# A three-dimensional momentum-space calculation of three-body bound state in
a relativistic Faddeev scheme

M. R. Hadizadeh  College of Engineering, Science, Technology and Agriculture,
Central State University, Wilberforce, OH 45384, USA,  Department of Physics
and Astronomy, Ohio University, Athens, OH 45701, USA,  M. Radin  Department
of Physics, K. N. Toosi University of Technology, Tehran, Iran.  K. Mohseni
Department of Physics, K. N. Toosi University of Technology, Tehran, Iran.

###### Abstract

In this paper, we study the relativistic effects in a three-body bound state.
For this purpose, the relativistic form of the Faddeev equations is solved in
momentum space as a function of the Jacobi momentum vectors without using a
partial wave decomposition. The inputs for the three-dimensional Faddeev
integral equation are the off-shell boost two-body $t-$matrices, which are
calculated directly from the boost two-body interactions by solving the
Lippmann-Schwinger equation. The matrix elements of the boost interactions are
obtained from the nonrelativistic interactions by solving a nonlinear integral
equation using an iterative scheme. The relativistic effects on three-body
binding energy are calculated for the Malfliet-Tjon potential. Our
calculations show that the relativistic effects lead to a roughly 2% reduction
in the three-body binding energy. The contribution of different Faddeev
components in the normalization of the relativistic three-body wave function
is studied in detail. The accuracy of our numerical solutions is tested by
calculation of the expectation value of the three-body mass operator, which
shows an excellent agreement with the relativistic energy eigenvalue.

###### keywords:

Faddeev equations, Three-dimensional scheme, Relativistic corrections

##  1 Introduction

The study of relativistic effects in quantum mechanical few-body systems has
been the subject of many studies of the bound [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
and scattering [11, 12, 13, 14, 15, 16] states. All these studies are mainly
performed using a partial wave decomposition, that leads to coupled equations
for the bound or scattering amplitudes which are expanded in the angular
momentum basis. The discrete partial wave summation of the resulting
amplitudes is dependent on the energy scale of the problem, and at
intermediate and high energy scales of few GeV, the convergence can be reached
very slowly. To avoid this problem, several studies have been performed in a
three-dimensional (3D) approach to formulate the bound and scattering
amplitudes as a function of the Jacobi momentum vectors by replacing the
discrete angular momentum quantum numbers with continuous angle variables.
While the number of mesh points for angular integrations in a 3D formalism
remains constant for all energy scales, it simultaneously considers the
contribution of all partial wave components. Consequently, the number of
equations in a 3D scheme is independent of the energy scale of the problem
[17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35,
36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48]. Thus, the 3D approach
would be a powerful tool to study the scattering problems in few GeV energy
scales when the partial wave formulation is much more complicated for the
evaluation of permutation operators, and the numerical convergence is
difficult to reach. On this basis, the 3D approach has been implemented in
relativistic calculations of three-body (3B) scattering problems [26, 28, 49,
50] and recently applied to the 3B bound states [44, 51]. It is shown that the
difference between the relativistic and the nonrelativistic dynamics is in i.)
the momentum dependence of the kinetic energy or free propagator, ii.) in the
relation of potential operators to the two-body (2B) $t-$matrices, iii.) and
in the form of the momentum basis. The last one leads to a non-identical
Jacobian of the momentum variable change and also a permutation coefficient in
the relativistic case. At the two-body level, the relativistic 2B interactions
designed in such a way that the deuteron properties and 2B observables are
preserved. The inputs for relativistic three- and four-body bound and
scattering calculations are relativistic 2B $t-$matrices. There are few
options to calculate the relativistic 2B $t-$matrices including the following
ones:

  * •

One option is to solve the relativistic Lippmann-Schwinger equation with
relativistic 2B interactions. The input relativistic interactions can be
obtained directly from the nonrelativistic one by solving a quadratic
equation. A number of computational methods have been developed to solve the
nonlinear equation, including the spectral expansion method [7, 2] and an
iterative approach [52]. In the spectral expansion method, a completeness
relation of the 2B bound and scattering states is inserted into the right side
of the quadratic equation and then the result is projected into the momentum
space. The powerful iterative scheme proposed by Kamada and Glöckle has been
successfully implemented in two- and three-body bound and scattering state
calculations using a partial wave decomposition. In this paper, we extend the
implementation of this iterative scheme to the 3B bound state calculations in
a 3D scheme by calculating the relativistic 2B $t-$matrices directly from the
relativistic 2B interactions.

  * •

The alternative option is the calculation of relativistic 2B $t-$matrices
directly from the nonrelativistic one. A two-step process is used to achieve
this. In the first step, the relativistic right-half-shell 2B $t-$matrices are
calculated analytically from the nonrelativistic right-half-shell 2B
$t-$matrices. Then the fully off-shell 2B $t-$matrices are calculated by
solving the first resolvent equation with the transformation of Coester-Piper-
Serduke [53, 13]. In this method, there is no need for the matrix elements of
the relativistic 2B interactions and it has been successfully implemented in
the 3B bound and scattering calculations in a 3D approach [26, 28, 49, 50, 44,
51].

The paper is organized as follows. In Sec. 2, a brief review to the formalism
of the relativistic Faddeev equations in a three-dimensional approach is
provided. Additionally, the formulation of the boost 2B interactions and
$t-$matrices, which are essential ingredients for our calculations, are
presented. In Sec. 3 our numerical results for boost 2B interactions and
$t-$matrices are given, and the contribution of different relativistic
corrections to the 3B binding energy is studied. As a test for our numerical
calculations, the expectation value of the 3B mass operator is calculated and
compared with the relativistic binding energy. In Sec. 4 we present our
conclusions and perspectives.

##  2 Relativistic Faddeev equation for three-body bound state

The 3B mass operator of three identical particles with mass $m$ and momentum
${\bf k}_{i}$, interacting with pairwise interactions, in the relativistic
quantum mechanics is defined as

$\displaystyle M=M_{0}+\sum_{i<j}V_{k}^{ij}.$ (1)

The free-body mass operator $M_{0}$ in the 3B Hilbert space is given by

$\displaystyle M_{0}=\sum_{i=1}^{3}(m^{2}+k_{i}^{2})^{\frac{1}{2}}.$ (2)

The 2B boost interactions $V_{k}^{ij}\equiv V_{k}$ embedded in the 3B Hilbert
space can be obtained in terms of the relativistic 2B interaction $V_{r}$ by
following nonlinear relation

$\displaystyle
V_{k}=\biggl{(}\bigl{(}\omega(\hat{p})+V_{r}\bigr{)}^{2}+\hat{k}^{2}\biggr{)}^{\frac{1}{2}}-\biggl{(}\omega^{2}(\hat{p})+\hat{k}^{2}\biggr{)}^{\frac{1}{2}},$
(3)

where ${\bf k}={\bf k}_{i}+{\bf k}_{j}$ is total momentum of the subsystem
$(ij)$, $\omega(\hat{p})=2\sqrt{m^{2}+\hat{p}^{2}}$, and ${\bf p}$ is the
relative momentum in the 2B subsystem $(ij)$. The $k$-dependence arises from
this fact that in a 3B system, the 2B subsystems are not at rest. Obviously,
for ${\bf k}={\bf 0}$ the boost interaction reduces to the relativistic
interaction, i.e. $V_{k}=V_{r}$. Since we work in the center of mass frame of
the 3B system, $k$ can be considered as the momentum of the spectator
particle. The Schrödinger equation for the bound state of three identical
particles interacting with pairwise forces is described by Faddeev equation

$\displaystyle\psi=G_{0}\,t\,P\,\psi,$ (4)

where $G_{0}=(M_{t}-M_{0})^{-1}$ is free propagator, $M_{t}=E_{t}+3m$ is the
3B mass eigenvalue. The permutation operator is defined as
$P=P_{ij}P_{jk}+P_{ik}P_{jk}$, and $t$ is 2B transition operator. In order to
solve the Faddeev equation (4) in momentum space, one needs the 3B basis
states, which are composed of two relativistic Jacobi momentum vectors $|{\bf
p}\,{\bf k}\rangle$, defined in details in Ref. [44]. The 3B Jacobi momentum
variables are defined by boosting the single-particle momenta to the three-
body rest frame and then boosting two-body subsystem momenta to the subsystem
rest frame. The free-body mass operator $M_{0}$, given in Eq. (2), is diagonal
in these basis states

$M_{0}|{\bf p}\,{\bf
k}\rangle=\biggl{(}\bigl{(}\omega^{2}(p)+k^{2}\bigr{)}^{\frac{1}{2}}+\Omega(k)\biggr{)}|{\bf
p}\,{\bf k}\rangle,$ (5)

where $\Omega(k)=\left(m^{2}+k^{2}\right)^{\frac{1}{2}}$. As shown in Ref.
[44] the representation of Faddeev equation (4) in the relativistic basis
states $|{\bf p}\,{\bf k}\rangle$ leads to the following three-dimensional
integral equation

$\psi({\bf p}\,,{\bf k})=\frac{1}{M_{t}-M_{0}(p,k)}\int
d^{3}k^{\prime}\,N({\bf k},{\bf k^{\prime}})\,T_{k}^{sym}\big{(}{\bf
p},\tilde{\boldsymbol{\pi}};\epsilon\big{)}\,\psi\bigl{(}\boldsymbol{\pi},{\bf
k^{\prime}}\bigr{)},$ (6)

where $\epsilon=M_{t}-\Omega(k)$ is 2B subsystem energy, $T_{k}^{sym}({\bf
p},{\bf p^{\prime}};\epsilon)$ is symmetrized boost 2B $t$-matrix defined as
$T_{k}^{sym}({\bf p},{\bf p^{\prime}};\epsilon)=T_{k}({\bf p},{\bf
p^{\prime}};\epsilon)+T_{k}({\bf p},-{\bf p^{\prime}};\epsilon)$, and the
Jacobian function $N({\bf k},{\bf k}^{\prime})$ is defined as

$N({\bf k},{\bf k}^{\prime})={\cal N}^{-1}(-{\bf k}-{\bf k}^{\prime},{\bf
k}^{\prime})\,{\cal N}^{-1}(-{\bf k}-{\bf k}^{\prime},{\bf k}),$ (7)

where ${\cal N}$ is the square root of the Jacobian which arises from going
from individual momenta ${\bf k}$ and ${\bf k^{\prime}}$ of the subsystem to
the relative momentum ${\bf p}$ and the total 2B momentum ${\bf k}+{\bf
k^{\prime}}$ [44],

${\cal N}({\bf k},{\bf k}^{\prime})=\left({\partial({\bf k},{\bf
k}^{\prime})\over\partial({\bf p},{\bf k}+{\bf
k}^{\prime})}\right)^{\frac{1}{2}}=\left({\omega(p)+\omega(p)\over\Omega(k)+\Omega(k^{\prime})}{\Omega(k)\Omega(k^{\prime})\over\omega(p)\omega(p)}\right)^{\frac{1}{2}}.$
(8)

In the nonrelativistic limit where the momenta are much smaller than the
masses, the Jacobin function $N$ reduces to one, and consequently, the
relativistic Jacobi momenta reduce to the nonrelativistic ones. The
relativistic shifted momentum arguments are defined as

