and Mostafa Kaveh. Behavioral analysis
+of anisotropic diffusion in image processing. IEEE Transactions on Image Processing,
+5(11):1539–1553, 1996.
+15
+

dtE1T4oBgHgl3EQfyAVc/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf,len=430
+page_content='Regularized Optimal Mass Transport with Nonlinear Diffusion  Kaiming Xu, Xinan Chen, Helene Benveniste, Allen Tannenbaum ∗†‡§  January 10, 2023  Abstract  In this paper, we combine nonlinear diffusion with the regularized optimal mass  transport (rOMT) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' As we will demonstrate, this new approach provides further  insights into certain applications of fluid flow analysis in the brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' From the point  of view of image processing, the anisotropic diffusion method, based on Perona-Malik,  explicitly considers edge information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Applied to rOMT analysis of glymphatic trans-  port based on dynamic contrast-enhanced magnetic resonance imaging data, this new  framework appears to capture a larger advection-dominant volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  1  Introduction  The theory of optimal mass transport(OMT) was first proposed by Gaspard Monge in 1781  and has since evolved into a unique scientific field which has had significant impact on  research in many disciplines [22, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Mass transport theory has been applied to diverse  fields including physics, biology, economics and engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' OMT defines a distance called  the Wasserstein distance, and thus creates a natural geometry on the space of probability  distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  Our study is based on a fluid dynamics reformulation of OMT [1] which  allows us to calculate the flow fields between two density distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  Regularized optimal mass transport (rOMT), an extension of fluid dynamics reformulation  of OMT, is a tool to study temporal flow fields as a physically inspired model of optical flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  It has the ability to capture the flow dynamics, handle noise and simulate diffusion [3, 5, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  rOMT utilizes an advection-diffusion equation as its flow-driven partial different equation  and is endpoint free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' A source term may be added to rOMT in which case the total mass  preservation condition can be circumvented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' This line of research will be pursued in other  work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  Anisotropic diffusion, a major tool for image segmentation, edge detection and image de-  noising, was first proposed by Perona and Malik [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Notably, instead of using a constant  diffusion coefficient, Perona and Malik considered a nonnegative function (conductivity  ∗K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Xu is with the Department of Applied Mathematics & Statistics, Stony Brook University, NY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' email:  kaiming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='xu@stonybrook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='edu  †X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Chen is with the Department of Medical Physics, Memorial Sloan Kettering Cancer Center, NY  ‡H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Benveniste is with the Department of Anesthesiology, Yale School of Medicine, CT  §A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Tannenbaum is with the Departments of Computer Science and Applied Mathematics & Statistics,  Stony Brook University, NY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' email: allen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='tannenbaum@stonybrook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='edu  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='03428v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='flu-dyn]  3 Dec 2022  coefficient) of the magnitude of the local density gradient;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' see equation (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' The authors  suggested two possible conductivity coefficients (see (9) and (10)), wherein the diffusion will  be very small near the edges, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' reflecting the fact that near edges images tend to have  very large intensity gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' In this work, we show that anisotropic diffusion enhances  the interpretation of glymphatic dynamic contrast-enhanced magnetic resonance imaging  (DCE-MRI) flow data and may be used in conjunction with the constant diffusion coefficient  approach [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' The anisotropic diffusion equation may be derived via the steepest descend  method for solving an energy minimization problem [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  The glymphatic system is involved in transporting waste products from the brain to the  meningeal lymphatic system which connects to the cervical lymph nodes [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' The function-  ing of the glymphatic and lymphatic systems decrease with age and have been implicated in  the pathophysiology of a wide range of neurodegenerative diseases including cerebral amy-  loid angiopathy [3, 24] and Alzheimer’s disease [4, 10, 13, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' We study glymphatic trans-  port using a temporal series of DCE-MRI data acquired from the rodent brain [6, 11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  Since the data are acquired at discrete time points, our work is motivated by the need to  find a dynamic physically based model of the transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Several different versions of OMT  [18] and rOMT [3, 5, 9] have been used to model the glymphatic flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  In the present work, we propose a new version of rOMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Specifically, we replace the lin-  ear diffusion in rOMT [3, 5, 9] with the Perona-Malik based anisotropic diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Here,  we argue that this gives us enhanced flexibility to study image-based flows inherent to  glymphatic transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Notably, many diffusion processes in fluids are better captured by  nonlinear models, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=', axisymmetric surface diffusion [2] and thin fluid films [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' We  utilize Lagrangian coordinates for visualizing the glymphatic transport pathlines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Several  properties of solute particle movement are computed along the pathlines such as speed and  the P´eclet number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Here we compare various parameters of the anisotropic diffusion coef-  ficient, and observe the impact of different values on several data metrics including P´eclet  plots which can map diffusion dominated versus advection dominated regions of the brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  We briefly summarize the contents of the present paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' In Section 2, we review the theory  of OMT, rOMT and nonlinear diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Section 3 introduces the algorithm and numerical  methods we employ for our current work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' In Section 4, we explicate the application of the  model to glymphatic DCE-MRI data and analyze the experimental results and we conclude  our paper in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  2  Model  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='1  OMT  In this section, we introduce OMT and its fluid dynamics formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' All the technical  details as well as a complete set of references may be found in [22, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  The original  formulation of OMT was given by Gaspard Monge and may be expressed as  inf  T {  �  Ω  c(x, T(x))ρ0(x)dx | T#ρ0 = ρ1},  (1)  where c(x, y) is the cost function of moving the unit mass from x to y, ρ0 and ρ1 are two  probability distributions in the domain Ω ⊆ Rd, T is the transport map, and T# is the  2  push-forward of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' This formulation assumes that ρ0 and ρ1 have the same total mass, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  �  Ω ρ0(x)dx =  �  Ω ρ1(x)dx and then seeks for the optimal transport map T to minimize the  total cost, the integral in equation (1), subject to the push-forward constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  Later, Leonid Kantorovich formulated a relaxed version of OMT as follows:  inf  π∈Π(ρ0,ρ1)  �  Ω×Ω  c(x, y)π(dx, dy),  (2)  where Π(ρ0, ρ1) denotes the set of all couplings (joint distributions) between the marginals  ρ0 and ρ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' From here on, the cost function c will be taken as the square of the Euclidean  distance c(x, y) = ∥x − y∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  Benemou and Brenier [1] proved that for c(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' y) = ∥x − y∥2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' the specific infimum of Monge-  Kantorovich formulation is equal to the result in following fluid dynamics formulation for  density/probability distributions with compact support:  inf  ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='v  � 1  0  �  Ω  ρ(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' x)|v(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' x)|2dxdt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  (3)  ∂ρ  ∂t + ∇ · (ρv) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  (4)  ρ(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' x) = ρ0(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  ρ(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' x) = ρ1(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  (5)  where ρ : [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' 1]×Ω → R≥0 is the family of density/probability distributions defining geodesic  path from ρ0 to ρ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' and v : [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' 1] × Ω → Rd is the velocity vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='2  rOMT  The regularized OMT model (rOMT) [5, 9] adds two assumptions: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' the image data we  use are noisy observations and thus we do not want to make the final density we calculate  coincide with the MR images;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' the flow is driven by an advection-diffusion equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  Based on these two assumptions, the rOMT formulation may be written as:  inf  ρ,v  � 1  0  �  Ω  ρ(t, x)|v(t, x)|2dxdt + β  �  Ω  (ρ(1, x) − ρ1(x))2dx,  (6)  ∂ρ  ∂t + ∇ · (ρv) = ∇ · (σ0∇ρ),  (7)  ρ(0, x) = ρ0(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  In this formulation, the final marginal condition is removed and a penalty of the error  between final density and ground truth is added in the objective function (6), where β is  the penalty parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Equation (7) is an advection-diffusion equation with a constant σ0  denoting the diffusion coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='3  Nonlinear diffusion  Instead of using linear diffusion in which σ0 is a constant, nonlinear diffusion seems to have  certain advantages that we will now describe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Perona and Malik proposed an anisotropic  3  diffusion [17], which is a useful tool for image segmentation, edge detection and image  denoising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' The anisotropic diffusion equation is  ∂ρ  ∂t = ∇ · (σ(|∇ρ|)∇ρ),  (8)  where σ(·) is a nonnegative strictly decreasing function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' If we consider a 3D problem, then  |∇ρ| =  �  ρ2x + ρ2y + ρ2z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' The proper diffusion should be large in smooth homogeneous areas  and become smaller near edges, the places where |∇ρ| is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  Perona and Malik [17]  suggested two versions of the diffusion (conductivity) coefficient:  σ(x) = σ0  1  1 + ( x  K )2 ,  (9)  σ(x) = σ0e−( x  K )2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  (10)  Both are 0 when x approaches ∞ and attend upper bound σ0 while x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' K is a constant  and controls the sensitivity to edges and can be tuned for different applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  Following [25], we may derive the anisotropic diffusion equation (8) via the steepest descent  from an energy minimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' More precisely, considering the following minimiza-  tion problem:  min  �  Ω  f(|∇ρ|)dΩ,  (11)  then the steepest descend equation may be computed to be  ∂ρ  ∂t = ∇ · (f′(|∇ρ| ∇ρ  |∇ρ|)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  (12)  Obviously, (12) is identical to (8) if  f′(x) = xσ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  (13)  For example, the corresponding f function of σ function (9) is  f(x) = σ0K2  2  ln[1 + ( x  K )2]  (14)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='4  rOMT with nonlinear diffusion  In this section, we present our new rOMT formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' We replace the diffusion in (7) by  anisotropic diffusion in (8) and obtain the following formulation:  inf  ρ,v  � 1  0  �  Ω  ρ(t, x)|v(t, x)|2dxdt + β  �  Ω  (ρ(1, x) − ρ1(x))2dx,  ∂ρ  ∂t + ∇ · (ρv) = ∇ · (σ(|∇ρ|)∇ρ),  (15)  ρ(0, x) = ρ0(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  One may employ various versions of the σ function and in this work, we choose the function  given in (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Note that, there are two parameters σ0 and K which may be tuned based on  the data we use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  4  Equation (15) may be written in conservation form as  ∂ρ  ∂t + ∇ · (ρ(v − σ(|∇ρ|)∇ log ρ)) = 0,  and after defining an augmented velocity  vaug = v − σ(|∇ρ|)∇ log ρ,  we derive a simple conservation form of equation (15)  ∂ρ  ∂t + ∇ · (ρvaug) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  The Lagrangian representation X = X(x, t) of the optimal trajectory for this rOMT with  nonlinear diffusion model is given by  X(x, 0) = x,  ∂X(x, t)  ∂t  = vaug  opt (X(x, t), t),  (16)  where  vaug  opt = vopt − σ(|∇ρopt|)∇ log ρopt,  (17)  and vopt and ρopt denote the optimal solution of the rOMT with nonlinear diffusion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  In Section 4, we exhibit the pathlines in Figure 2 and Figure 3 derived from the Lagrangian  coordinates (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  3  Numerical scheme  In this section, we focus on the numerical solution of the nonlinear diffusive rOMT model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  The pipeline that comes from [5, 9] is based on the Gauss-Newton method:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Give initial guess of v at each time and spatial point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Use v, ρ0 and the advection-diffusion equation (15) to calculate ρ at each subsequent  time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Calculate the objective function (6), which we will denote with Γ(v) as the discrete  form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Calculate the gradient g(v) and the Hessian matrix H(v) of Γ(v) with respect to v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Solve the descent direction s by solving H(v)s = −g(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Do line search to find l and update v by setting v = v + ls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Repeat step 2-6 until the results attain the final condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  Space is discretized into a cell-center grid of size nx × ny × nz with a total number of N  cells, each with width ∆x, height ∆y and depth ∆z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Time is divided into m intervals of  length ∆t with m + 1 time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Moreover, the superscript 0 corresponds to initial time  t = 0, M corresponds to final time t = 1 and dt × m = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' We use ρ = [(ρ0)T , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' , (ρm)T ]T  and v = [(v1)T , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' , (vm)T ]T to represent temporal density and velocity, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Note  that the velocity vi describes the velocity field from (i − 1)th time step to ith time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='1  Advection-diffusion equation  Here we describe the numerical scheme for equation (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  The discrete form of equation (15) between time tn and tn+1 is  ρn+1 − ρn  ∆t  + A(ρ, v) = D(ρ),  (18)  where A and D are discretizations of advective and diffusive terms, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' We will  describe these in greater detail below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Following the work of Steklova and Haber [21], we  split equation (18) into two parts,  ρadv − ρn  ∆t  + A(ρ, v) = 0,  (19)  ρn+1 − ρadv  ∆t  = D(ρ),  (20)  where ρadv is an auxiliary variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Simply by adding (19) and (20), we obtain the equation  (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  So far we have not chosen the time step of ρ in the advective part A(ρ, v) and  diffusive part D(ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' We use a standard forward scheme, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' ρ = ρn in our implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  Summarizing up to this point, to solve for the next time step density ρn+1, we first calculate  ρadv by solving equation (19) and use ρadv and ρn to calculate ρn+1 following equation  (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  For the advective part A(ρ, v), we utilize a particle-in-cell method which is also how Steklova  and Haber[21] dealt with their advective part to solve equation (19):  ρadv = S(v)ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  (21)  S(v) is the averaging matrix with respect to v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  The basic idea of particle-in-cell method is moving density the ρi in the cell center to  the target ρnew  i  according to its velocity vi and using its nearest neighbor cell centers to  interpolate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  The numerical techniques of solving equation (20) are based on hyperbolic conservation  laws and the theory of viscosity solutions [15, 19, 20], and we explicitly write D in the next  section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='2  Anisotropic diffusion  From now on, we explore in 3-dimension (d = 3), following [19, 25] and discretize the  anisotropic diffusion as follows:  D(ρi,j,k) = ∆x  −{σ[  �  (∆x  +ρi,j,k)2 + m2(∆y  +ρi,j,k, ∆y  −ρi,j,k) + m2(∆z  +ρi,j,k, ∆z  −ρi,j,k)]∆x  +ρi,j,k}  + ∆y  −{σ[  �  (∆y  +ρi,j,k)2 + m2(∆x  +ρi,j,k, ∆x  −ρi,j,k) + m2(∆z  +ρi,j,k, ∆z  −ρi,j,k)]∆y  +ρi,j,k}  + ∆z  −{σ[  �  (∆z  +ρi,j,k)2 + m2(∆x  +ρi,j,k, ∆x  −ρi,j,k) + m2(∆y  +ρi,j,k, ∆y  −ρi,j,k)]∆z  +ρi,j,k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  (22)  6  Here,  ∆x  −ai,j,k = ai,j,k − ai−1,j,k  ∆x  ,  ∆x  +ai,j,k = ai+1,j,k − ai,j,k  ∆x  ,  ∆y  −ai,j,k = ai,j,k − ai,j−1,k  ∆y  ,  ∆y  +ai,j,k = ai,j+1,k − ai,j,k  ∆y  ,  ∆z  −ai,j,k = ai,j,k − ai,j,k−1  ∆z  ,  ∆z  +ai,j,k = ai,j,k+1 − ai,j,k  ∆z  ,  m(a, b) = median(a, b, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  We note that the solution of equation (18) may be written recursively:  ρn+1 = S(vn)ρn + ∆tD(ρn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  (23)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='3  Objective function Γ(v)  A straightforward way [5, 9] to discretize the objective function Γ(v) in (6) is  hd ∗ ∆t ∗ ρT (Im ⊗ [IN|IN|IN])(v ⊙ v) + β|ρm − ρT |2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  (24)  Here hd = ∆x∗∆y∗∆z, ρ, v are column vectors, ⊗ is Kronecker product and ⊙ is Hadamard  product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='4  Gradient, hessian and sensitivity  In order to apply the Gauss-Newton minimization procedure such as described in Steklova  and Haber [21], we need expressions for the gradient g(v) and the Hessian H(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Taking  the gradient of (24) with respect to v, we find  g(v) = ∂Γ(v)  ∂v  = hd ∗ ∆t ∗ [2ρT Mdiag(v) + (M(v ⊙ v))T J] + β(ρm − ρ1)T ∂ρm  ∂v ,  (25)  where M = Im ⊗ [IN|IN|IN], matrix J = (Jk  j ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Here Jk  j = ∂ρk  ∂vj , k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' , m and j =  0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' , m − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  The Hessian matrix is  H(v) = ∂g  ∂v = hd∗∆t∗[2ρT ∇(Mdiag(v))+2∇(ρ)Mdiag(v)+M(v⊙v)∇J +∇[M(v⊙v)]J]  + β[(∂ρm  ∂v )T (∂ρm  ∂v ) + (ρm − ρ1)∂2ρm  ∂v2 ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  (26)  Numerically we approximate the Hessian by  H(v) = 2hd ∗ ∆t ∗ ρT ∇(Mdiag(v)) + β(∂ρm  ∂v )T (∂ρm  ∂v )  = 2hd ∗ ∆t ∗ diag(ρT M) + β(∂ρm  ∂v )T (∂ρm  ∂v ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  (27)  In the formulae for the gradient (25) and Hessian (27), we still need to know the sensitivity  of the density ρ with respect to the velocity v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' We recall equation (23)  ρn+1 = S(vn)ρn + ∆tD(ρn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  7  From that, the sensitivity can be calculated as below:  ∂ρk  ∂vj =  �  �  �  S(vk−1) ∂ρk−1  ∂vj  + ∆tD′(ρk−1) ∂ρk−1  ∂vj  k ≥ j + 2  ∂  ∂vj (S(vj)ρj)  k = j + 1  0  k ≤ j  (28)  4  Experimental results  In this section, we test our proposed methodology on 3D DCE-MRI data derived from  [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' In this dataset, rats were anesthetized, and a Gd-tagged tracer was injected into the  cerebrospinal fluid (CSF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' The rat underwent dynamic 3D MRI scanning every 5 minutes  to collect a total 29 3D brain images with a voxel size of 100×106×100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Post-processing of  the DCE-MRI data included head motion correction, intensity normalization, and voxel-by-  voxel conversion to percentage of baseline signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' In our experiment, we chose a 12-month-  old wild type rat for demonstrating the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  The new algorithm was run for data covering a 100-minute time period (60 minutes to  160 minutes) which includes 23 frames, and we used every other image as inputs to reduce  runtime, leaving 12 frames for the numerical experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  We use In, n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' , 12 to  represent these frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' To derive the interpolations, we applied our model between each of  two consecutive frames, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Ik and Ik+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' To ensure continuity, (except for the first step),  the initial density originates from the previous step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' For example, if we are considering  the problem between I2 and I3, and we will use the final density I′  2 calculated between I1  and I2 as the new initial density here and apply our model between I′  2 and I3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' One of the  metrics that can measure the model accuracy is the error between the final density I′  k and  the ground truth Ik at each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  Here we are using σ function (9) with σ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' The choice of σ0 follows [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' We tested  rOMT on the 3D DCE-MRI data set with σ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='00002, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='0002, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='002, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='02, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' The speed  maps in Figure 4 show a stable trend between σ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='00002 and σ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='002 and among  these three σ0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='00002,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='0002 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='002), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='002 has the minimal interpolation error (see  Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  We computed pathlines based on Lagrangian coordinates (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' We compared different K’s  and the results are shown in Figures 1-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Figure 1 shows the relative error  e = |I′ − I|2  |I|2  on each frame with different K’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' The x-axis represents the indices of frames and the y-  axis is the relative error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' From Figure 1, we observe that rOMT with anisotropic diffusion  has similar accuracy as the original rOMT model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Figure 2 compares the P´eclet number  along pathlines in the right lateral view plane for different K’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Further, Figure 3 shows  the ventral surface of the brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Red color represents larger P´eclet numbers (advection  dominant) and blue represents smaller P´eclet numbers (diffusion dominant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' As shown in  Figure 2 and Figure 3, a smaller K value results in more advection dominated transport  in ‘surface’ areas of the brain which corresponds to the CSF compartment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' When we set  K = ∞, then clearly σ(x) = σ0, since  lim  K→∞ σ0  1  1 + ( x  K )2 = σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  8  Figure 1: Relative interpolation error plot for different parameter K’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  Original means  constant diffusion coefficient, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' K = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  9  relativeinterpolationerror  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='15  original  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='1  K=10  K=100  K=1000  K=10000  K=100000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='05  K=1000000  0  1  2  3  4  5  6  7  8  9  10  11Figure 2: Pathlines endowed with P´eclet Number shown in the lateral view plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Parameter  K = 10, 100, 1000, 10000, 100000, ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' The maximal limit of color bar is 300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' When K is  small, the advective (red) pathline dominates in CSF rich areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  10  K=10  Pseudocolor  Var: PathPoint  300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='0  225.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='0  150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='0  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='000  Max: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='210e+13  Min: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='000K=100  Pseudocolor  Var: PathPoint  300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
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+page_content=' The first  three speed maps exhibit a stable trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
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+page_content='  Figure 5: Mean speed (blue line) and interpolation error (orange line) of different σ0’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' The  interpolation error is the relative error between interpolated frames and data image of the  last frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
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+page_content=' The updated model was tested  on glymphatic DCE-MRI data comparing different parameter K’s in the conductivity co-  efficient (σ) function and we observed that smaller K yields increased number of advective  pathlines in CSF rich areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' More uniform advective solutes flow in the CSF compartment  including at the level of the basal cisterns, ambient cistern and subarachnoid space above  the cerebellum may be more biologically realistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  This paper only applied the model on glymphatic DCE-MRI data, but it can be generally  applied to other types of biological imaging data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  In the future, we plan to apply our  approach to tumor vasculature imagery also derived from DCE-MRI, since the mass (tracer)  is injected and may leak, we also plan to explore an unbalanced version of rOMT with  nonlinear diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  Acknowledgments  This research was funded in part by AFOSR grant FA9550-20-1-0029, NIH grant R01-  AG048769, a grant from Breast Cancer Research Foundation BCRF-17-193, Army Research  Office grant W911NF2210292, and a grant from the Cure Alzheimer’s Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  References  [1] Jean-David Benamou and Yann Brenier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' A computational fluid mechanics solution to  the monge-kantorovich mass transfer problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Numerische Mathematik, 84(3):375–393,  2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  [2] Andrew J Bernoff, Andrea L Bertozzi, and Thomas P Witelski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Axisymmetric surface  diffusion: dynamics and stability of self-similar pinchoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Journal of statistical physics,  93(3):725–776, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  [3] Xinan Chen, Xiaodan Liu, Sunil Koundal, Rena Elkin, Xiaoyue Zhu, Brittany Monte,  Feng Xu, Feng Dai, Maysam Pedram, Hedok Lee, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Cerebral amyloid angiopathy  is associated with glymphatic transport reduction and time-delayed solute drainage  along the neck arteries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Nature Aging, 2(3):214–223, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  [4] Sandro Da Mesquita, Antoine Louveau, Andrea Vaccari, Igor Smirnov, R Chase Cor-  nelison, Kathryn M Kingsmore, Christian Contarino, Suna Onengut-Gumuscu, Emily  Farber, Daniel Raper, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Functional aspects of meningeal lymphatics in ageing and  alzheimer’s disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Nature, 560(7717):185–191, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  [5] Rena Elkin, Saad Nadeem, Eldad Haber, Klara Steklova, Hedok Lee, Helene Ben-  veniste, and Allen Tannenbaum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Glymphvis: visualizing glymphatic transport path-  ways using regularized optimal transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' In International Conference on Medical Im-  age Computing and Computer-Assisted Intervention, pages 844–852.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Springer, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  13  [6] Jeffrey J Iliff, Hedok Lee, Mei Yu, Tian Feng, Jean Logan, Maiken Nedergaard, He-  lene Benveniste, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Brain-wide pathway for waste clearance captured by contrast-  enhanced mri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' The Journal of clinical investigation, 123(3):1299–1309, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  [7] JR King.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Emerging areas of mathematical modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Philosophical Transactions of the  Royal Society of London.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Series A: Mathematical, Physical and Engineering Sciences,  358(1765):3–19, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  [8] JR King.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Two generalisations of the thin film equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Mathematical and Computer  modelling, 34(7-8):737–756, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  [9] Sunil Koundal, Rena Elkin, Saad Nadeem, Yuechuan Xue, Stefan Constantinou, Simon  Sanggaard, Xiaodan Liu, Brittany Monte, Feng Xu, William Van Nostrand, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  Optimal mass transport with lagrangian workflow reveals advective and diffusion driven  solute transport in the glymphatic system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Scientific reports, 10(1):1–18, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  [10] Benjamin T Kress, Jeffrey J Iliff, Maosheng Xia, Minghuan Wang, Helen S Wei, Dou-  glas Zeppenfeld, Lulu Xie, Hongyi Kang, Qiwu Xu, Jason A Liew, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Impairment  of paravascular clearance pathways in the aging brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Annals of neurology, 76(6):845–  861, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  [11] Hedok Lee, Kristian Mortensen, Simon Sanggaard, Palle Koch, Hans Brunner, Bjørn  Quistorff, Maiken Nedergaard, and Helene Benveniste.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Quantitative gd-dota uptake  from cerebrospinal fluid into rat brain using 3d vfa-spgr at 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='4 t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Magnetic resonance  in medicine, 79(3):1568–1578, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  [12] Hedok Lee, Lulu Xie, Mei Yu, Hongyi Kang, Tian Feng, Rashid Deane, Jean Logan,  Maiken Nedergaard, and Helene Benveniste.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  The effect of body posture on brain  glymphatic transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Journal of Neuroscience, 35(31):11034–11044, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  [13] Qiaoli Ma, Benjamin V Ineichen, Michael Detmar, and Steven T Proulx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Outflow of  cerebrospinal fluid is predominantly through lymphatic vessels and is reduced in aged  mice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Nature communications, 8(1):1–13, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  [14] Maiken Nedergaard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Garbage truck of the brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Science, 340(6140):1529–1530, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  [15] Stanley Osher and James A Sethian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Fronts propagating with curvature-dependent  speed: Algorithms based on hamilton-jacobi formulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Journal of computational  physics, 79(1):12–49, 1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  [16] Weiguo Peng, Thiyagarajan M Achariyar, Baoman Li, Yonghong Liao, Humberto  Mestre, Emi Hitomi, Sean Regan, Tristan Kasper, Sisi Peng, Fengfei Ding, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  Suppression of glymphatic fluid transport in a mouse model of alzheimer’s disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  Neurobiology of disease, 93:215–225, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  [17] Pietro Perona and Jitendra Malik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Scale-space and edge detection using anisotropic  diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' IEEE Transactions on pattern analysis and machine intelligence, 12(7):629–  639, 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  [18] Vadim Ratner, Yi Gao, Hedok Lee, Rena Elkin, Maiken Nedergaard, Helene Ben-  veniste, and Allen Tannenbaum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Cerebrospinal and interstitial fluid transport via the  glymphatic pathway modeled by optimal mass transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Neuroimage, 152:530–537,  2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  14  [19] Leonid I Rudin, Stanley Osher, and Emad Fatemi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Nonlinear total variation based  noise removal algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Physica D: nonlinear phenomena, 60(1-4):259–268, 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  [20] James A Sethian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' A review of recent numerical algorithms for hypersurfaces moving  with curvature dependent speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Differential Geometry, 31(31):131–161, 1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  [21] Klara Steklova and Eldad Haber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Joint hydrogeophysical inversion: state estimation  for seawater intrusion models in 3d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Computational Geosciences, 21(1):75–94, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  [22] C´edric Villani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Optimal transport: old and new, volume 338.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Springer, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  [23] C´edric Villani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Topics in optimal transportation, volume 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' American Mathematical  Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=', 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  [24] Jiajie Xu, Ya Su, Jiayu Fu, Xiaoxiao Wang, Benedictor Alexander Nguchu, Bensheng  Qiu, Qiang Dong, and Xin Cheng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Glymphatic dysfunction correlates with severity of  small vessel disease and cognitive impairment in cerebral amyloid angiopathy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' European  Journal of Neurology, 29(10):2895–2904, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  [25] Yu-Li You, Wenyuan Xu, Allen Tannenbaum, and Mostafa Kaveh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' Behavioral analysis  of anisotropic diffusion in image processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content=' IEEE Transactions on Image Processing,  5(11):1539–1553, 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}
+page_content='  15' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfyAVc/content/2301.03428v1.pdf'}

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@@ -0,0 +1,1307 @@
+1 
+ 
+Multifunctional Fiber-based Optoacoustic Emitter for Non-genetic 
+Bidirectional Neural Communication 
+Author Information 
+Nan Zheng1, Ying Jiang2, Shan Jiang3, Jongwoon Kim3, Yueming Li4, Ji-Xin Cheng2, 6, Xiaoting Jia3 * 
+and Chen Yang5, 6 * 
+Affiliations 
+1 Division of Materials Science and Engineering, Boston University, Boston, MA, USA 
+2 Department of Biomedical Engineering, Boston University, Boston, MA, USA 
+3 Bradley Department of Electrical and Computer Engineering, Virginia Tech, Blacksburg, VA, 
+USA 
+4 Department of Mechanical Engineering, Boston University, Boston, MA, USA 
+5 Department of Chemistry, Boston University, Boston, MA, USA 
+6 Department of Electrical and Computer Engineering, Boston University, Boston, MA, USA 
+Contributions 
+C.Y. conceived the project. N.Z. and S.J. performed fabrication and characterization of materials. N.Z. 
