Disentangling enantiosensitivity from dichroism using bichromatic fields

Abstract

We discuss how tensorial observables, available in photoelectron angular distributions resulting from interaction between isotropic chiral samples and cross polarized ω–2ω bichromatic fields, allow for chiral discrimination without chiral light and within the electric-dipole approximation. We extend the concept of chiral setup [A. F. Ordonez and O. Smirnova, Phys. Rev. A, 2018, 98, 063428], which explains how chiral discrimination can be achieved in the absence of chiral light, to the case of tensorial observables. We derive selection rules for the enantiosensitivity and dichroism of the b_l,m coefficients describing the photoelectron angular distribution valid for both weak and strong fields and for arbitrary ω–2ω relative phase. Explicit expressions for simple perturbative cases are given. We find that, besides the known dichroic non-enantiosensitive [R. E. Goetz, C. P. Koch and L. Greenman, J. Chem. Phys., 2019, 151, 074106], and dichroic-and-enantiosensitive b_l,m coefficients found recently [P. V. Demekhin, Phys. Rev. A, 2019, 99, 063406], there are also enantiosensitive non-dichroic b_l,m coefficients. These reveal the molecular enantiomer independently of the relative phase between the two colors and are therefore observable even in the absence of stabilization of the ω–2ω relative phase.

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1 Introduction

More than two centuries after the pioneering observations of Biot and Arago,¹ the interaction between light and chiral matter² remains a very active field of research.^3–7 This research effort is fueled not only by the interest in finding new ways of manipulating light and matter but also by the homochirality of life. While the molecule–molecule interactions that take place in biological systems are often strongly enantiosensitive,⁸ the enantiosensitive response in “traditional” light–matter interactions is usually very weak. This weakness, which limits the potential of light-based applications, is prevalent in situations where the electric–dipole approximation is well justified but the chiral effects appear as small corrections such as the magnetic-dipole interaction.⁹ Besides ingenious methods to cope with such situations,^10–12 weakly enantiosensitive responses can be avoided from the outset by relying on chiral effects occurring within the electric–dipole approximation.^13,14

Among the chiral electric–dipole effects, photoelectron circular dichroism (PECD)^15–19 is very well established and has been shown to consistently yield strongly enantiosensitive signals across many molecular species²⁰ and different photoionization regimes.^21,22 In PECD, isotropically oriented chiral molecules are photoionized using circularly polarized light and the photoelectron angular distribution displays a so-called forward–backward asymmetry (asymmetry with respect to the polarization plane). This asymmetry is both enantiosensitive (opposite for opposite enantiomers) and dichroic (opposite for opposite polarizations), and results from the lack of mirror symmetry of the chiral molecules (and thus of the light-matter system) with respect to the plane of polarization. For an isotropically oriented achiral molecule and within the electric-dipole approximation (i.e. ignoring light-propagation effects), the light-matter system is mirror-symmetric with respect to the polarization plane and therefore forward–backward asymmetric observables resulting from a single-molecule response are symmetry-forbidden. For a randomly oriented chiral molecule the mirror symmetry of the light-matter system is absent and there are usually no further symmetries preventing the emergence of forward–backward asymmetric observables. In the one-photon case, the forward–backward asymmetry results from the non-zero molecular rotational invariant describing the average circular polarization of the photoionization-dipole vector field.²³

Very recently, the response of chiral molecules to more elaborate field polarizations has been investigated in the multiphoton and strong-field regimes.^24–26 In these works, encompassing theory^24–26 and experiment,²⁶ a bichromatic field with frequencies ω and 2ω linearly polarized perpendicular to each other yields a new type of enantiosensitive and dichroic asymmetry in the photoelectron angular distribution. The Lissajous figure of this bichromatic field has an eight-like shape for particular phase relations between the two colors, and therefore “rotates” in opposite directions in the first and second halves of its cycle, with the direction of rotation of the field locked to the sign of its ω component. For example, the rotation is clockwise when the ω component is positive and counterclockwise when the ω component is negative. The observed asymmetry in the photoelectron angular distribution corresponds to a correlation between the so-called forward–backward direction and the up-down direction determined by the ω field (see Fig. 1 in ref. 24). So far, this asymmetry has been analyzed on the basis of a subcycle PECD-like picture where electrons detected in the upper hemisphere have an e.g. positive forward–backward asymmetry because they were produced by a field rotating e.g. clockwise, while the electrons detected in the lower hemisphere have a negative forward–backward asymmetry because they were produced by a field rotating counterclockwise.^24,26 Extensions of this reasoning to account for non-zero asymmetries for other relative phases between the ω and 2ω components have been considered in ref. 25 and 26.