$\displaystyle\tilde{\boldsymbol{\pi}}$ $\displaystyle=$ $\displaystyle{\bf
k}^{\prime}+\frac{1}{2}C({\bf k},{\bf k}^{\prime})\,{\bf k},$ (9)
$\displaystyle\boldsymbol{\pi}$ $\displaystyle=$ $\displaystyle{\bf
k}+\frac{1}{2}C({\bf k}^{\prime},{\bf k})\,{\bf k}^{\prime},$ (10)

where the permutation coefficients $C({\bf k},{\bf k}^{\prime})$ are defined
as [26]

$\displaystyle C({\bf k},{\bf k}^{\prime})\equiv
1+\frac{\Omega(k^{\prime})-\Omega(|{\bf k}+{\bf
k}^{\prime}|)}{\Omega(k^{\prime})+\Omega(|{\bf k}+{\bf
k}^{\prime}|)+\sqrt{\biggl{(}\Omega(k^{\prime})+\Omega(|{\bf k}+{\bf
k}^{\prime}|)\biggr{)}^{2}-k^{2}}}.$ (11)

In the nonrelativistic limit, when the momenta are much smaller than the
masses, the permutation coefficient $C({\bf k},{\bf k}^{\prime})$ is equal to
one. Consequently, the relativistic shifted momenta $\tilde{\boldsymbol{\pi}}$
and $\boldsymbol{\pi}$ reduce to the corresponding nonrelativistic ones. As we
mentioned earlier, in this paper we calculate the fully-off-shell boost 2B
$t-$matrices $T_{k}({\bf p},{\bf p^{\prime}};\epsilon)$ by solving the
following relativistic Lippmann-Schwinger equation

$\displaystyle T_{k}({\bf p},{\bf p^{\prime}};\epsilon)=V_{k}({\bf p},{\bf
p^{\prime}})+\int d^{3}p^{\prime\prime}\,\frac{V_{k}({\bf p},{\bf
p^{\prime\prime}})}{\epsilon-\biggl{(}\omega^{2}(p^{\prime\prime})+k^{2}\biggr{)}^{\frac{1}{2}}}\,T_{k}({\bf
p^{\prime\prime}},{\bf p^{\prime}};\epsilon).$ (12)

To solve the three-dimensional integral equation (6), we choose a coordinate
system where ${\bf k}$ is parallel to $Z-$axis, ${\bf p}$ is in the $X-Z$
plane, and the integration vector ${\bf k}^{\prime}$ is free in the space. In
a spherical coordinate system the radial distance, the cosine of polar angle,
and azimuthal angle for momentum vectors ${\bf k}$, ${\bf p}$, and ${\bf
k}^{\prime}$ are given by $(k,x_{k}=0,\phi_{k}=0)$, $(p,x_{p}\equiv
x,\phi_{p}=0)$, and $(k^{\prime},x_{k^{\prime}}\equiv
x^{\prime},\phi_{k^{\prime}}\equiv\phi^{\prime})$, correspondingly. Using this
coordinate system, Eq. (6) can be written explicitly as

$\displaystyle\psi(p,k,x)$ $\displaystyle=$
$\displaystyle\frac{1}{M_{t}-\sqrt{\omega^{2}(p)+k^{2}}-\Omega(k)}\,\int_{0}^{\infty}dk^{\prime}k^{\prime
2}\int_{-1}^{+1}dx^{\prime}\int_{0}^{2\pi}d\phi^{\prime}$ (13)
$\displaystyle\times$ $\displaystyle
N(k,k^{\prime},x^{\prime})\,T^{sym}_{k}\biggl{(}p,\tilde{\pi},x_{p\tilde{\pi}};M_{t}-\Omega(k)\biggr{)}\psi(\pi,k^{\prime},x_{\pi
k^{\prime}}).$ (14)

where the momentum and angle variables are defined as

$\displaystyle x$ $\displaystyle=$ $\displaystyle\hat{{\bf k}}\cdot\hat{{\bf
p}},$ (15) $\displaystyle x^{\prime}$ $\displaystyle=$ $\displaystyle\hat{{\bf
k}}\cdot\hat{{\bf k}}^{\prime},$ (16) $\displaystyle y$ $\displaystyle=$
$\displaystyle\hat{{\bf p}}\cdot\hat{{\bf
k}}^{\prime}=xx^{\prime}+\sqrt{1-x^{2}}\sqrt{1-x^{\prime
2}}\cos\phi^{\prime},$ (17) $\displaystyle\tilde{\pi}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{1}{4}C^{2}(k,k^{\prime},x^{\prime})k^{2}+k^{\prime
2}+C(k,k^{\prime},x^{\prime})kk^{\prime}x^{\prime}},$ (18) $\displaystyle\pi$
$\displaystyle=$
$\displaystyle\sqrt{k^{2}+\frac{1}{4}C^{2}(k^{\prime},k,x^{\prime})k^{\prime
2}+C(k^{\prime},k,x^{\prime})kk^{\prime}x^{\prime}},$ (19) $\displaystyle
x_{p\tilde{\pi}}$ $\displaystyle=$
$\displaystyle\frac{\frac{1}{2}C(k,k^{\prime},x^{\prime})kx+k^{\prime}y}{\tilde{\pi}},$
(20) $\displaystyle x_{\pi k^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{kx^{\prime}+\frac{1}{2}C(k^{\prime},k,x^{\prime})k^{\prime}}{\pi}.$
(21)

The integral equation (14) is an eigenvalue equation in the form
$\lambda\psi=K(M_{t})\cdot\psi$ with the eigenvalue $\lambda=1$. Since the
kernel of the integral equation $K(M_{t})$ is energy dependent, the solution
of the eigenvalue equation (14) can be started by an initial guess for the
energy $E_{t}$ and the search in the binding energy is stopped when
$|1-\lambda|<10^{-6}$.

The input for the Lippmann-Schwinger integral equation (12) is the 2B boost
potential $V_{k}({\bf p},{\bf p^{\prime}})$ which has been traditionally
calculated by the representation of Eq. (3) in momentum space by inserting the
completeness relation of bound and scattering eigenstates of the center of
mass Hamiltonian ($\omega(\hat{p})+V_{r}$) [7]. However, with the recent
successful application of the iterative scheme proposed by Kamada and Glöckle
[52] in the calculation of the relativistic interaction from the
nonrelativistic one in a 3D scheme [54], in this paper we utilize the same
method for the calculation of the boost potential. By following [52], one can
obtain the boost interaction from the nonrelativistic interaction by the
following quadratic equation

$\displaystyle
V_{nr}=\frac{1}{4m}\biggl{(}\omega_{k}(\hat{p})\,V_{k}+V_{k}\,\omega_{k}(\hat{p})+V_{k}^{2}\biggr{)},$
(22)

where
$\omega_{k}(\hat{p})=\biggl{(}\omega^{2}(\hat{p})+\hat{k}^{2}\biggr{)}^{\frac{1}{2}}$.
Obviously, for ${\bf k}={\bf 0}$ the Eq. (22) can be reduced to the connection
between the relativistic and nonrelativistic interactions as

$\displaystyle
V_{nr}=\frac{1}{4m}\biggl{(}\omega(\hat{p})V_{r}+V_{r}\omega(\hat{p})+V_{r}^{2}\biggr{)}.$
(23)

Equations (23) and (22) have same operator forms, where $V_{r}$ and
$\omega(\hat{p})$ are replaced by $V_{k}$ and $\omega_{k}(\hat{p})$. So the
procedure of calculation of boost potential $V_{k}$ from the nonrelativistic
potential $V_{nr}$ in Eq. (22) is similar to the calculation of the
relativistic potential $V_{r}$ from the nonrelativistic potential $V_{nr}$ in
Eq. (23). So we can obtain the boost potential matrix elements by taking the
nonrelativistic potential and by solving the following three-dimensional
integral equation

$\displaystyle V_{k}({\bf p},{\bf
p^{\prime}})+\frac{1}{\omega_{k}(p)+\omega_{k}(p^{\prime})}\int
d^{3}p^{\prime\prime}V_{k}({\bf p},{\bf p^{\prime\prime}})\,V_{k}({\bf
p^{\prime\prime}},{\bf p^{\prime}})=\frac{4m\,V_{nr}({\bf p},{\bf
p^{\prime}})}{\omega_{k}(p)+\omega_{k}(p^{\prime})}.$ (24)

The integral equation (24) is solved using an iterative scheme and the details
of the calculations are similar to the ones discussed in Ref. [54]. We would
like to mention that using an averaging scheme on two successive iterations
leads to a faster convergence in the calculation of the boost interaction.

##  3 Numerical Results

In the first step we solve the three-dimensional integral equation (24) to
calculate the matrix elements of the boost interaction
$V_{k}(p,p^{\prime},x^{\prime})$ from the nonrelativistic interaction
$V_{nr}(p,p^{\prime},x^{\prime})$ by an iterative scheme. The iteration starts
with an initial guess
$V_{k}^{(0)}(p,p^{\prime},x^{\prime})=\frac{4m\,V_{nr}(p,p^{\prime},x^{\prime})}{\omega_{k}(p)+\omega_{k}(p^{\prime})}$
and then continued to reach a convergence in the matrix elements of the boost
potential with a relative error of $10^{-6}$ at each set of points
$(p,p^{\prime},x^{\prime})$. In Table 1 we show an example of the convergence
of the matrix elements of the boost potential $V_{k}(p,p^{\prime},x^{\prime})$
as a function of the iteration number calculated for three different values of
the Jacobi momentum $k=1,\,5,\,10\,\text{fm}^{-1}$ for the Malfliet-Tjon-V
(MT-V) bare potential in the fixed points ($p=1.05$ fm-1, $p^{\prime}=2.60$
fm-1, $x^{\prime}=0.50$). As one can see for larger values of the Jacobi
momentum $k$ the convergence is reached faster. The data of Table 1 are also
illustrated in Fig. 1. In Fig. 2 the matrix elements of the nonrelativistic
and the boost interactions and also the difference between them are shown.

