Geometry of temporal chiral structures
Andres F. Ordonez Aycke Roos Pablo M. Maier David Ayuso Olga Smirnova










^1Department of Physics, Imperial College London, SW7 2BW London, United Kingdom

^2Department of Chemistry, Queen Mary University of London, E1 4NS London, United Kingdom

^3Max-Born-Institut, Max-Born-Str. 2A, 12489 Berlin, Germany




^4Technische Universität Berlin, Straße des 17. Juni 135, 10623 Berlin, Germany



§ ABSTRACT 


In non-relativistic physics the concepts of geometry and topology are usually applied to characterise spatial structures, or structures in momentum space.
We introduce the concept of temporal geometry, which encompasses the geometric and topological properties of temporal shapes, i.e. trajectories traced by a tip of a time-dependent vector in vector space.  We apply it to vectors of ultrafast electron current or induced polarization in chiral molecules. 

The central concepts of temporal geometry  – curvature and connection –  
emerge  as ubiquitous features of photoexcited, non-equilibrium, chiral electron dynamics. 

We demonstrate that curvature and connection  (i) rely on the interplay of molecular chirality and the polarization properties of light pulses, (ii) can be introduced for multiphoton processes, and (iii) control enantio-sensitive geometric observables via non-equilibrium electronic dynamics excited by tailored laser fields.
Our findings may open a way to ultrafast, topologically non-trivial, and enantio-sensitive chemical dynamics. 











Geometry of temporal chiral structures 
    Andres F. Ordonez^1,2, Aycke Roos ^3, Pablo M. Maier ^3, David Ayuso^1,2,3 and Olga Smirnova^3,4
    September 9, 2024
====================================================================================================



§ 1. INTRODUCTION

Synthetic chiral light <cit.> is an example of a locally chiral object: the Lissajous figure of its electric field vector draws a chiral three-dimensional trajectory in time at every point in space. Since its chirality is local, it can induce chiral interactions with electrons in molecules in the electric dipole approximation, promising improvement of the chiral response by orders of magnitude compared to standard methods <cit.> based on inefficient interactions of electrons with the magnetic field component of light. Recently, three-color locally chiral electromagnetic fields <cit.> have been used to harvest the electric-dipole chiral response in the microwave region <cit.>. 

If the locally chiral electromagnetic field  draws the same chiral Lissajous figure at every point in space in the interaction region,  then the field is also globally chiral. Since different frequencies of the multicolor light combine at different points in space with different phases, such global chirality is not  guaranteed a priory. 
While various set-ups can be constructed to create locally and globally chiral light <cit.>, the spatial variation of the relative phases of non-collinear multicolor fields constituting locally chiral light can also be used to shape the light's handedness in space in the near field from one-dimensional patterns of alternating handedness in space <cit.> to two-dimensional patterns forming a chiral vortex <cit.>. Such shaping of local handedness allows one to encode the handedness of a molecular medium in the far field emission patterns. 









Locally chiral temporal structures do not have to be limited to the one traced by the tip of the light polarization vector. Electronic polarization or electronic currents  induced in chiral molecules <cit.> can also trace temporal chiral structures. That is, the respective vectors  – current or induced polarization – trace a chiral trajectory in vector space as these vectors evolve in time. The chiral shape outlined by this trajectory is local because it is encoded in the temporal evolution of the vector, rather than in its spatial evolution. The specific chiral shape is defined by mutual orientations of the molecule and light's polarization vector: each orientation will give rise to a different temporal chiral shape. Thus, the temporally chiral shape is defined in the configuration space of molecular orientations.  If several such  mutual orientations are included into the measured response, such as e.g. photoionization from the current carrying superposition of states, the set of different temporal shapes corresponding to the underlying orientations may also combine into a non-trivial global structure.   


This poses several pertinent questions: (i) how is the temporal chiral geometry connected for two different orientations, 
and (ii) how does this connection quantify a global response for a full set of orientations? Here, we address these questions  by identifying the  connection and curvature associated with the temporal geometry of attosecond electronic response.


The realization of this program requires the implementation of geometric/topological concepts into the ultrafast electronic response in chiral molecules.

While topological aspects of nuclear dynamics at conical intersections in molecules are well-known <cit.>, 
the connection between topology and chirality in the light-induced electronic 
response of gas phase chiral molecules 
is an emergent topic <cit.>. In these pioneering works, the respective configuration or parameter spaces range from inverse lattice vectors (in case of periodic chiral arrangement of atoms)<cit.> to parameters of electromagnetic waves triggering the response <cit.> to propagation vectors of non-linear response <cit.>.  

The desire to invoke the geometric concepts  such as curvature, connection, and geometric phase is hardly surprising as it often enables one to identify novel properties of matter. Examples range from the emergence of the Fermi anti-symmetrization principle from the topology of configuration space <cit.>, to  topological materials <cit.>, anyons and topological quantum computing <cit.>, twisted light <cit.>, topological photonics and topological lasers <cit.>.

The topological aspects of the electronic response have so far been mainly harvested in condensed matter systems 
and have yet to find their room in the gas and liquid phases. 
Since topology has an attractive property of robustness to external perturbations and noise, one may be able to create efficient and robust new enantio-sensitive observables by combining chirality and topology in the ultrafast electronic and optical response of chiral gases and liquids. Relevant experimental imperfections in gas phase experiments involve fluctuations of laser parameters, sensitivity and instrument response of photoelectron detectors, challenges in sample preparation, dilute samples and small or fluctuating enantiomeric excess in a sample. In the combination of efficiency and robustness that we are aiming to achieve, the efficiency comes from driving an ultrafast non-linear response enabling chiral detection via electric-dipole interactions <cit.>, whereas robustness comes from topological concepts represented locally via curvature and connection in relevant configuration or parameter spaces. 







 
Among the first steps in this direction <cit.>
is a new manifestation of the Berry curvature in the photoionization of chiral molecules <cit.>.  Surprisingly, it enables a new class of extremely efficient enantio-sensitive photoionization observables <cit.>, which rely on excitation of ultrafast electronic or vibronic currents in chiral molecules, and serve as 
unique messengers of charge-directed reactivity <cit.>.  One such observable predicts and quantifies molecular orientation dichroism in photoionization (PI-MOCD) <cit.>.  In this case, the  curvature emerges due to the geometry of the photoelectron continuum states.

However, the Berry curvature derived in Ref. <cit.> is limited to adiabatic evolutions in the parameter space and one-photon ionization from a state with a stationary current.  

Here we generalize our approach to extend it to (i) multiphoton processes, (ii) excitations in bound
states of chiral molecules and (iii) non-adiabatic evolution. The first enables the opportunity to use tailored
fields, the second opens important applications in photoexcited non-equilibrium chiral dynamics, the third
removes the restriction to adiabatic evolutions of molecular degrees of freedom. We identify and describe new enantio-sensitive observables enabled by these genralizations. As a corollary of our approach, we
establish the geometric origin of the photo-excitation circular dichroism (PXCD)<cit.>  and its detection
using photoionization by linearly polarized light (PXECD)<cit.> .


Our findings establish geometrical concepts, such as the  curvature [Since the term Berry curvature is reserved for adiabatic evolution in parameter space, we shall use generic notation curvature and connection for the respective geometric concepts.] and the geometric phase, as the key quantities underlying the temporally chiral electronic response. It means that (i) there is a well-defined  functional dependence of the response on these quantities, (ii) the understanding of these quantities leads to a deeply illuminating understanding of the response, (iii) those quantities can be used as effective means of detection of molecular chirality. This may open a way to generate a topologically non-trivial electronic response in photoexcited bound states of chiral molecules via tailored light fields. 


 
 
 
 




The paper is organized as follows. In Section 2 we outline the concept of temporal geometry in the interaction of molecules with laser pulses and present  expressions for connection, curvature and vectorial and scalar observables relying on the curvature. 
 In Section 3 we use  these expressions  to provide a new general (i.e. using the connection and curvature  from the outset) derivation of the geometric effect – molecular orientation by photoionization (PI-MOCD)<cit.>. We show that the direction of the curvature defines molecular orientation and is a direct experimental observable. 
This approach shows how geometric quantities may appear in multiphoton processes, verifies our new method by comparison with the earlier results <cit.> and  
establishes the path that we follow to derive new results identifying the curvature in bound states in Section 4 and curvature in "mixed" bound and continuum states Section 4. 
Last but not least, the efficient enantio-sensitive observables generated by temporal geometry, such as enantio-sensitive molecular orientation,  can be controlled via excitation of non-equilibrium dynamics. In Sections 2, 3, 6, using the chiral molecule propylene oxide as an example, we demonstrate  that simple polarization control of laser pulses allows one to switch the direction of curvature from its bound to its continuum value and thus switch the direction of enantio-sensitive orientation in PI-MOCD. Both the demonstration of the control and its fundamental understanding are new.
Superscripts M and L denote that respective vectors belong to molecular or laboratory frame, respectively. Vectors contributing to scalar products will not be marked, because the pair of such vectors can be associated with any frame.








§ 2. TEMPORAL GEOMETRY IN CHIRAL MOLECULES


Light fields E⃗(t) interacting with molecules primarily couple to electrons and trigger their attosecond response. In the laboratory frame [To define the laboratory frame in the electric dipole approximation, we can not rely on the propagation vector of the laser field. Instead, we can define the laboratory frame using the two orthogonal polarization components of the laser field together with their cross product, necessitating circularly or elliptically polarized fields ] circularly polarized light interacts with randomly oriented molecules (Fig. <ref>(a)). Since the strength of such interaction depends on the orientation of a molecule with respect to the laser field, it inevitably couples electronic and rotational degrees of freedom. It means that the electronic current excited in  molecules oriented differently will actually be different.  It is easier to visualize it by switching to the molecular frame, in which the laser field would have all possible orientations (Fig. <ref>(b)). The coupling term now explicitly depends on the Euler angles, which characterize the orientation of the laser field in the molecular frame. 



 
 




To illustrate the geometric origin of such coupling, it is convenient to associate  the normal to the light polarization plane in the molecular frame with a unit radius vector on a sphere characterized by angles (θ, ϕ).  The polarization plane is tangent to this sphere. Therefore, the light's polarization vector 
in the molecular frame E⃗(t,θ, ϕ, α) depends on three angles, where 
 
α describes the orientation of the polarization ellipse [In case of circularly polarized field the angle α is redundant. We omit α in the following.] in the tangent plane (Fig. 1).  
Any change in the molecular orientation will "transport" the polarization vector along the surface of the sphere, while the electronic wave-function describing excited electrons will depend on this transport. At any point on the sphere, the laser field excites a geometrically different temporally chiral electronic response, 
because the geometry of light polarization vectors as seen by the molecule (Fig. 1(b,c)) changes from one point on the sphere to another. Is there a global geometric property associated with these chiral currents for different molecular orientations?