+and Y.J. performed the stimulation and recording experiments in vitro and in vivo. N.Z. and Y.L. 
+performed the in vivo biocompatibility evaluations. X.J. provided guidance on the multifunctional fiber 
+system. J.X.C. provided guidance on the design of fiber optoacoustic emitter. J.K. provided guidance on 
+optimization of recording and data analysis. The manuscript was written through contributions of all 
+authors. All authors have given approval to the final version of the manuscript. 
+Corresponding author 
+Correspondence to: Chen Yang ([email protected]) and Xiaoting Jia ([email protected]) 
+ 
+ 
+
+2 
+ 
+Abstract 
+A bidirectional brain interface with both “write” and “read” functions can be an important tool for 
+fundamental studies and potential clinical treatments for neurological diseases. Here we report a 
+miniaturized multifunctional fiber based optoacoustic emitter (mFOE) that first integrates simultaneous 
+non-genetic optoacoustic stimulation for “write” and electrophysiology recording of neural circuits for 
+“read”. The non-genetic feature addresses the challenges of the viral transfection required by optogenetics 
+in primates and human. The orthogonality between optoacoustic waves and electrical field provides a 
+solution to avoid the interference between electrical stimulation and recording. We first validated the non-
+genetic stimulation function of the mFOE in rat cultured neurons using calcium imaging. In vivo 
+application of mFOE for successful simultaneous optoacoustic stimulation and electrical recording of 
+brain activities was confirmed in mouse hippocampus in both acute and chronical applications up to 1 
+month.  Minimal brain tissue damage has been confirmed after these applications. The capability of non-
+genetic neural stimulation and recording enabled by mFOE opens up new possibilities for the 
+investigation of neural circuits and brings new insights into the study of ultrasound neurostimulation.  
+Introduction 
+Bidirectional communication with dynamic local circuits inside the brain of individual behaving animals 
+or humans has been an invaluable approach for fundamental studies of neural circuits and for effective 
+clinical treatment of neurological diseases, like epilepsy, Parkinson’ s disease, and depression1, 2.  
+Additionally, bidirectional neural interface paves the way for the closed-loop control, as  it could enable 
+more sophisticated, real-time control over neural dynamics3, behaviors4 and achieve effective therapeutic 
+effect in neurological disease5, 6. To achieve real time assessment of the stimulated outcome, neural 
+interfaces with ability to simultaneously manipulate and directly monitor the neural activities are 
+preferred. Among the technologies developed in past decades, electrical stimulation and 
+electrophysiology recording have been widely used and forms the basis of current implantable devices, 
+which has been applied to clinical applications7. For example, to restore both the motor and sensory 
+
+3 
+ 
+modalities, electric stimulation of the cortical surface is often associated with electrophysiology 
+recording8, 9, like electrocorticography (ECoG). Also, the bidirectional electrical stimulation has 
+demonstrated promising treatment effect in neurological diseases, such as epilepsy. The responsive focal 
+cortical stimulation (RNS), leveraging ECoG recording as the trigger to provide stimulation, showed a 
+statistically significantly greater reduction in seizure frequency and the benefits increased over time in a 
+two-year study10, 11. However, electrical stimulation has a limited spatial resolution due to current spread. 
+It also interferes with the electrical signals used for recording, leading to “contamination” in 
+electrophysiology recording2, 12. Although researchers are improving its performance through 
+technologies such as current steering13, novel electrode design14, and artifacts cancellation15, considering 
+the intrinsic physical properties of brain tissue16, the current spread, root cause of above-mentioned issues 
+is hard to be fully eliminated. Therefore, electrical stimulation for the bidirectional communication of 
+brain may not be the ideal candidate.  
+Being orthogonal with electrical recording, optical stimuli not only avoids the interference but 
+also enables a high spatial resolution. To take this advantage, early efforts developed so-called 
+optoelectrodes by simply assembling the optical fibers for optogenetics stimulation with the electrodes, 
+such as Utah arrays17-19, Michigan probes20, 21 and microwires22. Semiconductor fabrication techniques 
+and multiple material processing methods have recently been applied to improve the integration of those 
+bidirectional devices. New processing techniques not only make the device more compact but also 
+strengthen its functionality and biocompatibility. For example, monolithically integrated micro-light-
+emitting-diodes (µLEDs) were used to reduce the complexity of light-guide structures and significantly 
+boosted the number of stimulation sites and stimulation resolution 23, 24. Alternatively, a high-throughput 
+thermal drawing method has been used to integrate the function components, for example, electrodes, 
+microfluidic channels, and optical waveguides, to the flexible multifunctional polymer fiber 25, 26. 
+Through this approach, the flexible fiber probes showed low bending-stiffness and enabled 
+multifunctionalities, including optogenetics, electrical recording and drug delivery 27-29. Since 
+
+4 
+ 
+optogenetics relies on the expression of light-sensitive opsins in neurons through gene modification26, it is 
+challenging to apply optogenetics to non-human primates and human effectively and safely30.  
+Recently, our team showed non-genetic optoacoustic neural stimulation with a high spatial 
+resolution up to  single neuron level31, 32. In an optoacoustic process, the pulsed light is illuminated on an 
+absorber, causing transient heating and thermal expansion, and generating broadband acoustic pulses at 
+ultrasonic frequencies33, 34. As a light mediated neural modulation method, optoacoustic is an ideal 
+candidate to work with electrical recording for bidirectional neural communication. Compared with 
+existing technologies, it exhibited the advantages as a light mediated method, including a high spatial 
+resolution and minimal crosstalk noise with electrical recording. Importantly, the non-genetic 
+optoacoustic neurostimulation alleviates the challenges and safety concern in optogenetics since no viral 
+transfection is required.  
+Here, we developed a multifunctional fiber-based optoacoustic emitter (mFOE) as a miniaturized 
+bidirectional brain interface performing simultaneously non-genetic neural stimulation and electrical 
+recording of the neural activities. Through a thermal drawing process,25, 35 fabrication of mFOE integrated 
+an optical waveguide and multiple electrodes within a single fiber with a total diameter of 300 µm, 
+compatible to the typical size of silica fibers used in optogenetic studies. An optoacoustic coating was 
+selectively deposited to the tip of the core optical waveguide in the mFOE through a controlled micro-
+injection process. Upon nanosecond pulse laser delivered to the photoacoustic coating, the mFOE 
+generates a peak-to-peak pressure greater than 1 MPa, confirmed by the hydrophone measurement, which 
+is sufficient for successful neural stimulation in vitro and in vivo. By calcium imaging, the optoacoustic 
+stimulation function of the mFOE was validated in Oregon green-loaded rat primary neurons. 
+Importantly, we demonstrated the reliable functions of the chronic implanted mFOE for simultaneously 
+stimulating and recording neurons in mouse hippocampus. Chronic recording also demonstrated that the 
+embedded electrodes could provide long-term neural monitoring with a single-unit resolution. The 
+histological evaluation of the brain tissue response confirmed that our flexible mFOE established a stable 
+
+5 
+ 
+and biocompatible multifunctional neural interface. mFOE is the first device integrated both optoacoustic 
+stimulation with electrical recording for bidirectional neural communication. With the bidirectional 
+capabilities and excellent biocompatibility, it offers a non-generic tools probing brain circuits, alternative 
+to the optoelectrode devices, with improved feasibility in non-human primates and human. It also opens 
+up potentials for closed-loop neural stimulation and brain machine interface.    
+Results  
+Design, fabrication and characterization of mFOE  
+Towards bidirectional neural communication, we have designed the mFOE to utilize the optoacoustic 
+stimulation as “writing” and electrophysiological recording as “reading” of the neural interface (Fig. 1a). 
+Previously, fiber based optoacoustic emitters have been developed as a miniature invasive ultrasound 
+transducer for the biomedical applications, such as intravascular imaging and interventional cardiology36, 
+37. Recently, our work showed that fiber based optoacoustic emitters can also be applied to neural 
+stimulation in vitro and in vivo, with single neuron resolution and dual site capability32, 38. In these 
+studies, typically commercial silica fibers were used, together with optoacoustic coating. However, the 
+silica fiber, with Young’s modulus of ~70 GPa, is mismatched with mechanical properties of native 
+neural tissue (kilo- to mega pascals)2 and not easy to integrate with miniaturized electrodes for recording. 
+In this study, we took advantage of the fiber fabrication method developed by Anikeeva and Yoel25, and 
+utilized the polymer multifunctional fiber design as the base for the mFOE to delivering nanosecond laser 
+to the optoacoustic coating and to record electrical signals. Specifically, a multifunctional fiber with a 
+core optical waveguide and miniaturized electrodes was fabricated using the thermal drawing process 
+(TDP) as previously reported27 (Fig. 1b). The waveguide is made of polycarbonate core (PC, refractive 
+index nPC = 1.586, diameter = 150 µm) and polyvinylidene difluoride cladding (PVDF, refractive index 
+nPVDF = 1.426, thickness = 50 µm) as the core and the shell, respectively (Fig. 1c). BiSn alloy is used in 
+surrounding electrodes with diameters of 35 µm because of its conductivity and compatibility with TDP 
+
+6 
+ 
+(Fig. 1c).This multifunctional fiber showed broadband transmission across the visible range to near 
+infrared region and sub-megaohm impedance when it has been prepared into two centimetres long27, 39.  
+To integrate the optoacoustic converter to the multifunctional fiber, the optoacoustic coating, 
+composed of light absorbers and thermal expansion matrix, is needed to be selectively coated on the core 
+waveguide distal end while keeping the surrounding electrodes exposed and conducting. Compared to 
+previously reported FOE fabrication, here we took several innovative steps. First, a pressure-driven pico-
+litter injector was used to precisely deposit the optoacoustic materials to the core waveguide distal end. 
+The coating area was controlled through varying the injection volume (0.1 – 0.5 nL), which is controlled 
+by the regulated pressure (2-4 psi) over a set period of time (1-2 s, Supplementary Fig. S1) as described in 
+equation (1),  
+������������ = ������������ ∙ ������������������������������������������������������������������������
+3
+∙ ������������ ∙ ������������ 
+ 
+ 
+ 
+ 
+ 
+(1) 
+where ������������ is the injection volume, ������������ is a constant attributed to the unit conversion factors, effects of liquid 
+viscosity and the taper angle of micropipette, ������������������������������������������������������������������������ is the inner diameter of the pico-litter injector, ������������ is 
+the pressure, and ������������ is the deposition time. Two 3D translational stages with stereo microscopes were used 
+to precisely control the deposition localization.  Second, instead of using carbon nanotubes (CNT), we 
+used carbon black (CB) embedded polydimethylsiloxane (PDMS) as the composite optoacoustic material. 
+CB exhibited similar wideband light absorption40, assuring the sufficient photoacoustic conversion for 
+neural stimulation. Importantly, due to its relative low viscosity 41, 42, CB/PDMS composite shows much 
+higher injectability compared with CNT/PDMS, therefore more comparable to the pico-liter deposition 
+process.  Through these steps, we successfully coated 10-20 µm thick 10% w/w CB/PDMS composite 
+onto the 150 µm diameter core waveguide distal end while electrodes were still exposed as shown in Fig. 
+1e. Collectively mFOE with the photoacoustic emitter and multiple electrodes has been successfully 
+fabricated.  
+To characterize the optoacoustic performance of mFOE, a Q-switched 1030 nm pulsed 
+nanosecond laser was applied with pulse energies of 16.6 µJ, 27.3 µJ and 41.8 µJ, respectively.  The 
+
+7 
+ 
+generated acoustic waves were measured by a 40 µm needle hydrophone placed at about 100 µm away 
+from the fiber tip. Representative pulse acoustic pulse with a width of approximately 0.08 µs was 
+generated by a single laser pulse as shown in Fig. 1f. Higher input laser pulse energy led to larger acoustic 
+pressure. A peak-to-peak pressure of 1.0, 1.6 and 2.3 MPa were measured with the pulse energy of 16.6, 
+27.3 and 41.8 µJ, respectively.  The frequency spectrum shows the broadband characteristic of typical 
+optoacoustic waves34, and the peak frequencies are around 12.5 MHz (Fig. 1g). Based on previous work, 
+we expected that such pressure and frequency is capable to successfully stimulate neurons in vitro and in 
+vivo.  We also calculated the mechanical index (MI), a commonly used matrix, to evaluate the probability 
+of mechanical damage due to ultrasound generated. The MI of acoustic waves generated by 2.3 MPa is 
+0.198, lower than 1.9, the safety threshold suggested by the Food and Drug Administration (FDA) safety 
+guidelines.  
+
+8 
+ 
+Figure. 1 Design, fabrication and characterization of mFOE  
+a.  Schematic of mFOE for bidirectional communication with neurons. Input laser pulse (red) is used to 
+generate optoacoustic waves (black) by the converter and the neural activities are recorded by embedded 
+electrodes as the output electrical signal (blue). b. Illustration of the thermal drawing 
+process. c. Components of the multifunctional fiber, including a PC/PVDF waveguide, BiSn alloy 
+electrodes and PC sacrifice layer. d.  The selective deposition process for integrating the optoacoustic 
+converter to the core wave guide in the multifunctional fiber. A pressure-driven micro-injector is used to 
+control the volume of CS/PDMS deposited. 3D translation stages and microscope are used to control the 
+deposition location. Zoom-in: The micro pipette was aligned to the center of the fiber under the microscope. 
+e. Top view microscope image of the mFOE. Scale bar: 100 µm. f. Representative acoustic waveforms 
+under different laser pulse energy recorded by a needle hydrophone. g. Frequency spectrum of acoustic 
+waveforms shown in f. 
+mFOE stimulation of cultured primary neurons 
+
+a
+ptoacousticwave
+Pulsed laser
+Electrical signal
+q
+C
+d
+Injector on 3D stage
+Wave guide
+Micro pipette
+(PC/PVDF)
+Heat
+Fiber
+Sacrificelayer(PC)
+Vdrawing speed
+Electrode (BiSn alloy)
+Fiber holder
+on 3D stage
+e
+Optoacoustic
+g
+0.4
+41.8 μJ
+41.8 μJ
+converter
+1.5
+27.3 μJ
+27.3 μJ
+Pressure (MPa)
+16.6 μJ
+0.3
+16.6μJ
+(a.u
+Magnitude
+0.5
+0.2
+0
+0.1
+0.5
+Electrode
+0
+0
+0.1
+0.2
+0.3
+0.4
+0
+20
+40
+60
+80
+100
+Time (μs)
+Frequency (MHz)9 
+ 
+To investigate mFOE can directly trigger the neuronal activity, we examined the response of cultured 
+primary neurons under mFOE stimulation. Because of the presence of calcium channels in neuronal 
+membrane and their activation during the depolarization, calcium imaging has been widely used to 
+monitor the neuronal activities43, 44. Here, we cultured and loaded the rat cortical neurons (days in vitro 
+10-14) with a calcium indicator, Oregon Green™ 488 BAPTA-1 dextran (OGD-1)45 , and performed the 
+calcium imaging with an inverted wide-field fluorescence microscope (Supplementary Fig. S2). To 
+perform the optoacoustic stimulation, mFOE was placed approximately 50 µm above the in-focus target 
+neurons (Fig. 2a) by a micromanipulator under the microscope. 1030 nm 3 ns pulsed laser with a 
+repetition rate of 1.7 kHz was delivered to the mFOE through an optical fiber. The energy of laser pulse 
+was 41.8 µJ, corresponding to a peak-to-peak pressure of 2.3 MPa generated. Lower energy was tested 
+but did not induce calcium transient. The stimulation duration determined by each laser burst was 100 ms, 
+corresponding to 170 pulses (Supplementary Fig. S3). By applying 5 bursts of laser pulses with interval 
+of 1s, we investigated the reproducibility of the stimulation.  
+Using calcium imaging, we monitored the activities of all neurons in the field of view and divided 
+them into two groups: groups within the converter area (Fig. 2b) and outside the converter area (Fig. 2c). 
+For neurons within the converter area, i.e. the 100 µm from the center of the mFOE, Fig. 2b shows that 8 
+of 10 neurons showed successful and repeatable calcium transient (ΔF/F > 1%, the baseline standard 
+deviation) corresponding to each stimulation. Calcium transients are also repeatable for each burst applied 
+over the 1 s period, indicating the evoked neuronal activities and confirming the reliability and biosafety 
+of mFOE stimulation.  For neurons outside the converter area, only 2 of 10 neurons responded. This result 
+also suggested the mFOE with the 150 µm center waveguide with photoacoustic coating provided a 
+spatial precision of ~200 µm for stimulation in vitro. This observation is consistent with that fiber based 
+optoacoustic converters generate a confined ultrasound fields with sizes comparable with the radius of 
+converter31. 
+Next, to investigate the threshold of mFOE stimulation, we varied the stimulation duration from 5 
+ms, 50 ms, 100 ms to 200 ms on neurons in different cultures (N = 15) under the same laser pulse energy 
+
+10 
+ 
+of 41.8 µJ and the same repetition rate of 1.7 kHz. mFOE stimulation with duration of 5 ms did not 
+evoked any observable fluorescence change (n.s., p > 0.05) (Fig. 2g). Only when the duration was 50 ms 
+or longer, the mFOE successfully produced neural activation (ΔF/F > 1%, p < 0.01) as shown in Fig. 2d-f, 
+and Fig. 2h. Longer pulse durations leads to larger peak fluorescence changes, from 2.9 ± 1.1%, 6.0 ± 
+2.8% to 7.8 ± 1.3% corresponding to 50 ms, 100 ms and 200 ms, respectively. For the longest stimulation 
+duration of 200 ms tested, no obvious change on morphology or elevation of baseline fluorescence 
+intensity was detected in neurons after multiple stimulations (Supplementary Fig. 4), indicating the safety 
+of stimulation.  
+Laser only control experiment was also performed. Laser light with same pulse energy of 41.8 µJ 
+and duration (200 ms, 100 ms and 50 ms) was delivered to OGD-1 loaded neurons through 
+multifunctional fiber without optoacoustic coating. None of neuron culture showed detectable calcium 
+response, distinct from the observed in mFOE stimulated neurons (Supplementary Fig. 5).  
+To evaluate the photothermal effect of the mFOE stimulation and its potential impact on neurons, 
+we also characterized the thermal profile of the mFOE in PBS during the acoustic generation. 
+Temperature was measured by an ultrafast thermal sensor with a sampling rate of 2000 Hz placed in 
+contact with mFOE optoacoustic coating under the microscope. The laser conditions were consistent with 
+neural stimulation test, i.e., the pulse energy was maintained at 41.8 µJ and the burst duration was varied 
+from 50 ms, 100 ms to 200 ms. The temperature increase on the mFOE surface was found to be 1.23 ± 
+0.09 °C, 1.07 ± 0.08 °C, 0.96 ± 0.08 °C for 200, 100, 50 ms laser durations, respectively (Supplementary 
+Fig. 6). Such temperature increase is far below the previously reported threshold of thermal-induced 
+neural stimulation (ΔT > 5 °C)46, 47. Taken together, we conclude that activation of neurons was due to the 
+mFOE optoacoustic stimulation.  
+
+11 
+ 
+ 
+Figure 2. Calcium transients induced by mFOE in cultured primary neurons.   
+a. Calcium image of primary cultured neurons loaded with OGD-1. Twenty neurons within (orange) and 
+outside (blue) the optoacoustic converter area are circled and labelled. Scale bar: 100 µm. Solid circle: 
+area outside the converter area; dashed line circle: area within the optoacoustic converter area. b-c. 
+Calcium traces of neurons undergone repeated mFOE stimulations with a laser pulse train duration of 100 
+ms (red dots). Each pulse train was repeated 5 times. Colors and numbers of the traces are corresponding 
+to the neurons labelled in a. d-g. Average calcium traces of neurons triggered by mFOE stimulation with 
+durations of 200 ms (d), 100 ms (e), 50 ms (f) and 5 ms (g), respectively. Shaded area: the standard 
+deviation (SD). N=15 h. Average maximum ΔF/F of neurons stimulated by mFOE. N = 15. (n.s.: non-
+significant, p > 0.05; *: p < 0.05; **: p < 0.01; ***: p < 0.001, One-Way ANOVA and Tukey’s mean 
+comparison test) 
+In vivo simultaneous optoacoustic stimulation and electrophysiological recording 
+Since the animal experiment is a significant part of the study in neuroscience and neurological diseases, 
+we further investigated the performance of mFOE in the wild type C57BL/6J mice. In vivo optoacoustic 
+stimulation was performed by delivering pulsed laser to the implanted mFOE, and the optoacoustically 
+stimulated neuronal activities were recorded through electrodes in the mFOE (Fig. 3a).  Experimentally, 
+we implanted the mFOE into the hippocampus of mice (N =5). The chronically implanted mFOE allows 
+
+a
+b
+10
+hhyeh10
+C
+9
+9
+9
+8
+8
+Me
+40
+8
+6
+6
+5
+4
+3
+2
+2
+10%
+AA
+5s'
+d
+e
++
+g
+h
+0.12
+200 ms
+100 ms
+50 ms
+5 ms
+0.1
+0.1
+0.1
+0.1
+0.1
+T
+0.08
+F 0.05
+0.05
+0.05
+0.05
+F
+0.06
+A
+F
+0.04
+n.s.
+0.02
+0
+A
+0
+0
+0
+0
+0
+1
+2
+3
+0
+1
+2
+3
+0
+1
+2
+3
+0
+1
+2
+3
+-0.02
+Time (s)
+Time (s)
+Time (s)
+Time (s)
+200ms100ms50ms5ms12 
+ 
+mice to move freely after surgery (Fig 3b). During stimulation and recording tests, the mFOE was 
+coupled with the laser source and electrophysiological recording headstage through the standard ferrule 
+and pin connector, respectively. The stimulation and recording were conducted in the mice under 
+continuous anesthesia induced and maintained by isoflurane. Based on the threshold of optoacoustic 
+stimulation obtained in in vitro studies, 50 ms bursts of laser pulses with a pulse energy of 41.8 µJ were 
+delivered to the mFOE at 1Hz during the 5 second treatment period. The simultaneous 
+electrophysiological recording by mFOE electrodes was bandpass filtered to examine the local field 
+potential (LFP, 0.5-300 Hz). Simultaneous optoacoustic stimulation and electrophysiological recording 
+were performed at multiple time points, including 3 days, 7 days, 2 weeks and 1 month (Fig. 3c-f). Three 
+out of five mice tested showed successful simultaneous stimulation and recording functions for testing 
+periods of 3 days to one month. 
+The evoked brain activities corresponding to the optoacoustic stimulation were confirmed by 
+monitoring the LFP response. LFP response at two weeks after implantation was detected with latency of 
+7.19 ± 2.29 ms (N = 15, from three mice). The amplitude of LFP response varied at four time points.  The 
+largest and smallest responses occurred at 2 weeks and 1 month, respectively. A possible reason for this 
+observation may be the brain tissue injury and healing after the implantation. These results collectively 
+demonstrate the reliability of the optoacoustic stimulation and recording functions of the implanted 
+mFOE in the animals.  
+To eliminate the possibility that LFP response was induced by electrical noise or laser artifacts, 
+we also conducted two sham control experiments. In the light only control group, we implanted a 
+multifunctional fiber without optoacoustic coating to the mouse hippocampus and delivered the laser light 
+with the same condition. The LFP recorded didn’t correlate to the laser pulse train, indicating the 
+spontaneous brain activities were recorded and light only did not invoke the LFP response 
+(Supplementary Fig. 7a). In the dead brain control group, we tested the optoacoustic stimulation through 
+mFOE implanted to the euthanized mouse and did not observe the corresponding LFP response 
+
+13 
+ 
+(Supplementary Fig. 7b). These results collectively confirm the signals we detected from mFOE 
+stimulation were not artifacts.  
+We further evaluated the recording performance of implanted mFOE. To evaluate the ability of 
+mFOE for single unit recording, the electrophysiological signals recorded were bandpass filtered for spike 
+activity (0.5-3 kHz, Fig. 3g). Through a principal-component analysis (PCA) based spike sorting 
+algorithm, two spike clusters can be isolated from an endogenous neural recording (Fig. 3j). The cluster 
+quality was assessed by two common measures48, Lratio and isolation distance. Lratio is 0.0017 and isolation 
+distance has the value of 99.37. The first averaged spike shape (Fig. 3h) showed a narrower and larger 
+depolarization than that of the second spike shape (Fig. 3i). The different spike waveform and the cluster 
+analysis suggested that the action potentials were recorded from at least two different groups of neurons49, 
+50. Thus, the successfully spike sorted neural activities from CA3 confirmed the ability of mFOE 
+electrodes for the single-unit recording. 
+To examine the sensitivity of LFP recording, at one month after implantation we altered the 
+anesthesia level via adjusting the induced isoflurane concentration during the recording to see if the 
+characteristic anesthesia dosage-dependent changes can be observed (Fig. 3k). Initially, a low level of 
+anesthesia was maintained at 0.5% v/v isoflurane, and recorded LFP showed that spontaneous brain 
+activities occurred continuously (i in Fig. 3h. and Fig. 3l). Then a higher-level anesthesia (3% v/v 
+isoflurane) was applied for 3 minutes. After increased the isoflurane level, some spontaneous brain 
+activities were suppressed and a hyperexcitable brain state was induced, where the voltage alternation 
+(bursts) and isoelectric quiescence (suppression) appeared quasiperiodically27, 51 (ii in Fig. 3h and Fig. 
+3m). With maintaining 3% v/v isoflurane, a deep anesthesia state was induced in the animal. At the same 
+time, both respiration rate and responsiveness to toe pinch decreased due to the higher anesthetic level.
+ 
+Less voltage alternation occurred and for the most of time the LFP signal was a flat line 
+(suppression, iii in Fig. 3h and Fig. 3n). Compared with initial stage, γ band LFP activity in 30-100 Hz 
+was decreased due to the higher concentration of isoflurane as shown in the power spectrum52 (Fig. 3n). 
+Later, when the concentration of isoflurane was reduced to 0.5% v/v again, the LFP activity returned to a 
+
+14 
+ 
+similar level as measured in the initial stage. Taken together, this isoflurane dosage-dependent 
+characteristic confirmed the accuracy of LFP recording by mFOE. 