By having the matrix elements of the boost interactions
$V_{k}(p,p^{\prime},x^{\prime})$ we solve the Lippmann-Schwinger integral
equation (12) to calculate the off-shell boost $t-$matrices
$T_{k}(p,p^{\prime},x^{\prime};\epsilon)$ and then symmetrize it on the angle
variable to get $T_{k}^{sym}(p,p^{\prime},x^{\prime};\epsilon)$. In Fig. 3 we
show the angular dependencies of the symmetrized 2B boost $t-$matrices for
energies $\epsilon=-10,-50,-200\,\text{MeV}$ obtained for
$k=1\,\text{fm}^{-1}$ and compared with the corresponding nonrelativistic
$t-$matrices. As we expect for higher energies the difference between the
boost and the nonrelativistic $t-$matrices is more visible.

| $V_{nr}(p,p^{\prime},x^{\prime})$
---|---
| $1.1579249\cdot 10^{-2}$
| $V_{k}(p,p^{\prime},x^{\prime})$
Iteration # | $k=1\,\text{fm}^{-1}$ | $k=5\,\text{fm}^{-1}$ | $k=10\,\text{fm}^{-1}$
0 | -2.2688550 | -2.0244444 | -1.5822847
1 | -2.5977481 | -2.2796733 | -1.7267466
2 | -2.4833116 | -2.1967138 | -1.6871314
3 | -2.4880714 | -2.1984435 | -1.6861455
4 | -2.4886996 | -2.1988384 | -1.6861841
5 | -2.4887497 | -2.1988784 | -1.6861961
6 | -2.4887515 | -2.1988795 | -1.6861962
7 | -2.4887519 | -2.1988792 | -1.6861957
8 | -2.4887523 | -2.1988792 | -1.6861954
9 | -2.4887525 | -2.1988793 | -1.6861954
10 | -2.4887526 | -2.1988793 | –
11 | -2.4887527 | – | –
12 | -2.4887527 | – | –
Table 1: The convergence of the matrix elements of the boost potential
$V_{k}(p,p^{\prime},x^{\prime})$ (in units of MeV fm3) as a function of the
iteration number calculated using the MT-V bare potential in the fixed points
($p=1.05$ fm-1, $p^{\prime}=2.60$ fm-1, $x^{\prime}=0.50$) and for three
different values of the Jacobi momentum $k=1,\,5,\,10\,\text{fm}^{-1}$. The
value of the MT-V bare potential $V_{nr}(p,p^{\prime},x^{\prime})$ is also
given. Figure 1: The convergence of the matrix elements of the boost potential
$V_{k}(p,p^{\prime},x^{\prime})$ as a function of the iteration number
calculated for the MT-V bare potential in the fixed points ($p=1.05$ fm-1,
$p^{\prime}=2.60$ fm-1, $x^{\prime}=0.50$) with
$k=1,\,5,10\,\,\text{fm}^{-1}$. The left endpoint of each plot is the value of
the nonrelativistic MT-V potential
$V_{nr}(p,p^{\prime},x^{\prime})=1.1579249\cdot 10^{-2}\,\text{MeV\,fm}^{3}$.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 2:  The matrix elements of the nonrelativistic (a & d), the boost (b &
e) 2B potentials and their differences (c & f) calculated with the MT-V
potential. They are shown as a function of the 2B relative momenta $p$ and
$p^{\prime}$ for the angle $x^{\prime}=\frac{\sqrt{2}}{2}$ (upper panel) and
as a function of the 2B relative momenta $p=p^{\prime}$ and the angle between
them $x^{\prime}$ (lower panel). The boost potential is obtained with
$k=1\,\text{fm}^{-1}$. Figure 3: The angular dependency of the symmetrized
boost 2B $t-$matrix $T_{k}^{sym}(p_{0},p_{0},x,\epsilon)$ calculated for
$k=1\,\text{fm}^{-1}$ and energies $\epsilon=-10,-50,-200$ MeV, with
$p_{0}=\sqrt{m\cdot|\epsilon|}$. The diamond symbols indicate the
corresponding nonrelativistic $t-$matrices. The input boost potential is
obtained from the MT-V potential.

For the calculation of relativistic effects in the 3B binding energy, we solve
the three-dimensional integral equation (14). To this aim, for discretization
of continuous momentum and angle variables, we use Gauss-Legendre quadrature.
For angle variables, a linear mapping, and for Jacobi momenta, a hyperbolic-
linear mapping is used. The momentum cutoffs used for Jacobi momenta $p$ and
$k$ are $60\,\text{fm}^{-1}$ and $20\,\text{fm}^{-1}$, correspondingly. In
Table 2, we present our numerical results for the relativistic 3B binding
energy as a function of the number of mesh points for the Jacobi momenta $p$
and $k$. For discretization of polar and azimuthal angles, we use 40 mesh
points. The symmetry property of the Faddeev component on the angle $x$, i.e.
$\psi(p,k,x)=\psi(p,k,-x)$, is employed to save memory and computational time
in solving Eq. (14). As one can see, the relativistic 3B binding energy for
the MT-V potential converges to $-7.5852$ MeV.

$N_{p}$ | $N_{k}$ | $E^{3B}_{r}$
---|---|---
20 | 20 | -7.8056
30 | 30 | -7.6448
40 | 40 | -7.5958
50 | 50 | -7.5956
60 | 60 | -7.5883
70 | 70 | -7.5863
80 | 80 | -7.5856
90 | 90 | -7.5853
100 | 100 | -7.5853
110 | 110 | -7.5852
120 | 120 | -7.5852
130 | 130 | -7.5852
Table 2: The convergence of the relativistic 3B binding energy (in units of
MeV) as a function of the number of mesh points for Jacobi momenta $p$ and
$k$. The number of mesh points for polar and azimuthal angles is 40. The
results are obtained for the MT-V potential.

In Table 3, we present the contribution of different relativistic corrections
to the 3B binding energy. While the Faddeev calculations lead to the
nonrelativistic 3B binding energy of $-7.7380$ MeV, the contribution of
different relativistic corrections in the 3B binding energy is as

  * •

the Jacobian function $N$ decreases the 3B binding energy with about $0.06$
MeV,

  * •

the permutation coefficient $C$ increases the 3B binding energy with about
$0.05$ MeV,

  * •

the relativistic 3B free propagator $G_{0}$ increases the 3B binding energy
with about $0.18$ MeV,

  * •

the relativistic 2B $t-$matrices decreases the 3B binding energy with about
$0.28$ MeV.

So, two of the relativistic corrections, $N$ and 2B $t-$matrices, decrease the
3B binding energies, and the other two terms, $C$ and $G_{0}$, increase the 3B
binding energy. When we apply all the corrections together, we obtain a
relativistic 3B energy of $-7.5852$ MeV, which indicates a reduction of about
$-0.15$ MeV or a percentage difference of $2\%$.

As we discussed earlier, an alternative approach for the calculation of the
boost 2B $t-$matrices is solving a first resolvent equation, where the boost
2B $t-$matrices can be obtained directly from the nonrelativistic 2B
$t-$matrices. In this approach, there is no need for the matrix elements of
relativistic or boost potentials defined in Eqs. (22) and (23). The
implementation of this method to 3B bound state calculations leads to a
reduction of about 3.3% in the 3B binding energy for the MT-V potential [44].
As one can see, the calculation of the boost 2B $t-$matrices using two
different methods, i.e. solving the first resolvent equation discussed in Ref.
[44] and the solution of the relativistic Lippmann-Schwinger equation given in
Eq. (12), leads to a slightly different correction in the 3B binding energy.
Solving the relativistic Lippmann-Schwinger equation using the boost potential
of Eq. (24), as discussed in this paper, leads to a smaller reduction in the
3B binding energy, whereas the solution of the first resolvent equation leads
to a more reduction, about 1.3% larger. A comparison with the results of
Glöckle et al. in Ref. [2] for an $s-$wave projection of MT-V potential
indicates a reduction of about 2.7%, whereas the relativistic corrections for
different models of 2B interactions by Kamada et al. in Ref. [7] shows a wide
range of decrease between 2 to 6%.

Relativistic correction |  | $E^{3B}$ |  | value in MeV |  | $\left(\frac{E^{3B}_{app}-E^{3B}_{nr}}{E^{3B}_{nr}}\cdot 100\right)\%$
---|---|---|---|---|---|---
– |  | $E^{3B}_{nr}$ |  | $-7.7380$ |  | –
$N$ (the Jacobian function defined in Eq. 7) |  | $E^{3B}_{app}$ |  | $-7.6748$ |  | -0.8167
$C$ (the permutation coefficients $C$ defined in Eq. 11) |  | $E^{3B}_{app}$ |  | $-7.7900$ |  | +0.6720
$G_{0}^{rel}$ (the relativistic 3B free propagator) |  | $E^{3B}_{app}$ |  | $-7.9183$ |  | +2.3301
$T_{k}$ (the boost 2B $t-$matrix given in Eq. 12) |  | $E^{3B}_{app}$ |  | $-7.4540$ |  | -3.6702
Full relativistic ($N$ \+ $C$ \+ $G_{0}^{rel}$ \+ $T_{k}$) |  | $E^{3B}_{r}$ |  | $-7.5852$ |  | -1.9747
Table 3: The contributions of different relativistic corrections to the 3B
binding energy calculated for the MT-V potential.

By solving the three-dimensional integral equation (14) and obtaining the 3B
binding energy and the Faddeev component $\psi(p,k,x)$ one can calculate the
3B wave function $\Psi(p,k,x)$ by adding up the Faddeev components in three
different 3B clusters. The details of the calculation are addressed in Ref.
[44] and here we briefly represent the explicit form of the 3B wave function
as

$\Psi(p,k,x)=\sum_{i=1}^{3}\psi_{i}(p,k,x),$ (25)

where the Faddeev components $\psi_{1},\psi_{2},\psi_{3}$, corresponding to
the 3B clusters $(23,1),(31,2),(12,3)$, are given by

$\displaystyle\psi_{1}(p,k,x)$ $\displaystyle=$ $\displaystyle\psi(p,k,x),$
(26) $\displaystyle\psi_{2}(p,k,x)$ $\displaystyle=$
$\displaystyle\dfrac{{\cal N}({\bf k}_{2},{\bf k}_{3})}{{\cal N}({\bf k},{\bf
k}_{3})}\psi(p_{2},k_{2},x_{2}),$ (27) $\displaystyle\psi_{3}(p,k,x)$
$\displaystyle=$ $\displaystyle\dfrac{{\cal N}({\bf k}_{2},{\bf k}_{3})}{{\cal
N}({\bf k},{\bf k}_{2})}\psi(p_{3},k_{3},x_{3}).$ (28)

The Jacobian function ${\cal N}$ is defined in Eq. (8) and the shifted
momentum arguments $p_{2},k_{2},p_{3},k_{3}$ and the angle variables
$x_{2},x_{3}$ are defined as