Solving the time-dependent 
 Schrödinger equation (TDSE) locally for each  orientation characterized by a point ρ_0 = (θ_0, ϕ_0) ^3 on the sphere of Euler angles in the electric dipole approximation:  

    id/dtψ_el^0(r⃗,ρ_0,t)=[H_el+r⃗·E⃗(ρ_0,t)]ψ_el^0(r⃗,ρ_0,t),
 
we obtain an electron   
 wave-function ψ_el^0(t,E⃗(ρ_0),r⃗) [For simplicity we consider a single electron problem.], which is "agnostic" about other light field geometries.
 The temporally chiral shapes are generated by induced polarization P⃗(ρ,t) or current j⃗(ρ,t)=-∂P⃗/∂ t vectors: P⃗(ρ,t)≡ -⟨ψ_el^0(t,E⃗(ρ),r⃗)|r̂|ψ_el^0(t,E⃗(ρ),r⃗)⟩.
Thus the  question about the connection between the two currents at two different orientations reduces to the question about the connection between the two wave functions for two different ρ. Can these "local" solutions be connected in any way? Is there a curvature in the configuration space of molecular orientations, e.g. for randomly oriented molecules?


The comparison between the electron wave-function ψ_el^0(t,E⃗(ρ),r⃗)  describing the temporal geometry of chiral currents excited by the laser field fixed at a given point on the sphere ρ_0 = (θ_0, ϕ_0)  with its "global" counterpart ψ_el(t,E⃗(ρ),r⃗), "informed" about all different laser geometries,  encapsulates the essence of the "connection" between different local geometries. 

The TDSE for the electron wave-function ψ_el^0(t,E⃗(ρ_1),r⃗) at some other point on the sphere has the same general form, but the laser field, of course, has a different geometry leading to a different temporally chiral response. The issue arises due to the fact that these two TDSEs are completely independent, and thus each "local" electron wave-function ψ_el^0(t,E⃗,r⃗) is defined up to an arbitrary phase. It means that one may need to find a consistent procedure to "connect" all such solutions along a given cyclic path in the configuration space by introducing a common phase connecting all local solutions in a well-defined way and encoded in a single, consistent "full" solution ψ_ el(t):





    |ψ_ el(t)⟩   = e^-iα_ D(t) + iS(t)|ψ_ el^0(t)⟩ .
 
In general, the phase has a dynamical α_ D(t)

    α_ D(t)    = ∫_0^t dτ⟨ψ_ el(τ)|H(τ)|ψ_ el(τ)⟩ ,

and a geometric  S(t)

    S(t)    = i∫_0^t dτ⟨ψ_ el^0(τ)|τ|ψ_ el^0(τ)⟩

components. Our "local" solution ψ_ el^0(t) is known as a closed lift of a state vector defined as |ψ_ el(t)⟩⟨ψ_ el(t)| (see <cit.> and Appendix A). In Appendix B we show that, in the case of a periodic evolution of E⃗(t,ρ) in the space of Euler angles, the connection is a vector potential 

    A⃗(E⃗)=i⟨ψ_el^0(E⃗)|∇⃗_E⃗ψ_el^0(E⃗)⟩,

quantifying the change in the local wave-function in response to the change in the laser field geometry. 
It specifies the geometric phase S accumulated after a full cycle of evolution of E⃗(t,ρ) in the space of Euler angles ρ as detailed in Appendix B:

    S=∮ i⟨ψ_ el^0(E⃗)|∇⃗_E⃗ψ_ el^0(E⃗)⟩ dE⃗=∮A⃗(E⃗)· dE⃗.












This connection  leads to the geometric magnetic field (the curvature):

    Ω⃗=∇⃗_E⃗×A⃗= i⟨∇⃗_E⃗ψ_el^0(E⃗)|× |∇⃗_E⃗ψ_el^0(E⃗)⟩.

Eqs. (<ref>,<ref>) encapsulate the origin of the geometric magnetism, emerging due to the necessity to "connect" temporal  geometry at different points in configuration space. 
Eqs. (<ref>,<ref>) emphasize the dynamical, laser-driven origin of these geometric quantities. As derived, Eqs. (<ref>,<ref>) appear general and applicable to any light-driven process. 

In the next sections, we test the generality of Eq. (<ref>) by  applying it to three different types of two-photon processes  involving linearly and circularly polarized pump and probe pulses. First, we generalize the relationship between the curvature (Eq. (<ref>)) and the vectorial  observables expressing the 
light-induced enantio-sensitive molecular orientation by ionization of an initially randomly oriented ensemble of molecules, extending  our earlier work <cit.>, where we derived the curvature for one-photon ionization from a state with a stationary current. Second, we identify new  observables describing enantio-sensitive orientation by excitation, opening new  opportunities for controlling the direction of orientation. Third, we derive the relationship between the photoionization yield and the curvature Eq. (<ref>) in two-photon ionization by circularly polarized field. We show that it is similar to the one in one-photon ionization from a state with a stationary current <cit.>, confirming the generality of the link between the enantio-sensitive photoionization yield and the geometric phase. 

Specifically, we show that enantio-sensitive molecular orientation can be expressed using  Eq. (<ref>) as follows
(superscripts L and M stand for laboratory and molecular frame correspondingly):


    ⟨e⃗_Ω^L⟩= σ⟨(Ω⃗^L·ẑ^L)e⃗_Ω^L⟩=σR^(n,p)⟨Ω⃗^M·e⃗_Ω^M⟩ẑ^L.

Here ⟨e⃗_Ω^L⟩ is the orientationally averaged value of a unit vector directed along the curvature in the molecular frame (e⃗_Ω^M∥Ω⃗^M), σ=±1 is the direction of rotation of the light field,  







 
⟨Ω⃗^M·e⃗_Ω^M⟩ is the orientationally averaged value of a scalar product Ω⃗^M·e⃗_Ω^M,
R^(n,p) is a factor associated with the partial alignment of the molecular ensemble due to excitation or ionization, which depends on the number of absorbed photons n. Superscript p in R^(n,p) indicates that it also depends on polarizations of absorbed photons, which may or may not break the cylindrical symmetry of the set-up.  We establish the factor R^(n,p) by explicitly evaluating ⟨(Ω⃗^L·ẑ^L)e⃗_Ω^L⟩ and ⟨Ω⃗^M·e⃗_Ω^M⟩ using Eq. (<ref>).   

Eq. (<ref>) shows that (i) the vector ⟨e⃗_Ω^L⟩ orients itself along the direction of photon spin ẑ^L, and (ii)  the curvature controls the direction and the degree of molecular orientation. 
To demonstrate the generality of our approach, we use Eqs. (<ref>,<ref>) to evaluate enantio-sensitive molecular orientation for a range of two-photon processes and compare the resulting expressions with the direct evaluation, where the curvature is not employed.  We show that the results are equivalent. Our approach not only allows us to express enantio-sensitive molecular orientation in a compact way (Eq. (<ref>)), but also uncovers its fundamental origin, appearing to be invariant for different  multiphoton processes.

Moreover, we also show that the enantio-sensitive and dichroic part of photoionization yield in a two photon process for an oriented molecule is proportional to the projection of the curvature on the light propagation axis:

    W_σ(ρ)-W_-σ(ρ)   =σ(Ω⃗^L_σ(ρ)+Ω⃗^L_-σ(ρ)) ·ẑ^L.

This expression is essentially the same as in the one-photon case <cit.>, leading to a similar  relationship between the total, i.e. orientationally averaged yield, and the geometric phase in the multiphoton regime for circularly polarized pulses: 

    W_σ-W_-σ   =∫ dρ W_σ(ρ)-W_-σ(ρ)=σ/4π∫(Ω⃗^M_σ(k,θ,ϕ)+Ω⃗^M_-σ(k,θ,ϕ)) · dS⃗^M= σ/4π(Φ_σ^ES+Φ_-σ^ES).

Here ∫ dS⃗^M=e⃗^M_r∫_0^πsinθ dθ∫_0^2π dϕ, where e⃗^M_r is a unit radial vector in the molecular frame, corresponding to the orientation of the photon spin in the molecular frame, the third Euler angle α  (Fig. 1(d)) is not relevant for circularly polarized pulses. First, Eq. (<ref>) shows that the enantio-sensitive yield is proportional to the orientationally averaged radial component of the curvatures ⟨Ω_r,
σ^M(k)⟩ +⟨Ω_r,
-σ^M(k)⟩. Second, Eq. (<ref>) expresses the flux of the curvature (for a given σ) through the parameter space of field orientations (Fig. 1(d)), which by definition is equal to the geometric phase Φ_σ^ES. The superscript ES emphasizes the enantio-sensitive nature of the geometric phase. 


Thus the physical meaning of the curvature is the geometric magnetic field generated by electron current as it is "transported" in the configuration space of rotational degrees of freedom. Importantly, the transport is enabled and mediated by the laser field. This statement becomes evident as we rewrite the TDSE for the global wave-function for any value of E⃗(ρ(t),t) in the form emphasising the contribution of the geometric field via its vector potential and its coupling to the change of the orientation of the laser field in the molecular frame: 

    i∂/∂ tψ_el(r⃗,E⃗,t)=[H_el+r⃗·E⃗(ρ(t),t)-A⃗(E⃗)·Ė⃗̇]ψ_el(r⃗,E⃗,t).












§ 3. TEMPORAL GEOMETRY IN TWO-PHOTON IONIZATION

We can use Eq. (<ref>) to calculate the curvature for a two-photon process, in which a linearly polarized pulse excites a superposition of states in a molecule and a circularly polarized pulse photoionizes this superposition.  Since only spin carrying light fields induce temporal geometry, we use a linearly polarized pulse to excite the molecule to separate the curvature due to the continuum states from the curvature due to the bound states (see Section 4). If excitation and ionization are performed with the same circularly polarized pulse, then the contributions of the bound and continuum states to the curvature are mixed (see Section 5).

We  use perturbation theory to calculate |ψ(E⃗)⟩ explicitly: 
 
    ψ    =ψ_0+a_1ψ_1+a_2ψ_2+∫dΘ_ka_k⃗ψ_k⃗.

Here dΘ_k=sinθ_kdθ_kdϕ_k and θ_k, ϕ_k are angles characterizing the direction of the photoelectron momentum k⃗ for fixed k=|k⃗|, a_i and a_k⃗ are the amplitudes of bound (i=1,2)  and continuum states:




    a_i=i∫_0^t dt' d⃗_i·ℰ⃗_i(t')e^iω_i t'=id⃗_i·ℰ⃗_i,


    a_k⃗=-(D⃗_1·E⃗_k⃗1)(d⃗_1·ℰ⃗_1)e^-iω_1τ-(D⃗_2·E⃗_k⃗2)(d⃗_2·ℰ⃗_2)e^-iω_2τ.



 
 

Here d⃗_i are bound transition dipoles, and D⃗_i are photoionization dipoles from bound states (i=1,2) to the continuum state k⃗.
Writing E⃗_k⃗i=|E_ω_k⃗i|E⃗, with E⃗=1/√(2)(x̂+iσŷ)
 and taking the space spanned by the probe polarization E⃗ as the parameter space as suggested by Eq. (<ref>), the  curvature (Eq. (<ref>)) in the molecular frame yield






    Ω⃗_ion^M(k,ρ)
       =-|E_ω_k⃗1||E_ω_k⃗2|Im{(d⃗_1·ℰ⃗_1)^*(d⃗_2·ℰ⃗_2) e^iω_12τ∫dΘ_kD⃗_1^*M×D⃗_2^M}











where the subscript 'ion' emphasizes that the  curvature Ω⃗_ion^M(k,ρ) relies on the continuum states and manifests itself in photoionization.  
The superscript M emphasizes that vectors are expressed in the molecular frame. The direction of the curvature in the molecular frame is defined by the direction of 
the vector 𝖯⃗^+M_12(k), where 𝖯⃗^+M_12(k)≡1/2Im{ i∫dΘ_kD⃗_1^*M×D⃗_2^M}, superscript "+" indicates that only k⃗-even part of the vector product contributes to the integrated quantity (see also <cit.>). 