+ 
+Figure. 3 Simultaneous optoacoustic stimulation and electrophysiological recording by implanted 
+mFOE in mouse hippocampus.   
+a. Illustration of the mFOE enabled bidirectional neural communication using laser signal as input and 
+electrical signal as readout. b. mFOE was implanted into hippocampus of a wild type C57BL/6J mouse. 
+c-f. Simultaneous optoacoustic stimulation and electrophysiological recording performed at 3 days (c), 7 
+days (d), two weeks (e) and one month (f) after implantation. Blue dots the laser pulse trains. For each 
+laser train: 50 ms burst of pulses, pulse energy of 41.8 µJ, laser repetition rate 1.7 kHz.  g. Part of the 
+filtered spontaneous activity containing two separable units recorded by mFOE electrode at one month 
+after implantation. h-i. Spike shapes of two separable units in g. j. Principal-components analysis (PCA) 
+of the two units. k. Local field potential (LFP) recorded by mFOE one month after implantation with an 
+alternating anaesthesia level (0.5-3% v/v isoflurane). l-n. different LFP responses induced by varying the 
+concentration of isoflurane: l corresponds to the initial stage (0.5% of isoflurane level); m corresponds to 
+the burst/suppression transition stage (after increasing the isoflurane level to 3%); n corresponds to the 
+suppression stage (the isoflurane level was maintained at 3% and took effect).  
+
+a
+c
+d
+Optical
+Electrical
+0.5
+3 days
+0.5
+7 days
+input
+readout
+(mV)
+0.0
+Voltage (
+0.0
+-0.5
+-1.0-
+-1.0-
+0
+2
+4
+6
+10
+0
+4
+6
+8
+10
+Time (s)
+Time (s)
+b
+e
+ (mV)
+0.5
+2 weeks
+0.5-
+1 month
+(mV)
+Voltage (
+0.0
+Voltage
+0.0
+-0.5
+-0.5
+-1.0
+-1.0
+0
+2
+4
+6
+8
+10
+0
+2
+4
+6
+8
+10
+Time (s)
+Time (s)
+g
+h
+100
+40
+40
+20
+20
+Yoltage (μV)
+(Λ)
+50
+0
+0
+-20
+PC2
+40
+40
+-50
+60
+-60
+0
+0.5
+1
+1.5
+2
+0
+2 s
+0.5
+1
+1.5
+2
+-100
+Time (ms)
+Time(ms)
+-200
+-150
+-100
+-50
+0
+50
+100
+PC1
+k
+3 %
+i Initial stage
+m
+ii Burst/suppression
+n
+ii Suppression
+0.5 %
+M
+2 s
+N
+(ZH)
+100
+100
+Frequency
+(dB)
+Frequency
+(p)
+Frequency
+-50
+50
+-50
+50
+Power
+50
+100
+-100
+50
+-100
+150
+ii
+ili
+2
+4
+68101214
+2
+468101214
+2
+68101214
+50 s
+Time (s)
+Time (s)
+Time (s)15 
+ 
+Foreign body response comparison between mFOE and standard optical fiber using 
+immunohistochemistry 
+Foreign body response is a critical property of implantable neural interface to assure their usage in a safe 
+and chronic way, since the physical insertion into brain tissue commonly initiates a progressive 
+inflammatory tissue response53. To evaluate the biocompatibility of mFOE, we compared the foreign 
+body response of mouse brain to mFOE with the similar size standard silica optical fibers (diameter = 300 
+µm), which is widely used in optogenetic technologies54, 55. The immunohistochemistry analysis of 
+surrounding brain tissue was performed from mice (N = 3) implanted with the mFOE and a conventional 
+silica fiber 3 days and 1 month after implantation (Fig. 4a). The damage to surrounding neurons from 
+implant was assessed through evaluating neuronal density using the neuronal nuclei (NeuN) markers (Fig. 
+4b). Number of neurons was calculated by counting the NeuN-positive cells per field of view (650 × 650 
+µm). The presence of ionized calcium-binding adaptor molecule 1 (Iba1, Fig. 4c) and glial fibrillary 
+acidic protein (GFAP, Fig. 4d) were used as the markers for activated microglia and astrocytic response, 
+respectively.  
+Compared with the silica fiber, mFOE induced significantly less microglial response (p < 0.01, 
+Fig. 4c, f) and astrocyte reactivity (p < 0.001, Fig. 4d, g), but no significant difference was observed on 
+the neuronal density (Fig. 4b, e) 3 days after implantation. A decrease in foreign body response, 
+specifically, higher neuronal density and lower microglia and astrocytic response (Fig. 4e-g), was 
+observed from 3 days to 1 month after implantation of both mFOE and silica fiber and no significant 
+difference was observed between mFOE and silica fiber 1 month after implantation. Taken together, the 
+immunohistochemistry analysis confirmed that mFOE yielded less foreign body response in the short 
+period, i.e., 3 days, after implantation and showed similar biocompatibilty with silica fiber at longer 
+implantation time, i.e., 1 month.   
+
+16 
+ 
+ 
+Figure. 4 Foreign body response comparison of mFOE and silica fiber using 
+immunohistochemistry.    
+a-d. Immunohistochemistry images of mouse brains implanted with mFOE and silica fiber one month 
+after implantation (N = 3). Scale bar: 100 µm. Brain slices were labelled with the neuron-specific protein 
+(NeuN, cyan), ionized calcium-binding adaptor molecule 1 (Iba1, red) and glial fibrillary acidic protein 
+(GFAP, green).  e. Number of neurons in the field of view, calculated by counting the NeuN-positive cells 
+for mFOE and silica fiber at 3 days and 1 mon after implantation. f. Microglial reactivity, assessed by 
+counting the Iba-1 labelled area, for mFOE and silica fiber at 3 days and 1 mon after implantation. g. 
+Astrocyte reactivity, assessed by counting the GFAP labelled area, for mFOE and silica fiber at 3 days 
+and 1 mon after implantation. For each experimental group, two to four brain slices were used from each 
+
+a
+mFOE
+Silica Fiber
+800
+e
+n.s.
+700
+neurons
+Composite
+600
+500
+Number
+n.s.
+400
+300
+mFOE
+b
+Silica fiber
+200
+Day 3
+Day30
+TimePoint
+NeuN
+**
+mFOE
+3×104
+ Silica fiber
+n.s.
+2 ×10
+C
+10
+6×103
+Iba1
+Day 3
+Day30
+TimePoint
+g
+3×10
+mFOE
+Silica fiber
+d
+2 × 10
+GFAP area (μm"
+n.s.
+GFAP
+10
+6 ×10°
+Day 3
+Day30
+TimePoint17 
+ 
+mouse (N= 3). (n.s.: non-significant, p > 0.05; *: p < 0.05; **: p < 0.01; ***: p < 0.001, One-Way 
+ANOVA and Tukey’s mean comparison test) 
+ 
+Discussion  
+In this study, we designed and developed a miniaturized fiber-based device, i.e. mFOE, for 
+bidirectional neural communication. mFOE performs the “write” function, i.e.  non-genetic optoacoustic 
+stimulation and the “read” function, i.e. simultaneous electrophysiological recording. The broadband 
+acoustic wave with a broadband ultrasound pulse with pulse width about 0.1 µs and a center frequency at 
+12.5 MHz and a peak pressure of 2.3 MPa with pulse numbers >85  generated by mFOE successfully 
+stimulate neurons with a spatial resolution of approximately 200 µm in primary rat cortical neuron 
+culture. By implanting mFOE into mouse hippocampus, we demonstrated its ability for simultaneous 
+optoacoustic stimulation and electrophysiological recording and superior biocompatibility as a chronic 
+bidirectional neural interface.  Reliable stimulation and LFP recording have been achieved up to one 
+month post implantation. Recording quality has been demonstrated by single unit recording.  
+For the first time, combining this pico-liter deposition and thermal fiber pulling, we successfully 
+integrated an optoacoustic converter to the polymer multifunctional fiber. Different from the conventional 
+dip-coating method36, 56, the selective deposition through micro-injection allows the easy fabrication of 
+optoacoustic emitter in a volume and position-controlled way. Through the selective deposition, the 
+dimension of optoacoustic emitter is no longer limited by the tip sizes of optical fibers. Our choice of 
+CB/PDMS composite as the optoacoustic material is also essential as it is comparable with this deposition 
+process with a fine volume control at pico liter level. Besides the application in neural interface, such 
+design and fabrication method can also be applied to optical ultrasound probes used in imaging37, 57, for 
+example, in the tip engineering and the integration to photonics crystal fibers.  
+We introduced the optoacoustic stimulation as a new strategy for “writing” in the bidirectional 
+neural interface. Compared with previous optoelectrode devices based on optogenetics24, 25, 27 and 
+
+18 
+ 
+photothermal58, 59, the non-genetic optoacoustic stimulation enabled by mFOE reduces the barrier of 
+transgenic techniques for applications in primate and potentially human, and avoids the thermal toxicity. 
+At the same time, it offers the spatial precision benefit from the confined ultrasound field. It is orthogonal 
+to electrical recording, therefore minimizing crosstalk with electrical recording. As an emerging 
+neuromodulation method, the mechanism of optoacoustic stimulation is still not fully understood but 
+more studies indicated that mechanosensitive ion channels are responsible for the activation of neurons60, 
+61. 
+Bidirectional brain interfaces are important research tools to understand brain circuits, potential 
+treatments for neurological disease and bridges to brain computer interface for broad applications. New 
+features of mFOE compared to the previous fiber based interface, such as non-genetic and non-electrical 
+stimulation are critical to advance these applications. For example, closed-loop neuromodulation has been 
+demonstrated to be superior to the conventional open-loop system, as it can achieve more responsive and 
+real-time control over neural dynamics. In neurological diseases treatment, combining the detection and 
+in situ intervention improves the treatment effectiveness and safety. Because of its bidirectional 
+capabilities, mFOE has the potential to be used as a new brain interface with closed-loop capability. 
+Using epilepsy as an example, by implanting the mFOE into seizure foci, the continuous LFP recording 
+can guide the localized optoacoustic stimulation and intervene can be triggered at the early stage before 
+seizure progresses into a generalized seizure. The unique orthogonal non-electrical optoacoustic 
+stimulation and electrical recording prevents “contamination” of the recording signals, potentially 
+offering a more effective closed-loop strategy.  
+In comparison of the optoelectrodes fabricated through semiconductor fabrication process, the 
+recording and stimulation sites of the current mFOE design is fixed at the core waveguide and the number 
+of channels is limited because of the nature of multifunctional fiber. Some post processing methods have 
+been proposed to tackle this challenge, like the laser micromachining technique27. In addition, it is 
+possible to further engineer the fiber to offer multiple and selective stimulation sites62. With the further 
+
+19 
+ 
+development of multifunctional fiber strategy, we believe the bandwidth of mFOE would be improved 
+and open more opportunities in the research of neuroscience and neurological diseases. 
+ 
+Methods  
+Multifunctional fiber fabrication and optoacoustic emitter integration 
+Multifunctional fibers were fabricated from a preform fiber and then drawn into thin fibers through TDP 
+in a customized furnace. For the preform fiber, PVDF film (Mcmaster) and PC film (laminated plastics) 
+were rolled onto a PC rod (Mcmaster) and followed by a consolidation process in vacuum at 200 °C. 
+Next, four rectangular grooves (2 mm × 2 mm) were machined on the solid PC layer and inserted with the 
+BiSn (Indium Corporate) electrodes. Then, another PVDF layer was rolled over the rod to form an 
+insulation layer for the electrodes and followed by an additional PC as the sacrifice layer for the 
+convenience of TDP. The detailed fabrication process was discussed in the previous paper27.  
+A composite of 10% carbon black (diameter < 500 nm, Sigma Aldrich) and 90% 
+polydimethylsiloxane (PDMS, Sylgard 184, Dow Corning Corporation, USA) were used as the 
+optoacoustic material. The mixture was sonicated for 1 hour followed by degassing in vacuum for 30 
+minutes. The mixture was then filled in the glass micropipette (Inner diameter = 30 µm, TIP30TW1, 
+World Precision Instruments, USA) connected to the pico-liter injector (PLI-100A, Warner Instruments, 
+USA). Under the microscope, the glass micropipette was aligned with the core waveguide of 
+multifunctional fiber and the mixture was deposited to the surface of the core waveguide by controlling 
+the injection pressure and time. The deposited fiber was then cured vertically at room temperature for 2 
+days.  
+Before use, mFOE was further prepared for the optical coupling and electrodes connection. For 
+the optical coupling, a ceramic ferrule (Thorlabs, USA) was added and affixed to the end of the fiber by 
+the 5-min epoxy (Devcon, ITW Performance Polymers, USA). Then the end surface was polished by 
+
+20 
+ 
+optical polishing papers to reduce roughness from 30 µm to 1 µm. For the connection to electrodes 
+embedded in the multifunctional fiber, the electrodes were exposed manually along the side wall of the 
+fiber by using a blade and silver paint (SPI Supplies, USA). Then copper wires were wrapped around the 
+fiber at each exposure locations along the fiber and the silver paint were applied for the fixation and lower 
+resistance. The copper wires connected to fiber electrodes were soldered to the pin connector while a 
+stainless-steel wire was also soldered as the ground wire for later extracellular recording. In addition, the 
+5-min epoxy (Devcon, ITW Performance Polymers, USA) was applied to the connection interface for 
+strengthening affixation and better electrical insulation.  
+Optoacoustic wave characterization 
+To generate the optoacoustic signal, a compact Q-switched diode-pumped solid-state laser (1030 nm, 3 
+ns, 100 μJ, repetition rate of 1.7 kHz, RPMC Lasers Inc., USA) was used as the excitation laser source. 
+The laser was first connected to an optical fiber through a 200 µm fiber coupling module and then 
+connected to the mFOE with a SubMiniature version A (SMA) connector. The pulse energy was adjusted 
+through a fiber optic attenuator (varied gap SMA Connector, Thorlabs, Inc., USA). The acoustic signal 
+was measured through a homebuilt system including a needle hydrophone (ID. 40 µm; OD, 300 µm) with 
+a frequency range of 1–30 MHz (NH0040, Precision Acoustics Inc., Dorchester, UK), an amplifier and an 
+oscilloscope. The mFOE tip and hydrophone tip were both immersed in degassed water. The pressure 
+values were calculated based on the calibration factor provided by the hydrophone manufacturer. The 
+frequency data was obtained through a fast Fourier transform (FFT) calculation using the OriginPro 2019. 
+Embryonic neuron culture 
+All experimental procedures complied with all relevant guidelines and ethical regulations for animal 
+testing and research established and approved by Institutional Animal Care and Use Committee (IACUC) 
+of Boston University (PROTO201800534). Primary cortical neurons were isolated from embryonic day 
+15 (E15) Sprague−Dawley rat embryos of either sex (Charles River Laboratories, MA, USA). Cortices 
+
+21 
+ 
+were isolated and digested in TrypLE Express (ThermoFisher Scientific, USA). Then the neurons were 
+plated on poly-D-lysine (50 μgmL−1, ThermoFisher Scientific, USA)-coated glass bottom dish (P35G-
+1.5-14-C, MatTek Corporation, USA). Neurons were first cultured with a seeding medium composed of 
+90% Dulbecco’s modified Eagle medium (ThermoFisher Scientific, USA) and 10% fetal bovine serum 
+(ThermoFisher Scientific, USA) and 1% GlutaMAX (ThermoFisher Scientific, USA), which was then 
+replaced 24 h later by a growth medium composed of Neurobasal Media (ThermoFisher Scientific, USA) 
+supplemented with 1× B27 (ThermoFisher Scientific, USA), 1× N2 (ThermoFisher Scientific, USA), and 
+1× GlutaMAX (ThermoFisher Scientific, USA). Half of the medium was replaced with fresh growth 
+medium every 3 or 4 days. Cells cultured in vitro for 10−14 days were used for Oregon Green labelling 
+and PA stimulation experiments. 
+In vitro neurostimulation and calcium imaging 
+Oregon Green™ 488 BAPTA-1 dextran (OGD-1) (ThermoFisher Scientific, USA) was dissolved in 20% 
+Pluronic F-127 in dimethyl sulfoxide (DMSO) at a concentration of 1 mM as stock solution. Before 
+imaging, neurons were incubated with 2 µM OGD-1 for 30 min, followed by incubation with normal 
+medium for 30 min. Q-switched 1030 nm nanosecond laser was used to generate light and delivered to 
+mFOE. The pulse energy was adjusted through a fiber optic attenuator (varied gap SMA Connector, 
+Thorlabs, Inc., USA). Notably, 1030 nm is far from the excitation peak of Oregon Green (494 nm) and 
+pass band of emission filter (500-540 nm), therefore assuring no effect from direct excitation of OGD by 
+any light leak from the fiber. A 3D translational stage was used to position the mFOE approaching the 
+target neurons.  
+Calcium fluorescence imaging was performed on a lab-built wide-field fluorescence microscope 
+based on an Olympus IX71 microscope frame with a 20× air objective (UPLSAPO20X, 0.75NA, 
+Olympus, USA), illuminated by a 470 nm LED (M470L2, Thorlabs, USA), an emission filter (FBH520-
+40, Thorlabs, USA), an excitation filter (MF469-35, Thorlabs) and a dichroic mirror (DMLP505R, 
+Thorlabs, USA). Image sequences were acquired with a scientific CMOS camera (Zyla 5.5, Andor, 
+
+22 
+ 
+Oxfords Instruments, UK) at 20 frames per second. The fluorescence intensities, data analysis, and 
+exponential curve fitting were analyzed using ImageJ (Fiji) and MATLAB 2022. 
+Implantation surgery procedure 
+All surgery procedures complied with all relevant guidelines and ethical regulations for animal testing and 
+research established and approved by Institutional Animal Care and Use Committee (IACUC) of Boston 
+University (PROTO201800534). Eight to ten weeks old male wildtype C57BL/6-E mice (Charles River 
+Laboratories, US) were received and allowed to acclimate for at least 3 days before enrolling them in 
+experiments. All mice in experiments had access to food and water ad libitum and were kept in the BU 
+animal facility maintained for 12-h light/dark cycle. During the implantation surgery, mice were 
+anesthetized by isoflurane (5% for induction, 1-3.5% during the procedure) and positioned on a 
+stereotaxic apparatus (51500D, Stoelting Co., USA). After hair removal, a small incision was made by 
+sterile surgery scalpel at the target region and then a small craniotomy was made by using a dental drill. 
+Assembled mFOE was inserted into mice hippocampus (−2.0 mm AP, 1.5 mm ML, 2 mm DV) using the 
+manipulator with respect to the Mouse Brain Atlas. The ground stainless steel wire was soldered to a 
+miniaturized screw (J.I. Morris) on the skull. Finally, the whole exposed skull area was fully covered by a 
+layer of Metabond (C&B METABOND, Parkell, USA) and dental cement (51458, Stoelting Co., USA). 
+Buprenorphine SR was used to provide long effective analgesia after the surgery. 
+In vivo electrophysiology recording and optoacoustic stimulation 
+Extracellular recording was performed through an electrophysiology system (Molecular Devices, LLC, 
+USA). mFOE electrodes were connected to the amplifier (Multiclamp 700B, Molecular Devices, LLC, 
+USA) through the pin connector and headstages after the animals recovered from surgeries. The amplified 
+analog signal was then converted and recorded by the digitizer (Digidata 1550, Molecular Devices, LLC, 
+USA).  
+
+23 
+ 
+Q-switched 1030 nm nanosecond laser was used to generate light and delivered to mFOE. During the 
+extracellular electrophysiological recording, the preset trigger signal was generated by the digitizer and 
+used to trigger the Q-switch laser for optoacoustic stimulation. The pulse energy was adjusted through a 
+fiber optic attenuator (varied gap SMA Connector, Thorlabs, Inc., USA).  
+Data analysis was performed with Matlab and OriginPro and custom scripts were used to analyse the local 
+field potential and spike sorting. The raw extracellular recordings were first band filtered for local field 
+potential results (LFP, 0.5 – 300 Hz) and spike results (300 – 5000 Hz). A custom Matlab script was used 
+to create spectrograms to visually support the analysis of the LFPs in both the time domain and the 
+frequency domain. The spike sorting algorithm was implemented through several steps: first, individual 
+spike signals with length of 3 ms were picked up from the full recording through a standard amplitude 
+threshold method; then the dimensionality of each spike signal was reduced via the principal component 
+analysis (PCA) and unsupervised learning algorithms (K-means clustering) was used to separate out the 
+clusters. 
+Foreign body response assessment via immunohistochemistry 
+To compare the tissue response, animals were implanted with a silica optical fiber (diameter = 300 µm, 
+FT300EMT, Thorlabs, Inc, USA) and mFOE for 3 days or 4 weeks. Then at target timepoints, animals 
+were euthanized and transcardially perfused with phosphate-buffered saline (PBS, ThermoFisher 
+Scientific, USA) followed by 4% paraformaldehyde (PFA, ThermoFisher Scientific, USA) in PBS. The 
+fiber probes were carefully extracted before the extraction and then the brains were kept in 4% PFA 
+solution for one day at 4 °C. Brains were sectioned in the horizontal plane at 75 µm on a vibrating blade 
+vibratome. Free-floating brain slices were washed in PBS and blocked for 1 hour at room temperature in a 
+blocking solution consisting of 0.3% Triton X-100 (vol/vol) and 2.5% goat serum (vol/vol) in PBS. After 
+blocking, brain slices were incubated with the primary antibodies in the PBS solution with 2.5% goat 
+serum (vol/vol) for 24 hours at 4 °C. Primary antibodies used included rat anti-GFAP (Abcam Cat. # 
+
+24 
+ 
+ab279291, 1:500), chicken anti-NeuN (Millipore Cat. # ABN91, 1:500), and rabbit anti-Iba1 (Abcam Cat. 
+# ab178846, 1:500). Following primary incubation, slices were washed three times with PBS for 10 min 
+at room temperature. The brain slices were then incubated with secondary antibodies in the PBS solution 
+with 2.5% goat serum (vol/vol) for 2 hours at room temperature. Secondary antibodies used included goat 
+anti-rat Alexa Fluor 488 (Abcam Cat. # ab150157, 1:1000), goat anti-rabbit Alexa Fluor 568 (Abcam Cat. 
+# ab175471, 1:1000) and goat anti-chicken Alexa Fluor 647 (Abcam Cat. # ab150171, 1:1000). Slices 
+were then washed three times with PBS for 10 min at room temperature. Before imaging, slices were 
+stained with DAPI solution (1 µg/ml, Millipore, USA) for 15 minutes at room temperature. All 
+fluorescent images were acquired with a laser scanning confocal microscope (Olympus FV3000) with an 
+air 20× objective with a numerical aperture NA = 0.75 unless otherwise noted. Neuron density was then 
+calculated within the normalized area by counting NeuN labeled cell bodies using the cell counter plugin 
+(ImageJ). Area analysis of Iba1 and GFAP labeled cells was performed by creating binary layers of the 
+fluorescence images using the threshold function and quantified using the measurement tool (ImageJ). 
+Statistical information  
+Data shown are mean ± standard deviation. For the comparison on peak fluorescence change of in vitro 
+optoacoustic stimulation, one-way ANOVA and Tukey’s mean comparison test were conducted by using 
+OriginLab. 15 stimulation events were compared for each condition. For the comparison of foreign body 
+response between silica fiber and mFOE, N > 8 brain slices from 3 animals were analysed using one-way 
+ANOVA and Tukey’s mean comparison test. The p values were determined as n.s.: nonsignificant, p > 
+0.05; *: p < 0.05; **: p < 0.01; ***: p < 0.001. Statistic analysis were conducted using OriginPro.   
+Data Availability 
+The raw data that support the findings of this study are available from the corresponding author upon 
+request.  
+Code Availability 
+
+25 
+ 
+The MATLAB scripts for analysis are available from the corresponding author upon request.  
+Acknowledgements  
+This work was supported by National Institute of Health Brain Initiative R01 NS109794 to J-XC and CY.  
+Research reported in this publication was supported by the Boston University Micro and Nano Imaging 
+Facility and the Office of the Director, National Institutes of Health of the National Institutes of Health 
+under award Number S10OD024993 
+References 
+1. 
+Wilbrecht, L. & Shohamy, D. Neural Circuits can Bridge Systems and Cognitive Neuroscience. 
+Front Hum Neurosci 3, 81 (2010). 
+2. 
+Chen, R., Canales, A. & Anikeeva, P. Neural Recording and Modulation Technologies. Nat Rev 
+Mater 2 (2017). 
+3. 
+Sohal, V.S., Zhang, F., Yizhar, O. & Deisseroth, K. Parvalbumin neurons and gamma rhythms 
+enhance cortical circuit performance. Nature 459, 698-702 (2009). 
+4. 
+Paulk, A.C., Kirszenblat, L., Zhou, Y. & van Swinderen, B. Closed-Loop Behavioral Control 
+Increases Coherence in the Fly Brain. J Neurosci 35, 10304-10315 (2015). 
+5. 
+Rosin, B. et al. Closed-loop deep brain stimulation is superior in ameliorating parkinsonism. 
+Neuron 72, 370-384 (2011). 
+6. 
+Paz, J.T. et al. Closed-loop optogenetic control of thalamus as a tool for interrupting seizures 
+after cortical injury. Nat Neurosci 16, 64-70 (2013). 
+7. 
+Harmsen, I.E. et al. Clinical trials for deep brain stimulation: Current state of affairs. Brain Stimul 
+13, 378-385 (2020). 
+8. 
+Hughes, C., Herrera, A., Gaunt, R. & Collinger, J. Bidirectional brain-computer interfaces. Handb 
+Clin Neurol 168, 163-181 (2020). 
+9. 
+Zhou, A., Johnson, B.C. & Muller, R. Toward true closed-loop neuromodulation: artifact-free 
+recording during stimulation. Curr Opin Neurobiol 50, 119-127 (2018). 
+10. 
+Heck, C.N. et al. Two-year seizure reduction in adults with medically intractable partial onset 
+epilepsy treated with responsive neurostimulation: final results of the RNS System Pivotal trial. 
+Epilepsia 55, 432-441 (2014). 
+11. 
+Sun, F.T. & Morrell, M.J. The RNS System: responsive cortical stimulation for the treatment of 
+refractory partial epilepsy. Expert Rev Med Devices 11, 563-572 (2014). 
+12. 
+Miocinovic, S. et al. Experimental and theoretical characterization of the voltage distribution 
+generated by deep brain stimulation. Exp Neurol 216, 166-176 (2009). 
+13. 
+Paff, M., Loh, A., Sarica, C., Lozano, A.M. & Fasano, A. Update on Current Technologies for Deep 
+Brain Stimulation in Parkinson's Disease. J Mov Disord 13, 185-198 (2020). 
+14. 
+Buhlmann, J., Hofmann, L., Tass, P.A. & Hauptmann, C. Modeling of a segmented electrode for 
+desynchronizing deep brain stimulation. Front Neuroeng 4, 15 (2011). 
+15. 
+Qian, X. et al. A Method for Removal of Deep Brain Stimulation Artifact From Local Field 
+Potentials. IEEE Trans Neural Syst Rehabil Eng 25, 2217-2226 (2017). 
+
+26 
+ 
+16. 
+Koessler, L. et al. In-vivo measurements of human brain tissue conductivity using focal electrical 
+current injection through intracerebral multicontact electrodes. Hum Brain Mapp 38, 974-986 
+(2017). 
+17. 
+Zhang, J. et al. Integrated device for optical stimulation and spatiotemporal electrical recording 
+of neural activity in light-sensitized brain tissue. J Neural Eng 6, 055007 (2009). 
+18. 