$\displaystyle p_{2}$ $\displaystyle=$ $\displaystyle\left|\xi_{p}{\bf
p}+\xi_{k}{\bf
k}\right|=\sqrt{\xi_{p}^{2}\,p^{2}+\xi_{k}^{2}\,k^{2}+2\,\xi_{p}\,\xi_{k}\,p\,k\,x},$
(29) $\displaystyle k_{2}$ $\displaystyle=$ $\displaystyle\left|{\bf
p}+\alpha{\bf k}\right|=\sqrt{p^{2}+\alpha^{2}k^{2}+2\,\alpha\,p\,k\,x},$ (30)
$\displaystyle x_{2}$ $\displaystyle\equiv$ $\displaystyle\hat{\bf
p}_{2}\cdot\hat{\bf
k}_{2}=\dfrac{\xi_{p}\,p^{2}+\alpha\,\xi_{k}\,k^{2}+(\alpha\,\xi_{p}+\xi_{k})p\,k\,x}{p_{2}\,k_{2}},$
(31) $\displaystyle p_{3}$ $\displaystyle=$ $\displaystyle\left|\gamma_{p}{\bf
p}+\gamma_{k}{\bf
k}\right|=\sqrt{\gamma_{p}^{2}\,p^{2}+\gamma_{k}^{2}\,k^{2}+2\,\gamma_{p}\,\gamma_{k}\,p\,k\,x},$
(32) $\displaystyle k_{3}$ $\displaystyle=$ $\displaystyle\left|-{\bf
p}-\beta{\bf k}\right|=\sqrt{p^{2}+\beta^{2}k^{2}+2\,\beta\,p\,k\,x},$ (33)
$\displaystyle x_{3}$ $\displaystyle\equiv$ $\displaystyle\hat{\bf
p}_{3}\cdot\hat{\bf
k}_{3}=-\dfrac{\gamma_{p}\,p^{2}+\beta\,\gamma_{k}\,k^{2}+(\beta\,\gamma_{p}+\gamma_{k})p\,k\,x}{p_{3}\,k_{3}},$
(34)

where

$\displaystyle\alpha$ $\displaystyle=$
$\displaystyle\dfrac{1}{\omega(p)}\left(\dfrac{pkx}{\omega(p)+\sqrt{\omega(p)^{2}+k^{2}}}-\dfrac{1}{2}\omega(p)\right),$
(35) $\displaystyle\beta$ $\displaystyle=$ $\displaystyle 1+\alpha,$ (36)
$\displaystyle\gamma_{p}$ $\displaystyle=$
$\displaystyle\dfrac{\left(\dfrac{pkx+\beta
k^{2}}{\sqrt{\biggl{(}\Omega(k_{1})+\Omega(k_{2})\biggr{)}^{2}-k_{3}^{2}}+\Omega(k_{1})+\Omega(k_{2})}-\Omega(k_{1})\right)}{\sqrt{\biggl{(}\Omega(k_{1})+\Omega(k_{2})\biggr{)}^{2}-k_{3}^{2}}},$
(37) $\displaystyle\gamma_{k}$ $\displaystyle=$ $\displaystyle
1+\gamma_{p}\beta,$ (38) $\displaystyle\xi_{p}$ $\displaystyle=$
$\displaystyle-1+\dfrac{\left(\dfrac{-p^{2}-\alpha\beta
k^{2}-(\alpha+\beta)pkx}{\sqrt{\biggl{(}\Omega(k_{1})+\Omega(k_{3})\biggr{)}^{2}-k_{2}^{2}}+\Omega(k_{1})+\Omega(k_{3})}+\Omega(k_{3})\right)}{\sqrt{\biggl{(}\Omega(k_{1})+\Omega(k_{3})\biggr{)}^{2}-k_{2}^{2}}},$
(39) $\displaystyle\xi_{k}$ $\displaystyle=$
$\displaystyle\alpha(\xi_{p}+1)-\beta.$ (40)

The 3B wave function is normalized as

$\displaystyle\langle\Psi|\Psi\rangle=\sum_{i,j=1}^{3}\langle\psi_{i}|\psi_{j}\rangle=8\pi^{2}\int_{0}^{\infty}dp\,p^{2}\int_{0}^{\infty}dk\,k^{2}\int_{-1}^{+1}dx\,\Psi^{2}(p,k,x)=1,$
(41)

and the normalization of the Faddeev components is chosen as

$\displaystyle|\psi_{i}\rangle$ $\displaystyle=$
$\displaystyle\dfrac{|\psi_{i}\rangle}{\sum_{j=1}^{3}\langle\psi_{j}|\Psi\rangle},i=1,2,3.$
(42)

To study the contribution of different Faddeev components in the normalization
of the 3B wave function, we show the inner product of the Faddeev components,
i.e. $\langle\psi_{i}|\psi_{j}\rangle$, in Table 4. The calculated Faddeev
components defined in Eq. (42) satisfy the normalization of the 3B wave
function given in Eq. (41) with an error of the order of $10^{-12}$. While
three Faddeev components defined in Eq. (28) have completely different
momentum dependence, they are all normalized to almost the same value, i.e.
$\langle\psi_{i}|\psi_{i}\rangle=0.12658$. Moreover, the inner products of the
nonidentical Faddeev components leads to the same value of
$\langle\psi_{i}|\psi_{j}\rangle=0.10338$. Our numerical analysis confirms the
normalization condition of the 3B wave function
$\langle\Psi|\Psi\rangle=3\langle\psi|\psi\rangle+3\langle\psi|P|\psi\rangle=1$
with an accuracy of at least five significant digits.

| $j=1$ | $j=2$ | $j=3$
---|---|---|---
$i=1$ | 0.12657 | 0.10338 | 0.10338
$i=2$ | 0.10338 | 0.12658 | 0.10338
$i=3$ | 0.10338 | 0.10338 | 0.12658
Table 4: The inner product of the Faddeev components
$\langle\psi_{i}|\psi_{j}\rangle$ and their contribution in the normalization
of the 3B wave function $|\Psi\rangle$ defined in Eq. (41).

As a test for the numerical accuracy of our calculations, we calculate the
expectation value of the 3B mass operator $\langle\Psi|M|\Psi\rangle$ and
compared it with the 3B binding energy obtained from the solution of the
integral equation (14). By considering the definition of the 3B mass operator
$M$ in Eq. (1), the expectation value of $\langle M\rangle$ can be obtained as

$\langle\Psi|M|\Psi\rangle=\langle\Psi|M_{0}|\Psi\rangle+\langle\Psi|\sum_{i<j}V_{k}^{ij}|\Psi\rangle,$
(43)

where the expectation values of the free-body mass operator and the boost
potential are given by

$\langle\Psi|M_{0}|\Psi\rangle=8\pi^{2}\int_{0}^{\infty}dp\,p^{2}\int_{0}^{\infty}dk\,k^{2}\left(\Omega(k)+\sqrt{\omega^{2}(p)+k^{2}}\right)\int_{-1}^{1}dx\,\Psi^{2}(p,k,x),$
(44) $\displaystyle\langle\Psi|\sum_{i<j}V_{k}^{ij}|\Psi\rangle$
$\displaystyle\equiv$ $\displaystyle 3\langle\Psi|V_{k}|\Psi\rangle$ (45)
$\displaystyle=$ $\displaystyle 3\cdot
8\pi^{2}\int_{0}^{\infty}dp\,p^{2}\int_{0}^{\infty}dk\,k^{2}\int_{0}^{\infty}dp^{\prime}\,p^{\prime
2}\int_{-1}^{1}dx\int_{-1}^{1}dx^{\prime}\,\Psi(p,k,x)\,v_{k}(p,p^{\prime},x,x^{\prime})\,\Psi(p^{\prime},k,x^{\prime}),$
(46)

where $v_{k}(p,p^{\prime},x,x^{\prime})$ is the result of the azimuthal angle
integration on the matrix elements of the boost potential defined as

$v_{k}(p,p^{\prime},x,x^{\prime})=\int_{0}^{2\pi}d\phi\;V_{k}(p,p^{\prime},y=xx^{\prime}+\sqrt{1-x^{2}}\,\sqrt{1-x^{\prime
2}}\cos\phi).$ (47)

In Table 5, we present our numerical results for the expectation value of the
3B mass operator compared with the relativistic 3B binding energy. We also
show our results for the expectation value of the nonrelativistic 3B
Hamiltonian and the nonrelativistic 3B binding energy. As one can see the
expectation value of free-body mass operator is $\langle
M_{0}\rangle=3m+28.3876\,\text{MeV}$, while the expectation value of the boost
potential is $\langle V_{k}\rangle=-35.9716\,\text{MeV}$ which leads to the
expectation value of the 3B mass operator of $\langle
M\rangle=3m-7.5840\,\text{MeV}$. The difference between the relativistic 3B
binding energy, obtained from three-dimensional Faddeev integral equation
(14), and the expectation value of the 3B mass operator, obtained from Eq.
(43), is about $0.0012\,\text{MeV}$. Our numerical results for the
nonrelativistic solution of the three-dimensional Faddeev integral equation
leads to the binding energy of $E_{nr}^{3B}=-7.7380\,\text{MeV}$ and the
nonrelativistic expectation value of $\langle H\rangle=-7.7384\,\text{MeV}$
with a difference of $0.0004\,\text{MeV}$. As one can see for both
relativistic and nonrelativistic calculations, the calculated 3B binding
energy is in excellent agreement with the expectation value of the 3B
Hamiltonian.

|  | $\langle M_{0}\rangle-3m$ |  | $\langle V_{k}\rangle$ |  | $\langle M\rangle-3m$ |  | $E_{r}^{3B}$ |  | $\left|\langle M\rangle-3m-E_{r}^{3B}\right|$
---|---|---|---|---|---|---|---|---|---|---
Relativistic |  | 28.3876 |  | -35.9716 |  | -7.5840 |  | -7.5852 |  | 0.0012
|  | $\langle H_{0}\rangle$ |  | $\langle V_{nr}\rangle$ |  | $\langle H\rangle$ |  | $E_{nr}^{3B}$ |  | $\left|\langle H\rangle-E_{nr}^{3B}\right|$
nonrelativistic |  | 29.7789 |  | -37.5173 |  | -7.7384 |  | -7.7380 |  | 0.0004
Table 5: The expectation value of the 3B mass operator $\langle M\rangle$
obtained from free-body mass operator $\langle M_{0}\rangle$ and the boost
interaction $\langle V_{k}\rangle$ compared with the relativistic 3B binding
energy $E_{r}^{3B}$. The corresponding nonrelativistic results are also shown.
For both nonrelativistic and relativistic calculations, the MT-V potential is
used. All the numbers are given in the units of MeV.

In Table 6 we present the contribution of different Faddeev components in the
expectation value of the 3B mass operator
$\langle\psi_{i}|M|\psi_{j}\rangle-3m$. To this aim the expectation values of
the free-body mass operator $\langle\psi_{i}|M_{0}|\psi_{j}\rangle-3m$ and the
boost interaction $\langle\psi_{i}|V_{k}|\psi_{j}\rangle$ are calculated for
different Faddeev components. As one can see
$\sum_{i,j=1}^{3}\langle\psi_{i}|M_{0}|\psi_{j}\rangle-3m=28.3876\,\text{MeV}$,
$\sum_{i,j=1}^{3}\langle\psi_{i}|V_{k}|\psi_{j}\rangle=-35.9716\,\text{MeV}$,
and
$\sum_{i,j=1}^{3}\langle\psi_{i}|M|\psi_{j}\rangle-3m=-7.5840\,\text{MeV}$.

Panel A: $\langle\psi_{i}|M_{0}|\psi_{j}\rangle-3m$
---
|  $j=1$ |  $j=2$ |  $j=3$
$i=1$ |  4.58426 |  2.43931 |  2.43931
$i=2$ |  2.43931 |  4.58396 |  2.43911
$i=3$ |  2.43931 |  2.43911 |  4.58396

Panel B: $\langle\psi_{i}|V_{k}|\psi_{j}\rangle$
---
$i=1$ |  -6.88350 |  -4.87431 |  -4.87431
$i=2$ |  -4.87427 |  -2.54511 |  -2.25037
$i=3$ |  -4.87427 |  -2.25037 |  -2.54511
Panel C: $\langle\psi_{i}|M|\psi_{j}\rangle-3m$
$i=1$ |  -2.29924 |  -2.43500 |  -2.43500
$i=2$ |  -2.43496 |  2.03885 |  0.18873
$i=3$ |  -2.43496 |  0.18873 |  2.03885

Table 6: The contribution of different Faddeev components in the expectation
value of the 3B mass operator. Panel A: the expectation values of the free
mass operator $\langle\psi_{i}|M_{0}|\psi_{j}\rangle-3m$, Panel B: the
expectation values of the boost interaction
$\langle\psi_{i}|V_{k}|\psi_{j}\rangle$, Panel C: the expectation values of
the 3B mass operator $\langle\psi_{i}|M|\psi_{j}\rangle-3m$, calculated for
different Faddeev components. All the numbers are given in the units of MeV.