Here we approach the description of the photoelectron angular distribution taking explicitly into account its tensorial character, the role of tensorial detectors in forming the chiral setup required to distinguish between opposite enantiomers, and the symmetry of the light-matter system. In Section 2 we discuss general considerations regarding how to distinguish between opposite enantiomers in isotropic samples without relying on the chirality of a light field. In Section 3 we discuss how tensorial observables can be used for the construction of chiral setups. In Section 4 we consider the interaction of an ω–2ω cross polarized field with isotropic molecular samples and derive selection rules for multipolar observables. Then we consider the perturbative description of photoionization and exploit the technique presented in ref. 23 to obtain explicit formulas for some representative b̃_l,m coefficients of the photoelectron angular distribution displaying different combinations of dichroism and enantiosensitivity. Finally, we list our conclusions in Section 5. Analogous results but for the case of charge multipoles induced via excitation of bound states are presented in ref. 27.

2 Chiral probes

Distinguishing between the left version (L) and the right version (R) of a chiral object invariably requires interaction with another chiral object (say R′). The difference between the interactions L + R′ and R + R′ is the essence of any enantiosensitive phenomenon. Here we are interested in phenomena where light is used to distinguish between opposite enantiomers of isotropically oriented chiral molecules. In the simplest case, one lets circularly polarized light of a given handedness interact with a chiral sample and measures a scalar, namely the amount of light that is absorbed. In this situation, known as circular dichroism (CD), the chiral probe is the circularly polarized light and its handedness (a pseudoscalar) is given by its helicity, i.e. the projection of the photon spin (the rotation direction of the light at a given point in space, a pseudovector) on the propagation direction of the light (a vector). Another canonical example is optical activity, where one passes linearly polarized light through a chiral sample and measures the rotation of the polarization plane. In this case one measures an angle (a pseudovector). To measure it one must define a positive and a negative direction in the laboratory frame. Although such definition is just as arbitrary as defining what is left and what is right, it is a fundamental step in the measurement process. Once it has been defined, the handedness of the probe is given by the projection of the positive unit angle pseudovector on the propagation direction of the light (see Fig. 8 in ref. 14). That is, the chiral probe in this case is the chiral setup formed by the (achiral) light and the (achiral) detector (which encodes the definition of positive and negative rotations) together. This shows how a chiral setup may probe the handedness of a molecule in the absence of chiral light.

Circular dichroism and optical activity have in common that the handedness of the chiral probe relies on the propagation direction of the light. Since that propagation direction is immaterial within the electric-dipole approximation, unless one considers corrections to the electric-dipole approximation the probe ceases to be chiral and both effects vanish. To the extent that such corrections are typically small at the single molecule level, these effects are also correspondingly small. In order to obtain bigger enantiosensitive signals at the single-molecule level, a probe which is chiral within the electric-dipole approximation is required. Furthermore, if the result of the measurement is a scalar (like in circular dichroism), the light itself must be chiral. The concept of light which is chiral within the electric-dipole approximation, i.e. locally-chiral light, has been recently developed in ref. 6. If the result of the measurement is a polar vector, such as a net photoelectron current along a Cartesian axis, then the detector required to measure the vector must define (again, in an arbitrary fashion) a positive and a negative direction. In addition, if the polarization of the light allows the definition of a pseudovector, as is the case for example for circularly polarized light, where the pseudovector indicates the photon's spin, then one can define a chiral setup with its handedness given by the projection of the light pseudovector on the positive direction defined by the detector. This type of chiral setup is common to a series of recently discovered phenomena that range from rotational dynamics^28–30 to photoionization,^15–18 and it was recently discussed in ref. 13. In what follows we will extend the concept of chiral setups, to include those that rely on tensors of rank 2 (relevant for the results in ref. 24–26) and higher.

3 Tensor observables and chiral setups

Second and higher-rank tensors emerge naturally for observables which depend on a vector. Charge densities and photoelectron angular distributions (PADs) offer exactly such kind of observable and, as any other function which depends on a vector, they can be expanded aswhere k⃑ is the photoelectron momentum and we have chosen to do the expansion in terms of real spherical harmonics Ỹ^m_l. In the case of a charge distribution we replace the photoelectron momentum k⃑ by the position r⃑ and b̃_l,m could be a time dependent quantity.²⁷ The b̃_l,m coefficients not only encode all the information contained in W(k⃑) but are also examples of tensors of rank l (see e.g. Section 4.10 in ref. 31).