+Abaya, T.V., Blair, S., Tathireddy, P., Rieth, L. & Solzbacher, F. A 3D glass optrode array for optical 
+neural stimulation. Biomed Opt Express 3, 3087-3104 (2012). 
+19. 
+Wang, J. et al. Integrated device for combined optical neuromodulation and electrical recording 
+for chronic in vivo applications. J Neural Eng 9, 016001 (2012). 
+20. 
+Royer, S. et al. Multi-array silicon probes with integrated optical fibers: light-assisted 
+perturbation and recording of local neural circuits in the behaving animal. Eur J Neurosci 31, 
+2279-2291 (2010). 
+21. 
+He, Z. et al. A sapphire based monolithic integrated optrode. Annu Int Conf IEEE Eng Med Biol 
+Soc 2016, 6186-6189 (2016). 
+22. 
+Kravitz, A.V., Owen, S.F. & Kreitzer, A.C. Optogenetic identification of striatal projection neuron 
+subtypes during in vivo recordings. Brain Res 1511, 21-32 (2013). 
+23. 
+Wu, F. et al. Monolithically Integrated muLEDs on Silicon Neural Probes for High-Resolution 
+Optogenetic Studies in Behaving Animals. Neuron 88, 1136-1148 (2015). 
+24. 
+Voroslakos, M. et al. HectoSTAR muLED Optoelectrodes for Large-Scale, High-Precision In Vivo 
+Opto-Electrophysiology. Adv Sci (Weinh) 9, e2105414 (2022). 
+25. 
+Canales, A. et al. Multifunctional fibers for simultaneous optical, electrical and chemical 
+interrogation of neural circuits in vivo. Nat Biotechnol 33, 277-284 (2015). 
+26. 
+Park, S. et al. One-step optogenetics with multifunctional flexible polymer fibers. Nat Neurosci 
+20, 612-619 (2017). 
+27. 
+Jiang, S. et al. Spatially expandable fiber-based probes as a multifunctional deep brain interface. 
+Nat Commun 11, 6115 (2020). 
+28. 
+Antonini, M.J. et al. Customizing MRI-Compatible Multifunctional Neural Interfaces through 
+Fiber Drawing. Adv Funct Mater 31 (2021). 
+29. 
+Park, S. et al. Adaptive and multifunctional hydrogel hybrid probes for long-term sensing and 
+modulation of neural activity. Nat Commun 12, 3435 (2021). 
+30. 
+Bansal, A., Shikha, S. & Zhang, Y. Towards translational optogenetics. Nat Biomed Eng (2022). 
+31. 
+Jiang, Y. et al. Optoacoustic brain stimulation at sub-millimeter spatial precision. Nature 
+Communications (2020). 
+32. 
+Shi, L. et al. Non-genetic photoacoustic stimulation of single neurons by a tapered fiber 
+optoacoustic emitter. Light Sci Appl 10, 143 (2021). 
+33. 
+Wang, L.V. & Hu, S. Photoacoustic tomography: in vivo imaging from organelles to organs. 
+Science 335, 1458-1462 (2012). 
+34. 
+Lee, T., Baac, H.W., Li, Q. & Guo, L.J. Efficient Photoacoustic Conversion in Optical Nanomaterials 
+and Composites. Advanced Optical Materials 6 (2018). 
+35. 
+Abouraddy, A.F. et al. Towards multimaterial multifunctional fibres that see, hear, sense and 
+communicate. Nat Mater 6, 336-347 (2007). 
+36. 
+Colchester, R.J. et al. Laser-generated ultrasound with optical fibres using functionalised carbon 
+nanotube composite coatings. Applied Physics Letters 104 (2014). 
+37. 
+Noimark, S. et al. Carbon-Nanotube-PDMS Composite Coatings on Optical Fibers for All-Optical 
+Ultrasound Imaging. Advanced Functional Materials 26, 8390-8396 (2016). 
+38. 
+Chen, G. et al. High-precision neural stimulation by a highly efficient candle soot fiber 
+optoacoustic emitter. Front Neurosci 16, 1005810 (2022). 
+
+27 
+ 
+39. 
+Guo, Y. et al. Polymer-fiber-coupled field-effect sensors for label-free deep brain recordings. 
+PLoS One 15, e0228076 (2020). 
+40. 
+Han, D., Meng, Z., Wu, D., Zhang, C. & Zhu, H. Thermal properties of carbon black aqueous 
+nanofluids for solar absorption. Nanoscale Res Lett 6, 457 (2011). 
+41. 
+Hilarius, K. et al. Influence of shear deformation on the electrical and rheological properties of 
+combined filler networks in polymer melts: Carbon nanotubes and carbon black in 
+polycarbonate. Polymer 54, 5865-5874 (2013). 
+42. 
+Brigandi, P.J., Cogen, J.M., Reffner, J.R., Wolf, C.A. & Pearson, R.A. Influence of carbon black and 
+carbon nanotubes on the conductivity, morphology, and rheology of conductive ternary polymer 
+blends. Polymer Engineering & Science 57, 1329-1339 (2017). 
+43. 
+Smetters, D., Majewska, A. & Yuste, R. Detecting action potentials in neuronal populations with 
+calcium imaging. Methods 18, 215-221 (1999). 
+44. 
+Yang, W. & Yuste, R. In vivo imaging of neural activity. Nat Methods 14, 349-359 (2017). 
+45. 
+Nagayama, S. et al. In vivo simultaneous tracing and Ca(2+) imaging of local neuronal circuits. 
+Neuron 53, 789-803 (2007). 
+46. 
+Lyu, Y., Xie, C., Chechetka, S.A., Miyako, E. & Pu, K. Semiconducting Polymer Nanobioconjugates 
+for Targeted Photothermal Activation of Neurons. J Am Chem Soc 138, 9049-9052 (2016). 
+47. 
+Shapiro, M.G., Homma, K., Villarreal, S., Richter, C.P. & Bezanilla, F. Infrared light excites cells by 
+changing their electrical capacitance. Nat Commun 3, 736 (2012). 
+48. 
+Schmitzer-Torbert, N., Jackson, J., Henze, D., Harris, K. & Redish, A.D. Quantitative measures of 
+cluster quality for use in extracellular recordings. Neuroscience 131, 1-11 (2005). 
+49. 
+Heinricher, M.M. Principles of extracellular single-unit recording. Microelectrode Recording in 
+Movement Disorder Surgery 8 (2004). 
+50. 
+Henze, D.A. et al. Intracellular features predicted by extracellular recordings in the hippocampus 
+in vivo. J Neurophysiol 84, 390-400 (2000). 
+51. 
+Ching, S., Purdon, P.L., Vijayan, S., Kopell, N.J. & Brown, E.N. A neurophysiological-metabolic 
+model for burst suppression. Proc Natl Acad Sci U S A 109, 3095-3100 (2012). 
+52. 
+Hudetz, A.G., Vizuete, J.A. & Pillay, S. Differential effects of isoflurane on high-frequency and 
+low-frequency gamma oscillations in the cerebral cortex and hippocampus in freely moving rats. 
+Anesthesiology 114, 588-595 (2011). 
+53. 
+Kozai, T.D., Jaquins-Gerstl, A.S., Vazquez, A.L., Michael, A.C. & Cui, X.T. Brain tissue responses to 
+neural implants impact signal sensitivity and intervention strategies. ACS Chem Neurosci 6, 48-
+67 (2015). 
+54. 
+Aravanis, A.M. et al. An optical neural interface: in vivo control of rodent motor cortex with 
+integrated fiberoptic and optogenetic technology. J Neural Eng 4, S143-156 (2007). 
+55. 
+Ung, K. & Arenkiel, B.R. Fiber-optic implantation for chronic optogenetic stimulation of brain 
+tissue. J Vis Exp, e50004 (2012). 
+56. 
+Poduval, R.K. et al. Optical fiber ultrasound transmitter with electrospun carbon nanotube-
+polymer composite. Appl Phys Lett 110, 223701 (2017). 
+57. 
+Finlay, M.C. et al. Through-needle all-optical ultrasound imaging in vivo: a preclinical swine 
+study. Light Sci Appl 6, e17103 (2017). 
+58. 
+Yoo, M., Koo, H., Kim, M., Kim, H.I. & Kim, S. Near-infrared stimulation on globus pallidus and 
+subthalamus. J Biomed Opt 18, 128005 (2013). 
+59. 
+Abaya, T.V. et al. Characterization of a 3D optrode array for infrared neural stimulation. Biomed 
+Opt Express 3, 2200-2219 (2012). 
+60. 
+Yoo, S., Mittelstein, D.R., Hurt, R.C., Lacroix, J. & Shapiro, M.G. Focused ultrasound excites 
+cortical neurons via mechanosensitive calcium accumulation and ion channel amplification. Nat 
+Commun 13, 493 (2022). 
+
+28 
+ 
+61. 
+Shi, L., Jiang, Y., Zheng, N., Cheng, J.X. & Yang, C. High-precision neural stimulation through 
+optoacoustic emitters. Neurophotonics 9, 032207 (2022). 
+62. 
+Pisanello, F. et al. Multipoint-emitting optical fibers for spatially addressable in vivo 
+optogenetics. Neuron 82, 1245-1254 (2014). 
+ 
+Supplementary Information 
+ 
+Supplementary Figure 1. Microscope images of deposited carbon black and PDMS composite.  
+The coverage area was controlled through tuning the injection pressure and time. Injection time was 
+varied between 1 second and 2 seconds, and the pressure was varied from 2 psi, 3 psi and 4 psi. Scale bar: 
+50 µm.  
+ 
+
+2 psi
+3 psi
+4 psi
+2 s1030 nm
+Optical fiber
+pulsedlaser
+Function
+generator
+mFOE
+Neurons cultured
+Micromanipulator
+loadedwithOGD-1
+Objective
+Lens
+Lens
+470nm
+DM
+LED
+Lens
+CMOS
+Mirror
+camera29 
+ 
+Supplementary Figure 2. Schematic of in vitro mFOE stimulation and calcium imaging set up.  
+Stimulation: 1030 nm pulsed laser is triggered by a function generator and delivered to the mFOE through 
+an optical fiber. Calcium imaging: Oregon green is excited by 470 nm LED and the fluorescence signal is 
+detected through a CMOS camera. 
+ 
+Supplementary Figure 3. Illustration of the laser pulse train for 5 bursts with 100 ms duration at 
+1Hz.  
+ 
+
+Repetition rate: 1.7 kHz
+Pulse numbers: 170
+3 ns pulse width
+100 msPre
+Post30 
+ 
+Supplementary Fig. 4 Calcium imaging of neurons before and after mFOE stimulation. Scale bar: 
+100 µm. 
+ 
+Supplementary Fig. S5 Average calcium traces of laser only control groups. The laser duration was 
+same with three conditions tested in mFOE stimulation (200 ms, 100 ms and 50 ms). Laser light with 
+pulse energy of 41.8 µJ was triggered at the time point labelled by the red bar. Shaded areas: standard 
+deviation. (N=3) 
+ 
+ 
+
+200 ms
+100 ms
+50 ms
+0.1
+0.1
+0.1
+ 0.05
+F 0.05
+F 0.05
+△F/
+△F/
+△F/
+0
+1
+2
+3
+0
+1
+2
+3
+0
+1
+2
+3
+Time (s)
+Time (s)
+Time (s)1.5
+1.5
+1.5
+50 ms
+100 ms
+200 ms
+1
+1
+1
+(0。) .
+(0。)
+(0。)
+0.5
+0.5
+0.5
+△T
+△T
+△T
+0
+0
+0
+-0.5
+-0.5
+-0.5
+0
+2
+4
+0
+2
+4
+0
+2
+4
+Time (s)
+Time (s)
+Time (s)31 
+ 
+Supplementary Fig. S6 Temperature change of the optoacoustic emitter integrated on mFOE. The 
+pulse energy was maintained at 41.8 µJ and the burst duration was varied from 50 ms (blue), 100 ms 
+(yellow) to 200 ms (orange). Laser was trigger at 2.5 second as labelled by the red bar.  
+ 
+ 
+Supplementary Fig. S7 LFP recording of sham control stimulation experiments.  
+a. Electrophysiological recording under light only stimulations delivered through a bare multifunctional 
+fiber without optoacoustic emitter. b. Simultaneous optoacoustic stimulation and electrophysiological 
+recording of an euthanized mouse. Same laser condition was used: pulse energy of 41.8 µJ, 50 ms burst of 
+pulses, 1 Hz, blue dots indicate the laser onset. 
+ 
+
+b
+a
+0.5 -
+0.50
+0.25
+Voltage (mV)
+0.0
+Voltage (mV)
+0.00
+-0.5.
+-0.25
+-1.0.
+-0.50
+0
+2
+4
+6
+8
+10
+0
+2
+4
+6
+8
+10
+Time (s)
+Time (s)

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+arXiv:2301.13719v1  [math.GT]  31 Jan 2023
+FINITE SETS CONTAINING ZERO ARE MAPPING DEGREE SETS
+CRISTINA COSTOYA, VICENTE MU˜NOZ, AND ANTONIO VIRUEL
+Abstract. In this paper we solve in the positive the question of whether any finite set
+of integers A, containing the zero, is the mapping degree set between two oriented closed
+connected manifolds of the same dimension. We extend this question to the rational
+setting, where an affirmative answer is also given.
+1. Introduction
+In this paper, we settle in the positive various questions which have been raised about
+D(M, N), the set of mapping degrees between two oriented closed connected manifolds
+M and N of the same dimension:
+D(M, N) = {d ∈ Z | ∃f : M → N, deg(f) = d}.
+In [14, Problem 1.1], the authors discuss the problem of finding, for any set A ⊂
+Z containing the zero, two oriented closed connected manifolds M and N of the same
+dimension such that A = D(M, N). Note that 0 ∈ A is a necessary condition as the
+constant map M → N is of degree zero.
+A quick argument shows that when A is an infinite set, this problem is solved in the
+negative [14, Theorem 1.3]: there are uncountably many infinite subsets A ⊂ Z containing
+the zero, compared to the countably many mapping degree sets D(M, N) that exist for
+pairs of oriented closed connected manifolds of the same dimension. Hence not every
+infinite set, containing the zero, is realizable as the mapping degree set of manifolds.
+Thus, one might ask:
+Question 1.1 ([14, Problem 1.4]). Let A be a finite set of integers containing the zero.
+Is A = D(M, N) for some oriented closed manifolds M, N of the same dimension?
+Remark 1.2. It is important to notice that if {0} ⊊ A = D(M, N), A finite, for some
+manifolds M and N, then D(M, M) and D(N, N) must both be contained in {0, 1, −1}.
+Otherwise, if there exists g : M → M with | deg(g)| > 1, then for any non-zero degree
+f : M → N (which exists by assumption), the subset {deg(f ◦ gm) | m ∈ N} of D(M, N)
+is unbounded. This leads to a contradiction as A = D(M, N) is finite. The same follows
+for D(N, N).
+2020 Mathematics Subject Classification. 55M25, 57N65, 55P62, 55R10.
+Key words and phrases. Mapping degree sets, inflexible manifold, fiber bundle, unstable Adams
+operation.
+The first author was partially supported by MINECO (Spain) grants PID2020-115155GB-I00 and
+TED2021-131201B-I00. The second author was partially supported MINECO (Spain) grant PID2020-
+118452GB-I00. The third author was partially supported by MINECO (Spain) grant PID2020-118753GB-
+I00, and by PAIDI 2020 (Andalusia) grant PROYEXCEL-00827.
+1
+
+2
+C. COSTOYA, V. MU˜NOZ, AND A. VIRUEL
+An oriented closed manifold M satisfying D(M, M) ⊆ {0, 1, −1} is called an inflexible
+manifold [7, Definition 1.4].
+This condition is equivalent to asking that D(M, M) is
+bounded: since it is a multiplicative semi-group, if there exists any ℓ ∈ D(M, M) with
+|ℓ| > 1, then D(M, M) is unbounded. Simply connected inflexible manifolds are rare
+objects that have appeared quite recently in literature using rational homotopy theory
+and surgery theory (see [5] for an account on the simply connected inflexible manifolds
+that are known at present). Not surprisingly, and in lights of Remark 1.2, part of our key
+constructions will use rational homotopy methods.
+The main result in this work answers Question 1.1 positively:
+Theorem A. Let A be a finite set of integers containing the zero. Then, A = D(M, N)
+for some oriented closed connected 3-manifolds M, N.
+The proof of this theorem will be carried out at the end of Section 3. Appealing to [14,
+Example 1.5], we point out that the 3-dimension of the manifolds is the lowest possible.
+A second problem related to Question 1.1 is also treated in this paper. More precisely,
+let the rational mapping degree set between oriented closed connected n-manifolds M, N
+be the following set
+DQ(M, N) = {d ∈ Q | ∃f : (M(0), [M]Q) → (N(0), [N]Q), deg(f) = d},
+where [M] ∈ Hn(M; Z) denotes the cohomological fundamental class of M, [M]Q ∈
+Hn(M; Q) denotes the rational cohomological fundamental class of M, and M(0) the
+rationalization of M. Then, we raise the following question, which can be thought of
+as a rational version of [14, Problem 1.4]:
+Question 1.3. Let A be a finite set of rational numbers containing the zero. Is A =
+DQ(M, N) for some oriented closed connected manifolds M, N of the same dimension?
+In Section 4 we solve this problem in the positive by proving:
+Theorem B. Let A be a finite set of rational numbers containing the zero. Then A =
+DQ(M, N) for some oriented closed manifolds M, N. Moreover, given any integer k ≥ 1,
+the manifolds M, N above can be chosen (30k + 17)-connected.
+The proofs of Theorem A and Theorem B consist of mainly two steps:
+• Arithmetical decomposition of finite sets:
+In Section 2 we show how to decompose
+the candidate A to be realized as the mapping degree set of manifolds, as an in-
+tersection of sums over specifically designed sequences of integers SBi, i = 0, . . . , n
+(see Definition 2.1). Each of those sums gradually approaches A (Proposition 2.2,
+Corollary 2.3).
+• Spherical fibrations: In Sections 3 and 4, we use certain inflexible manifolds (resp.
+inflexible Sullivan algebras) as the basis of spherical fibrations where the total
+spaces are also inflexible manifolds (resp. inflexible Sullivan algebras). Relations
+between connected sums and mapping degree sets (see Propositions 3.3 and 4.3)
+allow one to consider iterated connected sums of the total spaces, in a first stage
+to realize the sums SBi above mentioned, and in a second stage to realize the
+candidate A.
+
+FINITE SETS CONTAINING ZERO ARE MAPPING DEGREE SETS
+3
+Looking at the connectivity, while manifolds from Theorem B are simply connected
+(indeed, they are as highly connected as desired) the ones from Theorem A have non-
+trivial fundamental group. In Section 5 we will use unstable Adams operations to prove
+the following results that guarantee that manifolds realizing finite sets of integers can be
+chosen simply connected:
+Theorem C. Suppose that there exists an oriented closed k-connected 2m-manifold Σ,
+m > 1, verifying that Σ(0) is inflexible and πj(Σ(0)) = 0 for j ≥ 2m − 1. Then any finite
+set of integers A containing the zero can be realized as A = D(M, N) for some oriented
+closed k-connected (4m − 1)-manifolds M, N.
+Examples of simply connected manifolds fulfilling the hypotheses of Theorem C can be
+found in [1, Example 3.8] and [7, Theorem 6.8]. Hence, the following holds:
+Corollary D. Any finite set of integers A containing the zero can be realized as A =
+D(M, N) for some oriented closed simply connected manifolds M, N.
+2. Some arithmetic combinatorics
+In this section we show that every finite set A ⊂ Z (resp. ⊂ Q) containing the zero can
+be expressed as the intersection of sums over certain sequences of integers, that gradually
+approach A. The sequences have an additional property (see Proposition 2.2) that will
+be crucial to prove Theorem C in Section 5 below.
+Following the notation in [7, Section II.1], [14, Section 3], given A, B ⊂ Z (resp. ⊂ Q)
+we write A + B := {a + b : a ∈ A, b ∈ B} ⊂ Z (resp. ⊂ Q).
+Definition 2.1. Let B be a finite sequence of not necessarily pairwise distinct non-zero
+integers (resp. rational numbers). We write
+SB :=
+�
+b∈B
+{0, b} ⊂ Z (resp. ⊂ Q),
+and we refer to it as the sum over the sequence B.
+Proposition 2.2. Let d1, . . . , dn be pairwise distinct non-zero integers. For any positive
+integer m ≥ 1, there exist finite sequences B(i), i = 0, . . . , n, of not necessarily pairwise
+distinct non-zero integers, such that
+{0, d1, . . . , dn} =
+n�
+i=0
+SB(i).
+Moreover, every element in B(i) can be written as a power ±km for some positive integer
+k coprime to m!.
+Proof. Fix m ≥ 1. Since the construction of B(i), i = 0, . . . , n, depends on the sign of
+the pairwise distinct di ∈ Z, i = 1, . . . , n, we write them as an ordered sequence
+{−ar < . . . < −a1 < 0 < e1 < . . . < es}
+where n = r + s. We assume a0 = 0 = e0.
+
+4
+C. COSTOYA, V. MU˜NOZ, AND A. VIRUEL
+In the first place, let B(0) be the sequence consisting of ar copies of −1 = −1m and es
+copies of 1 = 1m. Thus
+{−ar < . . . < es} ⊂ SB(0) = [−ar, es] ∩ Z.
+In the second place, for j = 1, . . . , s, choose a positive kj ∈ Z coprime with m! such that
+km
+j > max{es, ej + ar}. Then, let B(j) be the sequence consisting of km
+j − ej copies of
+−1 = −1m, ej−1 copies of 1 = 1m, and one copy of km
+j . Hence,
+{−ar < . . . < es} ⊂ SB(j) =
+�
+[−(km
+j − ej), ej−1] ∪ [ej, km
+j + ej−1]
+�
+∩ Z.
+Finally, for j = s + 1, . . . , n, choose a positive kj ∈ Z coprime with m! such that km
+j >
+max{ar, aj−s + es}. Then, let B(j) be the sequence consisting of km
+j − aj−s copies of
+1 = 1m, aj−s−1 copies of −1 = −1m, and one copy of −km
+j . Hence,
+{−ar < . . . < es} ⊂ SB(j) =
+�
+[−km
+j − aj−s−1, −aj−s] ∪ [−aj−s−1, km
+j − aj−s−1]
+�
+∩ Z.
+All of the above implies that
+{0, d1, . . . , dn} = {−ar < . . . < es} =
+n�
+i=0
+SB(i).
+□
+For A ⊂ Q and λ ∈ Q we write
+λA := {λa : a ∈ A}.
+Notice that if B(i) is a finite sequence of not necessarily pairwise distinct non-zero rational
+numbers, i = 0, . . . , n, for any λ ∈ Q, we have that
+λ
+� n�
+i=0
+SB(i)
+�
+=
+n�
+i=0
+SλB(i).
+Therefore, the following is a direct consequence of Proposition 2.2:
+Corollary 2.3. Let d1, d2, . . . , dn be pairwise distinct non-zero rational numbers. Then,
+there exist finite sequences B(i), i = 0, . . . , n, of not necessarily pairwise distinct non-zero
+rational numbers, such that
+{0, d1, . . . , dn} =
+n�
+i=0
+SB(i).
+3. Circle bundles over inflexible 2-manifolds: mapping degree set
+This section is devoted to prove Theorem A. As explained in the introduction (see
+Remark 1.2) if we want to realize a finite set of integer strictly containing the zero as a
+mapping degree set D(M, N), then both M and N need to be inflexible manifolds. We
+are going to consider circle bundles over certain inflexible 2-manifolds, with prescribed
+Euler class, whose total space is again an inflexible 3-manifold. These 3-manifolds will be
+used as building blocks to construct, be means of iterated connected sums, the manifolds
+M and N.
+We first collect from literature a couple of results that are needed:
+
+FINITE SETS CONTAINING ZERO ARE MAPPING DEGREE SETS
+5
+Lemma 3.1 ([5, Lemma 7.8], [14, Lemma 3.5]). Let M1, M2 and N be oriented closed
+connected n-manifolds. Then
+D(M1, N) + D(M2, N) ⊆ D(M1#M2, N)
+Moreover, if πn−1(N) = 0, then
+D(M1, N) + D(M2, N) = D(M1#M2, N).
+We reformulate [14, Lemma 4.3] as follows:
+Lemma 3.2. Let M and N1, N2 be oriented closed n-manifolds. Then
+D(M, N1#N2) ⊆ D(M, N1) ∩ D(M, N2).
+Using the previous two lemmas, we prove the following result:
+Proposition 3.3. Let M1, M2 and N1, N2 be oriented closed n-manifolds verifying that
+πn−1(Nj) = 0, j = 1, 2, and D(Mi, Nj) = {0}, for i ̸= j. Then
+D(M1#M2, N1#N2) = D(M1, N1) ∩ D(M2, N2).
+Proof. By combining Lemma 3.2 and Lemma 3.1, it follows directly that:
+D(M1#M2, N1#N2) ⊆ D(M1#M2, N1) ∩ D(M1#M2, N2) = D(M1, N1) ∩ D(M2, N2).
+Conversely, let fi : Mi → Ni, i = 1, 2, be maps both of the same degree d. Without
+loss of generality we may assume that fi is cellular, i = 1, 2.
+Therefore it induces a
+commutative diagram of cofibration sequences
+M[n−1]
+i
+Mi
+Sn
+N[n−1]
+i
+Ni
+Sn
+fi
+˜fi
+where X[n−1] stands for the (n−1)-skeleton of X, and ˜fi is a pointed map of degree d (the
+base points in Sn are the class represented by M[n−1]
+i
+and N[n−1]
+i
+). Hence, there exists a
+pointed homotopy deforming ˜fi to a pointed map ˜gi such that ˜gi stabilizes the equator
+Sn−1 ⊂ Sn and ˜gi|Sn−1 = g for some fixed g : Sn−1 → Sn−1 of degree d. Then, the pointed
+homotopy deforming ˜fi can be lifted to Mi and defines gi : Mi → Ni that induces a maps
+between disks gi : DMi → DNi such that gi|∂DMi = g, i = 1, 2.
+Finally, gluing M1#M2 along ∂DMi, i = 1, 2, and N1#N2 along ∂DNi, i = 1, 2, give
+rise to a well defined a map
+g1#g2: M1#M2 → N1#N2
+whose degree is precisely d by construction. Therefore
+D(M1, N1) ∩ D(M2, N2) ⊆ D(M1#M2, N1#N2)
+and we conclude the proof.
+□
+A rational version of Proposition 3.3 will be required in order to prove Theorem B. This
+will be done in Section 4. Although we will not give the details, previous results (Lemma
+3.1 and Lemma 3.2) can be easily generalized to finite iterated connected sums. Hence,
+following along the lines of the proof in Proposition 3.3 we obtain:
+
+6
+C. COSTOYA, V. MU˜NOZ, AND A. VIRUEL
+Corollary 3.4. Let Mi, Ni, i = 1, . . . , r, be oriented closed connected n-manifolds such
+that πn−1(Ni) = 0, i = 1, . . . , r, and D(Mi, Nj) = {0}, for i ̸= j. Then
+D(M1# · · · #Mr, N1# · · · #Nr) =
+r�
+i=1
+D(Mi, Ni).
+We now have all the ingredients to prove our main theorem.
+Proof of Theorem A. Let A = {0, d1, . . . , dn} be a finite set of pairwise distinct integers.
+We need to show that A is realized by two oriented closed 3-manifolds M, N in the sense
+that A = D(M, N).
+For this purpose, we consider an oriented closed hyperbolic surface of genus g > 1, Σg.