##  4 Summary

The inputs for the kernel of Faddeev and Yakubovsky integral equations, to
study the relativistic corrections in three- and four-body bound and
scattering states, are the boost 2B $t-$matrices. A number of theoretical
methods are developed to calculate the relativistic and boost 2B $t-$matrices.
The direct solution of the relativistic Lippmann-Schwinger equation using
relativistic interactions is one of the methods which has been successfully
implemented in traditional partial wave decomposition calculations. One of the
novel techniques for the calculation of the relativistic and boost
interactions from the nonrelativistic interactions is solving a quadratic
equation using an iterative scheme proposed by Kamada and Glöckle. In this
paper, we apply the direct calculation of the boost interaction from the
nonrelativistic interaction by solving a quadratic equation in a three-
dimensional approach. To speed up the convergence in the solution of the
quadratic equation, we use an averaging scheme. Then, by solving the Lippmann-
Schwinger integral equation with the input boost 2B interaction, the boost 2B
$t-$matrices are obtained as a function of the 2B relative momentum vectors,
mainly the magnitude of the initial and final relative momenta and the angle
between them. The 2B boost interactions have been then embedded into the
kernel of three-dimensional Faddeev integral equations to study the
relativistic corrections in the 3B binding energy. For a comprehensive study
of the relativistic effects, one must consider four different relativistic
corrections: the relativistic free propagator, the relativistic 2B
$t-$matrices, the Jacobian, and the permutation coefficient. The contribution
of each relativistic correction is studied in detail. We show that the
relativistic corrections lead to a reduction of about $2\%$ in the 3B binding
energy. As a test for the numerical accuracy of our calculations, we calculate
the expectation value of the 3B mass operator, which is in excellent agreement
with the relativistic 3B binding energy. It is shown that in the
nonrelativistic limit, where the momenta are much smaller than the masses, the
nonrelativistic results are being reproduced from the relativistic ones.

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## Acknowledgements

The work of M. R. Hadizadeh was supported by the National Science Foundation
under Grant Number NSF-HRD-1436702 with Central State University. M. Radin
acknowledges partial financial support from Iran National Science Foundation
(INSF).


# Birefringent spin-photon interface generates polarization entanglement

Nikita Leppenen<EMAIL_ADDRESS>Department of Chemical &
Biological Physics, Weizmann Institute of Science, Rehovot 7610001, Israel
Dmitry S. Smirnov  Ioffe Institute, 194021 St. Petersburg, Russia

###### Abstract

A spin-photon interface based on the luminescence of a singly charged quantum
dot in a micropillar cavity allows for the creation of photonic entangled
states. Current devices suffer from cavity birefringence, which limits the
generation of spin-photon entanglement. In this paper, we theoretically study
the light absorption and emission by the interface with an anisotropic cavity
and derive the maximal excitation and spin-photon entanglement conditions. We
show that the concurrence of the spin-photon state equal to one and complete
quantum dot population inversion can be reached for a micropillar cavity with
any degree of birefringence by tuning the quantum dot resonance strictly
between the cavity modes. This sweet spot is also valid for generating a
multiphoton cluster state, as we demonstrate by calculating the three-tangle
and fidelity with the maximally entangled state.

##  I Introduction

Spin entanglement is an essential resource for quantum information processing
due to the excellent isolation of spin degree of freedom from the environment
[1]. A basic example of the entanglement production is given by absorption of
a linearly polarized photon in GaAs-based nanostructures when a Bell state of
electron and hole spins is optically generated [2, 3]. However, flying qubits
naturally represented by photons are more useful for quantum information
transfer. Their polarization entanglement is routinely produced by biexciton
recombination [4, 5, 6, 7].

Photons can be coupled to the localized qubits — electron spins in quantum
dots (QDs), by zero-dimensional spin-photon interfaces [8], which can be
efficiently realized using micropillar cavities [9, 10, 9, 11, 12] (see Refs.
13, 14 for recent reviews). These interfaces allow for the photon polarization
manipulation by electron spin [15, 16, 17] and fast control of electron spin
by photons [18, 19, 20, 21]. But most importantly, electron-photon
entanglement can be created in these structures [22, 23, 24].

An auspicious task is the generation of entangled photon cluster states [25].
Among different approaches [26, 27, 28, 29] including parametric down
conversion [30], the most scalable one is the sequential entanglement of spin
of a photon and localized electron [31, 32, 33]. After the first realization
of the theoretical proposal [34], the intensive investigations in this
direction continue until now [35, 36].

An important factor limiting the generation of many entangled photons is the
intrinsic electron spin relaxation [37]. An approach for its investigation by
measurement of the equilibrium electron spin noise through the photon
correlation functions was proposed theoretically [38, 39] and recently
realized experimentally [40]. However, a significant obstacle for the spin
dynamics investigation and spin-photon entanglement generation remains due to
the splitting or birefringence of the micropillar cavity modes [41, 42, 43,
44, 45, 10, 17, 21] despite multiple attempts to reduce it [41, 46, 47, 48,
49].

Figure 1: (a) Sketch of the system: quantum dot (black pyramid) with an
electron with a spin (red arrow) in a micropillar cavity. Short optical pump
pulses (dark blue) transform inside the cavity into exponentially decaying
ones (lilac) and excite trion in the quantum dot. The photons emitted after
the trion recombination (cyan) are polarization entangled with each other and
the electron spin. (b) The spectrum of $H$ (blue) and $V$ (red) cavity modes
and the quantum dot resonance (magenta). (c) Energy levels of the quantum dot
(black and magenta lines) and transitions between them involving $H$ (blue
arrows) and $V$ (red arrows) photons.

In this work, we theoretically address this issue. After formulating the model
in Sec. II, we show proof of principle numerical calculation in Sec. III and
then describe optical excitation of the quantum dot in Sec. IV, spin-photon
entanglement generation in Sec. V, and multiphoton entanglement in Sec. VI. In
conclusion (Sec. VII), we stress that there is a ‘‘sweet spot’’ at which trion
can be efficiently excited and complete spin-photon entanglement can be
realized for arbitrary strength of the micropillar cavity birefringence.

##  II Model

We describe a micropillar cavity with an electron-charged quantum dot [Fig.
1(a)] by the following general Hamiltonian [18, 38, 39]:

${\cal H}=\sum_{H,V}{\cal H}^{\text{(cav)}}_{H,V}+\sum_{\pm}\quantity[{\cal
H}^{\text{(trion)}}_{\pm}+{\cal H}^{\text{(int)}}_{\pm}+{\cal
H}^{\text{(exc)}}_{\pm}].$ (1)

Let us describe each of these terms. The first one reads ${\cal
H}^{\text{(cav)}}_{H,V}=\omega_{H,V}c^{\dagger}_{H,V}c_{H,V}$. It describes
the energy of photons in two linearly polarized cavity modes, H and V, with
the corresponding annihilation operators $c_{H,V}$ and frequencies
$\omega_{H,V}$ (hereafter, we set $\hbar=1$). It is convenient to represent
these frequencies as $\omega_{H,V}=\omega_{c}\pm\Delta$, where $\omega_{c}$ is
the central frequency and $2\Delta$ is the splitting between cavity modes, as
shown in Fig. 1(b).

Absorption of a photon by a quantum dot with an electron leads to the
formation of a negatively charged trion. Its energy is given by ${\cal
H}^{\text{(trion)}}_{\pm}=\omega_{0}a^{\dagger}_{\pm 3/2}a_{\pm 3/2}$, where
$\omega_{0}$ is the trion resonance frequency and $a_{\pm 3/2}$ are the
annihilation operators of trions with the heavy hole spin $\pm 3/2$, which
correspond to the trion pseudospin $J_{z}=\pm 1/2$.

The third term in Eq. (1)

${\cal H}^{\text{(int)}}_{\pm}=g\quantity[a^{\dagger}_{\pm 3/2}c_{\pm}a_{\pm
1/2}+a^{\dagger}_{\pm 1/2}c_{\pm}^{\dagger}a_{\pm 3/2}],$ (2)

describes light-matter coupling with the strength $g$ in a circular basis with

$c_{\pm}=\frac{c_{H}\pm{i}c_{V}}{\sqrt{2}}.$ (3)

The operators $a_{\pm 1/2}$ destroy an electron in the ground state with the
spin $S_{z}=\pm 1/2$. The selection rules described by this Hamiltonian can
also be rewritten in the basis of H and V modes for the electron and trion
states $S_{x},J_{x}=\pm 1/2$. The corresponding transitions are shown in Fig.
1(c).

The last term of the general Hamiltonian ${\cal H}^{\text{(exc)}}_{\pm}={\cal
E}_{\pm}^{*}(t)c_{\pm}+{\cal E}_{\pm}(t)c^{\dagger}_{\pm}$ stands for the
resonant cavity excitation by external light pulse described by the amplitudes
${\cal E}_{\pm}(t)$ of two circular components. A train of pulses is needed to
generate many entangled photons. But if the intervals between the pulses are
long enough (longer than the trion recombination time $\gamma^{-1}$), one can
separately consider each pulse. We assume the length of the pulses to be so
short that their spectrum covers both cavity modes, and take ${\cal
E}_{\pm}(t)={\cal E}_{\pm}\delta(t)$ in what follows. Note that in the
opposite limit of the pump pulses longer than the inverse cavity decay rate,
the birefringence does not play a significant role; it only leads to the
polarization-dependent photon injection rate.

We consider only one, the most important incoherent process, the escape of
photons from the cavity with the polarization-independent rate $2\varkappa$.
It is assumed to be larger than the coupling strength, $\varkappa\gg g$. This
is the so-called fast cavity regime, which is realized in most structures.
Typically, in the experiments, the decay rates of the two cavity modes are
similar, so we leave the study of the effect of their difference for future
works. We neglect the incoherent trion decay, assuming it to be slower than
$\gamma$. For this reason, the trion state is represented by a narrow
Lorentzian in Fig. 1(b). The transmission spectrum of the system calculated
within the input-output approach is shown in Appendix A. In experiments, the
reflection geometry is commonly employed. Moving forward, our focus is on
studying spin entanglement with the photons emitted from the microcavity, with
the emission direction being inconsequential for the theoretical analysis.