In principle, the measurement of a particular b̃_l,m coefficient can be performed directly by using a detector with a structure that reflects the corresponding Ỹ^m_l. For example, a detector for b̃_0,0 would simply sum the counts in all directions. A detector for b̃_1,0 would have two plates arranged as in Fig. 1a and b, so that it would add all the counts on one of them (red) and subtract all the counts on the other (blue). The choice of which plate adds and which plate subtracts is what physically defines the ẑ direction of the laboratory frame. A detector for b̃_2,−2 would have four plates arranged as in Fig. 1c, with a pair of opposite plates adding counts and the other pair subtracting counts. In this case, the detector defines the directions that correspond to a positive correlation between x and y, and those that correspond to a negative correlation between x and y. Analogously, a detector for b̃_3,−2 has eight plates arranged as in Fig. 1d, and distinguishes positive from negative correlations of x, y, and z.

Fig. 1 Chiral (a, c and d) and achiral (b) setups consisting of a Lissajous figure and detectors. Counts on red/blue detectors are added/subtracted. The color of the Lissajous figure indicates the phase of the oscillation. (a) A field circularly polarized in the xy plane and a detector for b̃_1,0. (b), (c), and (d) A cross-polarized ω–2ω field in the xz plane and a detector for b̃_1,0, b̃_2,−2 and b̃_3,−2, respectively.

The combination of a b̃_l,m-specific detector and the Lissajous figure of the electric field can make up a chiral setup as shown in Fig. 1a, c and d. These setups are non-superimposable on their mirror images. This is particularly simple to see in Fig. 1a, c and d for reflections with respect to the polarization plane, which do not change the Lissajous figure but do swap blue and red plates. The chirality of these setups allows them to distinguish between opposite enantiomers of a chiral molecule. This is in contrast with the setup in Fig. 1b, which is symmetric with respect to reflection in the polarization plane and thus is not chiral and cannot distinguish between opposite enantiomers of a chiral molecule.

Of course, a chiral setup is only relevant if the light-matter system is indeed asymmetric enough that it can yield a corresponding non-zero b̃_l,m. In other words, the concept of a chiral setup answers the question of how we can distinguish opposite enantiomers without relying on the chirality of light, but it is the lack of symmetry of the light-matter system itself the deciding factor which determines the emergence of an enantiosensitive observable in the first place (see e.g.Fig. 2 in ref. 13). We now turn to the analysis of a specific family of Lissajous figures to illustrate this in detail.

Fig. 2 Lissajous curves of the field (2) for different values of ϕ. The position of the red arrow indicates t = 0.

4 ω–2ω cross polarized field

Consider a two-color field of the form

E⃑(t) = E_ω cos(ωt)x̂ + E_2ω cos(2ωt + ϕ)ẑ (2)

This field is illustrated in Fig. 2 for different phases†ϕ. If we denote rotations by π around the z axis by R̂^π_z and time shifts of π/ω by T_π/ω, then the joint system (field and isotropic chiral molecules) is invariant with respect to R̂^π_zT̂_π/ω. Clearly, the resulting observables must also be invariant with respect to R̂^π_zT̂_π/ω. That is, the symmetry-allowed b̃_l,m coefficients must satisfy R̂^π_zT̂_π/ωb̃_l,mỸ^m_l = b̃_l,mỸ^m_l. This corresponds to two scenarios: either R̂^π_zỸ^m_l = Ỹ^m_l and b̃_l,m contains only frequencies 2nω, or R̂^π_zỸ^m_l = −Ỹ^m_l and b̃_l,m contains only frequencies (2n + 1)ω, where n = 0, 1, 2, …. Furthermore, since a reflection [small sigma, Greek, circumflex]_y with respect to the y = 0 plane swaps the enantiomers while leaving the field invariant, an enantiosensitive b̃_l,m is associated to a Ỹ^m_l satisfying [small sigma, Greek, circumflex]_yỸ^m_l = −Ỹ^m_l. And, since a rotation R̂^π_x of π around the x axis changes the phase ϕ in the field by π while leaving the molecules invariant, a dichroic‡b̃_l,m is associated to a Ỹ^m_l satisfying R̂^π_xỸ^m_l = −Ỹ^m_l. These properties, which are valid independently of the ionization regime, are summarized in Tables 1 and 2.

Evidently, the values of l and m that determine whether a given b̃_l,m is symmetry-allowed and whether it is enantiosensitive and/or dichroic depends on how the field is oriented with respect to the axes. The choice made here [eqn (2)] yields the rather simple conditions in Tables 1 and 2. Conditions corresponding to other choices are given in Appendix A for the sake of comparison with ref. 24–26.