+Then, for any i ∈ Z, let Ki be the total space in the circle bundle
+S1 → Ki → Σg
+with Euler number e(Ki) = i. Observe that Ki, i ∈ Z, is an aspherical 3-manifold. The
+mapping degree set between these 3-manifolds is fully described in [14, Lemma 3.4]:
+D(Ki, Kj) =
+�
+{0, j/i},
+if i|j,
+{0},
+if i̸ |j.
+(1)
+According to Proposition 2.2, for any positive integer m > 0 that we fix, there exist
+finite sequences, B(i), i = 0, . . . , n, of not necessarily pairwise distinct non-zero integers,
+satisfying that
+A =
+n�
+i=0
+SB(i).
+Now, we choose particular pairwise distinct primes q0, q1, . . . , qn fulfilling the condition
+qi > max{|b| : b ∈ B(i)}, i = 0, . . . , n,
+and we denote
+αi = qi
+�
+b∈B(i)
+b, i = 0, . . . , n.
+Then, we construct the following “intermediate” manifolds (that will serve us to realize
+each of the sums SB(i)), for i = 0, . . . , n:
+Mi =
+#
+b∈B(i)
+Kαi/b
+Ni = Kαi.
+Because Kαi are aspherical 3-manifolds, for i = 0, . . . , n, we have that π2(Kαi) = 0,
+and conditions to apply Lemma 3.1 hold. Therefore:
+D(Mi, Nj) = D( #
+b∈B(i)
+Kαi/b, Kαj) =
+�
+b∈B(i)
+D(Kαi/b, Kαj).
+Using (1), we then get that, for i = 0, . . . , n,
+D(Mi, Ni) = SB(i) , and
+D(Mi, Nj) = {0}, for i ̸= j.
+
+FINITE SETS CONTAINING ZERO ARE MAPPING DEGREE SETS
+7
+Finally, we consider the following iterated connected sums:
+M = M0#M1# . . . #Mn,
+N = N0#N1# . . . #Nn,
+for which all the conditions to apply Corollary 3.4 plainly hold. Hence,
+D(M, N) =
+n�
+i=0
+SB(i) = A,
+and the proof of Theorem A is complete.
+□
+Remark 3.5. We end this section by pointing out that all the 3-manifolds involved in the
+previous theorem are inflexible (see also Remark 1.2). It is clear, by (1), that Ki, i ∈ Z,
+are inflexible. Now, proceeding along the lines of Theorem A, we apply repeatedly Lemma
+3.2 and Lemma 3.1 to get the inflexibility property. On the one hand, we obtain that
+D(Mi, Mj) = {0} for i ̸= j, and on the other hand
+D(M, M) ⊆
+n�
+i=0
+D(Mi, Mi).
+Also, by Lemma 3.2,
+D(Mi, Mi) = D(Mi,
+#
+b∈B(i)
+Kαi/b) ⊂
+�
+b∈B(i)
+D(Mi, Kαi/b)
+and using Lemma 3.1,
+D(Mi, Kαi/b) = D( #
+b′∈B(i)
+Kαi/b′, Kαi/b) =
+�
+b′∈B(i)
+D(Kαi/b′, Kαi/b).
+Now, by Equation (1), D(Kαi/b′, Kαi/b) is either {0} or {0, b′/b} whenever b|b′. Hence,
+D(Mi, Kαi/b) is bounded, and so D(Mi, Mi) and D(M, M) are bounded. Hence Mi and
+M are inflexible manifolds, i = 1, . . . , n. The same arguments work for N so we conclude.
+4. Spherical fibrations over inflexible Sullivan models:
+rational mapping degree set
+In this section we prove Theorem B, which can be thought of as the rational version
+of Theorem A. Rational homotoy theory provides an equivalence of categories between
+the category of simply connected rational spaces and the category of certain differential
+graded algebras, the so-called Sullivan minimal models. We refer to [8] for basics facts in
+Rational Homotopy Theory.
+More concretely, if V is a graded rational vector space, we write ΛV for the free com-
+mutative graded algebra on V . A Sullivan model (ΛV, ∂) is a commutative differential
+graded algebra (cdga for short) which is free as commutative graded algebra on a simply
+connected graded vector space V of finite dimension in each degree. It is minimal if in
+addition ∂(W) ⊂ Λ≥2W.
+Now, if M is an oriented closed simply connected manifold, then the cohomology of
+the associated minimal model AM coincides with the rational cohomology of M.
+In
+
+8
+C. COSTOYA, V. MU˜NOZ, AND A. VIRUEL
+particular AM has a cohomological fundamental class [AM] ∈ H∗(AM) ∼= H∗(M; Q)
+which is isomorphic to the rational cohomological fundamental class [M]Q of M.
+Ellipticity for a Sullivan minimal model (ΛV, ∂) means that both V and H∗(ΛV ) are
+finite-dimensional.
+Hence, the cohomology is a Poincar´e duality algebra [9] and one
+can easily compute the degree of its fundamental cohomological class [8, Theorem 32.6].
+In particular one can introduce the notion of mapping degree between elliptic Sullivan
+minimal models and also translate the notion of inflexibility:
+Let (ΛV, ∂) be an elliptic Sullivan minimal model. Let µ ∈ (ΛV )n be a representative of
+its cohomological fundamental class. Then (ΛV, ∂) is inflexible if for every cdga-morphism
+ϕ: (ΛV, ∂) → (ΛV, ∂)
+we have deg(ϕ) = 0, ±1, where H([µ]) = deg(ϕ)[µ].
+4.1. Rational mapping degree set and connected sums. The following results es-
+tablish, under certain restrictions, the relationship between rational mapping degree sets
+and connected sums of manifolds:
+Lemma 4.1 ([7, Lemma II.2]). Let M1, M2 and N be oriented closed n-manifolds with
+πn−1(N(0)) = 0. Then
+DQ(M1#M2, N) = DQ(M1, N) + DQ(M2, N).
+Proof. Under the same assumptions, in [7, Lemma II.2] is asserted that the following
+holds:
+D(M1#M2, N) ⊆ DQ(M1, N) + DQ(M2, N).
+However, a stronger result is demonstrated in the proof. Namely,
+DQ(M1#M2, N) ⊆ DQ(M1, N) + DQ(M2, N).
+Hence, it suffices to prove the other inclusion.
+To that end, one can apply the same
+arguments as in [5, Lemma 7.8]: let q(0) : (M1)(0)#(M2)(0) → (M1)(0) ∨ (M2)(0) denote the
+rationalization of the pinching map. Then for any given maps fi : (Mi)(0) → N(0), the
+composition
+(f1 ∨ f2) ◦ q: (M1)(0)#(M2)(0) → N(0)
+has degree deg(f1) + deg(f2) and the result follows.
+□
+A precise definition of connected sum in the world of cdga’s:
+Definition 4.2. Let Ai, i = 1, 2, be connected cdgas and let ai ∈ Ai, i = 1, 2, be elements
+of the same degree. The connected sum of the pairs (Ai, [ai]), i = 1, 2, is the dga
+(A1, [a1])#(A2, [a2])
+def
+:= (A1 ⊕Q A2)/I ,
+where A1 ⊕Q A2
+def
+:= (A1 ⊕ A2)/Q{(1, −1)}, and I ⊂ A1 ⊕Q A2 is the differential ideal
+generated by a1 − a2.
+
+FINITE SETS CONTAINING ZERO ARE MAPPING DEGREE SETS
+9
+Connected sums of cdgas provide rational models for connected sums of oriented mani-
+folds. Indeed, for Mi, i = 1, 2 oriented closed simply connected n-manifold, with Sullivan
+minimal model AMi, let mi be a representative of the cohomological fundamental class of
+AMi, for i = 1, 2. By [5, Theorem 7.12]
+(AM1, [m1])#(AM 2, [m2])
+(2)
+is a rational model of M1#M2.
+We use (2) above to prove the rational version of Proposition 3.3:
+Proposition 4.3. Let M1, M2 and N1, N2 be oriented closed simply connected n-manifolds
+such that πn−1(Nj) ⊗ Q = 0, j = 1, 2, and DQ(Mi, Nj) = {0}, i ̸= j. Then
+DQ(M1#M2, N1#N2) = DQ(M1, N1) ∩ DQ(M2, N2).
+Proof. According to Lemma 4.1 and the rational version of Lemma 3.2 (which can be
+proved following the same arguments as in [14, Lemma 4.3]), we get that
+DQ(M1#M2, N1#N2) ⊂ DQ(M1, N1) ∩ DQ(M2, N2).
+Conversely, let (AMi, [mi]) and (ANi, [ni]) be Sullivan minimal models of (Mi, [Mi]) and
+(Ni, [Ni]) respectively, i = 1, 2. For
+d ∈ DQ(M1, N1) ∩ DQ(M2, N2)
+there exists fi : ANi → AMi with fi(ni) = d · mi + αi and where αi is a coboundary,
+i = 1, 2.
+Because (AM1, [m1 + α1])#(AM2, [m2 + α2]) and (AN1, [n1])#(AN2, [n2]) are
+Sullivan minimal models for M1#M2 and N1#N2 respectively, then f1 and f2 give rise to
+a well defined cdga-morphism
+f1#f2 : (AN1, [n1])#(AN2, [n2]) → (AM1, [m1 + α1])#(AM2, [m2 + α2])
+defined by
+(f1#f2)(x) =
+�
+f1(x),
+if x ∈ AN1,
+f2(x),
+if x ∈ AN2
+and whose degree is deg(f1#f2) = d.
+□
+Remark 4.4. The previous result can be generalized to an arbitrary finite iterated con-
+nected sum, as in Corollary 3.4. Namely, if Mi, Ni, i = 1, . . . , r, are oriented closed simply
+connected n-manifolds such that πn−1(Nj) = 0, j = 1, . . . , r, and DQ(Mi, Nj) = {0}, for
+i ̸= j, then
+DQ(M1# · · · #Mr, N1# · · · #Nr) =
+r�
+i=1
+DQ(Mi, Ni).
+4.2. Inflexible Sullivan minimal models of inflexible manifolds. Following the
+same strategy as in Section 3, we consider spherical fibrations over certain elliptic and
+inflexible Sullivan minimal models (Definition 4.5), whose total spaces are the Sullivan
+minimal models of inflexible manifolds (see Lemma 4.7). These manifolds will be the
+building blocks to construct, by means of iterated connected sums, manifolds that realize
+finite sets of rational numbers.
+
+10
+C. COSTOYA, V. MU˜NOZ, AND A. VIRUEL
+Definition 4.5. Let (A, ∂) be an elliptic, inflexible Sullivan minimal model of formal
+dimension 2m, m ≥ 1, such that πj(A) = 0 for j ≥ 2m−1. Fix µ ∈ A a representative of
+its cohomological fundamental class. Then for any non-zero q ∈ Q, define the following
+Sullivan minimal model
+(Kq(A), ∂) := (A ⊗ Λ(y2m−1), ∂)
+that extends the differential of A by ∂(y2m−1) = qµ.
+Remark 4.6. Notice that (Kq(A), ∂) is the total space in the rational S2m−1-fiber sequence:
+(Λ(y2m−1), 0) ←− (Kq(A), ∂) ←− (A, ∂),
+whose Euler class is q[µ].
+Lemma 4.7. Let (A, ∂) be an elliptic, inflexible Sullivan minimal model of formal di-
+mension 2m, m ≥ 1, such that πj(A) = 0 for j ≥ 2m − 1. Fix µ ∈ A a representative
+of the fundamental class of A, and let x ∈ A such that ∂(x) = µ2. Then for any non-
+zero q ∈ Q,
+�
+Kq(A), [y2m−1µ − qx]
+�
+is the Sullivan minimal model of an oriented closed
+inflexible (4m − 1)-manifold MKq, with the same connectivity as (A, ∂).
+Proof. According to [6, Proposition 3.1], (Kq(A), ∂) is an elliptic Sullivan model of formal
+dimension 4m−1 where y2m−1µ−qx is a representative of its cohomological fundamental
+class. By [6, Lemma 3.2], (Kq(A), ∂) is an inflexible algebra because (A, ∂) is so. Now,
+since its formal dimension is 4m − 1 ≡ 3 mod 4, the obstruction theory of Sullivan [15,
+Theorem (13.2)] and Barge [2, Th´eor`eme 1] guarantees that (Kq(A), [y2m−1µ − qx]) is the
+Sullivan minimal model of an oriented closed simply-connected manifold MKq. Finally,
+by [4, Proposition A.1], MKq and (A, ∂) have the same connectivity.
+□
+We compute the rational mapping degree set between the manifolds appearing in the
+previous lemma:
+Lemma 4.8. For any non-zero q ∈ Q, let MKq be the oriented closed manifold from
+Lemma 4.7 whose Sullivan minimal model is (Kq(A), ∂) from Definition 4.5. Then
+DQ(MKp, MKq) = {0, q/p}.
+Proof. We follow the ideas in [6, Lemma 3.2].
+Let f : (Kq(A, ∂) → (Kp(A), ∂) be a
+morphism of non-trivial degree d ∈ Q, that is,
+f(y2m−1µ − qx) = d(y2m−1µ − px) + α
+(3)
+where α is a coboundary. By a degree argument, f induces a non-trivial degree morphism
+f|A : (A, ∂) → (A, ∂). On the one hand f(µ) = �dµ + β1 and f(x) = �d 2x + β2 where β1, β2
+are coboundaries, and �d ∈ {−1, 1}. On the other hand, f(y2m−1) = ay2m−1 + γ where
+a ∈ Q and γ is a coboundary.
+Because f(∂y2m−1) = ∂f(y2m−1), we get that ap = q �d and β1 = 0. Hence a = �d (q/p)
+and
+f(y2m−1µ − qx) =
+�
+(�d q/p y2m−1 + γ
+�
+(�dµ) − q(�d 2x + β2)
+= (�d 2q/p)(y2m−1µ − px) − qβ2
+= (q/p)(y2m−1µ − px) − qβ2
+(recall �d ∈ {−1, 1}).
+By comparing this equation to (3), we obtain that d = q/p and the proof is complete.
+□
+
+FINITE SETS CONTAINING ZERO ARE MAPPING DEGREE SETS
+11
+We illustrate the existence of elliptic, inflexible Sullivan minimal models satisfying the
+conditions from Definition 4.5 and Lemma 4.7:
+Definition 4.9. Let Γ be a connected finite simple graph with more that one vertex, i.e.,
+|V (Γ)| > 1. Given an integer k ≥ 1, let (Ak(Γ), ∂) be the (30k +17)-connected elliptic and
+inflexible Sullivan algebra constructed in [4, Definition 2.1], whose formal dimension is
+2m = 540k2+984k +396+|V (Γ)|(360k2+436k +132) and πj
+�
+Ak(Γ)
+�
+= 0 for j ≥ 2m−1.
+Fix µ ∈ Ak(Γ) a representative of the cohomological fundamental class. Then for any
+nonz-zero q ∈ Q, define the following Sullivan minimal model
+(Kq(Γ, k), ∂) := (Ak(Γ) ⊗ Λ(y2m−1), ∂)
+that extends the differential of Ak(Γ) by ∂(y2m−1) = qµ.
+Remark 4.10. Because conditions from Lemma 4.7 hold, (Kq(Γ, k), ∂) is a Sullivan model
+of an oriented closed (30k + 17)-connected inflexible (4m − 1)-manifold MKq(Γ,k), where
+2m = 540k2 + 984k + 396 + |V (Γ)|(360k2 + 436k + 132).
+Lemma 4.11. Let Γ1 and Γ2 be connected finite simple graphs with |V (Γ1)| = |V (Γ2)| > 1.
+Given a positive integer k ≥ 1, and a non-zero pi ∈ Q, i = 1, 2, consider the manifold
+MKpi(Γi,k), i = 1, 2 as in Remark 4.10. Then
+DQ(MKp1(Γ1,k), MKp2(Γ2,k)) =
+�
+{0, p2/p1},
+if
+Γ1 ∼= Γ2,
+{0},
+otherwise.
+Proof. Let (Kpi(Γi, k), ∂) = (Ak(Γi) ⊗ Λ(yi), ∂), introduced in Definition 4.9, be the Sul-
+livan model of the manifold MKpi(Γi,k), where ∂(yi) = piµi for µi a representative of the
+cohomological fundamental class of Ak(Γi), i = 1, 2. Recall from Lemma 4.7 that for
+xi ∈ Ak(Γi) satisfying ∂(xi) = µ2
+i , the element yiµi − pixi is a representative of the
+cohomological fundamental class of (Kpi(Γi, k), ∂), i = 1, 2.
+With these constructions in mind, we follow the ideas from [6, Lemma 3.2]. Consider
+a morphism of non-trivial degree d ∈ Q:
+f : (Kp2(Γ2, k), ∂) → (Kp1(Γ1, k), ∂).
+Then f(y2µ2 − p2x2) = d(y1µ1 − p1x1) + α with α a coboundary. By a degree argument,
+the morphism f induces a non-trivial degree morphism
+f|Ak(Γ2): (Ak(Γ2), ∂) → (Ak(Γ1), ∂).
+Focusing specifically on this former morphism, the arguments in [4, Lemma 2.12] (see
+also [6, Remark 2.8]), show that it is induced by a graph full monomorphism σ: Γ1 → Γ2.
+Now, since |V (Γ1)| = |V (Γ2)|, σ is indeed an isomorphism of graphs, and f(µ2) = µ1 + β1
+and f(x2) = x1 + β2 with β1, β2 coboundaries, by [4, Lemma 2.12].
+Finally, by another degree reasoning argument, one obtains that f(y2) = ay1 + γ where
+a is a non-zero rational number, and γ is a coboundary. We conclude as in the proof of
+Lemma 4.8.
+□
+
+12
+C. COSTOYA, V. MU˜NOZ, AND A. VIRUEL
+4.3. Proof of Theorem B. Let A = {0, d1, . . . , dn} where d1, d2, . . . , dn are pairwise
+different non-zero rational numbers. Fix an integer k ≥ 1. According to Corollary 2.3,
+there exist finite sequences of not necessarily pairwise distinct non-zero rational numbers
+B(i), i = 0, . . . , n, such that
+A =
+n�
+i=0
+SB(i).
+Choose Γ0, Γ1, . . . , Γn, pairwise non-isomorphic connected finite simple graphs, such
+that |V (Γi)| = |V (Γj)| > 1 for every i, j = 0, . . . , n. According to Remark 4.10, we define
+the (30k + 17)-connected manifolds
+Mi =
+#
+b∈B(i)
+MKb−1(Γi,k)
+Ni = MK1(Γi,k),
+for i = 0, . . . , n. By Lemmas 4.11 and the rational version of Lemma 3.2 (which can be
+proved following the same arguments as in [14, Lemma 4.3]), we have that
+DQ(Mi, Ni) = SB(i) , and
+DQ(Mi, Nj) = {0}, for i ̸= j.
+Finally, define
+M = M0#M1# . . . #Mn,
+N = N0#N1# . . . #Nn.
+and use Proposition 4.3 (see also Remark 4.4) to get
+DQ(M, N) =
+N�
+i=0
+SB(i) = A.
+5. From unstable Adams operations to mapping degree sets
+We recall the basics on unstable Adams operations following Jackowski-McCLure-
+Oliver’s work [10, 11]. Given a compact connected Lie group G, a self-map f : BG → BG
+is called an unstable Adams operation of degree r ≥ 0, if H2i(f; Q) is the multiplication
+by ri for each i > 0 [10, p. 183]. For a given simple Lie group G with Weyl group WG,
+an unstable Adams operation of degree r > 0 exists if and only if (r, |WG|) = 1, and
+moreover, this operation is unique [10, Theorem 2]. In particular, when G = SO(2m − 1)
+or G = SO(2m), m > 1, unstable Adams operations of degree r > 0 exist if r and m!
+are coprime numbers. In what follows, we denote by ϕr the unstable Adams operation
+of degree r > 0 on BSO(2m − 1) and BSO(2m). Notice that since they are unique, then
+ϕs ◦ ϕr = ϕrs.
+Henceforward, (Σ, [Σ]) is a fixed oriented closed connected 2m-manifold whose ratio-
+nalization (Σ(0), [Σ]Q) is inflexible and πj(Σ(0)) = 0 for j ≥ 2m − 1. Let (AΣ, ∂) be a
+Sullivan minimal model of Σ. Denote by π: Σ → S2m the map obtained by collapsing the
+(2m − 1)-skeleton of Σ.
+
+FINITE SETS CONTAINING ZERO ARE MAPPING DEGREE SETS
+13
+Lemma 5.1. Let X2m ∈ H2m(BSO(2m); Z) be the Euler class of the spherical fiber se-
+quence
+S2m−1 → BSO(2m − 1) → BSO(2m),
+thus X2m is a torsion free integral cohomology class [3, Theorem 1.5, Equation (2.1)], and
+S2m is thought of as an oriented closed manifold. There exists ι: S2m → BSO(2m), a tor-
+sion free element in π2m(BSO(2m)), and a non-zero integer κ ∈ Z such that H∗(ι; Z)(X2m) =
+κ[S2m].
+Proof. Recall that π2m(BSO(2m)) ∼= π2m−1(SO(2m)). By [13, Corollary IV.6.14] (see also
+[12, p. 161]), π2m−1(SO(2m)) contains a copy of Z inducing the p-local (thus rational)
+splitting SO(2m) ≃(p) SO(2m − 1) × S2m−1 [13, Corollary IV.6.21]. Let ι be a generator
+of such a copy of Z in π2m(BSO(2m)).
+By construction, H∗(ι; Q) is non-trivial on the Euler class of the rational fiber sequence
+S2m−1
+(0)
+→ BSO(2m − 1)(0) → BSO(2m)(0),
+which is just X2m ⊗Q 1. Therefore, H∗(ι; Z)(X2m) = κ[S2m] for some non-zero κ ∈ Z.
+□
+Definition 5.2. Given any integers r > 0, m > 1, with r coprime to m!, we define:
+(1) The oriented (4m − 1)-manifold Erm as the total space in the principal spherical
+SO(2m)-fiber bundle
+S2m−1
+S2m−1
+Erm
+BSO(2m − 1)
+Σ
+BSO(2m),
+⌟
+φr
+(4)
+where φr = ϕr ◦ ι ◦ π.
+(2) The oriented (4m−1)–manifold E−rm obtained by reversing the original orientation
+on the manifold Erm above introduced.
+Remark 5.3. The Euler class of the spherical fiber bundle over Σ given in diagram (4) is
+κrm[Σ] by construction.
+Recall from the beginning of this section that (Σ, [Σ]) is a fixed oriented closed connected
+2m-manifold where (AΣ, ∂) is its Sullivan minimal model.
+Lemma 5.4. Let Erm be the manifold introduced in Definition 5.2. A Sullivan mini-
+mal model of Erm is Kκrm(AΣ) as given in Definition 4.5. Therefore Erm is rationally
+equivalent to MKκrm, the manifold given in Lemma 4.7.
+Proof. As it was pointed out in Remark 4.6, Kκrm(AΣ) is a Sullivan minimal model for
+the total space in a rational S2m−1-fiber sequence whose Euler class is κrm[Σ]Q. It coin-
+cides with the Euler class of the rationalization of the spherical SO(2m)-fiber bundle in
+diagram (4). Therefore Kκrm(AΣ) is a Sullivan minimal model for Erm.
+□
+
+14
+C. COSTOYA, V. MU˜NOZ, AND A. VIRUEL
+Lemma 5.5. Let i, j, m be positive integers, m > 1, such that (i, m!) = (j, m!) = 1, and
+let Erm, r = i, j, be the (4m − 1)-manifold introduced in Definition 5.2. Then
+D(Eim, Ejm) =
+�
+{0, (j/i)m},
+if i|j,
+{0},
+if i̸ |j.
+Proof. By Lemma 5.4, the manifolds Erm and MKκrm are rationally equivalent, for every
+0 < r ∈ Z. Therefore:
+D(Eim, Ejm) ⊂ DQ(Eim, Ejm) ∩ Z = DQ(MKκim, MKκjm) ∩ Z
+= {0, (j/i)m} ∩ Z (by Lemma 4.8)
+=
+�
+{0, (j/i)m},
+if i|j,
+{0},
+if i̸ |j.
+The proof will be completed if we construct a map f : Eim → Ejm of degree (j/i)m
+when i|j. To this end, let us suppose that j = di, d ∈ Z, and recall that unstable Adams
+operations satisfy that ϕj = ϕd ◦ ϕi.
+Therefore, by construction (see Definition 5.2)
+φj = ϕd ◦ φi.
+Let f : Eim → Ejm be the map obtained by the universal property of
+pullbacks in the following commutative diagram:
+Eim
+BSO(2m − 1)
+Ejm
+BSO(2m − 1)
+Σ
+BSO(2m)
+BSO(2m)
+f
+ϕd
+⌟
+φi
+ϕd
+(5)
+Diagram (5) gives rise to a commutative diagram of spherical fiber sequences
+S2m−1
+S2m−1
+Eim
+Ejm
+Σ
+Σ,
+�f
+f
+(6)
+whose associated Serre spectral sequences (Sss) can be compared via the edge morphisms
+given by naturality: the Sss associated to the left (resp. right) side of diagram (6) is fully
+determined by the differential
+d2m([S2m−1]) = κim[Σ] (resp. d2m([S2m−1]) = κjm[Σ]),
+and since by naturality
+d2m
+�
+H∗( �f)([S2m−1])
+�
+= H∗(IdΣ)
+�
+d2m([S2m−1])
+�
+we obtain that deg( �f) = (j/i)m.
+
+FINITE SETS CONTAINING ZERO ARE MAPPING DEGREE SETS
+15
+Now, the cohomological fundamental class [Eim] (resp. [Ejm]) is represented by the
+class [S2m−1]⊗[Σ] in the E2m−1,2m
+∞
+-term of the Sss associated to the left (resp. right) fiber
+sequence in diagram (6). Hence by naturality
+H∗(f)([Ejm]) = H∗( �f)([S2m−1]) ⊗ H∗(IdΣ)([Σ])]
+=
+�
+(j/i)m[S2m−1]
+�
+⊗ [Σ]
+= (j/i)m[Eim]
+and therefore deg(f) = (j/i)m.
+□
+Remark 5.6. Notice that manifolds Erm and E−rm differ in just the orientation. Hence,
+for any other oriented closed connected (4m − 1)-manifold N, the mapping set degree is
+D(E−rm, N) = −D(Erm, N) and D(N, E−rm) = −D(N, Erm).
+Proof of Theorem C. Let Σ be an oriented closed k-connected 2m-manifold verifying
+that Σ(0) is inflexible and πj(Σ(0)) = 0 for j ≥ 2m − 1. Let A = {0, d1, . . . , dn} where
+d1, d2, . . . , dn are pairwise different non-zero integers.
+According to Proposition 2.2, there exist finite sequences B(i), i = 0, . . . , n, of not
+necessarily pairwise distinct non-zero integers, such that every element in B(i) can be
+written as ±rm for 0 < r ∈ Z with (r, m!) = 1, and
+A =
+n�
+i=0
+SB(i).
+Choose pairwise distinct prime numbers q0, q1, . . . , qn, in such a way that
+qj > max{|b| : b ∈ B(i), i = 0, . . . , n}
+and (qj, m!) = 1, for j = 0, . . . , n. Let αi = qm
+i
+�
+b∈B(i)
+b, for every i = 0, . . . , n. Notice
+that αi and αi/b, b ∈ B(i), are integers that can be written up to a sign as rm for some
+positive integer r such that (r, m!) = 1
+Following the notation in Definition 5.2, we define the following (4m − 1)-manifolds
+Mi =
+#
+b∈B(i)
+Eαi/b
+Ni = Eαi
+for i = 0, . . . , n. According to Lemma 5.5 and Lemma 3.1, we deduce that
+D(Mi, Ni) = SB(i) , and
+D(Mi, Nj) = {0}, for i ̸= j.