The dynamics of the system can be described on equal footing by the Lindblad
equation or by the method of quantum trajectories [50, 51]. In the latter
approach, the non-Hermitian Hamiltonian should be considered

${\cal H}_{\rm nh}={\cal H}-i\varkappa\sum_{\pm}c_{\pm}^{\dagger}c_{\pm}$ (4)

and the jump operators are $C_{\pm}=\sqrt{2\varkappa}c_{\pm}$. Averaging over
individual quantum trajectories gives the expectation values of physical
observables in the system.

##  III Numerical calculation

To illustrate the following analytical theory, we present a numerical
calculation for a particular set of parameters. In Fig. 2, we consider a
$\sigma^{+}$ circularly polarized excitation and initial electron spin
polarization along the $x$ axis, $S_{x}=1/2$. The other parameters are given
in the figure caption. For the simulations, we use the quantum Monte-Carlo
trajectory method [50] realized in QuTiP python framework [52] and perform
averaging over 300 individual quantum trajectories.

Panel (a) shows the numbers of $\sigma^{+}$ and $\sigma^{-}$ polarized photons
in the cavity mode as functions of time. The beatings with the period
$\pi/\Delta$ are related to the splitting between linearly polarized cavity
modes. After the initial injection of photons, their number exponentially
decays due to the escape of photons from the cavity.

Figure 2: (a) Numbers of $\sigma^{\pm}$ photons in the cavity mode (open
symbols). Solid green and blue lines are calculated after Eqs. (3) and (5).
The black dashed line is calculated after Eq. (13). (b) The population of the
trion state and electron-photon concurrence as functions of time. Magenta and
red dashed lines are calculated after Eqs. (9) and (15), respectively. The
parameters of the calculation are $g=0.15\varkappa$, $\Delta=\varkappa$,
$\omega_{0}=\omega_{c}$, ${\cal E}_{-}=0$, ${\cal E}_{+}=\pi\varkappa/g$ (a
finite pulse duration of $0.003/\varkappa$ is used).

During this time, a trion gets excited in the quantum dot. Its population
$N_{\rm
tr}=\expectationvalue{a_{+3/2}^{\dagger}a_{+3/2}+a_{-3/2}^{\dagger}a_{-3/2}}$
is shown in Fig. 2(b) by the red circles. It starts from zero and then quickly
($t\sim 1/\varkappa$) increases almost to one, corresponding to the $\pi$
pulse. Then, the trion starts to decay slowly, giving rise to a small
population of the cavity mode at the long time scales. The black circles show
the concurrence between a photon emitted from the cavity at the given time and
the spin of the electron left in the quantum dot. It is close to one, which
demonstrates complete spin-photon entanglement.

The analytical expressions for these quantities are given below. However, the
main results can already be seen from the figure: (i) Despite splitting of the
cavity modes and their detuning from the trion resonance, a complete
population inversion ($\pi$ pulse) is possible. (ii) Even strong birefringence
of the microcavity does not spoil spin photon entanglement so that concurrence
reaches one. Below, we describe both results analytically and in more detail.

##  IV Trion excitation

Light quantization does not play a role in the weak coupling regime
($g\ll\varkappa$). Then, the average amplitudes of the linearly polarized
light components in the cavity obey the classical equations

$\expectationvalue{c_{H,V}(t)}=\expectationvalue{c_{H,V}(0)}e^{-i\omega_{H,V}t-\varkappa
t},$ (5)

with the initial conditions determined by the pump pulse:
$\expectationvalue{c_{H}(0)}=\frac{i}{\sqrt{2}}({\cal E}_{+}+{\cal E}_{-})$,
$\expectationvalue{c_{V}(0)}=\frac{1}{\sqrt{2}}({\cal E}_{+}-{\cal E}_{-})$.
These expressions are used to calculate the blue and green solid lines in Fig.
2(a). The total number of photons decays at the rate of $2\varkappa$.

The trion decay can be neglected during the photon lifetime in the cavity so
that the trion excitation can be described by the Schrodinger equation [53].
The quantum dot state is characterized by the four quantum amplitudes
$\psi_{\pm 1/2}(t)$ and $\psi_{\pm 3/2}(t)$ of the corresponding electron and
trion spin states. From the Hamiltonian (1), one can see that the Schrodinger
equation for them reads

${i}\dot{\psi}_{\pm 3/2}=\omega_{0}\psi_{\pm
3/2}+g\expectationvalue{c_{\pm}(t)}\psi_{\pm 1/2},$ (6a) ${i}\dot{\psi}_{\pm
1/2}=g\expectationvalue{c_{\pm}^{\dagger}(t)}\psi_{\pm 3/2},$ (6b)

where the dot denotes the time derivative. One can see that the excitations of
the trions with $J_{z}=\pm 1/2$ are independent of each other. The solution of
these equations for $t\to\infty$ gives the trion state as a function of the
initial electron state, amplitude, and polarization of the exciting light.

In the general case, an analytical solution of these equations is complicated.
We solve them numerically for the different splittings of the cavity modes,
$\Delta$, and different trion resonance frequencies, $\omega_{0}$. For each
$\Delta$ and $\omega_{0}$, we find the maximum (as a function of the pump
amplitudes ${\cal E}_{+}$ and ${\cal E}_{-}$) occupancy of the trion state
$N_{\rm tr}^{\rm max}$ for the initial electron in-plane state. This gives the
average probability of excitation of the two trion states from the
corresponding electron states.

Figure 3: (a) Maximal possible trion population as a function of the trion
resonance frequency and splitting between cavity modes for the initial in-
plane electron spin orientation obtained from the numerical solution of Eqs.
(6). The cyan lines show the levels of $N_{\rm tr}^{\rm max}=0.9$. (b)
Electron-photon concurrence calculated after Eq. (20) (c) Three-tangle of the
electron and two photons in the cluster-like state calculated after Eq. (22).

The result is shown in Fig. 3(a). Surprisingly, the trion can be excited with
the probability close to one for the most range of the parameters $\Delta$ and
$\omega_{0}$. The cyan curves show the levels of $N_{\rm tr}^{\rm max}=0.9$.
Several limiting cases can explain this result.

First, if the splitting between the modes is zero, then the quantum dot
represents a pair of two-level systems. If the quantum dot frequency coincides
with the cavity frequency, $\omega_{0}=\omega_{c}$, the populations of the
trion states experience Rabi oscillations as functions of the pump amplitudes
[53]:

$\expectationvalue{a_{\pm 3/2}^{\dagger}a_{\pm 3/2}}=\sin^{2}\left(g|{\cal
E}_{\pm}|/\varkappa\right).$ (7)

So the $\pi$ pulse is realized, for example, by a linearly polarized
excitation with ${\cal E}_{\pm}=\pi\varkappa/(2g)$. If the trion resonance is
detuned from the cavity frequency, one can show that the maximum trion
population is given by

$N_{\rm tr}^{\rm
max}=\frac{1}{2}\left[1+\sech\left(\frac{\pi(\omega_{0}-\omega_{c})}{2\varkappa}\right)\right].$
(8)

This expression describes the trion population along the line $\Delta=0$ in
Fig. 3(a). For small trion detunings, it reaches one, and for large detunings,
it tends to one-half. The trion population for the strong pump pulses
experiences the Rabi oscillations from zero to one. Still, when the light
amplitude inside the cavity decays according to Eq. (5), the amplitude of the
oscillations decreases, so a finite trion population $N_{\rm tr}<1$ is left in
the QD after the escape of all photons from the cavity.

Another simple limit is realized for $\omega-\omega_{c}=\pm\Delta$ [diagonals
of Fig. 3(a)]. The trion resonance coincides, in this case, with one of the
two cavity modes. So, the pump polarization along this mode can realize the
complete population inversion. Now, it becomes natural that in the whole
region of $|\omega-\omega_{c}|<|\Delta|$, the $\pi$ pulses can be realized for
some amplitudes and polarizations of the pump pulse.

The most important case is the trion resonance exactly between the cavity
modes, $\omega_{0}=\omega_{c}$. Due to the detuning from both modes, one could
expect only a limited maximum possible trion population. However, we find that
if the pump amplitudes ${\cal E}_{+}$ and ${\cal E}_{-}$ are shifted in phase
by $\pi/2$ (${\cal E}_{+}=|{\cal E}_{+}|$, ${\cal E}_{-}=-{i}|{\cal E}_{-}|$),
the solution of Eq. (6) reads

$\psi_{\pm 3/2}(t)=\psi_{\pm 1/2}(0)\sin(g\tilde{\cal
E}_{\pm}(t)\over\Delta^{2}+\varkappa^{2}),$ (9a) where $\tilde{\cal
E}_{\pm}(t)=\absolutevalue{{\cal E}_{\pm}}\varkappa\mp\absolutevalue{{\cal
E}_{\mp}}\Delta+\left[(\pm\absolutevalue{{\cal
E}_{\mp}}\Delta-\absolutevalue{{\cal E}_{\pm}}\varkappa)\cos(\Delta
t)\right.\\\ \left.+(\absolutevalue{{\cal E}_{\pm}}\Delta+\absolutevalue{{\cal
E}_{\mp}}\varkappa)\sin(\Delta t)\right]e^{-\varkappa t}.$ (9b)

From this, one can see the conditions for complete trion excitation ($\pi$
pulses):

$\sin^{2}\quantity[g\frac{\absolutevalue{{\cal
E}_{\pm}}\varkappa\mp\absolutevalue{{\cal
E}_{\mp}}\Delta}{\Delta^{2}+\varkappa^{2}}]=1.$ (10)

This proves that $N_{\rm tr}^{\rm max}=1$ along the vertical line in the
middle of Fig. 3(a), and this will be important for the spin-photon
entanglement in the next section.

For example, at $\Delta=\varkappa$, Eq. (10) is fulfilled for circularly
polarized light with ${\cal E}_{-}=0$, ${\cal E}_{+}=\pi\varkappa/g$. This
particular case was used for the numerical calculations in Fig. 2.

Finally, we note that the $\pi$ pulses completely transfer populations of
electron spin-up and spin-down states to the trion spin-up and -down states,
respectively. This means that after excitation, the trion spin $J_{z}$ equals
the initial electron spin $S_{z}$. Also, for the electron in-plane spin
orientation, the trion spin will also be completely polarized in the $(xy)$
plane.

##  V Spin-photon entanglement

Having described the trion excitation at the short time scales, we now analyze
the slow trion radiative recombination and the spin-photon entanglement
generation. Making use again of the weak coupling regime, we use the
Schrödinger equation with the non-Hermitian Hamiltonian (4) to describe the
trion recombination [15, 38]. We consider a trion state described by the
amplitudes $\psi_{+3/2,0}$ and $\psi_{-3/2,0}$ (the subscript $0$ indicates
the absence of photons in the cavity mode). Then the Schrödinger equation
gives us

${i}\dot{\psi}_{\pm 1/2,\pm}=\left(\omega_{c}-{i}\varkappa\right)\psi_{\pm
1/2,\pm}+\Delta\psi_{\pm 1/2,\mp}+g\psi_{\pm 3/2,0},$ (11a)
${i}\dot{\psi}_{\pm 1/2,\mp}=\left(\omega_{c}-{i}\varkappa\right)\psi_{\pm
1/2,\mp}+\Delta\psi_{\pm 1/2,\pm},$ (11b)

where the first subscript $\pm 1/2$ in the wave functions refers to the
electron spin, and the second subscript $\pm$ denotes a single $\sigma^{\pm}$
photon in the cavity.