We now introduce the polar unit vector e⃗_Ω^M along the direction of the curvature Ω⃗_ion^M(k,ρ) (i.e. along 𝖯⃗^+M_12(k)) and obtain the averaged value of ⟨Ω⃗_ion·e⃗_Ω⟩ [Here and in the following we omit indices M, L in scalar products, since the scalar products are rotationally invariant and can be associated with any frame. ]  (see Appendix D):

    ⟨Ω⃗_ion(k)·e⃗_Ω⟩
       =-1/3C(d⃗_1·d⃗_2)(𝖯⃗^+_12(k)·e⃗_Ω)sin(ω_12τ)=-1/3υ C(d⃗_1·d⃗_2)|𝖯⃗^+_12(k)|sin(ω_12τ),

where C≡|E_ω_k⃗1||ℰ_ω_1||E_ω_k⃗2||ℰ_ω_2|.
Note that 𝖯⃗^+_12(k)·e⃗_Ω=υ|𝖯⃗^+_12(k)|, where υ=±1 is molecular pseudoscalar.

Using Eq. (<ref>), we obtain the  dichroic part of the photoionization yield  for a fixed molecular orientation ρ:

 


 
  



    W^↕,↺(k,ρ)=1/2σ(Ω⃗_ion^L(k,ρ)·ẑ^L),

Here the direction of rotation of E⃗ is characterized by σ=± 1, and ẑ^𝖫 denotes the respective axis of the laboratory frame,  
the superscript "↕,↺" emphasises the polarization sequence of the pulses: linearly polarized pump, circularly polarized probe. 

It is convenient to separate the  curvature into two parts using the Binet-Cauchy identity:






    (d⃗_1·ℰ⃗_1^*)(d⃗_2·ℰ⃗_2)=(ℰ⃗_1^*·ℰ⃗_2)(d⃗_1·d⃗_2)-(d⃗_1×ℰ⃗_2)·(d⃗_2×ℰ⃗_1^*).

The first term represents the part corresponding to the purely isotropic excitation and the second term is associated with the alignment of randomly oriented molecular ensemble after the absorption of a photon. The isotropic component of the  curvature  is also rotationally invariant and thus represents its isotropic component in the molecular frame:

    Ω⃗^𝖬_cont,iso(k,ρ)=-2|E_ω_k1||E_ω_k2|(ℰ⃗_1^*·ℰ⃗_2)(d⃗_1·d⃗_2)𝖯⃗^+_12(k)sin(ω_12τ).



The contribution of Ω⃗^M_cont,iso to the orientation-averaged dichroic part of the photoelectron yield 
can be written as the flux of the  curvature through the sphere in molecular frame uniting the points defined by the Euler angles θ,ϕ [The third Euler angle γ describing the rotation around the radial vector falls out for the isotropic component, see Eqs. (21-25) in Methods in Ref. <cit.>]: 



    W^↕,↺_iso(k)   =∫ W^↕,↺_iso(k,ρ)dρ=σ/8π∫Ω⃗^𝖬_cont,iso(k,ρ) · dS⃗^𝖬=σ/8πΦ_B^ion,


where ρ≡(ϕθγ) is the molecular orientation and (∫ dρ≡1/8π^2∫_0^πsinθ dθ∫_0^2π dϕ∫_0^2πdγ).

The enantio-sensitive molecular orientation by ionization (PI-MOCD) for the linear pump - circular probe excitation is associated with the preferential photoionization of the molecules with the curvature  oriented along the photon spin. 
Using Eq. (<ref>) together with Eq. (<ref>) we can calculate (see Appendix D) the orientation averaged value of ⟨e⃗_Ω^L⟩:
 
    ⟨e⃗_Ω^L⟩^↕,↺=σ⟨(Ω⃗_ion^L·ẑ^L)e⃗_Ω^L⟩=σR^(2,↕,↺)⟨Ω⃗_ion·e⃗_Ω⟩ẑ^L,

where 

    R^(2,↕,↺)≡2/5[1 - 1/2(d⃗_1·E⃗_)(d⃗_2·𝖾⃗_Ω)/(d⃗_1·d⃗_2)].
 
  Up to notations Eq. (<ref>)  reproduces our earlier result for PI-MOCD <cit.>,
validating the general approach  (i.e. calculating  ⟨e⃗_Ω^L⟩ using Eqs. (<ref> and <ref>)  developed here.
 Vector ⟨e⃗_Ω^L⟩ can be used to find the expectation value of cosβ, where β is the angle between the unit vector along the curvature e⃗_Ω^L and the direction of photon spin.  The value of cosβ (see Fig. <ref>(c)) is given by the normalized (to the total photoionization yield averaged over the pump-probe delay)  magnitude of ⟨e⃗_Ω^L⟩.























The dichroic and enantio-sensitive component of the  photoionization yield  W≡∫ |a_k⃗|^2 dΘ_k can be expressed as the flux of the curvature through the sphere in the space of light orientations in the molecular frame (see Appendix E):

    W^ES_↕,↺   =σ/4π∫Ω⃗_ion(k,θ,ϕ) · dS⃗^𝖬=σ/4π⟨Ω^r_ion(k)⟩= σ/4πΦ_^↕,↺^ES.





§ 4. TEMPORAL GEOMETRY IN PHOTOEXCITATION
 
The curvature can also be associated with the excitation of bound states. Importantly, the general definition Eq. (<ref>) describes both cases, even though the specific expressions are different. This surprising result demonstrates the importance of laser pulse polarization geometry: applying a circularly polarized pulse either to photoionize or to excite the molecule, one can address these two different "faces" of the curvature.  


Let a circularly polarized pump pulse excite a superposition of states in a chiral molecule. We now show that the subsequent photoionization by a linearly polarized probe pulse will lead to enantio-sensitive orientation of ions (PI-MOCD). Exchanging E⃗_ki and ℰ⃗_i in Eq. (<ref>), using Eq. (<ref>) and following the same steps as in the previous section we obtain the analog of Eq. (<ref>):

    Ω⃗_exc^M(k,ρ)
    =-|E_ω_1||E_ω_2|Im{∫dΘ_k(D⃗_1·ℰ⃗_k⃗1)^*(D⃗_2·ℰ⃗_k⃗2) e^iω_12τ}( d⃗_1^M×d⃗_2^M).

Here the subscript 'bound' emphasizes the nature of the  curvature, which now relies on bound states. 










 


The orientationally averaged expression for the scalar product  Ω⃗·e⃗_Ω, where e⃗_Ω is oriented along the direction of ( d⃗_1×d⃗_2) in the molecular frame, can be derived in the same way as in the previous section:

    ⟨Ω⃗_exc(k)·e⃗_Ω⟩=-1/3C
    Re{∫dΘ_kD⃗_1^*·D⃗_2}
    
    ( d⃗_1×d⃗_2)·e⃗_Ωsin(ω_12τ),

where C≡|ℰ_ω_k⃗1||E_ω_1||ℰ_ω_k⃗2||E_ω_2| and  we took into account that for the time-reversal-
invariant excited states ψ_1 and ψ_2, the time-reversal symmetry implies that (see Appendix C):

    Im{∫dΘ_kD⃗_1^*·D⃗_2} = 0.
 
Note that ( d⃗_1×d⃗_2)·e⃗_Ω=υ| d⃗_1×d⃗_2|e⃗_Ω, meaning that Eq. (<ref>) can also be rewritten as

    ⟨Ω⃗_exc(k)·e⃗_Ω⟩=-1/3υ C
    Re{∫dΘ_kD⃗_1^*·D⃗_2}
    
    | d⃗_1×d⃗_2|sin(ω_12τ).



Using Eq. (<ref>) together with Eq. (<ref>) we can calculate (see Appendix D) the orientation averaged value of 
 
    ⟨e⃗_Ω^L⟩^↺,↕=σ⟨(Ω⃗_exc^L·ẑ^L)e⃗_Ω^L⟩=σR^(2,↺,↕)⟨Ω⃗_exc·e⃗_Ω⟩ẑ^L,

where 

    R^(2,↺,↕)≡2/5[1 - 1/2Re{∫ d Θ_k(D⃗^*_1·E⃗_)(D⃗_2·𝖾⃗_Ω)}/Re{∫ d Θ_k(D⃗^*_1·D⃗_2)}].
 
 Evaluating it explicitly 

    ⟨e⃗_Ω^L⟩^↺,↕    =∫dϱ W^↕,↺(k,ρ)e⃗^L_Ω,

we get the expression equivalent to Eq. (<ref>),   up to rearrangement of terms:

    ⟨e⃗_Ω^L⟩^↺,↕ =iσ/60C∫dΘ_k[4(D⃗_2^*·D⃗_1)e⃗_Ω^M-(D⃗_2^*·e⃗_Ω)D⃗_1^M-(D⃗_1·e⃗_Ω)D⃗_2^*M](d⃗_2^M×d⃗_1^M)e^iω_21tẑ+c.c.


Eq. (<ref>) together with Eq. (<ref>)  show that  PI-MOCD can also emerge due to  curvature in bound states: molecules, in which the curvature is  oriented along the spin of the laser light, are excited preferentially. Thus, photoionization following photoexcitation creates enantio-sensitive orientation of both neutrals and cations. The expectation value of cosβ, where β is the angle between the unit vector along the curvature e⃗_Ω^L and the direction of photon spin,  is given by the normalized (to the total photoionization yield averaged over the pump-probe delay) magnitude of ⟨e⃗_Ω^L⟩.



The dichroic and enantio-sensitive component of the  photoionization yield  W≡∫ |a_k⃗|^2 dΘ_k can be expressed as the flux of the curvature through the sphere in the space of light orientations in the molecular frame (see Appendix D):

    W^ES_↺,↕   =σ/4π∫Ω⃗_exc(k,θ,ϕ) · dS⃗^𝖬=σ/4π⟨Ω^r_exc(k)⟩= σ/4πΦ_^↺,↕^ES.














§ 5. INTERPLAY OF BOUND AND CONTINUUM TEMPORAL GEOMETRIES


 

In the previous two sections, we considered photoexcitation followed by photoionization induced by pairs of pump-probe pulses with linear - circular or circular-linear polarizations correspondingly. In the first case, the curvature has emerged due to the temporal geometry in photoelectron continuum states, in the second case it has emerged due to the temporal geometry in bound states. What happens if both the pump and the probe pulses are circularly polarized? 