Tables 1 and 2 reveal two important features. First, the spatial structure of the response depends on whether it oscillates with a frequency which is an even (including zero) or an odd multiple of the fundamental frequency (see also ref. 32). Second, in contrast to a circularly polarized field where dichroism goes hand in hand with enantiosensitivity, the symmetry of the field (2) leads to four types of signals:

Table 1 Conditions imposed by symmetry on the b̃_l,m coefficients describing the photoelectron angular distribution (1) resulting from the interaction between an isotropic molecular sample and the field (2)

b̃ l,m	(l,m) condition
Symmetry-allowed at 2nω	Even m
Symmetry-allowed at (2n + 1)ω	Odd m
Enantiosensitive	m < 0
Dichroic	[(Odd l − m) and (m ≥ 0)] or
	(even l − m) and (m < 0)

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Table 2 (l,m) conditions derived from Table 1 for non-vanishing enantiosensitive and/or dichroic b̃_l,m coefficients

Symmetry-allowed b̃l,m	(l,m) condition at 2nω	(l,m) condition at (2n + 1)ω
Enantiosensitive and dichroic	(Even l) and (even m < 0)	(Odd l) and (odd m < 0)
Enantiosensitive non-dichroic	(Odd l) and (even m < 0)	(Even l) and (odd m < 0)
Dichroic non-enantiosensitive	(Odd l) and (even m > 0)	(Even l) and (odd m > 0)

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(i) Non-dichroic and non-enantiosensitive.

(ii) Dichroic and enantiosensitive.

(iii) Dichroic and non-enantiosensitive.

(iv) Enantiosensitive and non-dichroic.

Type (i) and type (ii) signals are well known from traditional CD and PECD. Type (iii) signals are well known in atoms subject to light fields whose Lissajous figure is not symmetric under spatial inversion.³³ They have also been recently calculated for the case of randomly oriented chiral molecules³⁴ for parallel polarizations of ω and 2ω. Here, type (iii) signals are due to the asymmetry of the field (2) along the z direction (see Fig. 2). Type (iv) signals are more exotic but apparently they can also occur in the context of magnetic effects beyond the electric-dipole approximation,^35,36 where to the best of our knowledge they remain to be confirmed by experiment. As we will show next, photoionization may prove to be a better candidate for experimentally measuring type (iv) signals.

4.1 Photoionization

The photoelectron angular distribution accumulated over many cycles of the field (2) corresponds to a signal with zero frequency and must satisfy the conditions given in Table 1 for frequencies 2nω. For convenience, we list the properties of the symmetry-allowed b̃_l,m coefficients for l up to four in Table 3. From this table we can see that e.g. b̃_1,0 is dichroic but not enantiosensitive, b̃_2,−2 is enantiosensitive and dichroic, and b̃_3,−2 is enantiosensitive but not dichroic. Since Ỹ⁰₁(k̂) ∝ k_z, Ỹ⁻²₂(k̂) ∝ k_xk_y, and Ỹ⁻²₃(k̂) ∝ k_xk_yk_z, then b̃_1,0 is associated to asymmetry along the direction of the 2ω field (ẑ), b̃_2,−2 is associated to correlations between the direction of the ω field (x̂) and the direction perpendicular to the polarization plane (ŷ), and b̃_3,−2 is associated to correlations of the three directions corresponding to the ω field (x̂), the 2ω field (ẑ), and the perpendicular to the polarization plane (ŷ).

Table 3 Dichroism and enantiosensitivity of symmetry-allowed b̃_l,m coefficients describing the photoelectron angular distribution (1) resulting from interaction of an isotropic molecular sample and the field (2) for l up to 4. N = No and Y = Yes

Symmetry-allowed	b̃ 0,0	b̃ 1,0	b̃ 2,−2	b̃ 2,0	b̃ 2,2	b̃ 3,−2	b̃ 3,0	b̃ 3,2	b̃ 4,−4	b̃ 4,−2	b̃ 4,0	b̃ 4,2	b̃ 4,4
Enantiosensitive	N	N	Y	N	N	Y	N	N	Y	Y	N	N	N
Dichroic	N	Y	Y	N	N	N	Y	Y	Y	Y	N	N	N

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In ref. 23 we showed that, in general, a b̃_l,m coefficient is enantiosensitive if and only if it results from interference between pathways with N₁ and N₂ photons, respectively, and l + N₁ + N₂ is odd; or if it results from a direct pathway and l is odd. These conditions together with Table 2 tell us that a dichroic non-enantiosensitive b̃_l,m (with odd l) can only occur as the result of interference between pathways involving an even and an odd number of photons, respectively (so that N₁ + N₂ is odd and l + N₁ + N₂ is even). For example, b̃_1,0 contributes to the photoelectron peak where absorption of two ω photons interferes with absorption of one 2ω photon (see Fig. 3a). The same condition (odd N₁ + N₂) applies for a dichroic and enantiosensitive b̃_l,m (even l) such as b̃_2,−2. In contrast, an enantiosensitive non-dichroic b̃_l,m (odd l) can occur as the result of either a direct pathway involving at least one ω photon and one 2ω photon, or as the result of interference between two pathways both with an even or both with an odd number of photons (even N₁ + N₂). For example, b̃_3,−2 contributes to the photoelectron peak corresponding to absorption of one ω photon followed by absorption of one 2ω photon (see Fig. 3b) and vice versa, or as the result of interference between absorption of two 2ω photons and four ω photons.