The solution of these equations reads

$\psi_{\pm 1/2,\pm}=\frac{g(\omega_{0}-\omega_{c}+{i}\varkappa)\psi_{\pm
3/2,0}}{(\omega_{0}-\omega_{c}+{i}\varkappa)^{2}-\Delta^{2}},$ (12a)
$\psi_{\pm 1/2,\mp}=\frac{g\Delta\psi_{\pm
3/2,0}}{(\omega_{0}-\omega_{c}+{i}\varkappa)^{2}-\Delta^{2}},$ (12b)

where we took into account that the trion wave functions $\psi_{\pm 3/2,0}$
have a phase factor of $\mathrm{e}^{-{i}\omega_{0}t}$. In Appendix B, we show
that the same result can be obtained from the solution of the Lindblad
equation for the density matrix of the system.

The numbers of photons in the cavity modes are given by

$\expectationvalue{c_{\pm}^{\dagger}c_{\pm}}=|\psi_{+1/2,\pm}^{2}|+|\psi_{-1/2,\pm}^{2}|,$
(13)

and this expression is shown by the black dashed line in Fig. 2(a). It allows
us to calculate the trion (amplitude) decay rate as:

$\gamma=\varkappa\frac{|\psi_{\pm 1/2,+}^{2}|+|\psi_{\pm
1/2,-}^{2}|}{|\psi_{\pm 3/2,0}^{2}|},$ (14)

because it equals the photon escape rate. Thus we obtain

$\gamma=\frac{1}{2}\sum_{H,V}\frac{g^{2}\varkappa}{(\omega_{0}-\omega_{H,V})^{2}+\varkappa^{2}},$
(15)

which is generally of the order of $g^{2}/\varkappa$ or smaller. The slow
trion decay $N_{\rm tr}=\exp(-2\gamma t)$ is shown by the red dashed line in
Fig. 2(b).

Because of the assumption of equal decay rates of the two cavity photon modes,
the trion decay rate does not depend on the trion spin. So, the wave function
of an electron in the quantum dot and an emitted photon is also given by Eqs.
(12). It can be written (up to a phase factor) as

$\Psi=\psi_{+3/2,0}\ket{\uparrow,\tilde{+}}+\psi_{-3/2,0}\ket{\downarrow,\tilde{-}},$
(16)

where $\uparrow$ and $\downarrow$ denote the electron spin state, and the
photon states are

$\ket{\tilde{\pm}}=\cos\alpha\ket{\pm}-{i}\sin\alpha\mathrm{e}^{{i}\beta}\ket{\mp}$
(17)

with $\ket{\pm}$ denoting $\sigma^{\pm}$ photons and

$\tan\alpha=\frac{\Delta}{\sqrt{(\omega_{0}-\omega_{c})^{2}+\varkappa^{2}}},$
(18a) $\tan\beta=\frac{\omega_{0}-\omega_{c}}{\varkappa}.$ (18b)

Both angles $\alpha$ and $\beta$ are in the range $\left(-\pi/2,\pi/2\right)$
and depend on the trion resonance frequency and cavity mode splitting.

We stress that the electron-photon state is pure. Moreover, it has the same
form as for the optically isotropic cavity [32] (with $\Delta=0$) but with the
rotated photon states (17). So, we can easily calculate spin-photon
concurrence

$\mathcal{C}=2\sqrt{J_{x}^{2}+J_{y}^{2}}\mathcal{F}_{c}$ (19)

with

$\mathcal{F}_{c}=\sqrt{1-\sin^{2}(2\alpha)\sin^{2}\beta}.$ (20)

One can see that the spin-photon entanglement requires in-plane trion spin
polarization as usual [32]. The cavity birefringence is accounted for by the
factor $\mathcal{F}_{c}\leq 1$, shown in Fig. 3(b) as a function of the system
parameters. It is related to the scalar product of the two modified photon
states as
$\mathcal{F}_{c}^{2}=1-|\left<\tilde{+}\middle|\tilde{-}\right>^{2}|$, so the
largest concurrence requires the minor overlap of the emitted photon states.
The overlap vanishes in the absence of cavity mode splitting, $\Delta=0$, and
one arrives at $\mathcal{C}=2\sqrt{J_{x}^{2}+J_{y}^{2}}$ [39].

But strikingly, at $\omega_{0}=\omega_{c}$ we obtain $\beta=0$,
$\left<\tilde{+}\middle|\tilde{-}\right>=0$, and $\mathcal{F}_{c}=1$ also. So,
despite the cavity birefringence, the perfect spin-photon entanglement is
achieved when the trion resonance is located exactly between the two cavity
modes. Qualitatively, perfect spin-photon entanglement requires zero overlap
between two possible emitted photon states $\ket{\tilde{\pm}}$, which is
possible for the equal probabilities of photon emission to H and V modes only.
From Eq. (15), one can see that this corresponds exactly to
$\omega_{0}=\omega_{c}$, which was used already in Ref. 36.

We note that complete trion excitation and in-plane spin polarization are also
realized in this case, as shown in the previous section. For this reason, we
call the condition $\omega_{0}=\omega_{c}$ a ‘‘sweet spot’’ for the
entanglement generation in the birefringent microcavities. This is the main
result of our work.

##  VI Multiphoton entanglement

Spin-photon entanglement is useful by itself and indispensable for generating
many-body entangled photon cluster states. The form of the electron-photon
wave function (16) shows that the protocol of Ref. 32 can be successfully
applied to the birefringent micropillar cavities. But in the result, the
circularly polarized $\sigma^{\pm}$ photon states should be replaced with
$\ket{\tilde{\pm}}$.

For example, an electron spin rotation by $\pi/2$ between the two excitation
pulses leads to the entangled state of an electron and two photons

$\Psi_{3}=\frac{\ket{\uparrow\tilde{+}\tilde{+}}+\ket{\downarrow\tilde{-}\tilde{+}}+\ket{\downarrow\tilde{-}\tilde{-}}-\ket{\uparrow\tilde{+}\tilde{-}}}{2}.$
(21)

The straightforward calculation shows that the three-tangle [54] of this state
is given by

$\tau=\mathcal{F}_{c}^{4}.$ (22)

It is shown in Fig. 3(c). It has more narrow regions with high entanglement
than the concurrence in panel (b) but generally has the same form. In
particular, it reaches one at the sweet spot $\omega_{0}=\omega_{c}$.

The three tangle does not reach zero for any system parameters. This suggests
that the cluster-like states generated by the birefringent cavity are always
genuine entangled. In particular, localizable entanglement [55] between any
two emitted photons in the cluster state equals $\mathcal{F}_{c}$. In this
sense, the entanglement length of the multiphoton state is infinite for any
system parameters.

To quantify the degree of entanglement, we introduce the orthogonal photon
states

$\ket{\accentset{\approx}{\pm}}=\cos(\theta/2)\ket{\pm}-{i}\sin(\theta/2)\ket{\mp},$
(23)

where the angle $\theta$ is defined by

$\tan\theta=\frac{2\varkappa\Delta}{(\omega_{0}-\omega_{c})^{2}+\varkappa^{2}-\Delta^{2}}.$
(24)

These states are the closest orthogonal states to $\ket{\tilde{\pm}}$ (see
Appendix C), so they determine the optimal basis for the measurement of
multiphoton entanglement.

We denote the cluster-like states with $n$ photons produced by the
birefringent cavity as $\Psi_{n}$, and the ideal cluster states in the basis
of $\ket{\accentset{\approx}{\pm}}$ states as $\Psi_{n}^{(0)}$. Then, the
fidelity between these states is

$\left|\left<\Psi_{n}\middle|\Psi_{n}^{(0)}\right>\right|^{2}=\left(\frac{1+\mathcal{F}_{c}}{2}\right)^{n}.$
(25)

In particular, for $n=1$, the fidelity equals $(1+\mathcal{F}_{c})/2$ in
agreement with the general relations between concurrence and fidelity [56,
57]. Eq. (25) shows that the fidelity exponentially decreases with the number
of photons, except for the sweet spot, where the maximally entangled
multiphoton states are generated.

##  VII Conclusion

We have developed a microscopic theory of the optical resonant excitation of a
charged quantum dot in a birefringent micropillar cavity and its radiative
recombination. The results show that a sweet spot exists for the system when
the frequency of the trion resonance is placed exactly between the two cavity
modes. At this point, even for the large cavity modes splitting, a complete
trion excitation ($\pi$ pulse) and in-plane spin polarization are possible, as
well as the complete spin-photon entanglement after trion recombination. It
shows the way to increase the fidelity of the present spin-photon interfaces.

###### Acknowledgements.

We thank A. A. Toropov for fruitful discussions and the Foundation for the
Advancement of Theoretical Physics and Mathematics ‘‘BASIS.’’ The derivation
of the analytical expressions for the spin-photon entanglement by D.S.S was
supported by the Russian Science Foundation Grant No. 21-72-10035.

##  Appendix A Transmission spectra

Here, we study the transmission of the continuous wave coherent light with
frequency $\omega$ through the birefringent spin-photon interface. We modify
the input-output relations [58] for the case of the asymmetric cavity using
the Hamiltonian (1)

$\begin{cases}\dot{\hat{c}}_{+}=-i\omega_{c}\hat{c}_{+}-i\Delta\hat{c}_{-}-\varkappa\hat{c}_{+}-ig\hat{a}_{+1/2}^{\dagger}\hat{a}_{3/2}+\sqrt{\varkappa}\hat{c}^{\rm
in}_{+}\\\
\dot{\hat{c}}_{-}=-i\omega_{c}\hat{c}_{-}-i\Delta\hat{c}_{+}-\varkappa\hat{c}_{-}+\sqrt{\varkappa}\hat{c}^{\rm
in}_{-}\\\
\dot{\hat{a}}_{+3/2}=-i\omega_{0}\hat{a}_{+3/2}-ig\hat{c}_{+}\hat{a}_{+1/2}\\\
\dot{\hat{a}}_{+1/2}=-ig\hat{c}^{\dagger}_{+}\hat{a}_{+3/2}\\\
\hat{c}_{+}^{\rm out}=\sqrt{\varkappa}\hat{c}_{+}\\\ \hat{c}_{-}^{\rm
out}=\sqrt{\varkappa}\hat{c}_{-}\end{cases}.$ (26)

Here $\hat{c}^{\rm in,out}_{\pm}$ are input and output $\sigma^{\pm}$ light
field operators, which are proportional to $\mathrm{e}^{-{i}\omega t}$, and we
assume the electron and trion spin-down states to be not populated. These
equations can be solved in the first order in the amplitude of the incident
light by replacing $a_{+1/2}^{\dagger}a_{+1/2}$ and
$a_{+3/2}^{\dagger}a_{+1/2}$ with one and zero, respectively [15].