To answer this question, we need to apply Eqs. (<ref>)  to equations (<ref>-<ref>), where the Fourier components of  linearly polarized fields ℰ⃗_2 and ℰ⃗_1 are substituted by the Fourier components of circularly polarized fields:  ℰ⃗_2=>E⃗_2 and
ℰ⃗_1=>E⃗_1. Note that there is a difference  in using Eq. (<ref>) in this case. As detailed in Appendix B, the gradients ∇⃗_E⃗ in Eq. (<ref>) should be taken only with respect to fields carrying spin angular momentum, because only these fields uniquely define the orientation of the laboratory frame with respect to the molecular frame. 
That is why in sections 3,4 the gradients ∇⃗_E⃗  were  taken only with respect to the circularly polarised Fourier components of the field (i.e. probe field  in Section 3 and pump field in Section 4). Here, we have to evaluate the gradients with respect to both the pump field components and the probe field components. It leads to the following expression for the curvature (see Appendix F):

    Ω⃗^M(k,ρ)
       =Ω⃗_exc^M(k,ρ)+Ω⃗^M_ion(k,ρ)+Ω⃗^M_mix(k,ρ)

 where 
 
    Ω⃗^M_exc(k,ρ)
       =-2CIm{∫dΘ_k(D⃗_1·E⃗__k⃗1)^*(D⃗_2·E⃗__k⃗2) e^iω_12τ}( d⃗_1^M×d⃗_2^M),
    Ω⃗^M_ion(k,ρ)
       =-2CIm{(d⃗_1·E⃗__1)^*(d⃗_2·E⃗__2) e^iω_12τ∫dΘ_kD⃗_1^*M×D⃗_2^M}.
    Ω⃗^M_mix(k,ρ)=
       -2CIm{∫dΘ_k(d⃗_1·E⃗__1)(D⃗_2·E⃗__k⃗2)^*( D⃗_1^*M×d⃗_2^M)e^iω_12τ}
       -2CIm{∫dΘ_k(D⃗_1·E⃗__k⃗1)^*(d⃗_2·E⃗__2) ( d⃗_1^M×D⃗_2^M)e^iω_12τ},

Here C≡|E_ω_k⃗1||E_ω_1||E_ω_k⃗2||E_ω_2|. Eq. (<ref>) combines bound and continuum curvatures as detailed in  Eqs. (<ref>, <ref>),  and also contains "mixed" terms, relying on both bound and continuum curvatures.

The orientationally averaged scalar product of the curvatures and a unit vector e⃗_Ω are:

    ⟨Ω⃗_exc(k,ρ)·e⃗_Ω⟩
       =-1/3C Re{∫dΘ_k(D⃗_1^*·D⃗_2) }( d⃗_1×d⃗_2)·e⃗_Ωsin(ω_12τ),
    ⟨Ω⃗_ion(k,ρ)·e⃗_Ω⟩
       =-1/3C(d⃗_1·d⃗_2)𝖯⃗^+_12(k)·e⃗_Ωsin(ω_12τ).
    ⟨Ω⃗_mix(k,ρ)·e⃗_Ω⟩=
       -1/3CIm{∫dΘ_k(d⃗_1·D⃗_2^*) ( D⃗^*_1×d⃗_2)·e⃗_Ωe^iω_12τ}
       -1/3CIm{∫dΘ_k(D⃗_1^*·d⃗_2) ( d⃗_1×D⃗_2)·e⃗_Ωe^iω_12τ},

One can verify that using the general expression for expectation value of ⟨e⃗_Ω^L⟩^↺,↺:

    ⟨e⃗_Ω^L⟩^↺,↺=σ⟨(Ω⃗^L·ẑ^L)e⃗_Ω^L⟩

 one obtains the same results as by evaluating ⟨e⃗_Ω^L⟩^↺ explicitly without involving the curvature:

    ⟨e⃗_Ω^L⟩^↺    =iσ/30C∫dΘ_k{(d⃗_2·d⃗_1)[e⃗_Ω·(D⃗_2^*×D⃗_1)]+(D⃗_2^*·D⃗_1)[e⃗_Ω·(d⃗_2×d⃗_1)]
        +(d⃗_2·D⃗_1)[e⃗_Ω·(D⃗_2^*×d⃗_1)]+(D⃗_2^*·d⃗_1)[e⃗_Ω·(d⃗_2×D⃗_1)]}
     e^iω_21tẑ+c.c.

Comparing this result to ⟨Ω⃗_mix·e⃗_Ω⟩ we find that the factor R^(2,↺,↺)=1/10  is purely numerical due to preserved cylindrical symmetry:

    ⟨e⃗_Ω^L⟩^↺,↺=σ⟨(Ω⃗^L·ẑ^L)e⃗_Ω^L⟩=σ R^(2,↺,↺)⟨Ω⃗·e⃗_Ω⟩ẑ^L.



Finally, one can verify that the expression for the enantio-sensitive and dichroic yield W^ES_↺,↺-W^ES_↻,↻ via geometric phase also remains invariant, because the sum of the curvatures (and the sum of the geometric phases) only contains terms independent of σ, just like the curvature and the geometric phase in the one-photon case: 

    W^ES_↺,↺-W^ES_↻,↻   =σ/4π∫(Ω⃗^M_↺,↺(k,θ,ϕ)+Ω⃗^M_↻,↻(k,θ,ϕ)) · dS⃗^𝖬= σ/4π(Φ_↺,↺^ES+Φ_↻,↻^ES).




§ 6. OPPORTUNITIES FOR CONTROL OF GEOMETRIC OBSERVABLES


PI-MOCD is the first geometric observable that is directly proportional to the curvature, relying on the geometry of  either  bound or  continuum states. Importantly, the curvature is a dynamical property, as it relies on the excitation of currents and reflects their temporal geometry. Dynamics offers several opportunities for the control. The direction of enantio-sensitive molecular orientation is defined by the direction of the  curvature vector in the molecular frame. The curvature vectors relying on the vector product of bound or continuum dipoles naturally have  different directions: Ω⃗_exc∥d⃗_1 ×d⃗_2 and  Ω⃗_ion∥Re{∫dΘ_kD⃗_1^*×D⃗_2}. Moreover,  both orientations are state-specific (see Figs. <ref>,<ref> (b) for propylene oxide). It presents several opportunities for controlling the enantio-sensitive direction of molecular orientation.


First, by changing the frequency of the pump pulse, one can address different bound states and thus control both the magnitude and direction of the curvature. Second,  controlling the polarization of the pump-probe pulse sequence also allows us to control the magnitude and the direction of the enantio-sensitive molecular orientation by switching the curvature between its form defined by the geometry of the excited bound states to the one defined by the geometry of the continuum states.  Figs. <ref> and <ref> illustrate how the direction of the curvature and respective molecular orientation changes if one switches polarizations of pump and probe pulses from circular (linear) to linear (circular).  Third, by changing the frequency of the probe pulse in case of linear-circular pulse sequence one can change the direction of the curvature as shown in Fig. <ref>(b).   

For a given set of intermediate bound states excited by a pump pulse and continuum states populated by the probe pulse,  the magnitudes of Ω⃗_ion and  Ω⃗_exc   differ only in their dipole factors. The bound transition dipole contributions to the magnitudes of Ω⃗_ion and  Ω⃗_exc are d⃗_⃗1⃗·d⃗_⃗2⃗=-7.3× 10^-2 a.u. and |d⃗_1 ×d⃗_2|=4.9× 10^-2 a.u., respectively. The continuum dipole contributions Re{∫dΘ_kD⃗_1^*·D⃗_2} and |Re{∫dΘ_kD⃗_1^*×D⃗_2}| change as a function of the photoelectron energy and are shown in Fig. <ref>.
These two parameters determine the k-dependence of   expectation value of the degree of orientation  in Figs. <ref>,<ref>(c).





















 




§ OUTLOOK

Geometric magnetism in chiral molecules appears to be a ubiquitous phenomenon, which can be controlled by tailoring light pulses and by addressing various molecular states. The linear-circular and circular-linear pump-probe sequences have been used experimentally to observe time-resolved photoelectron circular dichroism (TR-PECD) and photoexcitation circular dichroism (PXCD) in chiral molecules via detection of angular distribution of photoelectrons. We show that in both cases, the detection of molecular fragments leads to a complementary geometric observable, directly proportional to the  curvature in continuum or bound states. 

Our results show that the temporal evolution of angular distribution of fragments (Eqs. (<ref>,<ref>)) and angular distribution of photoelectrons produced after inducing  the photo-excitation circular dichroism    (circular-linear pump-probe sequence)<cit.> and controlled by the term  ∫dΘ_kk⃗(D⃗_1^*·D⃗_2) have fundamentally different properties.

The former is proportional to the current in the bound states ∝sin(ω_12τ) and thus vanishes at zero pump-probe delay τ=0, while the latter also includes terms ∝cos(ω_12τ) and does not vanish at τ=0.
This property establishes PI-MOCD as a direct probe of charge-directed reactivity: PI-MOCD is proportional to the  curvature (either in bound or in continuum states) and as such relies on time-reversal symmetry breaking, which can be introduced by exciting current prior to photoionization. The direction of current at the moment of ionization defines the orientation of molecular cations and therefore the direction of molecular fragmentation, highlighting the ability of the ultrafast electron dynamics to control subsequent dynamics of the nuclei. 


Simulations <cit.> show that enantio-sensitive fragment asymmetry resulting from excitation of electronic wave-packet in Rydberg states in methyl lactate molecule in linear-circular polarization sequence  reaches  20%, confirming strong enantio-sensitivity of curvature-driven geometric observables. 


The control over the directions of enantio-sensitive orientations of molecular ions suggests new opportunities. For example, in linear-circular sequence of the pulses the tip of the curvature vector traces the orange curve in space (Fig. <ref>(b)) as a function of photoelectron energy. This orange curve illustrates the sequence of orientations of the curvature vector achieved for different final photoelectron momenta from 0.25 to 2 a.u. Since enantio-sensitive molecular orientation follows the orientation of the curvature vector,   polarization controlled pulses at free electron lasers (FELs), such as chirped circularly polarized pulses, can be used to force molecular ions to draw the trajectory shown in Fig. <ref>(b) (see orange trajectory) within a single experiment. Alternately, the set of experiments in which the frequency of the probe circularly polarized light  is tuned within several tens of eV will produce a variety of well controlled molecular orientations. 



The geometric magnetism in bound states suggests new recipes for merging topological and enantio-sensitive  response  in chiral molecules. In particular, the flux of mixed curvature in the configurations space of laser field orientations emerging in  the excitation by circularly polarized pump and probe pulses is proportional to the enantio-sensitive and dichroic part of the photoionization yield (Eqs. <ref>, <ref>).  In analogy to the  so-called Thouless charge pump, one would expect to realize a quantized rate of enantio-sensitive charge transfer via diabolic points in rotational states. 



§ ACKNOWLEDGMENTS

We gratefully acknowledge many enlightening discussions with Prof. Misha Ivanov. We gratefully acknowledge Prof. Vladimir Chernyak for discussions and lectures on the topic.

A.F.O. and D.A. acknowledge funding from the Royal Society (URF/R1/201333). O.S., A.R. P.M.M. gratefully acknowledge ERC-2021-AdG project ULISSES, grant agreement No 101054696.
 