Fig. 3 (a) Simplest scheme to generate a non-zero b̃_1,0 (dichroic non-enantiosensitive) and b̃_2,−2 (dichroic and enantiosensitive) using the field (2). (b) Simplest scheme to generate a non-zero b̃_3,−2 (enantiosensitive non-dichroic) using the field (2). |0〉 and |j〉 are bound states and |k⃑〉 is a continuum state.

As an example, let us calculate explicit expressions for b̃_1,0, b̃_2,−2, and b̃_3,−2. For the process depicted in Fig. 3a, the interference between the two pathways gives rise to a non-zero orientation-averaged b̃_1,0 given by (see Appendix B)

b̃_1,0(k) = A^(1)*A⁽²⁾g_1,0f_1,0 + c.c., (3)

where A⁽¹⁾ and A⁽²⁾ are complex-valued constants depending on detunings and pulse envelopes, c.c. denotes the complex conjugate, g_1,0 is a (complex-valued) molecular rotational invariant, and f_1,0 is a setup (i.e. field + detector) rotational invariant. g_1,0 is a scalar§ (in contrast to a pseudoscalar) and therefore b̃_1,0 is not enantiosensitive. The expression for f_1,0 reads aswhere E⃑_2ω ≡ E_2ωe^−iϕẑ, E⃑_ω ≡ E_ωx̂. Eqn (4) shows that the setup rotational invariant f_1,0 is a scalar involving the field vectors E⃑_2ω and E⃑_ω, and the ẑ axis. As discussed in Section 3, the axis vector ẑ is defined by the detector needed to measure b̃_1,0. From eqn (4) it is evident that f_1,0 changes sign when ϕ is shifted by π, and therefore b̃_1,0 is dichroic.

Similarly, the interference between the two pathways in Fig. 3a gives rise to a non-zero orientation-averaged b̃_2,−2 given by (see Appendix C)

b̃_2,−2 = A^(1)*A⁽²⁾g_2,−2f_2,−2 + c.c., (5)

where the molecular rotational invariant g_2,−2 is a (complex-valued) pseudoscalar and therefore b̃_2,−2 is enantiosensitive. The setup rotational invariant, given byis also a pseudoscalar and it involves the x̂ and ŷ axes, which are defined by the detector needed to measure b̃_2,−2 (see Section 3). Just like g_2,−2 is a pseudoscalar that distinguishes between molecules with opposite handedness, f_2,−2 is a pseudoscalar that distinguishes between chiral setups of opposite handedness. Note that the detector for b̃_2,−2 does not directly tell apart a positive x from a negative x, or a positive y from a negative y. It only tells apart positive correlations of x and y from negative correlations of x and y. This is consistent with the invariance of f_2,−2 with respect to a simultaneous inversion of x̂ and ŷ. Furthermore, eqn (6) shows that f_2,−2 changes sign when ϕ is shifted by π, and therefore b̃_2,−2 is dichroic. We remark, that although the right hand side of eqn (4) and (6) look exactly the same after performing the vector operations, f_1,0 and f_2,−2 emerge from different setup rotational invariants of a different physical nature: one is a scalar involving only the ẑ axis, and the other is a pseudoscalar involving correlation between the x̂ and ŷ axes.

We consider now the simplest process leading to the enantiosensitive but not dichroic term b̃_3,−2, i.e. we consider the absorption of one ω photon and one 2ω photon, resonantly enhanced through a bound state as shown in Fig. 3b. In this case we get (see Appendix D)

b̃_3,−2(k) = |A⁽²⁾|²g_3,−2f_3,−2, (7)

where the rotational invariant g_3,−2 is a (real-valued) pseudoscalar, which makes b̃_3,−2 enantiosensitive. The setup rotational invariant f_3,−2 is given by

f _3,−2 is also a pseudoscalar, however it does not record ϕ and b̃_3,−2 is therefore not dichroic. The robustness of f_3,−2 against changes of ϕ means that recording b̃_3,−2 allows distinguishing opposite enantiomers in the absence of stabilization of the ω–2ω phase shift ϕ.