Without the cavity mode splitting, $\Delta=0$, we arrive at the previously
established results [15, 18, 39]: Transmission of $\sigma^{-}$ polarized light
is given by

$t_{0}=\frac{\hat{c}_{-}^{\rm out}}{\hat{c}_{-}^{\rm
in}}=\frac{i\varkappa}{\omega-\omega_{c}+i\varkappa},$ (27)

which corresponds to the transmission when the optical selection rules forbid
photon interaction with the QD, while for the $\sigma^{+}$ light polarization,
we have

$t_{1}=\frac{\hat{c}_{+}^{\rm out}}{\hat{c}_{+}^{\rm
in}}=\frac{i\varkappa}{\omega-\omega_{c}+i\varkappa-\frac{g^{2}}{\omega-\omega_{0}}},$
(28)

relevant for the case when the QD can absorb the circularly polarized photons.

In the birefringent cavite case, $\Delta\neq 0$, from the solution of Eqs.
(26) we obtain

$t_{++}=\frac{t_{1}}{1+\Delta^{2}t_{1}t_{0}/\varkappa^{2}},\quad
t_{--}=\frac{t_{0}}{1+\Delta^{2}t_{1}t_{0}/\varkappa^{2}},\\\
t_{-+}=-\frac{{i}\Delta}{\varkappa}t_{++}t_{0},\qquad
t_{+-}=-\frac{i\Delta}{\varkappa}t_{--}t_{1},$ (29)

where $t_{\alpha\beta}=\hat{c}_{\beta}^{\rm out}/\hat{c}_{\alpha}^{\rm in}$
for $\alpha,\beta=\pm$. As one can see, the mode splitting results in the
light polarization conversion. We note that these expressions are valid in
both weak and strong coupling regimes.

Figure 4: Intensity transmission spectrum for the unpolarized light calculated
after Eq. (30) for the parameters $\Delta=3\varkappa$,
$\omega_{0}-\omega_{c}=-\varkappa$, $g=0.15\varkappa$.

The intensity transmission coefficient for unpolarized light is

$T=\frac{1}{2}\left(\left|t_{++}^{2}\right|+\left|t_{+-}^{2}\right|+\left|t_{-+}^{2}\right|+\left|t_{--}^{2}\right|\right).$
(30)

It does not depend on the electron spin. The same expression describes the
transmission of polarized light for unpolarized electron spin. Its frequency
dependence is shown in Fig. 4. One can see that in the fast cavity regime
$\varkappa\gg g$, the transmission has two Lorentzians peaks at the cavity
eigenfrequencies $\omega_{c}\pm\Delta$ and an asymmetric Fano-like peak at the
trion resonance frequency $\omega_{0}$.

##  Appendix B Trion recombination after master equation

Let us describe trion recombination studied in Sec. V within a different
approach similar to Ref. 59. The Lindblad equation for the density matrix of
the system $\rho$ accounting for the photon escape from the cavity can be
written in the form

$\derivative{\rho}{t}=-i\quantity({\cal H}_{\rm nh}\rho-\rho{\cal H}_{\rm
nh}^{\dagger})-\sum_{\pm}C_{\pm}\rho C_{\pm}^{\dagger},$ (31)

with the non-Hermitian Hamiltonian and jump operators introduced in Sec. II.

In the fast cavity regime, the density matrix elements between trion and
electron with photon states are determined by the trion density matrix only:

$\dot{\rho}_{\Uparrow,\uparrow+}=-[i(\omega_{0}-\omega_{c})+\varkappa]\rho_{\Uparrow,\uparrow+}+\\\
+i\Delta\rho_{\Uparrow,\uparrow-}+ig\rho_{\Uparrow,\Uparrow},$ (32a)
$\dot{\rho}_{\Uparrow,\uparrow-}=-[i(\omega_{0}-\omega_{c})+\varkappa]\rho_{\Uparrow,\uparrow-}+i\Delta\rho_{\Uparrow,\uparrow+}.$
(32b)

Equations for $\rho_{\Uparrow,\downarrow-}$ and $\rho_{\Uparrow,\downarrow+}$
can be obtained by replacing in these equations in the second subscript
$\uparrow$, $\Uparrow$, and $\pm$ with $\downarrow$, $\Downarrow$, and $\mp$,
respectively. Similarly, the equations for the density matrix elements
$\rho_{\Downarrow,\uparrow+}$, $\rho_{\Downarrow,\uparrow-}$,
$\rho_{\Downarrow,\downarrow-}$, and $\rho_{\Downarrow,\downarrow+}$ can be
obtained from the equations for $\rho_{\Uparrow,\uparrow+}$,
$\rho_{\Uparrow,\uparrow-}$, $\rho_{\Uparrow,\downarrow-}$, and
$\rho_{\Uparrow,\downarrow+}$ simply by replacing $\Uparrow$ with $\Downarrow$
in the first subscript. The rest of the off-diagonal density matrix elements
with the trion states follow from the Hermitian conjugation.

We solve these equations in the steady state and substitute the solution to
the equations for the $4\times 4$ density submatrix $\rho_{\rm e-ph}$ in the
basis of states $\ket{\uparrow+}$, $\ket{\uparrow-}$, $\ket{\downarrow+}$,
$\ket{\downarrow-}$. This gives

$\dot{\rho}_{\rm e-ph}=-2\varkappa\rho_{\rm e-ph}-i[{\cal
H}_{\Delta},\rho_{\rm e-ph}]+\rho_{\rm rec},$ (33)

where ${\cal
H}_{\Delta}=\Delta(\hat{c}_{-}^{\dagger}\hat{c}_{+}+\hat{c}_{+}^{\dagger}\hat{c}_{-})$
is a part of the Hamiltonian (1) proportional to the cavity modes splitting
and the last term $\rho_{\rm rec}$ depends on the trion density matrix

$\rho_{\rm rec}=\frac{2\varkappa
g^{2}}{\Delta^{2}+(\varkappa+{i}(\omega_{0}-\omega_{c}))^{2}}\times\\\
\times\matrixquantity[{\cal
A}\rho_{\Uparrow,\Uparrow}&\frac{{i}\Delta}{2\varkappa}\rho_{\Uparrow,\Uparrow}&\frac{{i}\Delta}{2\varkappa}\rho_{\Uparrow,\Downarrow}&{\cal
A}\rho_{\Uparrow,\Downarrow}\\\ -\frac{{i}\Delta}{2\varkappa}{\cal
B}\rho_{\Uparrow,\Uparrow}&0&0&-\frac{{i}\Delta}{2\varkappa}{\cal
B}\rho_{\Uparrow,\Downarrow}\\\ -\frac{{i}\Delta}{2\varkappa}{\cal
B}\rho_{\Downarrow,\Uparrow}&0&0&-\frac{{i}\Delta}{2\varkappa}{\cal
B}\rho_{\Downarrow,\Downarrow}\\\ {\cal
A}\rho_{\Downarrow,\Uparrow}&\frac{{i}\Delta}{2\varkappa}\rho_{\Downarrow,\Uparrow}&\frac{{i}\Delta}{2\varkappa}\rho_{\Downarrow,\Downarrow}&{\cal
A}\rho_{\Downarrow,\Downarrow}],$ (34)

with

${\cal
A}=\frac{\Delta^{2}+(\varkappa+{i}(\omega_{0}-\omega_{c}))^{2}}{\Delta^{2}+\varkappa^{2}+(\omega_{0}-\omega_{c})^{2}}{\cal
B}=\\\
=\frac{\Delta^{2}+\varkappa^{2}+(\omega_{0}-\omega_{c})^{2}}{\Delta^{2}+(\varkappa-{i}(\omega_{0}-\omega_{c}))^{2}}.$
(35)

Finally, we solve Eq. (33) in the steady state and normalize the result,
obtaining

$\rho_{\rm
e-ph}=\frac{(\omega_{0}-\omega_{c})^{2}+\varkappa^{2}}{(\omega_{0}-\omega_{c})^{2}+\Delta^{2}+\varkappa^{2}}\times\\\
\times\matrixquantity[\rho_{\Uparrow,\Uparrow}&{\cal
G}\rho_{\Uparrow,\Uparrow}&{\cal
G}\rho_{\Uparrow,\Downarrow}&\rho_{\Uparrow,\Downarrow}\\\ {\cal
G}^{*}\rho_{\Uparrow,\Uparrow}&\absolutevalue{\cal
G}^{2}\rho_{\Uparrow,\Uparrow}&\absolutevalue{\cal
G}^{2}\rho_{\Uparrow,\Downarrow}&{\cal G}^{*}\rho_{\Uparrow,\Downarrow}\\\
{\cal G}^{*}\rho_{\Downarrow,\Uparrow}&\absolutevalue{\cal
G}^{2}\rho_{\Downarrow,\Uparrow}&\absolutevalue{\cal
G}^{2}\rho_{\Downarrow,\Downarrow}&{\cal G}^{*}\rho_{\Downarrow,\Downarrow}\\\
\rho_{\Downarrow,\Uparrow}&{\cal G}\rho_{\Downarrow,\Uparrow}&{\cal
G}\rho_{\Downarrow,\Downarrow}&\rho_{\Downarrow,\Downarrow}],$ (36)

where ${\cal G}=\Delta/(\omega_{0}-\omega_{c}-{i}\varkappa)$. One can readily
check that this is the density matrix of the pure state given in Eq. (16).

##  Appendix C Closest orthogonal photon states

The electron-photon state (16) has the same form as for the optically
isotropic micropillar cavity except for the replacement of circular photon
states $\ket{\pm}$ with $\ket{\tilde{\pm}}$, see Eq. (17). These states are
generally not orthogonal. The Stockes parameters [60] for them are

$\xi_{1,\pm}=\mp\cos\beta\sin 2\alpha,\quad\xi_{2,\pm}=\pm\cos 2\alpha,\\\
\xi_{3,\pm}=\sin\beta\sin 2\alpha.$ (37)

The latter parameter is the same for the two states, which evidences their
overlap. Generally, by the appropriate rotation of the polarization basis of
the Poincaré sphere, one can obtain the Stokes parameters of the form

$\xi_{1^{\prime},\pm}=0,\qquad\xi_{2^{\prime},\pm}=\pm\xi,\qquad\xi_{3,\pm}=\xi_{3}$
(38)

Where the parameter $\xi=\sqrt{\xi_{1,\pm}^{2}+\xi_{2,\pm}^{2}}$ and the
polarization basis is rotated about the axis $3$ by the angle $\vartheta$
which is defined by

$\cos\vartheta=\frac{\xi_{2,+}}{\sqrt{\xi_{2,+}^{2}+\xi_{1,+}^{2}}}=\frac{1}{\sqrt{1+\cos^{2}\beta\tan^{2}(2\alpha)}}.$
(39)

The closest orthogonal states $\ket{\accentset{\approx}{\pm}}$ are defined by
the conditions
$\bra{\accentset{\approx}{\pm}}\ket{\accentset{\approx}{\mp}}=0$ and the
overlap $\bra{\accentset{\approx}{\pm}}\ket{\tilde{\pm}}$ is maximal.
Therefore they should satisfy $\xi_{1^{\prime},\pm}=0$,
$\xi_{2^{\prime},\pm}=\pm 1$, and $\xi_{3^{\prime},\pm}=0$. Such states are
given by Eq. (23).

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