§ APPENDICES



 §.§ Appendix A: Closed lift for a state vector |ψ_ el(t)⟩⟨ψ_ el(t)|

The closed lift is a concept from fiber bundle theory that allows one to identify the geometric phase outside the approximation of an adiabatic evolution.


In the Aharonov-Anandan fiber bundle theory, one introduces the base space in which all pure state vectors |ψ_ el(t)⟩⟨ψ_ el(t)| live. By definition, the state vectors do not possess any phase. Thus, when the system evolves along the closed contour in the base space, the state vector remains the same after this cyclic evolution along the contour γ : |ψ_ el(t)⟩⟨ψ_ el(t)|=|ψ_ el(t+T)⟩⟨ψ_ el(t+T)|.

Wave-functions corresponding to state vectors |ψ_ el(t)⟩⟨ψ_ el(t)| at every point in base space |ψ_ el(t)⟩e^iϕ "live" on a fiber, which "grows vertically" from the base space while the phase ϕ is taking all possible values along 
this vertical direction (the fiber). These wave-functions are said to constitute "lifts" of the state vector |ψ_ el(t)⟩⟨ψ_ el(t)|  from the base space into the fiber space (at every point on the contour γ in the base space). The wave-function that does not accumulate a phase as a result of cyclic evolution of a state vector on the contour γ such that |ψ^0_ el(t)⟩=|ψ^0_ el(t+T)⟩ is called a closed lift.

We can show that the wave-function ψ_el^0(t,E⃗(ρ),r⃗) defined by Eq. (<ref>) does not accumulate a phase as a result of cyclic evolution of rotational degrees of freedom and therefore presents the  closed lift of the state vector |ψ_ el(t)⟩⟨ψ_ el(t)|.

In our case, one cycle of evolution corresponds to one full rotation of the molecular frame ψ_el^0(t,E⃗(ρ),r⃗) from some initial  orientation characterised by Euler angles ρ_0={θ_0,ϕ_0,α_0} with respect to the lab frame  to the exact  same orientation ρ_0+2π={θ_0,α_0+2π,ϕ_0+2π}.
The wave-functions ψ_el^0(t,E⃗(ρ_0),r⃗) and ψ_el^0(t,E⃗(ρ_0+2π),r⃗) can be obtained from the same lab-frame wave-function using an appropriate sequence of unitary transformations, such as e.g. rotation around n̂-axes by the angle ξ: ψ_el^0(t,E⃗(ρ_0+2π),r⃗)=e^-iξ(n̂·L̂)ψ_el^0(t,r⃗)=ψ_el^0(t,E⃗(ρ_0),r⃗). Since these unitary transformations do not impart any relative phases on the transformed wave-functions ψ_el^0(t,E⃗(ρ_0+2π),r⃗) and 
ψ_el^0(t,E⃗(ρ_0),r⃗), and the same arguments remain valid for every point ρ_0, the wave-function ψ_el^0(t,E⃗(ρ),r⃗) presents a closed lift. 





 §.§ Appendix B: Derivation of Eq. (<ref>)

Previously, we have derived Eqs. (<ref>, <ref>) using the concept of adiabatic evolution of the rotational degree of freedom, introducing two different time scales for the evolution of electronic and rotational degrees of freedom. We now extend our derivation to a general case, including non-adiabatic evolution.
We follow the arguments of Ref. <cit.> and introduce a rotational Hamiltonian H_rot(t), responsible for the cyclic non-adiabatic evolution of the rotational wave-function ψ_rot(ρ,t): 

    id/dtψ_rot(ρ,t)=H_rot(t)ψ_rot(ρ,t).

For example, H_rot(t) could encapsulate a sequence of microwave fields driving  resonant transitions in three rotational levels of a chiral molecule.
Since our focus is laser-induced coupling between electronic and rotational degrees of freedom, which arises due to the sensitivity of the electronic dynamics to the mutual orientation between the molecule and the laser field, we omit the standard Coriolis coupling between electronic and rotational degrees of freedom in the zero approximation.

Staring from the TDSE for the full wave-function including the electronic and rotational degrees of freedom:

    id/dtΨ(r⃗,ρ,t)=[H_el+r⃗·E⃗(ρ,t)+H_rot(t)]Ψ(r⃗,ρ,t)

and using the ansatz:

    Ψ(r⃗,ρ,t)=ψ_el(r⃗,ρ,t)ψ_rot(ρ,t),

we first obtain

    iψ_rot(ρ,t)d/dtψ_el(r⃗,ρ,t)+iψ_el(r⃗,ρ,t)d/dtψ_rot(ρ,t)=[H_el+r⃗·E⃗(ρ,t)+H_rot(t)]ψ_el(r⃗,ρ,t)ψ_rot(ρ,t).

Multiplying the TDSE (Eq. (<ref>)) by ψ_rot^*(ρ,t) from the left and integrating over ρ yields the following equation:

    i∫ dρ|ψ_rot(ρ,t)|^2d/dtψ_el(r⃗,ρ,t)+i∫ dρψ_el(r⃗,ρ,t)ψ_rot^*(ρ,t)d/dtψ_rot(ρ,t)   =   
    ∫ dρψ_rot^*(ρ,t)[H_el+r⃗·E⃗(ρ,t)]ψ_el(r⃗,ρ,t)ψ_rot(ρ,t)
    -i∫ dρd/dt[ψ_rot^*(ρ,t)]ψ_el(r⃗,ρ,t)ψ_rot(ρ,t),

where we have used Eq. (<ref>) in the last term applying the rotational Hamiltonian H_rot(t) to the bra-vector on the left. Note, that in doing so we have avoided the assumption of the adiabatic decoupling of the electronic and rotational degrees of freedom. Indeed, the adiabatic decoupling implies that H_rot does not affect the electronic wave-function, 
while here it clearly does so via the last term in Eq. (<ref>). In the following we shall use the adiabatic decoupling to derive the equation for the closed lift solution (see Eqs. (<ref>-<ref>)).


It is instructive to introduce a  "global" electron wave-function  sensitive to  evolution of rotational degrees of freedom, suggested by the structure of Eq. <ref>:

    Ψ_el(r⃗,T,t)def=∫ dρ |ψ_rot(ρ,T)|^2ψ_el(r⃗,ρ,t).


We find that it is instructive to introduce the second time scale T emphasising the evolution of the rotational degree of freedom. However, this step is not necessary and in the next section we perform the derivation without introducing the second time scale.

Expanding the laser field around some local orientation ρ_0: E_i(ρ,t)=E_i(ρ_0,t)+∇_ρ_0E_i(ρ_0,t)(ρ-ρ_0), and expanding ψ_el(r⃗,ρ,t) in the same way, we shall see that the interaction term linearly depends on ρ. Integrating the interaction term  over ρ we will obtain  average orientation ρ(T)=∫ dρ |ψ_rot(ρ,T)|^2ρ.  Finally  choosing ρ_0=ρ(T), we cancel out the gradient terms and obtain TDSE for the global wavefunction Ψ_el(r⃗,T,t):

    id/dtΨ_el(r⃗,T,t)+id/dTΨ_el(r⃗,T,t)=[H_el+r⃗·E⃗(ρ(T),t)]Ψ_el(r⃗,T,t).

The choice of ρ_0 guarantees that in Eq. <ref> we have included the gradients wrt to laser field up to the first order.
Note that ρ(T)=∫ dρ |ψ_rot(ρ,T)|^2ρ follows the evolution of the center of the rotational wave-packet and therefore is controlled by H_rot(T).

Comparing the TDSE (Eq. <ref>) for the "local" wavefunction ψ_el^0(r⃗,ρ,t) at fixed geometry ρ=ρ(T) and TDSE (Eq. <ref>) for the "global" wavefunction Ψ_el(r⃗,T,t), we find that they only differ by the term id/dTΨ_el(r⃗,T,t). Thus, ignoring  id/dTΨ_el(r⃗,T,t) in Eq. <ref> gives us the equation for ψ_el^0(r⃗,ρ(T),t).
We can now show that the local and global solutions only differ by phase: 

    Ψ_el(r⃗,T,t)= e^iS(T)ψ_el^0(r⃗,ρ(T),t).

Substituting Eq. <ref> into Eq. <ref>, and taking into account that ∂/∂ tS(T)=0 (see derivation below) we find that  S(T)=∫ dT ⟨ψ_el^0(r⃗,T,t)|∂/∂ Tψ_el^0(r⃗,T,t)⟩. 
Indeed, Eq. <ref> takes the following form:

    ie^iS(T)∂/∂ tψ_el^0(r⃗,ρ(T),t)-e^iS(T)ψ_el^0(r⃗,ρ(T),t)∂/∂ TS(T)+ie^iS(T)∂/∂ Tψ_el^0(r⃗,ρ(T),t)=
    [H_el+r⃗·E⃗(ρ(T),t)]e^iS(T)ψ_el^0(r⃗,ρ(T),t).

Using Eq. <ref>,  multiplying Eq. <ref> by ψ_el^0*(r⃗,ρ(T),t) and integrating over r⃗, we obtain:

    S(T)=i∫ dT⟨ψ_el^0(r⃗,ρ(T),t)|∂/∂ Tψ_el^0(r⃗,ρ(T),t)⟩.

Since T enters the local Hamiltonian (Eq. <ref>) only via E⃗(ρ(T),t), S(T) can be expressed in an equivalent form: S(T)=∫ i⟨ψ_el^0(r⃗,E⃗,t)|∇⃗_E⃗ψ_el^0(r⃗,E⃗,t)⟩ dE⃗, emphasising its geometric origin. 

Since in our derivation of Eq.<ref> we expanded up to the first order around ρ(T), we can also perform such expansion in Eq.<ref>, yielding:

    Ψ_el(r⃗,T,t)= ψ_el(r⃗,ρ(T),t)+𝒪(ρ^2(T)-ρ^2(T)).

Thus, Eq.<ref> connects the full solution for the electron wave-function ψ_el(r⃗,ρ(T),t)  (see ansatz <ref>) with its local approximation ψ_el^0(r⃗,ρ(T),t).

    ψ_el(r⃗,ρ(T),t)= e^iS(T)ψ_el^0(r⃗,ρ(T),t).