Eqn (3)–(8) confirm our expectations based on general symmetry arguments according to which b̃_1,0 is dichroic and non-enantiosensitive, b̃_2,−2 is dichroic and enantiosensitive, and b̃_3,−2 is enantiosensitive and non-dichroic. In addition, these equations show that b̃_1,0, b̃_2,−2, and b̃_3,−2 are in general not zero for the specific processes considered here. This is important because although a given b̃_l,m may be symmetry allowed according to the general symmetry analysis above, further “hidden” symmetries¶ involved in a specific process may forbid it. For example, although the geometry of a monochromatic elliptical field allows for a non-zero b̃_3,−2, the presence of the triple product in eqn (8) means that opposite photon orderings contribute to b̃_3,−2 with opposite signs (see Appendix E). Such photon orderings are degenerate in the monochromatic case and thus the two contributions cancel each other out (see Appendix F). However, it must be kept in mind that the hidden symmetry preventing a non-zero b̃_3,−2 for elliptical fields is specific to the 2-photon process we investigated here, so it may be broken in higher order processes or in the strong field regime. That is, photoionization induced by elliptically polarized strong fields may indeed yield a non-zero enantiosensitive and non-dichroic b̃_3,−2 coefficient. This would be analogous to how b̃_2,−2 is symmetry-forbidden in the elliptical one-photon case²³ but it is symmetry allowed in the elliptical strong-field case.³⁸

5 Conclusions

We discussed how tensorial observables offer new opportunities for constructing chiral setups able to distinguish between opposite enantiomers in isotropic samples without relying on the chirality of light. As a concrete example we considered the interaction of an ω–2ω cross polarized field with isotropic samples and we found selection rules for l and m that specify if a multipole coefficient b̃_l,m is symmetry allowed, if it is dichroic (sensitive to changes of π in the ω–2ω relative phase), and if it is enantiosensitive. These b̃_l,m coefficients are relevant in the description of photoelectron angular distributions and in the description of induced multipoles of bound charge distributions (discussed in ref. 27). We found that the enantiosensitivity and the dichroism of a b̃_l,m coefficient are in general independent of each other. That is, in addition to the usual types of b̃_l,m coefficients in isotropic samples, namely: (i) non-dichroic and non-enantiosensitive, (ii) dichroic and enantiosensitive, and (iii) dichroic and non-enantiosensitive, we found the more exotic possibility of (iv) enantiosensitive and non-dichroic b̃_l,m coefficients.

We derived analytic expressions for the lowest rank b̃_l,m coefficients in photoelectron angular distributions corresponding to types (ii)–(iv) for the case of one- vs. two-photon absorption and for the case of ω + 2ω absorption. Unlike type (ii) coefficients, type (iv) coefficients allow enantiomeric discrimination in the absence of stabilization of the ω–2ω relative phase and remain to be numerically calculated and experimentally observed. Such verification is within reach with current theoretical³⁹ and experimental^40,41 capabilities which provide access to the full photoelectron momentum distribution. Imaging of the type (iv) coefficient b̃_3,−2 discussed here with a velocity map imaging detector as e.g. in ref. 26 is complicated by two facts: (a) the lack of cylindrical symmetry prevents Abel inversion and (b) projection of the Ỹ_3,−2 contribution to the momentum distribution on the xy (or yz) plane cancels exactly (see Fig. 1c). However, re-orientation of the detector so as to project the momentum distribution on e.g. the x = z plane, as shown in Fig. 4, should facilitate imaging the non-zero b̃_3,−2 coefficient. Experimentally, one can keep the detector fixed and instead rotate the polarizations of the fields. That is, measurement of the b̃_3,−2 coefficient only requires rotating ω and 2ω by 45° around their propagation axis in ref. 24 and 26.

Fig. 4 Proposed velocity map imaging setup to measure the coefficient b̃_3,−2: ω (red wave) and 2ω (violet wave) are polarized along x and z, respectively. The detector projects the distribution on the x = z plane. Only the contribution of b̃_3,−2 to the photoelectron angular distribution is shown. In red/blue regions there is an excess/deficit of photoelectrons.

Finally, it is also possible to obtain the type (iv) coefficient b̃_3,−2 using monochromatic light with elliptical polarization. However, a “hidden” symmetry associated to the degeneracy in photon orderings in the monochromatic case prevents this coefficient in the case of a two-photon process. The possibility of this hidden symmetry to be violated for higher-order or strong-field processes, or in a more refined description of photoionization remains to be investigated.

Conflicts of interest

There are no conflicts to declare.

Appendix

A Selection rules for an alternative orientation of the field

In Section 4 we chose the orientation of the field so as to simplify as much as possible the selection rules for the b̃_l,m coefficients. Here, for the sake of comparison, we consider the alternative orientation|| used in ref. 24–26, namely

E⃑(t) = E_ω cos(ωt)ŷ + E_2ω cos(2ωt + ϕ)x̂. (9)

Tables 4–6 show the analogues of Tables 1–3 for the orientation in eqn (9). In this case, we see that e.g. b̃_1,1, b̃_2,−1, and b̃_3,−2 are dichroic non-enantiosensitive, enantiosensitive and dichroic, and enantiosensitive non-dichroic, respectively. These are just the correspondingly rotated versions of the coefficients in Section 4. Note that b̃_2,−1 ∝ Im{b_2,1} is indeed the coefficient discussed in ref. 25.