+Finally, we construct
+M = M0#M1# . . . #Mn,
+N = N0#N1# . . . #Nn,
+and, according to Corollary 3.4, we obtain that
+D(M, N) =
+N
+�
+i=0
+SB(i) = A.
+
+16
+C. COSTOYA, V. MU˜NOZ, AND A. VIRUEL
+References
+[1] M. Amann. Degrees of self-maps of simply connected manifolds. Int. Math. Res. Not., 2015(18):8545–
+8589, 2015.
+[2] J. Barge. Structures diff´erentiables sur les types d’homotopie rationnelle simplement connexes. Ann.
+Sci. ´Ec. Norm. Sup´er. (4), 9:469–501, 1976.
+[3] E. H. j. Brown. The cohomology of BSOnand BOnwith integer coefficients. Proc. Am. Math. Soc.,
+85:283–288, 1982.
+[4] C. Costoya, D. M´endez, and A. Viruel. Homotopically rigid Sullivan algebras and their applications.
+In An alpine bouquet of algebraic topology, volume 708 of Contemp. Math., pages 103–121. Amer.
+Math. Soc., Providence, RI, 2018.
+[5] C. Costoya, V. Mu˜noz, and A. Viruel. On Strongly Inflexible Manifolds. International Mathematics
+Research Notices, 03 2022. rnac064.
+[6] C. Costoya and A. Viruel. Every finite group is the group of self-homotopy equivalences of an elliptic
+space. Acta Math., 213(1):49–62, 2014.
+[7] D. Crowley and C. L¨oh. Functorial seminorms on singular homology and (in)flexible manifolds.
+Algebr. Geom. Topol., 15(3):1453–1499, 2015.
+[8] Y. F´elix, S. Halperin, and J.-C. Thomas. Rational homotopy theory, volume 205 of Springer-Verlag.
+Springer-Verlag, New York, 2001.
+[9] S. Halperin. Finiteness in the minimal models of sullivan. Trans. Amer. Math. Soc., 230:173–199,
+1977.
+[10] S. Jackowski, J. E. McClure, and B. Oliver. Homotopy classification of self-maps of BG via G-actions.
+I. Ann. Math. (2), 135(1):183–226, 1992.
+[11] S. Jackowski, J. E. McClure, and B. Oliver. Homotopy classification of self-maps of BG via G-actions.
+II. Ann. Math. (2), 135(2):227–270, 1992.
+[12] M. A. Kervaire. Some non-stable homotopy groups of Lie groups. Ill. J. Math., 4:161–169, 1960.
+[13] M. Mimura and H. Toda. Topology of Lie groups, I and II. Transl. from the Jap. by Mamoru Mimura
+and Hirosi Toda, volume 91 of Transl. Math. Monogr. Providence, RI: American Mathematical
+Society, 1991.
+[14] C. Neofytidis, S. Wang, and Z. Wang. Realising sets of integers as mapping degree sets. Bull. Lond.
+Math. Soc. (to appear), 2022.
+[15] D. Sullivan. Infinitesimal computations in topology. Publ. Math., Inst. Hautes ´Etud. Sci., 47:269–331,
+1977.
+CITIC, Departamento de Computaci´on, Universidade da Coru˜na, 15071-A Coru˜na, Spain.
+Email address: [email protected]
+Departamento de ´Algebra, Geometr´ıa y Topolog´ıa, Universidad Complutense de Madrid,
+Plaza de las Ciencias, 3, 28040-Madrid, Spain
+Email address: [email protected]
+Departamento de ´Algebra, Geometr´ıa y Topolog´ıa, Universidad de M´alaga, Campus
+de Teatinos, s/n, 29071-M´alaga, Spain
+Email address: [email protected]
+

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@@ -0,0 +1,2306 @@
+Spatially Varying Anisotropy for Gaussian Random Fields
+in Three-Dimensional Space
+Martin Outzen Berild∗and Geir-Arne Fuglstad
+Department of Mathematical Sciences,
+Norwegian University of Science and Technology, Norway
+Abstract
+Isotropic covariance structures can be unreasonable for phenomena in
+three-dimensional spaces. We construct a class of non-stationary anisotropic
+Gaussian random fields (GRFs) in three dimensions through stochastic par-
+tial differential equations allowing for Gaussian Markov random field approx-
+imations. The class is proven in a simulation study where we explore the
+amount of data required to estimate these models. Then, we apply it to an
+ocean mass outside Trondheim, Norway, based on simulations from a numer-
+ical ocean model. And our model outperforms a stationary anisotropic GRF
+on predictions using in-situ measurements collected with an autonomous
+underwater vehicle.
+Keywords: Spatial non-stationarity; spatially-varying anisotropy; stochastic par-
+tial differential equations; Gaussian Markov random fields.
+∗Corresponding author, [email protected]
+1
+arXiv:2301.01372v1  [stat.ME]  3 Jan 2023
+
+1
+Introduction
+Gaussian random fields (GRFs) are a powerful tool for spatial and spatio-temporal
+geostatistical modeling (Diggle et al., 1998; Cressie and Wikle, 2015). When the
+key goal is predictions at unobserved locations, i.e., kriging, isotropic covariance
+functions often perform well, and more flexible covariance structures should be
+used with care (Fuglstad et al., 2015b). However, the screening effect in kriging
+(Stein, 2002) is not relevant in other settings where the primary goal is the esti-
+mated covariance structure. E.g., to describe internal variability in a climate model
+ensemble (Castruccio et al., 2019), or to produce a spatial prior based on numerical
+simulations that will later be used to guide autonomous sampling (Fossum et al.,
+2021; Foss et al., 2021). For the former, Fuglstad and Castruccio (2020); Hu et al.
+(2021) demonstrated that flexible covariance structures can perform better than
+stationary covariance structures.
+There are many approaches to constructing flexible covariance structures (Samp-
+son, 2010; Salvaña and Genton, 2021; Schmidt et al., 2011). Some early approaches
+are the deformation method (Sampson and Guttorp, 1992) and kernel convolutions
+(Paciorek and Schervish, 2006), but they both involve the covariances between any
+pair of locations. This means standard implementations are infeasible for large
+datasets. There are many ways to overcome such computational issues in spatial
+statistics and some are applicable for flexible covariance structures (Heaton et al.,
+2019). The stochastic partial differential equation (SPDE) approach (Lindgren
+et al., 2011) is interesting because it directly gives rise to computationally efficient
+models and easily extends to non-stationary covariance models.
+However, increasing the degree of flexibility in the covariance structure requires
+increasing the number of parameters. The common isotropic Matérn covariance
+functions (Stein, 2012) are parametrized through 3 parameters: marginal vari-
+ance, range, and smoothness. Flexible models can have 100s or more parameters
+2
+
+(Fuglstad et al., 2015b). An appealing way to reduce dimensionality is to describe
+the covariance structure through covariates (Schmidt et al., 2011; Neto et al., 2014;
+Ingebrigtsen et al., 2014, 2015; Risser and Calder, 2015).
+The aforementioned works are all considering flexible covariance structures in
+two-dimensional space, and while the methods can be extended to three-dimensional
+space, the literature is sparse. For example, the SPDE approach has been used
+for simple anisotropic covariance structures in the context of fMRI data from the
+brain (Sidén et al., 2021), and more complex covariance structures in the context of
+astronomy (Lee and Gammie, 2021), though this was two-dimensional space and
+time treated as three-dimensional space. However, spatially varying anisotropy
+in the SPDE approach (Fuglstad et al., 2015a) has not been extended to three-
+dimensional space.
+The aim of this paper is to develop a new method for spatially varying anisotropy
+in three-dimensional space through the SPDE approach.
+A key advantage is
+that the formulation as an SPDE guarantees a valid covariance structure, and
+the main challenge is how to describe and parametrize non-stationary covariance
+structures. Fuglstad et al. (2015a) used one vector field to describe spatially vary-
+ing anisotropy, but in three dimensions, two spatially varying orthogonal vector
+fields are necessary for full generality.
+In a simulation study, we investigate how much data is necessary to recover
+parameters for three different model complexities: stationary isotropic, station-
+ary anisotropic, and non-stationary anisotropic. We then estimate GRF priors to
+encode knowledge about the ocean from a numerical forecast generated by the nu-
+merical model SINMOD by SINTEF. A stationary GRF prior and a non-stationary
+GRF prior are updated based on in-situ measurements by an autonomous under-
+water vehicle (AUV), and we evaluate the predictive ability during a mission in
+Trondheimsfjorden, Norway, on May 27, 2021. Improved predictions are key, for
+3
+
+example, in autonomous sampling of the oceans (Fossum et al., 2019, 2021), but
+current approaches in autonomous ocean sampling are limited to stationary GRFs.
+In Section 2, we describe how to model anisotropy and non-stationarity in three
+dimensions using SPDEs. Then in Section 3, we describe how to perform inference
+for the new model in a computationally efficient way. In Section 4, we describe
+the simulation study and discuss the results, and continue with the application to
+sampling in the ocean in Section 5. We end with a discussion in Section 6.
+2
+Constructing SPDEs with spatially varying anisotropy
+2.1
+Existing models
+The Matérn covariance function on R3 is given by
+r(s1, s2) =
+σ2
+2ν−1Γ(ν)(κ||s1 − s2||)νKν(κ||s1 − s2||),
+s1, s2 ∈ R3,
+(1)
+where ||·|| is the Euclidean distance in R3, σ > 0 is the marginal standard deviation,
+Kν is the modified Bessel function of the second kind and order ν > 0, and κ > 0
+is an inverse spatial scale parameter. As discussed in Lindgren et al. (2011), GRFs
+with this covariance function is the stationary solutions of the SPDE
+(κ2 − ∇ · ∇)α/2(τu(s)) = W(s),
+s ∈ R3,
+(2)
+where α = ν + 3/2, τ =
+√
+8πκ/σ, ∇ · ∇ is the Laplacian, and W is a standard
+Gaussian white noise process.
+Lindgren et al. (2011) proposed to introduce non-stationarity by allowing κ
+and τ to vary in space (Ingebrigtsen et al., 2014, 2015) or by deformations of space
+(Hildeman et al., 2021). Fuglstad et al. (2015a,b) consider a version of the SPDE,
+where the Laplacian is replaced by an anisotropic Laplacian where the direction
+and degree of anisotropy vary spatially. This was further extended to spherical
+4
+
+geometry in Fuglstad and Castruccio (2020); Hu et al. (2021). However, all of
+these works were in two-dimensional base spaces, and only simpler models have
+been applied for three-dimensional base spaces (Sidén et al., 2021).
+The key idea in Fuglstad et al. (2015a) was to replace ∇ · ∇ by ∇ · H(s)∇,
+where H(s) is everywhere a symmetric positive definite 2 × 2 matrix that controls
+the strength and direction of anisotropy. The matrix-valued function was specified
+as H(s) = γ(s)I2 + v(s)v(s)T, s ∈ R2, where γ(·) is a positive function and v(·)
+is a vector field. This allows γ(·) to control the baseline strength of dependence in
+all directions, and v(·) to control the strength and direction of additional spatial
+dependence. However, the same parametrization in R3 is not sufficiently general
+to control anisotropy fully.
+2.2
+Stationary anisotropy in R3
+We follow the idea in Fuglstad et al. (2015a) for R2, and change the SPDE in
+Equation (2) to
+(κ2 − ∇ · H∇)u(s) = W(s), s ∈ R3,
+(3)
+where ∇·H∇ is an anisotropic Laplacian and the symmetric positive definite 3×3
+matrix H controls the anisotropy. The parameter τ has been dropped since κ and
+H together control both marginal variance and correlation.
+As shown in Appendix A.1, the resulting marginal variance is
+σ2
+m =
+1
+8πκ
+�
+det(H)
+(4)
+and the covariance function is explicitly known as
+r(s1, s2) =
+1
+8πκ
+�
+det(H)
+exp
+�
+−κ||H−1/2(s1 − s2)||)
+�
+(5)
+for s1, s2 ∈ R3. The latter is derived in Appendix A.2. This corresponds to geo-
+metric anisotropy in the Matérn covariance function with smoothness ν = 1/2. To
+5
+
+understand the behavior of the covariance function, it is useful to think about H
+in terms of its eigenvalue decomposition. Let ˜v1, ˜v2, and ˜v3 be orthonormal eigen-
+vectors corresponding to eigenvalues λ1, λ2 and λ3, respectively. Then Figure 1
+shows an example of the 0.37 level iso-correlation surface that will arise from the
+covariance function in Equation (5). The semi-axes of the ellipsoid in the figure
+are v1 = (√λ1/κ)˜v1, v2 = (√λ2/κ)˜v2, and v3 = (√λ3/κ)˜v3, which by evaluating
+the covariance function with either of these semi-axes will yield the relationship
+and the iso-correlation level r(v)/σ2
+m = e−1 ≈ 0.37.
+v
+v
+v
+1
+2
+3
+Figure 1: Iso-correlation surface at the ∼0.37 level of Equation (5), where v1, v2,
+and v3 are the eigenvectors of H with lengths √λ1/κ, √λ2/κ and √λ3/κ.
+We generalize the parametrization described in Section 2.2 and H is decom-
+posed as
+H = γI3 + vvT + ωωT.
+(6)
+where v = (vx, vy, vz)T ∈ R3 and w = (ωx, ωy, ωz)T ∈ R3, v ⊥ ω, and γ > 0.
+The eigenvalue decomposition of H has eigenvalues λ1 = γ, λ2 = γ + ||v||2 and
+6
+
+λ3 = γ + ||w||2 with the corresponding eigenvectors v1 = v × ω, v2 = v and
+v3 = ω, respectively. We construct ω by a linear combination of two orthogonal
+vectors in the plane with v as normal vector. First, let ω1 = (−vy, vx, 0)T, which
+satisfies v ⊥ ω1. Second, let ω2 = v × ω1 = (−vzvx, −vzvy, v2
+x + v2
+y)T, which also
+satisfies v ⊥ ω2. We parametrize ω through
+ω = ρ1
+ω1
+||ω1|| + ρ2
+ω2
+||ω2||,
+(7)
+where ρ1, ρ2 ∈ R which works whenever vx = vy ̸= 0. An alternative solution is to
+use Euler-Rodrigues parametrization (Euler, 1771; Rodrigues, 1840) to obtain both
+v and ω; however, in this case, the parameters are less interpretable and the issue
+is simply nullified by numerical optimization with appropriate initial parameter
+values.
+The above parametrization for H uses six parameters, γ, vx, vy, vz, ρ1, and
+ρ2, to describe all forms of geometric anisotropy. The parameterization is inter-
+pretable: 1) γ controls the isotropic effect, 2) vx, vy, and vz controls one anisotropy
+in one direction, and 3) ρ1 and ρ2 controls anisotropy in a second direction orthog-
+onal to the first. Lastly, κ simultaneously controls scaling of spatial dependence
+equally in all directions, and the variance of the GRF together with the six other
+parameters as seen in Equation (4).
+2.3
+Spatially varying anisotropy on bounded domain D ⊂ R3
+Non-stationarity and spatially varying anisotropy is achieved by making the coef-
+ficients in Equation (3) spatially varying,
+(κ(s)2 − ∇ · H(s)∇)u(s) = W(s),
+s ∈ R3,
+(8)
+where κ(·) is a positive function, and H is a spatially varying symmetric positive
+definite 3 × 3 matrix.
+Heuristically, one can imagine that the SPDE is gluing
+7
+
+together different local behavior described by ellipsoids, as discussed in Section
+2.2, to a valid non-stationary covariance structure.
+In practice, we need to limit Equation (8) to a bounded domain to parametrize
+the non-stationarity. The SPDE we propose is
+(κ(s)2 − ∇ · H(s)∇)u(s) = W(s),
+s ∈ D ⊂ R3,
+(9)
+where D is bounded, and we enforce the boundary condition
+(H(s)∇u(s))Tn(s),
+s ∈ ∂D,
+where n(s) is the outward normal vector of D. This corresponds to no flux through
+the boundary. The effect of the boundary conditions is increased marginal variance
+on the boundary and increased spatial dependency due to the “reflective” boundary
+condition. As discussed in Lindgren et al. (2011); Fuglstad et al. (2015b), one can
+extend the domain D outside the area with observations to reduce boundary effects,
+or one can consider the boundary effects a feature that the non-stationary model
+can adjust for if necessary.
+3
+Estimating SPDEs with spatially varying anisotropy
+3.1
+Parameterizing the non-stationarity
+Before using the SPDE in Equation (8) in inference, we parametrize the non-
+stationarity through a finite number of parameters.
+This involves expanding
+log(κ(·)), log(γ(·)), vx(·), vy(·), vz(·), ρ1(·), and ρ2(·) in basis functions.
+The
+log-transform is used for κ(·) and γ(·) since they must be positive functions.
+Let g : R3 → R denote a generic function that we want to expand in a basis,
+and let p > 0 the number of basis functions.
+We use basis splines similar to
+Fuglstad et al. (2015b), and set
+g(s) = f(s)Tαg,
+(10)
+8
+
+where αg ∈ Rp, and f(s) = (f1(s), . . . , fp(s))T is a p-dimensional vector with the
+basis functions evaluated at location s.
+In this paper, we will use rectangular domains D = [A1, B1]×[A2, B2]×[A3, B3],
+and a basis constructed as a tensor product of three one-dimensional B-splines.
+This means that p = m3, where m > 0 is the number of basis functions used in each
+dimension. We use clamped splines where the derivative is 0 at each boundary,
+and the construction of the clamped one-dimensional B-splines is discussed in
+Appendix A.3.
+Figure 2 shows an example of the resulting basis functions in
+1-dimension.
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Figure 2: Clamped B-spline basis with three basis functions in 1D.
+Let Bx,i denote the i-th basis function of the second-order basis in the x-
+dimension, and similarly By,j and Bz,k for the y- and z-dimension. The resulting
+tree-dimensional basis is then
+fijk (s) = Bx,i(s1) · By,j(s2) · Bz,k(s3),
+s = (s1, s2, s3)T ∈ D,
+(11)
+for all combinations i, j, k ∈ {1, . . . , m}. This means that αg ∈ Rm3, and m3
+parameters must be estimated for each of the seven functions described at the
+start of the section.
+9
+
+Figure 3: Parameterized function representation with B-splines in 3D.
+In Sections 4 and 5, we use m = p3 = 33 = 27. For a total of 189 parameters
+in the seven functions. When data is sparse, such a model can easily result in
+overfitting (Fuglstad et al., 2015b), and it is necessary to introduce penalties on
+the seven functions. In Fuglstad et al. (2015b), this was achieved by a hierarchical
+model where
+τg∆g(s) = Wg(s),
+s ∈ D,
+together with Neumann boundary conditions of zero derivatives on the boundary
+of the domain.
+However, this requires selecting a reasonable value for τg > 0
+for each of the seven functions and is computationally expensive if it is done
+using cross-validation. However, in the context of this paper, we are constructing
+a stochastic model that mimics the behavior of a densely “observed�� numerical
+simulation model and does not include penalties beyond the restriction of using
+27 basis functions. We demonstrate the ability of this model to be estimated in
+our context in the simulation study in Section 4, and also investigate the amount
+of data needed to estimate the model.
+10
+
+40
+35
+30
+25
+20
+L
+15
+10
+10
+15
+0
+25
+25
+35
+AO3.2
+Hierarchical model and discretization
+Consider a bounded domain D ⊂ R3, and observations y = (y1, y2, . . . , yn) made
+at locations s1, s2, . . . , sn ∈ D. We assume a Gaussian observation model
+yi|η(si), σ2
+N ∼ N(η(si), σ2
+N),
+i = 1, . . . , n,
+where σ2
+N > 0 is the nugget variance and
+η(s) = x(s)Tβ + u(s),
+s ∈ D,
+describes true spatial variation as a combination of covariates and a GRF. Here
+x(·) is a spatially varying vector of k covariates, β ∈ Rk are the coefficients of
+the covariates, and u(·) is a GRF with spatially varying anisotropy as presented
+in Section 2.
+As described in Appendix B, the GRF u(·) is discretized using a regular grid
+with l cells, and we get a Gaussian Markov random field w = (w1, . . . , wl)T. Let
+θ be the vector of all parameters controlling u(·), then
+w|θ ∼ Nl(0, Q−1),
+where dependence on θ is suppressed for Q, and Q is a l × l precision matrix
+with a three-dimensional spatial sparsity structure.
+The vector w is linked to
+u(·) through a linear transformation u(s) = a(s)Tw, where a has only one non-
+zero entry corresponding to which grid cell location s belongs. This gives u =
+(u(s1), . . . , u(sn))T = Aw, where the n × l matrix A only has one non-zero entry
+on each row.
+The coefficients of the fixed effect, β, is assigned the weak penalty β ∼
+NK(0, V IK) for a fixed V > 0. Thus we can write y as
+y = Xβ + Aw + ϵ,
+(12)
+11
+
+where X is the design matrix of covariates, and ϵ ∼ Nn(0, Inσ2
+N) is an n-dimensional
+vector of random noise. This gives rise to the hierarchical formulation
+y|β, w, σ2
+N ∼ Nn(Xβ + Aw, σ2
+NIn),
+β ∼ Nk(0, V Ik),
+w|θ ∼ Nl(0, Q−1).
+Let s∗ ∈ D be an unobserved location. After parameters ˆθ and ˆ
+σ2
+N are esti-
+mated, one can predict the underlying value η(s∗) = x(s∗)Tβ + a(s∗)Tw or a new
+observation y∗ = x(s∗)Tβ + a(s∗)Tw + ϵ∗, where ϵ∗ ∼ N(0, ˆ
+σ2
+N) is a new nugget.
+The predictions are made using the conditional distributions η(s∗)|y, θ = ˆθ, σ2
+N =
+ˆ
+σ2
+N and y∗|y, θ = ˆθ, σ2
+N = ˆ
+σ2
+N. The estimation of parameters is detailed in the next
+section.
+3.3
+Parameter inference
+Simplify notation by letting z = (uT, βT)T. Then
+z|θ ∼ N(0, Q−1
+z ), where Qz =
+�
+�Q
+0
+0
+V Ik
+�
+� .
+Let S =
+�
+A
+X
+�
+, then the observation model can be rewritten as
+y|z, σ2
+N ∼ Nn(Sz, Inσ2
+N).
+(13)
+Using this notation the log-likelihood can be expressed as
+log π(θ, σ2
+N|y) = Const + log π(θ, σ2
+N) + 1
+2 log det (Qz) − n
+2 log(σ2
+N)
+− 1
+2 log det (QC) − 1
+2µT
+CQCµC −
+1
+2σ2
+N
+(y − SµC)T(y − SµC).
+(14)
+Here dependence on θ is suppressed for µC, Qz and QC, and π(θ, σ2
+N) can be used
+to assign a penalty on θ, e.g., like the random-walk penalty used in Fuglstad et al.
+12
+
+(2015b). The conditional precision matrix QC is
+QC = Qz + STS/σ2
+N
+(15)
+and µC is the conditional mean,
+µC = Q−1
+C STy/σ2
+N.
+(16)
+Parameter inference is done by maximizing Equation (14) with respect to θ
+and σ2
+N. The parameter vector θ includes all coefficients for the basis functions,
+and when using 27 basis functions for each function,
+θ =
+�
+αlog(κ2), αlog γ, αvx, αvy, αvz, αρ1, αρ2
+�
+,
+has 189 parameters. The parameter space is challenging to search and we use an
+analytical expression for the gradient in the optimization algorithm. The deriva-
+tion of the analytical gradient involves many nested chain rules and a technique
+to calculate a partial inverse of sparse matrices (Rue and Held, 2010), see Ap-
+pendix A.5 for a complete description.
+4
+Simulation study
+In this section, we perform a simulation study to investigate the amount of data
+required to acquire reasonable parameter estimates of models with varying com-
+plexity that are specified through the SPDE. A comparison of these estimates is
+made from simulated data generated from three different parametrizations of the
+covariance structures.
+The observation model for the different parametrizations is
+ymod = Awmod + ϵ,
+(17)
+13
+
+where wmod is the GMRF controlled by the parameters θmod in the respective
+models, and ϵ is the independent noise term with mean zero and standard deviation
+σN = 0.1 which is identical for all the parametrizations. Furthermore, the models
+are discretized on the same domain with a grid of size (M, N, P) = (30, 30, 30)
+resulting in a total of 27000 grid nodes where the center of which is our spatial
+locations s ∈ D = [A1, B1] × [A2, B2] × [A3, B3] = [0, 40] × [0, 40] × [0, 40].
+The first and simplest model is a Stationary Isotropic (SI) model which has a
+covariance structure controlled by the three parameters θSI = (log κ2, log γ, log σ2
+N),
+that is assigned to the values κ2 = 0.2, γ = 2.5 and σN = 0.1. The resulting spatial
+range is 10.59 with a marginal variance of 0.023.
+The second is a Stationary Anisotropic (SA) model composed of the 8 parame-
+ters θSA = (log κ2, log γ, vx, vy, vz, ρ1, ρ2, log σ2
+N) set to κ2 = 0.35, γ = 0.5, vx = 1.9,
+vy = 1.4, vz = 0.4, ρ1 = 1.4, ρ2 = 0.6 and σN = 0.1. This results in spatial ranges
+of 10.08 along the x-dimension, 6.75 along y, and 3.88 along z with a marginal
+variance of 0.023.
+The parameters of these first two models are simply assigned some reason-
+able value; however, the third and most complex model with a non-stationary
+anisotropic covariance and a total of 190 parameters, they are much more trou-
+blesome to select. Therefore, functions are chosen to assign the parameter val-
+ues in θNA throughout the domain D such that the dependency directions im-
+itate a vortex.
+Using these functions and evaluating them at the spatial lo-
+cations in the discretization the parameters of the B-splines, described in Sec-
+tion 3.1, are found by optimization. These aforementioned parameters are θNA =
+�
+αlog(κ2), αlog γ, αvx, αvy, αvz, αρ1, αρ2, log σN
+�
+with σN = 0.1, and the resulting
+covariance structure can be viewed in Figure 4.
+We will now examine the extent of data required to fit back the parameters
+of the three models described above. First, we simulate multiple datasets from
+14
+
+(a) Correlation
+(b) Marginal Variance
+Figure 4: Spatial correlation at location [26,26,20] (a) and variance of the spatial
+effect (b) in the non-stationary anisotropic model.
+the observation model, Equation (17), with a different number of observed spatial
+locations and realizations (replicated observations of these spatial locations). The
+number of spatial locations varies between 100, 10000, and 27000 (all), and the
+number of realization range between 1, 10, and 100, so nine different combinations
+of dataset sizes. Furthermore, we want to perform 100 different trials for each of
+these combinations, and thereby have 900 total datasets per model. Also, note
+that the observed spatial locations are randomly chosen in each trial. From this,
+some statistics can be recovered about the model estimates that can give insight
+into the applicability of the different parameterizations.
+Table 1 shows the root mean square error (RMSE) between the set parameter
+values in each model and their values inferred by the different datasets. This was
+obtained using the inference method described in Section 3.3 with the observation
+model in Equation (17) for each respective parametrization and trial. The columns
+describe the different number of observation locations (No. loc.) and the number
+15
+
+40
+0.8
+30
+0.6
+- 40
+0.4
+- 30
+20
+X
+0.2
+10
+a
+0
+y0.09
+40
+0.08
+30
+0.07
+0.06
+20-
+0.05
+10
+0.04
+&o -
+40
+0.03
+30-
+30
+20
+y
+20
+x
+0.02
+10
+10
+0
+0.01Table 1: The Root Mean Square Error (RMSE) of parameter estimates in the
+stationary isotropic, stationary anisotropic, and non-stationary anisotropic model
+from 100 independent trials for each combination of dataset sizes; the number of
+observed locations (No. loc.) and the number of replicated observations of these
+locations (No. real.).