Note that ∂/∂ tS(T)=0.
To prove it we can explicitly calculate the derivative 

    ∂/∂ tS(T)=i∫ dT⟨∂/∂ tψ_el^0(r⃗,ρ(T),t)|∂/∂ Tψ_el^0(r⃗,ρ(T),t)⟩+i∫ dT⟨ψ_el^0(r⃗,ρ(T),t)|∂/∂ T∂/∂ tψ_el^0(r⃗,ρ(T),t)⟩=
     2∫ dT Im{⟨ψ_el^0(r⃗,ρ(T),t)|∂/∂ T∂/∂ tψ_el^0(r⃗,ρ(T),t)⟩}=2∫ dT Im{⟨ψ_el^0(r⃗,ρ(T),t)|∂/∂ THψ_el^0(r⃗,ρ(T),t)⟩},

where we introduced a short-hand notation for H≡ H_el+r⃗·E⃗(ρ(T),t) and used Eq. <ref>.
The last integral can be evaluated by parts :

    ∫ dT Im{⟨ψ_el^0(r⃗,ρ(T),t)|∂/∂ THψ_el^0(r⃗,ρ(T),t)⟩}=∫Im{⟨ψ_el^0(r⃗,ρ(T),t)|d [Hψ_el^0(r⃗,ρ(T),t)]⟩}=
    Im{⟨ψ_el^0(r⃗,ρ(T),t)| Hψ_el^0(r⃗,ρ(T),t)⟩}-∫ dT Im{⟨∂/∂ Tψ_el^0(r⃗,ρ(T),t)| Hψ_el^0(r⃗,ρ(T),t)⟩}

The last equation can be simplified to yield:

    2∫ dT Im{⟨ψ_el^0(r⃗,ρ(T),t)|∂/∂ THψ_el^0(r⃗,ρ(T),t)⟩}=Im{⟨ψ_el^0(r⃗,ρ(T),t)| Hψ_el^0(r⃗,ρ(T),t)⟩}=0

Note that the last term is equal to zero due to Hermiticity of the Hamiltonian H. It is always zero for cyclic processes. 

Eq. (<ref>) takes the  following form:

    i∫ dρ|ψ_rot(ρ,t)|^2d/dtψ_el(r⃗,ρ,t)+i∫ dρψ_el(r⃗,ρ,t)d/dt|ψ_rot(ρ,t)|^2   =   
    ∫ dρψ_rot^*(ρ,t)[H_el+r⃗·E⃗(ρ,t)]ψ_el(r⃗,ρ,t)ψ_rot(ρ,t).

We shall now assume that the  rotational wave-packet is narrow. This assumption can be realized in an experiment by preparing a narrow rotational wavepacket with a pump pulse and using a short
probe pulse with time-dependent polarization, rotating with respect to the frame established
by the pump pulse.
Introducing ρ̇(t)≡∫ dρd/dt|ψ_rot(ρ,t)|^2ρ=d/dt∫ dρ |ψ_rot(ρ,t)|^2ρ=d/dtρ(t) and expanding E_i(ρ,t)=E_i(ρ_0,t)+∇_ρ_0E_i(ρ_0,t)(ρ-ρ_0), and ψ_el(r⃗,ρ,t) up to the first order around ρ(t), we obtain for the integral ∫ dρψ_el(r⃗,ρ,t)d/dt|ψ_rot(ρ,t)|^2 the following expansion:

    ∫ dρψ_el(r⃗,ρ,t)d/dt|ψ_rot(ρ,t)|^2= ∫ dρψ_el(r⃗,ρ(t),t)d/dt|ψ_rot(ρ,t)|^2+∇_ρψ_el(r⃗,ρ(t),t)·ρ̇(t).

Here we took into account that ∫ dρ |ψ_rot(ρ,t)|^2=1, which leads to d/dt∫ dρ |ψ_rot(ρ,t)|^2=0. Combining this term with the other terms, we obtain the following TDSE for the "global" wave-function ψ_el(r⃗,ρ(t),t):

    i∂/∂ tψ_el(r⃗,ρ(t),t)+i∇_ρψ_el(r⃗,ρ(t),t)·ρ̇(t)=[H_el+r⃗·E⃗(ρ(t),t)]ψ_el(r⃗,ρ(t),t).

Note that ρ(t)=∫ dρρ |ψ_rot(ρ,t)|^2 follows the evolution of the center of the rotational wave-packet and therefore is controlled by H_rot(t).

Let's derive  an approximate solution ψ_el^0(r⃗,ρ,t) for the case when the rotational Hamiltonian itself does not affect the electronic dynamics, i.e.  

    H_rot(t)ψ_el^0(r⃗,ρ,t)ψ_rot(ρ,t)=ψ_el^0(r⃗,ρ,t)H_rot(t)ψ_rot(ρ,t).

Applying this condition to  Eq. (<ref>) yields

    iψ_rot(ρ,t)d/dtψ_el^0(r⃗,ρ,t)=[H_el+r⃗·E⃗(ρ,t)]ψ_el^0(r⃗,ρ,t)ψ_rot(ρ,t),

demonstrating that ψ_el^0(r⃗,ρ,t) solves the "local" TDSE for fixed ρ:

    id/dtψ_el^0(r⃗,ρ,t)=[H_el+r⃗·E⃗(ρ,t)]ψ_el^0(r⃗,ρ,t).

Comparing Eq. (<ref>) with Eq. (<ref>)  we find  that the "global" and the "local" solutions are related as follows:

    ψ_el(r⃗,ρ(t),t)= e^iS(ρ(t))ψ_el^0(r⃗,ρ(t),t).

Substituting Eq. (<ref>) to Eq. (<ref>), we find:

    S(ρ(t))=i∫_ρ(t_0)^ρ(t) dρ' ⟨ψ_el^0(r⃗,ρ',t)|∇_ρ'ψ_el^0(r⃗,ρ',t)⟩,



Note that, by the virtue of equations Eq. (<ref>) and Eq. (<ref>), the only reason why ψ_el^0(r⃗,ρ,t) depends on ρ is because the laser field in the molecular frame depends on ρ, i.e. ψ_el^0(r⃗,ρ,t)≡ψ_el^0(r⃗,E⃗(ρ),t).

Indeed, ψ_el^0(r⃗,ρ,t) satisfies the TDSE for the electronic degrees of freedom
Eq.<ref>. This equation, by definition, has no information about the rotational degree of freedom.  Thus, the gradient of  ψ_el^0(r⃗,ρ,t) with respect to ρ can be rewritten in an equivalent form,  ∇_ρψ_el^0(r⃗,ρ,t)=∇_E⃗ψ_el^0(r⃗,ρ,t)∂E⃗/∂ρ, as long as the laser field E⃗ can be used to unambiguously define the laboratory frame with respect to which the molecular orientation ρ is expressed. For example, circularly (elliptically) polarized field E⃗ satisfies this requirement. Hence the derivative should be taken with respect to E⃗. This leads  to the following equivalent expression for the geometric phase: 

    S(ρ(t))=i∫_E⃗(t_0)^E⃗(t)⟨ψ_el^0(r⃗,ρ',t)|∇_E⃗ψ_el^0(r⃗,ρ',t)⟩ dE⃗≡∫_E⃗(t_0)^E⃗(t)A⃗(E⃗)· dE⃗,

where A⃗(E⃗)=⟨ψ_el^0(r⃗,E⃗,t)|∇_E⃗ψ_el^0(r⃗,E⃗,t)⟩.

Finally, since ∇_ρψ_el(r⃗,ρ(t),t)=e^iS(ρ(t))ψ_el^0(r⃗,ρ(t),t)∇_ρS= ψ_el(r⃗,ρ(t),t)A⃗, the Eq.<ref> takes the following form:

    i∂/∂ tψ_el(r⃗,ρ(t),t)+A⃗(ρ(t))·ρ̇(t)ψ_el(r⃗,ρ(t),t)=[H_el+r⃗·E⃗(ρ(t),t)]ψ_el(r⃗,ρ(t),t).





















 §.§ Geometric phase associated with  the path in the space of rotational densities

We are now going to consider and alternative mechanism of  geometric phase accumulation by the electronic wave-function. 
In this example, we select a manifold of initial conditions for rotational wave-functions ψ_rot^(n)(t_0), where index n enumerates different initial state vectors. 
We now consider a snapshot of this manifold at some later time t, which corresponds to the manifold of states ψ_rot^(n)(t) evolved from these different initial conditions according to Eq. <ref>. We shall derive the relationship between the local electronic wave-functions, which ignore the presence of multiple rotational wave-packets, and the global electronic wave-function.
We start with the  ansatz similar to Eq.<ref>:

    Ψ(r⃗,ρ,t)=ψ_el(r⃗,ρ,t)ψ_rot^(n)(ρ,t).






 §.§ Appendix C: Time-reversal symmetry
 
To prove Eq. (<ref>) we extend the approach established in <cit.> and introduce an operator Ĝ:

    ∫dΘ_kD⃗_1^*(k⃗)·D⃗_2(k⃗)	    =∫dΘ_k⟨ψ_k⃗|r⃗|ψ_1⟩^*·⟨ψ_k⃗|r⃗|ψ_2⟩
        =∫dΘ_k⟨ψ_1|r⃗|ψ_k⃗⟩·⟨ψ_k⃗|r⃗|ψ_2⟩
        =∫dΘ_k⟨ψ_1|r_iP_k⃗r_i|ψ_2⟩
        =⟨ψ_1|r_iP_kr_i|ψ_2⟩
        =⟨ψ_1|Ĝ|ψ_2⟩

We can now test the properties of this operator with respect to time-reversal symmetry.

    Ĝ≡r̂_iP̂_kr̂_i



    Ĝ^†	=r̂_i^†P̂_k^†r̂_i^†
    	=r̂_iP̂_kr̂_i
    	=Ĝ



    [P̂_k,T̂]=[r̂_i,T̂]=0⇒[Ĝ,T̂]=0



    ⟨α|Ĝ|β⟩=⟨α̃|Ĝ|β̃⟩^*


For |α⟩=|α̃⟩ and |β⟩=|β̃⟩, we get ⟨α|Ĝ|β⟩=⟨α|Ĝ|β⟩^*, thus ⟨α|Ĝ|β⟩∈ℝ. This results yields Eq. (<ref>). 



 §.§ Appendix D: Orientational averaging, derivation of Eq. (<ref>) and Eq. (<ref>) 

We shall use the following expressions:

    ∫ dϱ l_iα l_jβ l_k γ l_l δ   = 
        1/30( 
        [ ijkl ikjl iljk ]) 
        ( 
        [  4 -1 -1; -1  4 -1; -1 -1  4 ]) 
        ( 
        [ αβγδ; αγβδ; αδβγ ]) 
       = 
        1/30[ 
        4ijklαβγδ -ikjlαβγδ -iljkαβγδ
       -ijklαγβδ +4ikjlαγβδ -iljkαγβδ
       -ijklαδβγ -ikjlαδβγ +4iljkαδβγ],
 

i.e., 

    ∫ dρ 
        (a⃗^L ·u⃗^L) (b⃗^L ·v⃗^L) 
        (w⃗^L ·c⃗^L) x⃗^L 
       = 
        1/30[ 
        4(a⃗^L·b⃗^L)c⃗^L (u⃗^M·v⃗^M)(w⃗^M·x⃗^M) 
       -(a⃗^L·c⃗^L)b⃗^L (u⃗^M·v⃗^M)(w⃗^M·x⃗^M) 
       - (b⃗^L·c⃗^L)a⃗^L (u⃗^M·v⃗^M)(w⃗^M·x⃗^M) 
       -(a⃗^L·b⃗^L)c⃗^L (u⃗^M·w⃗^M)(v⃗^M·x⃗^M) 
       +4(a⃗^L·c⃗^L)b⃗^L (u⃗^M·w⃗^M)(v⃗^M·x⃗^M) 
       - (b⃗^L·c⃗^L)a⃗^L (u⃗^M·w⃗^M)(v⃗^M·x⃗^M) 
       -(a⃗^L·b⃗^L)c⃗^L (u⃗^M·x⃗^M)(v⃗^M·w⃗^M) 
       -(a⃗^L·c⃗^L)b⃗^L (u⃗^M·x⃗^M)(v⃗^M·w⃗^M) 
       +4 (b⃗^L·c⃗^L)a⃗^L (u⃗^M·x⃗^M)(v⃗^M·w⃗^M) 
        ].