Table 4 Same as Table 1 for the orientation (9)

b̃ l,m	(l,m) condition
Non-zero 2nω	(Even l − m and m ≥ 0) or
	(odd l − m and m < 0)
Non-zero (2n + 1)ω	(Odd l − m and m ≥ 0) or
	(even l − m and m < 0)
Enantiosensitive	Odd l − m
Dichroic	[(Odd l) and (m ≥ 0)] or
	[(even l) and m < 0]

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Table 5 Same as Table 2 for the orientation of the field used in eqn (9)

b̃ l,m	(l,m) condition at 2nω	(l,m) condition at (2n + 1)ω
Enantiosensitive and dichroic	(Even l) and (odd m < 0)	(Odd l) and (even m ≥ 0)
Enantiosensitive non-dichroic	(Odd l) and (even m < 0)	(Even l) and (odd m ≥ 0)
Dichroic non-enantiosensitive	(Odd l) and (odd m ≥ 0)	(Even l) and (even m < 0)

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Table 6 Same as Table 3 for the orientation of the field used in eqn (9)

Symmetry-allowed	b̃ 0,0	b̃ 1,1	b̃ 2,−1	b̃ 2,0	b̃ 2,2	b̃ 3,−2	b̃ 3,1	b̃ 3,3	b̃ 4,−3	b̃ 4,−1	b̃ 4,0	b̃ 4,2	b̃ 4,4
Enantiosensitive	N	N	Y	N	N	Y	N	N	Y	Y	N	N	N
Dichroic	N	Y	Y	N	N	N	Y	Y	Y	Y	N	N	N

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B Derivation of b̃_1,0 in eqn (3)

The process depicted in Fig. 3a yields a b̃_1,0 coefficient given by²³where c.c. denotes the complex conjugate, L and M indicate vectors and functions in the laboratory (L) and molecular (M) frames, k⃑ is the photoelectron momentum, k̂ ≡ k⃑/k, ρ ≡ αβγ is the molecular orientation specified by the Euler angles αβγ, is the normalized integral over all molecular orientations, is the integral over all photoelectron directions k̂^M, is the photoelectron angular distribution in the molecular frame for a particular orientation ρ. is the probability amplitude of the state |k⃑^M〉, A⁽¹⁾ and A⁽²⁾ are complex-valued constants depending on detunings and pulse envelopes, d⃑_i,j ≡ 〈i|d⃑|j〉 is the dipole transition matrix element between states |i〉 and |j〉, |k⃑〉 is the scattering state describing an outgoing plane wave with photoelectron momentum k⃑, and E⃑_ω = E_ωx̂ and E⃑_2ω = E_2ωe^−iϕẑ are the Fourier amplitudes of the field (2) at frequencies ω and 2ω, respectively. The vectors k̂^L and d⃑^L_i,j, depend on the molecular orientation ρ according to v⃑^L = S(ρ)v⃑^M, where S(ρ) is the rotation matrix taking vectors from the molecular frame to the laboratory frame. Note that only the interference between the two pathways in Fig. 3 contributes to b̃_1,0 (see Section 4).

The integral over orientations yields^23,42

where g⃑⁽⁴⁾ and f⃑⁽⁴⁾ are vectors of molecular and setup rotational invariants, respectively,

Replacing eqn (11)–(14) in eqn (10) we obtain eqn (3), where f_1,0 is given by eqn (4) and g_1,0 is given by

We remark that eqn (3), (4) and (15) are not valid for arbitrary polarizations of E⃑_ω and E⃑_2ω. We have kept the vectorial form of the rotational invariant in eqn (4) to emphasize the scalar vs. pseudoscalar character (see discussion in Section 4.1). As can be seen from eqn (14), to derive eqn (3), (4) and (15) we explicitly used E⃑^L_2ω‖ẑ^L and E⃑^L_ω‖x̂^L.

C Derivation of b̃_2,−2 in eqn (5)

Similarly, for the b̃_2,−2 coefficient we have²³The integral over orientations can be performed analogously to what we did for b̃_1,0.⁴² Making explicit use of E⃑^L_2ω‖ẑ^L and E⃑^L_ω‖x̂^L yields eqn (5), with f_2,−2 given by eqn (6) and g_2,−2 given by