+No. loc.
+100
+10000
+27000
+No. real.
+1
+10
+100
+1
+10
+100
+1
+10
+100
+Stat. Iso.
+log κ
+0.763 0.168 0.047 0.123
+log γ
+0.626 0.164 0.062 0.032
+log τ
+2.670 0.674 0.182 0.049
+Stationary Anisotropic
+log κ
+0.876
+0.195
+0.081 0.094 0.038
+log γ
+8.289
+5.601
+0.463 0.228 0.079
+|vx|
+1.208
+0.785
+0.440 0.200 0.070
+|vy|
+1.040
+0.679
+0.354 0.152 0.035
+|vz|
+1.091
+0.498
+0.214 0.075 0.027
+|ρ1|
+0.977
+0.801
+0.249 0.129 0.038
+|ρ2|
+1.337
+0.489
+0.275 0.078 0.027
+log τ
+1.977
+1.352
+0.182 0.189 0.028
+Non-Stationary Anisotropic
+log κ
+2.572 0.811 0.356 0.269
+log γ
+2.615 1.173 0.694 0.585
+|vx|
+1.929 0.742 0.531 0.509
+|vy|
+2.699 0.668 0.453 0.432
+|vz|
+1.591 0.610 0.343 0.296
+|ρ1|
+0.144 0.714 0.287 0.210
+|ρ2|
+0.420 0.604 0.376 0.344
+log τ
+1.152 0.017 0.005 0.005
+16
+
+of realizations (No. real.), and the different blocks represent the different models.
+The columns highlighted in bold for each respective model are the ones we have
+deemed as reasonable parameter estimates. Also, note that some parts of the table
+are omitted to simplify the presentation of the results for the reader as the full
+table does not affect the conclusion of this study. From Table 1 we observe that the
+(simple) stationary models, SI and SA, require very little data. In fact, observing
+under 1% of the grid for 10 realizations or more is good enough for the SI and the
+SA only requires some more realizations to attain similar parameter accuracy.
+On the other hand, the most flexible parameterization, the NA model, requires
+much more data and only reaches reasonable parameter accuracy when the whole
+grid is observed with 10 or more realizations. Now there is a large discrepancy
+between 10000 observed points ( 37%) and 27000 (100%), so it could be interesting
+to investigate where in this range reasonable estimates are obtained. However, we
+have not chosen to explore this here. We also want to note that these estimates
+will change with the complexity of the covariance structure and with the initial
+values in the optimization.
+5
+GRF prior for statistical sampling of the ocean
+5.1
+Aim
+Forecasts produced by numerical ocean models describe realistic behavior for the
+ocean, but local behavior such as plumes created by freshwater discharge from a
+river into the ocean are hard to accurately forecast. However, we can construct
+a prior based on the numerical ocean model that informs prior beliefs about the
+ocean, which can aid AUVs to more effectively sample the ocean. In this paper,
+the goal is to determine the three-dimensional extent of a freshwater plume in the
+ocean, and we assume operation time is short enough to justify a purely spatial
+17
+
+prior that does not assume dynamical changes in time.
+There are two steps in our approach. Step 1 is to estimate a stationary GRF
+prior and a non-stationary GRF prior based on a simulation from the numerical
+ocean model as described in Section 5.2. Step 2 is to combine each of the estimated
+priors with an observation model, and evaluate the predictive ability on in-situ
+observations from AUV as described in Section 5.3. The GRFs that we estimate
+based on the numerical ocean model can be viewed as statistical emulators of the
+ocean.
+5.2
+The numerical ocean model and the GRF prior
+The model training data used in this application is from a forecast produced by
+the ocean model SINMOD. Data is provided by SINTEF Ocean which developed
+and ran the simulation. SINMOD is a three-dimensional numerical ocean model
+based on primitive equations that are solved using finite difference methods on a
+regular grid with horizontal cell sizes of 20km×20km and is nested in several steps
+down to 32m × 32m. Moreover, it uses z* vertical layers which allow for varying
+grid resolutions depending on the depth and help capture the higher variability of
+the surface. SINMOD is driven by atmospheric forces, freshwater outflows, and
+tides, and it provides numerical simulations of multiple variables such as salinity,
+temperature, and currents.
+The reader is referred to Slagstad and McClimans
+(2005) for a more detailed description of the method.
+The area of operation is located in Trondheimsfjorden at Ladehammaren just
+outside of Trondheim, Norway, and the operation date, the time measurements are
+collected with the AUV, is May 27, 2021, between 10:30 and 14:30. The outlined
+area in Figure 5 indicates the operational area which covers 1408m × 1408m in
+the horizontal plane. At the southeast side of this field, the Nidelva river flows
+into the fjord.
+This causes a very dynamic salinity field that is unfeasible to
+18
+
+Figure 5: The area of operation in Trondheimsfjorden at Ladehammaren just
+outside of Trondheim, Norway. The compass shows the cardinal directions relative
+to the map.
+describe with a stationary covariance model. Therefore, we will use the numerical
+simulations from SINMOD to estimate a non-stationary GRF. As demonstrated
+in the simulation study, complex covariance structures can reliably be estimated
+based on such dense data.
+In this application, we will focus on univariate modeling of the salinity and
+we choose the fine-scale horizontal grid sizes hx = 32 m hy = 32 m, which in total
+gives N = 45 and M = 45 grid nodes for both the numerical and the statistical
+model. Moreover, in the vertical plane, we use 1-meter increments between the
+depth layers, i.e., hz = 1 m.
+To avoid any major effects of the boundaries in
+this direction P = 11 depth layers are used resulting in a depth range of 0.5m
+19
+
+Ostmarkneset
+Munkholmen
+Korsvika
+Ladehammaren
+6668
+Trondheim
+Traante
+6668
+Reina
+Brattora
+Trondheim sentralstasjon
+706
+moen
+6692
+706
+Sjobadet
+706
+6690
+Skansen
+Kuhauoer
+6650
+6666
+6650
+6650
+Rosenheto 10.5m. SINMOD outputs zt, t = 0, 1, 2, . . . , 143, which are vectors of salinity
+values in all cells in the three-dimensional grid at different time points throughout
+the whole May 27, 2021. The timesteps are 10 minutes, and Figure 6 shows five
+timesteps from SINMOD for the top six depth layers during the operation. Note
+Figure 6: Five timesteps of the dataset simulated with the numerical ocean model
+SINMOD for May 27, 2021. The timestamps are displayed over their respective
+timesteps. The N-arrow is the cardinal north.
+that the varying vertical layers in the numerical model are either with 0.5m or 1m
+increments, so the SINMOD simulations don’t require any additional modification
+to fit within our statistical model.
+We first estimate the model
+zt = Φzt−1 + ϵt,
+t = 1, . . . , 143,
+where Φ is a diagonal matrix of AR(1) coefficients. The diagonal entries of Φ are
+estimated with maximum likelihood separately for each spatial location such that
+ˆΦii = �143
+t=1 zt,izt−1,i/ �143
+t=1 z2
+t−1,i for i = 1, . . . , NMP, where zt,i is the value in cell
+i at time t. We then compute empirical innovations ˆϵt = zt− ˆΦzt−1, t = 1, . . . , 143.
+20
+
+10:30
+Depth:
+0.5
+30
+N
+1.5
+25
+20
+2.5
+15
+3.5
+10
+4.5
+5
+5.5
+014:30
+Depth:
+0.5
+30
+N
+1.5
+25
+20
+2.5
+15
+3.5
+10
+4.5
+5
+5.5
+013:30
+Depth:
+0.5
+30
+N
+1.5
+25
+20
+2.5
+15
+3.5
+10
+4.5
+5
+5.5
+012:30
+Depth:
+0.5
+30
+N
+1.5
+25
+20
+2.5
+15
+3.5
+10
+4.5
+5
+5.5
+011:30
+Depth:
+0.5
+30
+N
+1.5
+25
+20
+2.5
+15
+3.5
+10
+4.5
+5
+5.5
+0These empirical innovations describe the spatial covariance structure for short-term
+changes in salinity.
+We fit the flexible non-stationary anisotropic model with 190 parameters, ˆθNA =
+(αlog κ, αlog γ, αvx, αvy, αvz, αρ1, αρ2, log σ2
+N), and the stationary anisotropic model
+with 8 parameters, ˆθSA = (log κ2, log γ, vx, vy, vz, ρ1, ρ2, log σ2
+N), to the assumed in-
+dependent realization from a GRF ˆϵ1, . . . .ˆϵ143. Note that there are NMP = 22275
+spatial locations and the 144 empirical innovations cover the whole day of May 27,
+2021. Figures 7b show the resulting variance of the spatial effect and Figure 7c
+the spatial correlation with location (x, y, z) = (22, 10, 0) of the non-stationary
+anisotropic model. The same figures of the stationary anisotropic model can be
+found in Appendix C, Figure S3.
+(a) SINMOD prior
+(b) Marginal Variance
+(c) Correlation
+Figure 7:
+Prior field (a) found from SINMOD simulations, the variance of the
+spatial effect (b) and spatial correlation of point [22,10,0] (marked) (c) in the
+non-stationary anisotropic model. The N-arrow is the cardinal north.
+21
+
+Depth:
+0.5
+N
+0.8
+1.5
+2.5
+0.6
+3.5
+0.4
+4.5
+0.2
+5.5
+0Depth:
+0.5
+N
+30
+1.5
+25
+2.5
+20
+3.5
+15
+4.5
+10
+5.5
+5Depth:
+0.5
+N
+0.7
+1.5
+0.6
+0.5
+2.5
+0.4
+3.5
+0.3
+4.5
+0.2
+5.5
+0.1In the next step, we construct the expected value of the GRF using the time
+average of the whole day, µ = �143
+t=0 zt/144. The mean is shown in Figure 7a
+and shows the overall tendency for freshwater near the river outlet and saltwater
+further out in the ocean. We choose the prior
+η = µ + e,
+(18)
+where we combine the fixed mean vector, µ, with a new realization, e, of the
+estimated stationary anisotropic model or the non-stationary anisotropic model.
+This is a spatial prior on a 32 m × 32 m × 1 m resolution.
+5.3
+In-situ data collection and emulator evaluation
+In-situ measurements were made with the AUV on May 27, 2021, between 10:30
+and 14:30. The AUV followed 9 pre-planned paths within the area of operation:
+two intersects at 0.5m depth one northbound and one north-westbound starting
+from the river, two zig-zags in each depth layer (0.5m,2m,5m), and one up and
+down pattern in depth ranging from 0.5m to 10.5m moving north-westbound start-
+ing from the river. Figure 8 displays the locations of the measurements in the top
+5 layers of the field.
+The AUV is moving at 1.5 m/s and continuously samples the salinity. This
+means that multiple measurements are made within each 32 m × 32 m × 1 m grid
+cell. Measurements are represented as yi, i = 1, . . . , nobs, whereby yi is the average
+value measured in grid cell i. We combine these measurements with the prior in
+Equation (18) using
+yi|η, σ2
+N
+ind
+∼ N(aT
+i η, σ2
+meas),
+i = 1, . . . , nobs,
+η ∼ N(µ, Q−1
+Prior),
+where ai selects the correct grid cell, Q−1
+Prior is the estimated precision matrix
+for the GMRF, and the Gaussian likelihood with nugget variance σ2
+meas describes
+22
+
+Figure 8: Measurement locations of the AUV in the top 6 depth layers of the spatial
+field on May 27th, 2021, in Trondheimsfjorden at Ladehammaren just outside of
+Trondheim, Norway. The N-arrow is the cardinal north.
+measurement noise and sub-grid variation. In general, we would estimate σ2
+meas
+using a trial run, but in this case, we estimated σ2
+meas using the average empirical
+variance over all observed grid cells in the total dataset. Note that we have not
+accounted for the uncertainty in the AUVs positions in these models. As the AUV
+dive, it loses its GPS signal and only relies on estimated location.
+When the
+GPS signal is returned a linear interpolation is made to account for drift but no
+uncertainty is included.
+We evaluated the two priors, or emulators, by randomly ordering the 9 seg-
+ments and then sequentially including more and more observations for predicting
+the remaining hold-out data. The random permutation of the segments was done
+repeatedly to determine the variation in scores over different paths. This scheme
+23
+
+0.5m
+Nevaluates the AUVs’ ability to predict future observations while maintaining the
+sequential structure of measurements.
+Figure 9 shows that the non-stationary
+model provides a better prior for the salinity in the ocean than the stationary
+model. The differences are largest when little data is available, which is consistent
+with the idea that the prior is most important in this case. The non-stationary
+model can leverage knowledge about which areas are most uncertain using the
+spatially varying marginal variance and update the prior based on expected simi-
+larities from the spatially varying anisotropy. The improvements are seen both in
+point predictions through RMSE and in predictive distributions as measured by
+CPRS (Gneiting and Raftery, 2007).
+6
+Discussion
+We extend the class of SPDE-based GRFs introduced in Fuglstad et al. (2015a)
+to three-dimensional space by overcoming two key issues: parametrization and
+computation. For the former, we developed a specification of spatially varying
+anisotropy through a spatially varying baseline isotropic dependence, and two
+orthogonal spatially varying vector fields that describe extra dependence. This
+allows for an interpretable description of the 3×3 positive definite matrix describing
+anisotropy. For the latter, we use a finite volume method to construct a GMRF
+that approximates the solution of the SPDE.
+The specification of spatially varying marginal variance and spatially varying
+anisotropy requires specifying 7 spatially varying real functions. In this paper,
+we expand each function with a clamped B-spline basis. If each function uses
+P 3 basis functions, this gives in total 7P 3 coefficients. As demonstrated in the
+simulation study, an unpenalized estimation of these parameters requires a densely
+observed area and multiple realizations. Application of the new models in data-
+24
+
+Figure 9: The root mean square error (RMSE, top) and the continuous ranked
+probability score (CRPS, bottom) of predictions from the stationary anisotropic
+(orange) and non-stationary anisotropic models (blue) given different proportions
+of observed data (5%, 95%). The error bars are the standard deviations of the
+different measures under random permutations of the 9 segments.
+sparse situations will require penalties that restrict the regularity of the 7 spatially
+varying functions. However, more research is needed to come up with a practical
+25
+
+2.5
+2.0
+1.0
+0.5
+1.2
+1.0
+CRPS
+B'0
+0.6
+0.4
+0.2
+5%10%15%20%25%30%35%40%45%50%55%60%65%70%75%80%85%90%95%
+StationaryAnisotropic
+Non-stationaryAnisotropicway to determine the appropriate strength of penalization for each of the functions.
+While we did not experience any practical issues with the chosen way to de-
+scribe the two orthogonal vector fields, the construction has a “gimbal lock” type
+issue. If one vector field points exactly along the z-axis, there is no unique choice
+for the second vector field. A potential way to avoid this issue is by describing
+the orientation of the two orthogonal vector fields through quaternions or Euler-
+Rodrigues parameters.
+Moving from two-dimensional space to three-dimensional space introduces an
+asymptotically higher computation cost as a function of grid size. For a regular
+three-dimensional grid with N nodes, the computational cost is O(N 2) compared
+to O(N 3/2) in two-dimensional space. This increased computational cost arises
+from increased fill-in in the Cholesky factor. However, the application demon-
+strates that the use of a grid size of N = 22275 is unproblematic even for real-time
+updates on an AUV.
+For the predictions of salinity in the Trondheim’s fjord, we see the highest
+improvement of the complex GRF prior compared to an isotropic GRF, for sparse
+in-situ measurements. As more data is collected, the difference between the models
+decreases. This suggests that the key advantage of training the more complex
+GRF is to encode prior physical knowledge so that we can more effectively update
+knowledge about unobserved locations. Salinity was used as an example, but in
+general, the same approach could be used to map other biologically interesting
+quantities such as phytoplankton (Fossum et al., 2019). The GRFs developed in
+this paper are a step forward in quantifying beliefs about unobserved regions in
+the ocean, which is essential for optimal decisions and more effective autonomous
+sampling (Fossum et al., 2021).
+In future work, it would be interesting to add a dynamic component to the
+model to capture physical processes such as diffusion and advection. However,
+26
+
+this substantially increases computational cost, and it is not clear to which degree
+an advection field from a numerical model should be trusted and which boundary
+conditions are best in an advection-dominated problem. The new class of GRFs
+shows great promise for encoding prior knowledge about a phenomenon in a com-
+putationally efficient way. However, overfitting is an important issue, and we must
+consider ways to penalize the complexity. In particular, we need to consider ways
+to allow flexibility in an area where it is needed such as a river outlet, and restrict
+flexibility in areas where we expect stationarity.
+Acknowledgments
+Berild and Fuglstad are supported by the Research Council of Norway, project
+number 305445. The authors are grateful to Ingrid Ellingsen and SINTEF for
+providing the simulations from the numerical ocean model SINMOD.
+27
+
+A.
+General properties
+A.1
+Marginal Variance
+Here, we will derive the expression for the marginal variance in a general sense and
+then specify it for three-dimensional spaces with exponential covariance functions.
+The SPDE considered in this work is
+(κ2 − ∇ · H∇)α/2u(s) = W(s),
+(S1)
+where s ∈ D ⊆ Rd a spatial location in the domain of dimension d and α = ν +d/2
+where ν > 0 is the smoothness. Any solution of this SPDE is a Matérn field and
+let σm > 0 be its marginal standard deviation; then, its covariance function is
+r(s1, s2) =
+σ2
+m
+2ν−1Γ(ν)(κ||H−1/2(s1 − s2)||)νKν(κ||H−1/2(s1 − s2)||).
+(S2)
+The transfer function of the SPDE is
+g(w) = (κ2 + wTHw)−α/2.
+Using this and by including the spectral density of standard Gaussian white noise
+in Rd is (2π)−d, the spectral density of the solution of the SPDE is
+fS(w) = (2π)−d(κ2 + wTHw)−α.
+Lastly, to find the marginal variance of the field the integral of the spectral density
+is made over Rd as
+σ2
+m =
+�
+Rd fS(w)dw.
+Including the change of variables w = κH−1/2z the expression becomes
+σ2
+m = (2π)−d
+�
+Rd(κ2 + κ2zTz)−α det(κH−1/2)dz
+= (2π)−d
+�
+Rd κd−2α(1 + zTz)−α det(H)−1/2dz
+α=ν+d/2
+=
+(2π)−dκ−2ν det(H)−1/2
+�
+Rd(1 + zTz)−αdz,
+(S3)
+28
+
+which by specifying a exponential covariance in R3 with α = 2, ν = 1/2 and d = 3
+is
+σ2
+m =
+1
+8πκ
+�
+det(H)
+.
+Note that the integral in Equation (S3) is solved by converting to polar coordinates
+as
+�
+R3
+1
+(1 + zTz)2dz =
+� π
+0
+sin(φ)dφ
+� 2π
+0
+dθ
+� ∞
+0
+ρ2
+(1 + ρ2)2dρ = π2.
+A.2
+Covariance function
+Evaluating Equation (S2) at ν = 1/2 and including the expression for the marginal
+variance the covariance function can be formalized as
+r(s1, s2) =
+�
+2
+π
+1
+8πκ
+�
+det(H)
+�
+κ||H−1/2(s1 − s2)||K 1
+2(κ||H−1/2(s1 − s2)||).
+Then, consider the modified Bessel function of the second kind
+Kn(z) =
+� π
+2z
+e−z
+(n − 1
+2)!
+� ∞
+0
+e−ttn−1/2
+�
+1 − t
+2z
+�n−1/2
+dt,
+and evaluate this at order 1/2 gives
+K 1
+2(z) =
+� π
+2ze−z.
+The covariance function can then be formalized as
+r (s1, s2) =
+�
+2
+πσ2
+m
+�
+κ||H−1/2(s1 − s2)||
+×
+�
+π
+2 · κ||H−1/2(s1 − s2)|| exp
+�
+−κ||H−1/2(s1 − s2)||
+�
+=σ2
+m exp
+�
+−κ||H−1/2(s1 − s2)||
+�
+.
+(S4)
+A.3
+One-dimensional clamped B-splines
+We illustrate the construction of 1-dimensional splines B-splines using the interval
+[A, B] ∈ R. Let A = t0 < t1 < · · · < tm = B be the knot points. Then the
+29
+
+zero-order B-splines are constructed recursively as
+Bi,0(t) =
+�
+�
+�
+�
+�
+1,
+ti ≤ t ≤ ti+1,
+0,
+otherwise,
+,
+t ∈ [A, B],
+for i = 0, . . . , p − 1.
+Let r denote the order of the B-splines.
+The first- and
+second-order basis splines are constructed as
+Bi,r(t) =
+t − ti
+ti+r − ti
+Bi,r−1(t) +
+ti+r+1 − t
+ti+r+1 − ti+1
+Bi+1,r−1(t),
+t ∈ [A, B],
+for i = 0, . . . , p − r − 1.
+Using the r-order B-spline basis, we construct a function g : [A, B] → R by
+g(t) =
+p−r−1
+�
+i=0
+αiBi,r(t).
+where α0, . . . , αp−r−1 ∈ R are coefficients. We use a clamped spline where g′(A) =
+g′(B) = 0 and need the additional requirement that α0 = α1 and αp−r−2 = αp−r−1.
+A.4
+Integrated likelihood
+The distribution of z = (u, β) is given by
+z|θ ∼ N(0, Q−1
+z ),
+and the observation model is
+y|z, θ, σ2
+N ∼ Nn(Sz, Inσ2
+N).
+30
+
+From this the distribution of z given some observations y is
+π(z|θ, σ2
+N, y) ∝ π(z, θ, σ2
+N, y)
+= π(θ, σ2
+N)π(z|θ)π(y|θ, σ2
+N, z)
+∝ exp
+�
+−1
+2zTQzz − 1
+2(y − Sz)TInσ−2
+N (y − Sz)
+�
+∝ exp
+�
+−1
+2
+�
+zT �
+Qz + σ−2
+N STS
+�
+z − 2zTSTy · σ−2
+N
+��
+∝ exp
+�
+−1
+2(z − µC)TQC(z − µC)
+�
+⇓
+z|θ, σ2
+N, y ∼ Nn
+�
+µC, Q−1
+C
+�
+Here, QC = Qz +STS·σ−2
+N is the conditional precision matrix and µC = Q−1
+C STy·
+σ−2
+N is the conditional mean.
+Then, integrating out z from the joint distribution gives
+π(θ, σ2
+N, y) = π(θ, z, σ2
+N, y)
+π(z|θ, σ2
+N, y)
+= π(θ, σ2
+N)π(z|θ)π(y|θ, σ2
+N, z)
+π(z|θ, σ2
+N, y)
+,
+where the left-hand side does not depend on z such that it may be evaluated for
+any given value. Let us evaluate it for z = µC such that
+π(θ, σ2
+N, y) ∝π(θ, σ2
+N)π(z = µC|θ)π(y|θ, σ2
+N, z = µC)
+π(z = µC|θ, σ2
+N, y)
+∝π(θ)|Qz|1/2|In · σ−2
+N |1/2
+|QC|1/2
+exp
+�
+−1
+2µT
+CQzµC
+�
+× exp
+�
+−1
+2(y − SµC)TIn · σ−2
+N (y − SµC)
+�
+.
+The last term π(z|θ, σ2
+N, y) is removed since it is equal to 1. Thereby, conditioning
+31
+
+on y and taking the log we have the log-likelihood
+log(π(θ, σ2
+N|y)) =Constant + log(π(θ, σ2
+N)) + 1
+2 log(det(Qz)) + n
+2 log(σ−2
+N )
+− 1
+2 log(det(QC)) − 1
+2µT
+CQzµC −
+1
+2 · σ2
+N
+(y − SµC)T(y − SµC).
+(S5)
+A.5
+Gradient of the log-likelihood
+This section is similar to the derivation of the gradient presented in the supple-
+mentary material of Fuglstad et al. (2015b).
+log(π(θ, τN|y)) =Constant + log(π(θ, τN)) + 1
+2 log(det(Qz)) + n
+2 log(σ−2
+N )
+− 1
+2 log(det(QC)) + 1
+2µT
+CQCµC − τN
+2 yTy.
+Note that the last two terms are rewritten for simplicity in the gradient calculation
+and that the variance of the Gaussian noise term, σ2
+N is re-parametrized with its
+inverse τN = 1/σ2
+N (precision). Derivatives of the log-likelihood are taken with
+respect to θi, the elements of θ, and the precision on log scale as log(τN).
+The first term is a constant and therefore its derivative is zero with respect to
+any of the parameters. The next term, the penalty or the prior of the parameters,
+is not used in this paper and otherwise depends on the choice of penalty so gradient
+calculation is not specified for this term.
+To continue note the derivatives of the precision matrix
+∂QC
+∂θi
+= ∂Qz
+∂θi
+and
+∂QC
+∂ log(τN) = STSτN,
+which is used in the following derivations. First, the derivatives with respect to θi
+are considered. The derivative of the log determinant terms are
+∂
+∂θi
+(log(det(Q)) − log(det(QC))) =Tr
+�
+Q−1∂Q
+∂θi
+�
+− Tr
+�
+Q−1
+C
+∂Q
+∂θi
+�
+=Tr
+�
+(Q−1 − Q−1
+C )∂Q
+∂θi
+�
+,
+32
+
+and the derivative of the quadratic terms are
+∂
+∂θi
+�1
+2yTyτN + 1
+2µT
+CQCµC
+�
+= ∂
+∂θi
+�1
+2µT
+CQCµC
+�
+= − 1
+2yTτNSQ−1
+C
+�∂QC
+∂θi
+�
+Q−1
+C STτNy
+= − 1
+2µT
+C
+�∂Q
+∂θi
+�
+µC.
+Then, combining these the derivative of the log-likelihood with respect to θi is
+∂
+∂θi
+log(π(θ, τN|y)) = ∂
+∂θi
+log(π(θ, τN))+Tr
+�
+(Q−1 − Q−1
+C )∂Q
+∂θi
+�
+−1
+2µT
+C
+�∂Q
+∂θi
+�
+µC
+Next, the derivative with respect to the log precision, log τN, is considered.
+The derivative of the log determinant terms are
+∂
+∂ log(τN)
+�n
+2 log(τN) − 1
+2 log(det(QC))
+�
+=n
+2 − 1
+2Tr
+�
+Q−1
+C
+∂
+∂ log(τN)QC
+�
+=n
+2 − 1
+2Tr
+�
+Q−1
+C STS · τN
+�
+Further, the derivative of 1/2yTy · τN with respect to log(τN) is just the same
+expression so the remaining quadratic term becomes
+∂ 1
+2µT
+CQCµC
+∂ log(τN)
+=∂ 1
+2yTτNSQ−1
+C STτNy
+∂ log(τN)
+=yTτNSQ−1
+C ST
+∂τN
+∂ log(τN)y − 1
+2yTτNSQ−1
+C
+∂QC
+∂ log(τN)Q−1
+C STτNy
+=µT
+CSTτNy − 1
+2µT
+CSTSµCτN,
+and then, by adding the last quadratic term, the expression simplifies to
+−1/2yTy · τN + µT
+CSTy · τN − 1
+2µT
+CSTSµC · τN = −1
+2(y − SµC)T(y − SµC) · τN.
+Finally, combining all these terms we have the derivative of the log-likelihood with
+respect to log(τN):
+∂ log(π(θ, τN|y))
+∂ log(τN))
+=∂ log(π(θ, τN)
+∂ log(τN)
++ n
+2 − 1
+2Tr
+�
+Q−1
+C STS · τN
+�
+− 1
+2(y − SµC)T(y − SµC) · τN
+33
+
+Note that the derivative of QC can be calculated quickly and it is derived from
+a series of chain rules; first on QC, then on A and AH, and finally within H. The
+most computationally heavy calculation in the gradient of the log-likelihood is to
+calculate the inverses in the difference Q−1 − Q−1
+C . However, since this term is
+multiplied with the derivative of Q with respect to θi, which carries the non-zero
+structure of Q, only elements of Q−1 and Q−1
+C which correspond to the non-zero
+structure of Q need to be calculated. This is done by calculating a partial inverse
+of two matrices as described in Rue and Held (2010).