We  derive Eq. (<ref>), which we list here for completeness: 

    ⟨e⃗_Ω^L⟩^↕,↺=σ⟨(Ω⃗_ion·ẑ^L)e⃗_Ω^L⟩=σRυ⟨Ω⃗_ion·e⃗_Ω⟩ẑ^L,

where

    Ω⃗_ion(k,ρ)
        =-|E_ω_k⃗1||E_ω_k⃗2|(d⃗_1·ℰ⃗_1)^*(d⃗_2·ℰ⃗_2)𝖯⃗^+_12(k)sin(ω_12τ),

(Eq. (<ref>)).
Substituting  Eq. (<ref>) into Eq. (<ref>)  we obtain:

    ⟨e⃗_Ω^L⟩^↕,↺=-|E_ω_k⃗1||E_ω_k⃗2|sin(ω_12τ)∫ dϱ(d⃗_1^L·ℰ⃗_1^L)^*(d⃗_2^L·ℰ⃗^L_2)(𝖯⃗^+L_12(k)·ẑ^̂L̂)e⃗_Ω^L

Applying Eq. (<ref>) to Eq. (<ref>) we obtain 

    ⟨e⃗_Ω⟩^↕,↺    =σ/30C[4(d⃗_2·d⃗_1)e⃗_Ω^M-(d⃗_2·e⃗_Ω^M)d⃗_1-(d⃗_1·e⃗_Ω^M)d⃗_2]·𝖯⃗^+_12(k)sin(ω_12τ)ẑ^L,

which is up to notations 𝖯⃗^+_12(k)=υ|𝖯⃗^+_12(k)|e⃗_Ω is equivalent to Eq. (<ref>).

Starting from

    Ω⃗_ion(k,ρ)·e⃗_Ω^M
        =-|E_ω_k⃗1||E_ω_k⃗2|(d⃗_1·ℰ⃗_1)^*(d⃗_2·ℰ⃗_2)(𝖯⃗^+_12(k)·e⃗_Ω^M)sin(ω_12τ),

we shall now evaluate ⟨Ω⃗_ion·e⃗_Ω^M⟩.  Note that we only need to average the part that depends on mutual orientations of the pump pulse and molecule:

    ∫ dϱ(d⃗_1·ℰ⃗_1)^*(d⃗_2·ℰ⃗_2)= 1/3(d⃗_1·d⃗_2)(ℰ⃗_1^*·ℰ⃗_2)=1/3(d⃗_1·d⃗_2)|ℰ_ω_1||ℰ_ω_2|,

yielding Eq. (<ref>):

    ⟨Ω⃗_ion(k)·e⃗_Ω⟩
       =-1/3συ C(d⃗_1·d⃗_2)|𝖯⃗^+_12(k)|sin(ω_12τ).




 §.§ Appendix E: Connection between the curvature and the photoionization yield

In this section, we calculate the photoionization yield W≡∫ |a_k⃗|^2 dΘ_k for two-photon processes considered  in this work. For example, consider a sequence of linear and circularly polarized pulses:

    ℰ⃗^L_i=ℰ_ω_ix̂^L,    E⃗^L_j=E_ω_jk1/√(2)(x̂^L+iσŷ^L)

The photoionization amplitude |a_k⃗|^2 can be written as:

    |a_k⃗|^2=1/2[|E_ω_1k|^2|(d⃗^L_1·ℰ⃗^L_1) |^2[|D⃗_1^L·x̂^L|^2+|D⃗^L_1·ŷ^L|^2+iσ(D⃗_1^*L×D⃗^L_1)·ẑ^L]
    
      +1/2[|E_ω_2k|^2|(d⃗^L_2·ℰ⃗^L_2) |^2[|D⃗^L_2·x̂^L|^2+|D⃗^L_2·ŷ^L|^2+iσ(D⃗_2^*L×D⃗^L_2)·ẑ^L] 
    
      +Re{E_ω_1k^*E_ω_2k(d⃗^L_1·ℰ⃗^L_1)^*·(d⃗^L_2·ℰ⃗^L_2) [(D⃗_1^*L·x̂^L)(D⃗_2^L·x̂^L)+(D⃗_1^*L·ŷ^L)(D⃗_2^L·ŷ^L)+iσ(D⃗_1^*L×D⃗_2^L)·ẑ^L]}

The dichroic (i.e. proportional to σ) photoionization yield is

    ∫|a_k⃗|^2 dΘ_k=σRe{E_ω_1k^*E_ω_2k(d⃗^L_1·ℰ⃗^L)^*·(d⃗^L_2·ℰ⃗^L) [e^iω_12τ∫ i(D⃗_1^*L×D⃗_2^L)dΘ_k]}·ẑ^L=σΩ⃗^L·ẑ^L,

where Ω⃗ is given by Eq. (<ref>).

Next, consider a sequence of two circularly polarized pulses of the form

    E⃗^L_j=E_ω_j1/√(2)(x̂^L+iσŷ^L),    E⃗^L_k⃗j=E_ω_k⃗j1/√(2)(x̂^L+iσŷ^L) .

The photoionization amplitude |a_k⃗|^2 can be written as:

    |a_k⃗|^2 = |d^L_1 ·E^L_1|^2 |D^L_1 ·E^L_k⃗1|^2 + |d^L_2 ·E^L_2|^2 |D^L_2 ·E^L_k⃗2|^2    + 2Re{(D^L_1 ·E^L_k⃗1)^* (d^L_1 ·E^L_1)^* (D^L_2 ·E^L_k⃗2) (d^L_2 ·E^L_2)}
    
        = |d^L_1 ·E^L_1|^2 |D^L_1 ·E^L_k⃗1|^2 + |d^L_2 ·E^L_2|^2 |D^L_2 ·E^L_k⃗2|^2    + 1/2Re{(D^L_1 ·E^L_k⃗1)^* (d^L_1 ·E^L_1)^* (D^L_2 ·E^L_k⃗2) (d^L_2 ·E^L_2)}
       +1/2Re{(d^L_1 ·E^L_1)^* (D^L_1 ·E^L_k⃗1)^* 
        (d^L_2 ·E^L_2) (D^L_2 ·E^L_k⃗2)}

The  part of the  photoionization yield proportional to σ or σ^2 is

    W_σ = ∫ dΘ_k C/2{   [(d_1^*L·x̂^L)(d^L_2·x̂^L)+(d_1^*L·ŷ^L)(d^L_2·ŷ^L)]iσ (D_1^*×D^L_2) ·ẑ^Le^iω_12τ
     +    [ (D_1^*·x̂^L)(D^L_2·x̂^L)+(D_1^*·ŷ^L)(D^L_2·ŷ^L)] iσ (d_1^*L×d^L_2) ·ẑ^Le^iω_12τ
    
        +    [ (d_2^*L·x̂^L)(D^L_1·x̂^L)+(d_2^*L·ŷ^L)(D^L_1·ŷ^L)]iσ (D_2^*L×d^L_1) ·ẑ^Le^iω_12τ
     +    [ (D_2^*L·x̂^L)(d^L_1·x̂^L)+(D_2^*L·ŷ^L)(d^L_1·ŷ^L)]iσ (d_2^*L×D^L_1) ·ẑ^Le^iω_12τ
    
        -   σ^2[(d_1^*L×d^L_2) ·ẑ^L] [(D_1^*L×D^L_2) ·ẑ^L]e^iω_12τ
    
        -   σ^2[(D_2^*L×d^L_1) ·ẑ^L] [(d_2^*L×D^L_1) ·ẑ^L]e^iω_12τ
    }+    c.c.
 
where C≡|E_ω_k⃗1||E_ω_1||E_ω_k⃗2||E_ω_2|. 
 
Let us prove Eq. (<ref>) for circular pump and probe fields, using the yield (Eq. (<ref>)) and the curvature (Eq. (<ref>)). To this end, we calculate the expression

    σΩ^L(ρ)·ẑ^L    = C/2∫ dΘ_k{
       +[(d^L_1·x̂^L)(d^L_2·x̂^L)+(d^L_1·ŷ^L)(d^L_2·ŷ^L)] iσ(D_1^*L×D^L_2) ·ẑ^L e^iω_12τ
       +[(D_1^*L·x̂^L)(D^L_2·x̂^L)+(D_1^*L·ŷ^L)(D^L_2·ŷ^L)] iσ(d^L_1×d^L_2) ·ẑ^L e^iω_12τ
       +[(d^L_1·x̂^L)(D^L_2·x̂^L)+(d^L_1·ŷ^L)(D^L_2·ŷ^L) ]iσ( D⃗_1^*L×d⃗^L_2)·ẑ^L e^iω_12τ
       +[(D_1^*L·x̂^L)(d^L_2·x̂^L)+(D_1^*L·ŷ^L)(d^L_2·ŷ^L) ]iσ( d⃗^L_1×D⃗^L_2)·ẑ^L e^iω_12τ
       -2σ^2(d^L_1×d^L_2)·ẑ^L(D_1^*L×D^L_2) ·ẑ^L e^iω_12τ
       -2σ^2 (D_1^*L×d_2) ·ẑ^L( d⃗_1×D⃗^L_2)·ẑ^L e^iω_12τ
         }+c.c.

The yield in Eq. (<ref>) satisfies the equation 

    W_σ - W_-σ 
        = 
        C∫ dΘ_k {   [(d_1^*L·x̂^L)(d^L_2·x̂^L)+(d_1^*L·ŷ^L)(d^L_2·ŷ^L)]iσ (D_1^*L×D^L_2) ·ẑ^Le^iω_12τ
     +    [ (D_1^*L·x̂^L)(D^L_2·x̂^L)+(D_1^*L·ŷ^L)(D^L_2·ŷ^L)] iσ (d_1^*L×d^L_2) ·ẑ^Le^iω_12τ
    
        +    [ (d_2^*L·x̂^L)(D^L_1·x̂^L)+(d_2^*L·ŷ^L)(D^L_1·ŷ^L)]iσ (D_2^*L×d^L_1) ·ẑ^Le^iω_12τ
     +    [ (D_2^*L·x̂^L)(d^L_1·x̂^L)+(D_2^*L·ŷ^L)(d^L_1·ŷ^L)]iσ (d_2^*L×D^L_1) ·ẑ^Le^iω_12τ.
 
On the other hand we have 

    σΩ_σ^L(ρ)·ẑ^L - (-σ)Ω_-σ^L(ρ)·ẑ^L 
           = C∫ dΘ_k{
       +[(d^L_1·x̂^L)(d^L_2·x̂^L)+(d^L_1·ŷ^L)(d^L_2·ŷ^L)] iσ(D_1^*L×D^L_2) ·ẑ^L e^iω_12τ
       +[(D_1^*L·x̂^L)(D^L_2·x̂^L)+(D_1^*L·ŷ^L)(D^L_2·ŷ^L)] iσ(d^L_1×d^L_2) ·ẑ^L e^iω_12τ
       +[(d^L_1·x̂^L)(D^L_2·x̂^L)+(d^L_1·ŷ^L)(D^L_2·ŷ^L) ]iσ( D⃗_1^*L×d⃗^L_2)·ẑ^L e^iω_12τ
       +[(D_1^*L·x̂^L)(d^L_2·x̂^L)+(D_1^*L·ŷ^L)(d^L_2·ŷ^L) ]iσ( d⃗^L_1×D⃗^L_2)·ẑ^L e^iω_12τ
         }+c.c.,

such that

    W_σ - W_-σ   = C
        σΩ_σ^L(ρ)·ẑ^L - (-σ)Ω_-σ^L(ρ)·ẑ^L 
       = 
        σ(Ω_σ^L(ρ) + Ω_-σ^L(ρ))·ẑ^L.




 §.§ Appendix E: Symmetric contributions of the  curvature in bound and continuum states to enantio-sensitive orientation 

A symmetric contribution of the bound and continuum curvature to the enantio-sensitive orientation of molecular cations emerges in the case where both the excitation and ionization is performed by circularly polarized pulses and involve a single complex intermediate bound state. In this case, an oriented molecular vector v⃗: 

    <v⃗>     =|A^(2)|^2∫dΩ_k^M∫dϱ (v⃗^L·ẑ^L)|D⃗^L·F⃗^L|^2|d⃗^L·F⃗^L|^2.

Here <...> corresponds to orientational averaging, D stands for photoionization dipole matrix element, while d stands for photoexcitation dipole matrix element.

    |d⃗^L·F⃗^L|^2    =|(d⃗_r^L+id⃗_i^L)·F⃗^L|^2
        =(d⃗_r^L·F⃗^L+id⃗_i^L·F⃗^L)^*(d⃗_r^L·F⃗^L+id⃗_i^L·F⃗^L)
        =|d⃗_r^L·F⃗^L|^2+|d⃗_i^L·F⃗^L|^2+i(d⃗_r^L·F⃗^L*)(d⃗_i^L·F⃗^L)-i(d⃗_i^L·F⃗^L*)(d⃗_r^L·F⃗^L)
        =|d⃗_r^L·F⃗^L|^2+|d⃗_i^L·F⃗^L|^2+i(d⃗_r^L×d⃗_i^L)·(F⃗^L*×F⃗^L)
        =|d⃗_r^L·F⃗^L|^2+|d⃗_i^L·F⃗^L|^2+1/2(d⃗^L*×d⃗^L)·(F⃗^L*×F⃗^L)


After orientation averaging the first two terms read as:


    ∫dϱ (v⃗^L·ẑ^L)|D⃗^L·F⃗^L|^2|d⃗_r^L·F⃗^L|^2    =1/30d_r^2{[2v⃗-(v⃗·d̂_r)d̂_r]·(D⃗^*×D⃗)}{[ẑ·(F⃗^*×F⃗)]|F⃗|^2}



    ∫dϱ (v⃗^L·ẑ^L)|D⃗^L·F⃗^L|^2|d⃗_i^L·F⃗^L|^2=1/30d_i^2{[2v⃗-(v⃗·d̂_i)d̂_i]·(D⃗^*×D⃗)}{[ẑ·(F⃗^*×F⃗)]|F⃗|^2}


Their sum yields


    ∫dϱ (v⃗^L·ẑ^L)|D⃗^L·F⃗^L|^2|d⃗_r^L·F⃗^L|^2+∫dϱ (v⃗^L·ẑ^L)|D⃗^L·F⃗^L|^2|d⃗_i^L·F⃗^L|^2
        =1/30{[2|d⃗|^2v⃗-Re{(v⃗·d⃗^*)d⃗}]·(D⃗^*×D⃗)}{[ẑ·(F⃗^*×F⃗)]|F⃗|^2}


Orientation averaging <cit.> of the third term in Eq. (<ref>)
yields


    1/2∫dϱ (v⃗^L·ẑ^L)|D⃗^L·F⃗^L|^2[(d⃗^L*×d⃗^L)·(F⃗^L*×F⃗^L)]=1/2g⃗^(4)· M^(4)f⃗^(4)


where (see Eq. (19) in <cit.>)


    g⃗^(4)=([ (v⃗·D⃗^*)[D⃗·(d⃗^*×d⃗)]; (v⃗·D⃗)[D⃗^*·(d⃗^*×d⃗)]; [v⃗·(d⃗^*×d⃗)](D⃗^*·D⃗) ])



    M^(4)=1/30([  4 -1 -1; -1  4 -1; -1 -1  4 ])



    f⃗^(4)=([ (ẑ·F⃗^*)[F⃗·(F⃗^*×F⃗)]; (ẑ·F⃗)[F⃗^*·(F⃗^*×F⃗)]; [ẑ·(F⃗^*×F⃗)](F⃗^*·F⃗) ])=([                   0;                   0; [ẑ·(F⃗^*×F⃗)]|F⃗|^2 ])



    g⃗^(4)· M^(4)f⃗^(4)    =1/30([ (v⃗·D⃗^*)[D⃗·(d⃗^*×d⃗)]; (v⃗·D⃗)[D⃗^*·(d⃗^*×d⃗)]; [v⃗·(d⃗^*×d⃗)](D⃗^*·D⃗) ])·([ -1; -1;  4 ])[ẑ·(F⃗^*×F⃗)]|F⃗|^2
        =1/30{-(v⃗·D⃗^*)[D⃗·(d⃗^*×d⃗)]-(v⃗·D⃗)[D⃗^*·(d⃗^*×d⃗)]
        +4[v⃗·(d⃗^*×d⃗)](D⃗^*·D⃗)}[ẑ·(F⃗^*×F⃗)]|F⃗|^2


We can factorize either v⃗, as in


    1/30{ -[D⃗·(d⃗^*×d⃗)]D⃗^*-[D⃗^*·(d⃗^*×d⃗)]D⃗+4(d⃗^*×d⃗)|D⃗|^2}·v⃗
        =1/302{ 2(d⃗^*×d⃗)|D⃗|^2-Im{[D⃗·(d⃗^*×d⃗)]D⃗^*}}·v⃗


or d⃗^*×d⃗, as in 


    1/30{ -(v⃗·D⃗^*)D⃗-(v⃗·D⃗)D⃗^*+4|D⃗|^2v⃗}·(d⃗^*×d⃗)
    
    =    1/302{ 2|D⃗|^2v⃗-Re{(v⃗·D⃗^*)D⃗}}·(d⃗^*×d⃗)


The later is analogous to the sum of the first two terms in Eq. (<ref>).
Putting everything together we get 


    <v⃗ >=|A^(2)|^2{∫dΩ_kG(k⃗)}{[ẑ·(F⃗^*×F⃗)]|F⃗|^2}



    G     =[2|d⃗|^2v⃗-Re{(v⃗·d⃗^*)d⃗}]·(D⃗^*×D⃗)
        +[2|D⃗|^2v⃗-Re{(v⃗·D⃗^*)D⃗}]·(d⃗^*×d⃗)

The two terms correspond to a linear-circular photon absorption, and circular-linear photon absorption. 
In both terms,  the first term in square brackets represents the isotropic term, while the second term reflects the alignment due to absorption of the first photon. Note that in this expression d⃗ and D⃗ appear on the
same footing. 




 §.§ Appendix F: Berry curvature for circular pump and probe fields

The wave function expressed in first order perturbation theory is

    |ψ⟩ = |0⟩ + i ∑_jd⃗_j ·E⃗_j |k⟩ - ∑_j ∫ dΘ_k (D⃗_j ·E⃗_kj)(d⃗_j ·E⃗_j) e^-iω_jτ|k⃗⟩.

The corresponding curvature is 

    Ω = i ⟨∇_E ψ|×|∇_E ψ⟩



    |∇_E ψ⟩ = i ∑_j |E_ω_j|d⃗_j |k⟩ - ∑_j |E_ω_j||E_ω_k⃗j|∫ dΘ_k[D⃗_j (d⃗_j ·E⃗_j) + (D⃗_j ·E⃗_kj)d⃗_j] e^-iω_jτ|k⃗⟩

Using the orthogonality of the basis, we get

    -iΩ⃗   = ∑_j |E_ω_j|d⃗_j^* ×d⃗_j 
          + ∑_jl 
        |E_ω_j||E_ω_k⃗j||E_ω_l||E_ω_k⃗l| 
        ∫ dΘ_k[D⃗_j^* (d⃗_j ·E⃗_j)^* + (D⃗_j ·E⃗_kj)^*d⃗_j^*] ×[D⃗_l (d⃗_l ·E⃗_l) + (D⃗_l ·E⃗_kl)d⃗_l] e^iω_jlτ

Since the dipole moments d⃗_l are real the connection simplifies to

    -iΩ⃗ = ∑_jl|E_ω_j||E_ω_k⃗j||E_ω_l||E_ω_k⃗l|∫ dΘ_k[D⃗_j^* (d⃗_j ·E⃗_j^*) + (D⃗_j ·E⃗_kj)^*d⃗_j] ×[D⃗_l (d⃗_l ·E⃗_l) + (D⃗_l ·E⃗_kl)d⃗_l] e^iω_jlτ

We employ the identity

    ∫ dΘ_kD⃗_j ×D⃗_j^* = 0 ,

such that terms satisfying j = l vanish, 

    -iΩ⃗ = ∑_j≠ l|E_ω_j||E_ω_k⃗j||E_ω_l||E_ω_k⃗l| 
        ∫ dΘ_k[
           (d⃗_j ·E⃗_j^*)(d⃗_l ·E⃗_l)(D⃗_j^*×D⃗_l)
    
        +   (d⃗_j ·E⃗_j^*)(D⃗_l ·E⃗_kl)(D⃗_j^*×d⃗_l)
    
        +   (D⃗_j ·E⃗_kj)^*(d⃗_l ·E⃗_l)(d⃗_j×D⃗_l )
    
        +   (D⃗_j ·E⃗_kj)^*(D⃗_l ·E⃗_kl)(d⃗_j×d⃗_l )
        ] e^iω_jlτ.

Due to the anti-symmetry of the cross product, the curvature  can be expressed as 

    Ω⃗ = -2C∫ dΘ_k[
           Im{(d⃗_1 ·E⃗_1^*)(d⃗_2 ·E⃗_2)(D⃗_1^*×D⃗_2)e^iω_12τ}
    
        +   Im{(d⃗_1 ·E⃗_1^*)(D⃗_2 ·E⃗_k2)(D⃗_1^*×d⃗_2)e^iω_12τ}
    
        +   Im{(D⃗_1 ·E⃗_k1)^*(d⃗_2 ·E⃗_2)(d⃗_1×D⃗_2 )e^iω_12τ}
    
        +   Im{(D⃗_1 ·E⃗_k1)^*(D⃗_2 ·E⃗_k2)(d⃗_1×d⃗_2 )e^iω_12τ}] ,

where we defined 

    C = |E_ω_1||E_ω_k⃗1||E_ω_2||E_ω_k⃗2| .
 