+B.
+Derivation
+B.1
+Discretization
+To find the local solution of the SPDE the domain D = [A1, B1]×[A2, B2]×[A3, B3]
+is divided into equally sized rectangular cubes or cells. We use M cells to divide
+[A1, B1] in the x-direction, N cells on [A2, B1] in y-direction and P cells on [A3, B3]
+in z-direction. The cells have sides parallel to each axis of size hx = (B1 − A1)/M,
+hy = (B2 − A2)/N, and hz = (B3 − A3)/P. The cells are assigned an index with
+regards to their cell number along each axes starting from number 0; i ∈ [0, M]
+along x, j ∈ [0, N] along y, and k ∈ [0, P] along z. For a specific cell, its domain
+can be denoted as
+Ei,j,k = [ihx, (i + 1)hx] × [jhy, (j + 1)hy] × [khz, (k + 1)hz],
+and Figure S1 shows this cell and its closest neighbors. Furthermore, as a regular
+grid is employed the volume of a cell is V = hxhyhz.
+To further define the local solution of the SPDE we denote the faces of a grid
+cell as σF
+i,j,k (front), σB
+i,j,k (back), σL
+i,j,k (left), σR
+i,j,k (right), σU
+i,j,k (up) and σD
+i,j,k
+(down) with their respective face centers si,j−1/2,k, si,j+1/2,k, si−1/2,j,k, si+1/2,j,k,
+34
+
+Figure S1: One cell Ei,j,k in the discretization with its closest neighbours; Ei+1,j,k,
+Ei−1,j,k, Ei,j+1,k, Ei,j−1,k, Ei,j,k+1, and Ei,j,k−1.
+si,j,k+1/2 and si,j,k−1/2. Figure S2 describes the different faces of a cell.
+B.2
+Local solution of the SPDE
+Note that this description is an extension to three dimensions of the derivation
+described in Fuglstad et al. (2015a), and the reader is referred to there for fur-
+ther details. To locally solve the SPDE a finite volume scheme is derived. First,
+35
+
+I
+1Figure S2: One cell Ei,j,k of the discretization with all its faces; σF
+i,j,k (front),
+σB
+i,j,k (back), σL
+i,j,k (left), σR
+i,j,k (right), σU
+i,j,k (up), and σD
+i,j,k (down) each with its
+respective face centres.
+Equation (S1) is integrated over a cell Ei,j,k as
+�
+Eijk
+κ2(s)ds −
+�
+Eijk
+∇ · H(s)∇u(s)ds =
+�
+Eijk
+W(s)ds,
+(S6)
+where ds is a volume element. The integral of the Gaussian white noise on the
+right-hand side is a Gaussian variable with mean zero and variance equal to the
+volume of a cell which is independent of neighboring cells. Let zijk be an standard
+Gaussian variable; then, Equation (S6) becomes
+�
+Eijk
+κ2(s)ds −
+�
+Eijk
+∇ · H(s)∇u(s)ds =
+√
+V zijk.
+36
+
+Then, applying the divergence theorem to the second integral with the divergence
+operator gives
+�
+Eijk
+κ2(s)ds −
+�
+∂Eijk
+(H(s)∇u(s))Tn(s)dσ =
+√
+V zijk.
+The first integral is approximated by letting k2
+ijk be the average value of the con-
+tinuous function κ2(s) within a cell, i.e. κ2
+ijk = 1/V
+�
+Eijk κ2(s)ds, resulting in
+V κ2
+ijkuijk −
+�
+∂Eijk
+(H(s)∇u(s))Tn(s)dσ =
+√
+V zijk.
+(S7)
+To describe the solution of the second integral it is divided into integrals over
+each surface as
+�
+∂Eijk
+(H(s)∇u(s))Tn(s)dσ = W L
+ijk + W R
+ijk + W B
+ijk + W F
+ijk + W U
+ijk + W D
+ijk,
+(S8)
+or W dir
+ijk =
+�
+σdir
+ijk(H(s)∇u(s))Tn(s)dσ, where dir denotes the surface; R (posi-
+tive x-direction), L (negative x-direction), B (positive y-direction), F (negative
+y-direction), U (positive z-direction), and D (negative z-direction). Now, an ap-
+proximation of this surface integral over each face is required. It is assumed that
+the gradient of u(s) is constant over each face and equal to the value at the center
+of each face. The resulting scheme for the gradient on each face is described in
+Table S1.
+Furthermore, let H be approximated by its value at the center of the
+face, and then, we have the approximation
+W dir
+ijk =
+�
+σdir
+ijk
+∇u(s)TH(s)n(s)dσ
+≈∇u(cdir
+ijk)TH(cdir
+ijk)n(cdir
+ijk)
+�
+σdir
+ijk
+dσ
+=∇u(cdir
+ijk)TH(cdir
+ijk)n(cdir
+ijk)A(σdir
+ijk),
+(S9)
+where cdir
+ijk is the center of face dir in the cell Eijk, and A(σdir
+ijk) is the area of the
+face. Combining Equation (S9) with the scheme of ∇u(cdir
+ijk) from Table S1, and
+37
+
+Face
+Scheme
+σR
+i,j,k
+∂
+∂xu(si+1/2,j,k) ≃
+1
+hx (u(si+1,j,k) − u(si,j,k))
+∂
+∂yu(si+1/2,j,k) ≃
+1
+4hy (u(si+1,j+1,k) + u(si,j+1,k) − u(si+1,j−1,k) − u(si,j−1,k))
+∂
+∂zu(si+1/2,j,k) ≃
+1
+4hz (u(si+1,j,k+1) + u(si,j,k+1) − u(si+1,j,k−1) − u(si,j,k−1))
+σL
+i,j,k
+∂
+∂xu(si−1/2,j,k) ≃
+1
+hx (u(si,j,k) − u(si−1,j,k))
+∂
+∂yu(si−1/2,j,k) ≃
+1
+4hy (u(si,j+1,k) + u(si−1,j+1,k) − u(si,j−1,k) − u(si−1,j−1,k))
+∂
+∂zu(si−1/2,j,k) ≃
+1
+4hz (u(si,j,k+1) + u(si−1,j,k+1) − u(si,j,k−1) − u(si−1,j,k−1))
+σB
+i,j,k
+∂
+∂xu(si,j+1/2,k) ≃
+1
+4hx (u(si+1,j+1,k) + u(si+1,j,k) − u(si−1,j+1,k) − u(si−1,j,k))
+∂
+∂yu(si,j+1/2,k) ≃
+1
+hy (u(si,j+1,k) − u(si,j,k))
+∂
+∂zu(si,j+1/2,k) ≃
+1
+4hz (u(si,j+1,k+1) + u(si,j,k+1) − u(si,j+1,k−1) − u(si,j,k−1))
+σF
+i,j,k
+∂
+∂xu(si,j−1/2,k) ≃
+1
+4hx (u(si+1,j,k) + u(si+1,j−1,k) − u(si−1,j,k) − u(si−1,j−1,k))
+∂
+∂yu(si,j−1/2,k) ≃
+1
+hy (u(si,j,k) − u(si,j−1,k))
+∂
+∂zu(si,j−1/2,k) ≃
+1
+4hz (u(si,j,k+1) + u(si,j−1,k+1) − u(si,j,k−1) − u(si,j−1,k−1))
+σU
+i,j,k
+∂
+∂xu(si,j,k+1/2) ≃
+1
+4hz (u(si+1,j,k+1) + u(si+1,j,k) − u(si−1,j,k+1) − u(si−1,j,k))
+∂
+∂yu(si,j,k+1/2) ≃
+1
+4hy (u(si,j+1,k+1) + u(si,j+1,k) − u(si,j−1,k+1) − u(si,j−1,k))
+∂
+∂zu(si,j,k+1/2) ≃
+1
+hx (u(si,j,k+1) − u(si,j,k))
+σD
+i,j,k
+∂
+∂xu(si,j,k−1/2) ≃
+1
+4hz (u(si+1,j,k) + u(si+1,j,k−1) − u(si−1,j,k) − u(si−1,j,k−1))
+∂
+∂yu(si,j,k−1/2) ≃
+1
+4hy (u(si,j+1,k) + u(si,j+1,k−1) − u(si,j−1,k) − u(si,j−1,k−1))
+∂
+∂zu(si,j,k−1/2) ≃
+1
+hx (u(si,j,k) − u(si,j,k−1))
+Table S1: Numerical scheme of the partial derivative with respect to x, y and z of
+uijk on the different faces of cell Eijk.
+denoting the components of H as
+H(s) =
+�
+����
+H11(s)
+H12(s)
+H13(s)
+H21(s)
+H22(s)
+H23(s)
+H31(s)
+H32(s)
+H33(s)
+�
+����
+38
+
+the approximations for each face become
+ˆW R
+i,j,k =
+hyhz
+�
+H11(si+1/2,j,k)u(si+1,j,k) − u(si,j,k)
+hx
+�
++
+hyhz
+�
+H21(si+1/2,j,k)u(si+1,j+1,k) + u(si,j+1,k) − u(si+1,j−1,k) − u(si,j−1,k)
+4hy
+�
++
+hyhz
+�
+H31(si+1/2,j,k)u(si+1,j,k+1) + u(si,j,k+1) − u(si+1,j,k−1) − u(si,j,k−1)
+4hz
+�
+,
+ˆW L
+i,j,k =
+hyhz
+�
+H11(si−1/2,j,k)u(si−1,j,k) − u(si,j,k)
+hx
+�
++
+hyhz
+�
+H21(si−1/2,j,k)u(si,j−1,k) + u(si−1,j−1,k) − u(si,j+1,k) − u(si−1,j+1,k)
+4hy
+�
++
+hyhz
+���
+H31(si−1/2,j,k)u(si,j,k−1) + u(si−1,j,k−1) − u(si,j,k+1) − u(si−1,j,k+1)
+4hz
+�
+,
+ˆW B
+i,j,k =
+hxhz
+�
+H12(si,j+1/2,k)u(si+1,j+1,k) + u(si+1,j,k) − u(si−1,j+1,k) − u(si−1,j,k)
+4hx
+�
++
+hxhz
+�
+H22(si,j+1/2,k)u(si,j+1,k) − u(si,j,k)
+hy
+�
++
+hxhz
+�
+H32(si,j+1/2,k)u(si,j+1,k+1) + u(si,j,k+1) − u(si,j+1,k−1) − u(si,j,k−1)
+4hz
+�
+,
+ˆW F
+i,j,k =
+hxhz
+�
+H12(si,j−1/2,k)u(si−1,j,k) + u(si−1,j−1,k) − u(si+1,j,k) − u(si+1,j−1,k)
+4hx
+�
++
+hxhz
+�
+H22(si,j−1/2,k)u(si,j−1,k) − u(si,j,k)
+hy
+�
++
+hxhz
+�
+H32(si,j−1/2,k)u(si,j,k−1) + u(si,j−1,k−1) − u(si,j,k+1) − u(si,j−1,k+1)
+4hz
+�
+,
+39
+
+ˆW U
+i,j,k =
+hxhy
+�
+H13(si,j,k+1/2)u(si+1,j,k+1) + u(si+1,j,k) − u(si−1,j,k+1) − u(si−1,j,k)
+4hx
+�
++
+hxhy
+�
+H23(si,j,k+1/2)u(si,j+1,k+1) + u(si,j+1,k) − u(si,j−1,k+1) − u(si,j−1,k)
+4hy
+�
++
+hxhy
+�
+H33(si,j,k+1/2)u(si,j,k+1) − u(si,j,k)
+hz
+�
+,
+ˆW D
+i,j,k =
+hxhy
+�
+H13(si,j,k−1/2)u(si−1,j,k) + u(si−1,j,k−1) − u(si+1,j,k) − u(si+1,j,k−1)
+4hx
+�
++
+hxhy
+�
+H23(si,j,k−1/2)u(si,j−1,k) + u(si,j−1,k−1) − u(si,j+1,k) − u(si,j+1,k−1)
+4hy
+�
++
+hxhy
+�
+H33(si,j,k−1/2)u(si,j,k−1) − u(si,j,k)
+hz
+�
+,
+ˆW T
+i,j,k =
+hxhy
+�
+H13(si,j,k+1/2)u(si+1,j,k+1) + u(si+1,j,k) − u(si−1,j,k+1) − u(si−1,j,k)
+4hx
+�
++
+hxhy
+�
+H23(si,j,k+1/2)u(si,j+1,k+1) + u(si,j+1,k) − u(si,j−1,k+1) − u(si,j−1,k)
+4hy
+�
++
+hxhy
+�
+H33(si,j,k+1/2)u(si,j,k+1) − u(si,j,k)
+hz
+�
+,
+ˆW B
+i,j,k =
+hxhy
+�
+H13(si,j,k−1/2)u(si−1,j,k) + u(si−1,j,k−1) − u(si+1,j,k) − u(si+1,j,k−1)
+4hx
+�
++
+hxhy
+�
+H23(si,j,k−1/2)u(si,j−1,k) + u(si,j−1,k−1) − u(si,j+1,k) − u(si,j+1,k−1)
+4hy
+�
++
+hxhy
+�
+H33(si,j,k−1/2)u(si,j,k−1) − u(si,j,k)
+hz
+�
+.
+Next, a vectorization of the discretization is made; first moving along the z-
+direction, then along x-direction, and lastly along the y-direction. Let us denote
+this with the common index l = j · M · P + i · P + k so sijk = sj·M·P+i·P+k = sl
+which gives u(sijk) = ul and κ2(sijk) = κ2
+l , and let the last index be L = (N −
+40
+
+1)MP + (M − 1)P + P − 1. Further, the vectorization results in the linear system
+of equations
+(DV Dκ2 − AH)u = D1/2
+V z,
+(S10)
+where DV = V · IMNP, Dκ2 = [κ2
+0, . . . , κ2
+l , . . . , κ2
+L] IMNP, and z ∼ N(0, IMNP).
+For simplicity the indices of the neighbors are denoted kp = k + 1, kn = k − 1,
+jp = j + 1, jn = j − 1, ip = i + 1, and in = i − 1. The development of AH is done
+by the sum ˆW L
+ijk + ˆW R
+ijk + ˆW B
+ijk + ˆW F
+ijk + ˆW U
+ijk + ˆW D
+ijk and accounting for the index
+in uijk to form the linear relationship. In the following, non-zero elements of the
+(jMN + iP + k)-th row of AH are formalized, and the index in (AH)_ denotes
+the column being assigned. The resulting coefficient with the point itself is
+(AH)j·M·P+i·P+k = − hyhz
+hx
+�
+H11(si+1/2,j,k) + H11(si−1/2,j,k)
+�
+− hxhz
+hy
+�
+H22(si,j+1/2,k) + H22(si,j−1/2,k)
+�
+− hxhy
+hz
+�
+H33(si,j,k+1/2) + H22(si,j,k−1/2)
+�
+,
+with the six closest neighbors are
+(AH)j·M·P+i·P+kp =hxhy
+hz
+H33(si,j,k+1/2)
++ hy
+4
+�
+H31(si+1/2,j,k) − H31(si−1/2,j,k)
+�
++ hx
+4
+�
+H32(si,j+1/2,k) − H32(si,j−1/2,k)
+�
+(AH)j·M·P+i·P+kn =hxhy
+hz
+H33(si,j,k−1/2)
+− hy
+4
+�
+H31(si+1/2,j,k) − H31(si−1/2,j,k)
+�
+− hx
+4
+�
+H32(si,j+1/2,k) − H32(si,j−1/2,k)
+�
+(AH)j·M·P+ip·P+k =hzhy
+hx
+H11(si+1/2,j,k)
++ hy
+4
+�
+H12(si,j,k+1/2) − H12(si,j,k−1/2)
+�
++ hz
+4
+�
+H13(si,j+1/2,k) − H13(si,j−1/2,k)
+�
+41
+
+(AH)j·M·P+in·P+k =hzhy
+hx
+H11(si−1/2,j,k)
+− hy
+4
+�
+H12(si,j,k+1/2) − H12(si,j,k−1/2)
+�
+− hz
+4
+�
+H13(si,j+1/2,k) − H13(si,j−1/2,k)
+�
+(AH)jp·M·P+i·P+k =hxhz
+hy
+H22(si,j+1/2,k)
++ hx
+4
+�
+H23(si,j,k+1/2) − H23(si,j,k−1/2)
+�
++ hz
+4
+�
+H21(si+1/2,j,k) − H21(si−1/2,j,k)
+�
+(AH)jn·M·P+i·P+k =hxhz
+hy
+H22(si,j−1/2,k)
+− hx
+4
+�
+H23(si,j,k+1/2) − H23(si,j,k−1/2)
+�
+− hz
+4
+�
+H21(si+1/2,j,k) − H21(si−1/2,j,k)
+�
+,
+and with the twelve closest diagonals are
+(AH)j·M·P+ip·P+kp = hy
+4
+�
+H31(si+1/2,j,k) + H13(si,j,k+1/2)
+�
+,
+(AH)j·M·P+in·P+kn = hy
+4
+�
+H31(si−1/2,j,k) + H13(si,j,k−1/2)
+�
+,
+(AH)j·M·P+in·P+kp = −hy
+4
+�
+H31(si−1/2,j,k) + H13(si,j,k+1/2)
+�
+(AH)j·M·P+ip·P+kn = −hy
+4
+�
+H31(si+1/2,j,k) + H13(si,j,k−1/2)
+�
+,
+(AH)jp·M·P+i·P+kp = hx
+4
+�
+H32(si,j+1/2,k) + H23(si,j,k+1/2)
+�
+,
+(AH)jn·M·P+i·P+kn = hx
+4
+�
+H32(si,j−1/2,k) + H23(si,j,k−1/2)
+�
+,
+(AH)jn·M·P+i·P+kp = −hx
+4
+�
+H32(si,j−1/2,k) + H23(si,j,k+1/2)
+�
+,
+(AH)jp·M·P+i·P+kn = −hx
+4
+�
+H32(si,j+1/2,k) + H23(si,j,k−1/2)
+�
+,
+(AH)jp·M·P+ip·P+k = hz
+4
+�
+H21(si+1/2,j,k) + H12(si,j+1/2,k)
+�
+,
+42
+
+(AH)jn·M·P+in·P+k = hz
+4
+�
+H21(si−1/2,j,k) + H12(si,j−1/2,k)
+�
+,
+(AH)jn·M·P+ip·P+k = −hz
+4
+�
+H21(si+1/2,j,k) + H12(si,j−1/2,k)
+�
+,
+(AH)jp·M·P+in·P+k = −hz
+4
+�
+H21(si−1/2,j,k) + H12(si,j+1/2,k)
+�
+.
+Note that the corner points are not included in this scheme.
+Denoting A =
+DV Dκ2 − AH, Equation (S10) can be written as
+z = D−1/2
+V
+Au,
+and thus, the joint distribution of u is
+π(u) ∝ π(z) ∝ exp
+�
+−1
+2zTz
+�
+π(u) ∝ exp
+�
+−1
+2uTATD−1
+V Au
+�
+π(u) ∝ exp
+�
+−1
+2uTQu
+�
+.
+Here, Q = ATD−1
+V A which is a sparse matrix of 93 non-zero elements per row. This
+corresponds to the point, the 18 closest neighbors, and their 18 closest neighbors.
+Then removing duplicates results in 93 points.
+C.
+Additional figures
+In the application, Section 5, we estimate the parameters of a non-stationary
+anisotropic and stationary anisotropic model on a simulated dataset from the nu-
+merical ocean model SINMOD. The resulting properties of the non-stationary
+model are presented in Figure 7 in Section 5.2 since this is the main focus of the
+applications. The properties of the stationary anisotropic model fit on the same
+dataset are presented in Figure S3. The marginal variance in Figure S3b, which
+should be constant for this stationary model, shows some variability caused by the
+43
+
+boundary conditions. Notice that this boundary effect is also bigger in the direc-
+tion of the strongest dependency directions seen in the south and north corners.
+Notice also the large discrepancies between the correlations in these two models,
+Figure S3c and Figure 7c, as the stationary anisotropic model kind of captures an
+average correlation within the field.
+(a) SINMOD prior
+(b) Marginal Variance
+(c) Correlation
+Figure S3:
+Prior field (a) found from SINMOD simulations, the variance of the
+spatial effect (b) and spatial correlation of point [22,10,0] (c) in the stationary
+anisotropic model. The N-arrow shows the cardinal north.
+44
+
+Depth:
+0.5
+N
+0.7
+1.5
+0.6
+2.5
+0.5
+3.5
+0.4
+4.5
+0.3
+5.5
+0.2Depth:
+0.5
+N
+0.8
+1.5
+2.5
+0.6
+3.5
+0.4
+4.5
+0.2
+5.5
+0Depth:
+0.5
+N
+30
+1.5
+25
+2.5
+20
+3.5
+15
+4.5
+10
+5.5
+5References
+Castruccio, S., Hu, Z., Sanderson, B., Karspeck, A., and Hammerling, D. (2019).
+Reproducing internal variability with few ensemble runs. Journal of Climate,
+32(24):8511–8522.
+Cressie, N. and Wikle, C. K. (2015). Statistics for spatio-temporal data. John
+Wiley & Sons.
+Diggle, P. J., Tawn, J. A., and Moyeed, R. A. (1998). Model-based geostatistics.
+Journal of the Royal Statistical Society: Series C (Applied Statistics), 47(3):299–
+350.
+Euler, L. (1771). Problema algebraicum ob affectiones prorsus singulares memo-
+rabile. Novi Commentarii academiae scientiarum Petropolitanae, pages 75–106.
+Foss, K. H., Berget, G. E., and Eidsvik, J. (2021). Using an autonomous underwa-
+ter vehicle with onboard stochastic advection-diffusion models to map excursion
+sets of environmental variables. Environmetrics, page e2702. In press.
+Fossum, T. O., Fragoso, G. M., Davies, E. J., Ullgren, J. E., Mendes, R., Johnsen,
+G., Ellingsen, I., Eidsvik, J., Ludvigsen, M., and Rajan, K. (2019). Toward adap-
+tive robotic sampling of phytoplankton in the coastal ocean. Science Robotics,
+4(27):eaav3041.
+Fossum, T. O., Travelletti, C., Eidsvik, J., Ginsbourger, D., and Rajan, K. (2021).
+Learning excursion sets of vector-valued Gaussian random fields for autonomous
+ocean sampling. The Annals of Applied Statistics, 15(2):597 – 618.
+Fuglstad, G.-A. and Castruccio, S. (2020). Compression of climate simulations
+with a nonstationary global spatiotemporal spde model. The Annals of Applied
+Statistics, 14(2):542–559.
+45
+
+Fuglstad, G.-A., Lindgren, F., Simpson, D., and Rue, H. (2015a). Exploring a
+new class of non-stationary spatial gaussian random fields with varying local
+anisotropy. Statistica Sinica, pages 115–133.
+Fuglstad, G.-A., Simpson, D., Lindgren, F., and Rue, H. (2015b).
+Does non-
+stationary spatial data always require non-stationary random fields?
+Spatial
+Statistics, 14:505–531.
+Gneiting, T. and Raftery, A. E. (2007). Strictly Proper Scoring Rules, Prediction,
+and Estimation. Journal of the American Statistical Association, 102(477).
+Heaton, M. J., Datta, A., Finley, A. O., Furrer, R., Guinness, J., Guhaniyogi,
+R., Gerber, F., Gramacy, R. B., Hammerling, D., Katzfuss, M., Lindgren, F.,
+Nychka, D. W., Sun, F., and Zammit-Mangion, A. (2019). A case study compe-
+tition among methods for analyzing large spatial data. Journal of Agricultural,
+Biological and Environmental Statistics, 24:398–425.
+Hildeman, A., Bolin, D., and Rychlik, I. (2021). Deformed spde models with an
+application to spatial modeling of significant wave height. Spatial Statistics,
+42:100449.
+Hu, W., Fuglstad, G.-A., and Castruccio, S. (2021). A stochastic locally diffusive
+model with neural network-based deformations for global sea surface tempera-
+ture. Stat. In press.
+Ingebrigtsen, R., Lindgren, F., and Steinsland, I. (2014).
+Spatial models with
+explanatory variables in the dependence structure. Spatial Statistics, 8:20–38.
+Ingebrigtsen, R., Lindgren, F., Steinsland, I., and Martino, S. (2015). Estimation
+of a non-stationary model for annual precipitation in southern norway using
+replicates of the spatial field. Spatial Statistics, 14:338–364.
+46
+
+Lee, D. and Gammie, C. F. (2021). Disks as inhomogeneous, anisotropic gaussian
+random fields. The Astrophysical Journal, 906(1):39.
+Lindgren, F., Rue, H., and Lindström, J. (2011). An explicit link between Gaussian
+fields and Gaussian Markov random fields: the stochastic partial differential
+equation approach. Journal of the Royal Statistical Society: Series B (Statistical
+Methodology), 73(4):423–498.
+Neto, J. H. V., Schmidt, A. M., and Guttorp, P. (2014). Accounting for spatially
+varying directional effects in spatial covariance structures. Journal of the Royal
+Statistical Society: Series C: Applied Statistics, pages 103–122.
+Paciorek, C. J. and Schervish, M. J. (2006). Spatial modelling using a new class
+of nonstationary covariance functions. Environmetrics: The official journal of
+the International Environmetrics Society, 17(5):483–506.
+Risser, M. D. and Calder, C. A. (2015). Regression-based covariance functions for
+nonstationary spatial modeling. Environmetrics, 26(4):284–297.
+Rodrigues (1840). Des lois géométriques qui régissent les déplacements d’un sys-
+tème solide dans l’espace, et de la variation des coordonnées provenant de ces
+déplacements considérés indépendamment des causes qui peuvent les produire.
+Journal de Mathématiques Pures et Appliquées, pages 380–440.
+Rue, H. and Held, L. (2010). Markov Random Fields. Chapman & Hall/CRC
+Handbooks of Modern Statistical Methods, pages 171–200.
+Salvaña, M. L. O. and Genton, M. G. (2021). Lagrangian spatio-temporal non-
+stationary covariance functions. In Advances in Contemporary Statistics and
+Econometrics, pages 427–447. Springer.
+47
+
+Sampson, P. D. (2010).
+Constructions for nonstationary spatial processes.
+In
+Gelfand, A. E., Diggle, P. J., Fuentes, M., and Guttorp, P., editors, Handbook
+of Spatial Statistics, chapter 9, pages 119–130. CRC Press, Boca Rotan, FL.
+Sampson, P. D. and Guttorp, P. (1992). Nonparametric estimation of nonstation-
+ary spatial covariance structure. Journal of the American Statistical Association,
+87(417):108–119.
+Schmidt, A. M., Guttorp, P., and O’Hagan, A. (2011). Considering covariates in
+the covariance structure of spatial processes. Environmetrics, 22(4):487–500.
+Sidén, P., Lindgren, F., Bolin, D., Eklund, A., and Villani, M. (2021). Spatial 3D
+Matérn priors for fast whole-brain fMRI analysis. Bayesian Analysis. In press.
+Slagstad, D. and McClimans, T. A. (2005). Modeling the ecosystem dynamics
+of the Barents sea including the marginal ice zone: I. Physical and chemical
+oceanography. Journal of Marine Systems, 58(1):1–18.
+Stein, M. L. (2002). The screening effect in kriging. The Annals of Statistics,
+30(1):298–323.
+Stein, M. L. (2012). Interpolation of spatial data: some theory for kriging. Springer
+Science & Business Media.
+48
+