We point out that when dealing with integrals over orientations involving five or more scalar products, the number of rotational invariants that can be formed is such that they are no longer linearly independent from each other.⁴² As a result it is possible to write the result of the integral in several different ways that, although perfectly equivalent, are not evidently related to each other at first sight. For example, by changing the order of the scalar products in such a way that exchanges its position with (k̂^L·x̂^L) in eqn (15) one obtains

whereand again we assumed that E⃑^L_2ω‖ẑ^L and E⃑^L_ω‖x̂^L. Writing the explicit expressions E⃑_ω = E_ωx̂ and one can show that . And using standard vectorial algebra relations one can show that . The latter equality reflects the fact that, for four arbitrary vectors a⃑, b⃑, c⃑, d⃑, the three different rotational invariants r₁ ≡ [a⃑·(b⃑ × c⃑)](b⃑·d⃑), r₂ ≡ [a⃑·(b⃑ × d⃑)](c⃑·b⃑), and r₃ ≡ [a⃑·(c⃑ × d⃑)](b⃑·b⃑) can be written as a linear combination of the two rotational invariants r₁ and r₄ ≡ [b⃑·(c⃑ × d⃑)](b⃑·a⃑). Care must therefore be taken when looking for interpretations that depend on the particular ordering of the vectors appearing in the rotational invariants.

D Derivation of b̃_3,−2 for ω + 2ω [eqn (7)]

The process depicted in Fig. 3b yields a b̃_3,−2 coefficient given by²³The integral over orientations can be performed analogously to what we did for b̃_1,0.⁴² Making explicit use of E⃑^L_2ω‖ẑ^L and E⃑^L_ω‖x̂^L yields eqn (7), with f_3,−2 given by eqn (8) and g_3,−2 given by**

E Derivation of b̃_3,−2 for 2ω + ω: symmetry in photon ordering

First we rewrite the integral over orientations in eqn (21) asFor the opposite photon ordering we havethat is, reversing the photon ordering is equivalent to the exchanging x̂^L and ẑ^L in eqn (23). On the other hand, from the triple product in eqn (8) it follows that

I_ω,2ω(ẑ^L,ŷ^L,x̂^L) = −I_ω,2ω(x̂^L,ŷ^L,ẑ^L) (25)

and thus

I_2ω,ω(x̂^L,ŷ^L,ẑ^L) = −I_ω,2ω(x̂^L,ŷ^L,ẑ^L). (26)

This means that the expression for b̃_3,−2 in the photon ordering 2ω + ω is exactly the same as for the photon ordering ω + 2ω up to a minus sign. However, since for a fixed molecular spectrum the coupling coefficient A⁽²⁾ strongly depends on detunings (and therefore photon ordering), if one of the photon orderings is resonant it will dominate. Furthermore, in practice each photon ordering might be resonant with different transitions, and one should therefore use different transition dipole matrix elements for each ordering.

Since the derivation we just presented is actually independent of the frequencies, we have that

I_ω1,ω2 = −I_ω2,ω1, (27)

provided ω₁ and ω₂ have linear polarizations perpendicular to each other. We can therefore conclude that b̃_3,−2 → 0 as ω₁ → ω₂. Note that this is analogous to the corresponding result for the polarization in sum-frequency generation.⁴³

F Vanishing of b̃_3,−2 for elliptical light

Using relation (27) it is simple to establish why, although a monochromatic elliptical field has a geometry that allows for a non-zero b̃_3,−2 in an ω + 2ω process, the “hidden” symmetry in eqn (27) forces it to vanish. To see this, note that for an elliptical field we have

E⃑(t) = E⃑_ωe^−iωt + c.c., (28)

with E⃑_ω = E_ωx̂ + iE_yŷ and therefore [cf.eqn (21)]that is, we can decompose the process into six pathways. The first two of them involve either two x-polarized photons or two y-polarized photons and the associated geometrical symmetry prevents them from contributing to b̃_3,−2. The next two terms involve absorption of one x-polarized photon and one y-polarized photon and their lower geometrical symmetry allows for contributions to b̃_3,−2. However, since these two terms correspond to opposite photon orderings satisfying ω₁ → ω₂, the photon-ordering symmetry in eqn (27) implies that their contributions will cancel each other exactly. The remaining terms describe PECD from the state |j〉 [cf. eqn (10) in ref. ¹³] after excitation via either an x- or a y-polarized photon. These terms make a null contribution to the integral over orientations in b̃_3,–2 because all possible setup rotational invariants that can be formed either with the set {x̂^L, ŷ^L, ẑ^L, (E_xx̂^L × Ê_yŷ^L), E_xx̂^L, E_xx̂^L} or with the set {x̂^L, ŷ^L, ẑ^L, (E_xx̂^L × Ê_yŷ^L), E_yŷ^L, E_yŷ^L} vanish.⁴²

Acknowledgements

A. F. O. and O. S. gratefully acknowledge the MEDEA Project, which has received funding from the European Union's Horizon 2020 Research and Innovation Programme under the Marie Skłodowska-Curie Grant Agreement No. 641789. A. F. O. and O. S. gratefully acknowledge support from the DFG SPP 1840 “Quantum Dynamics in Tailored Intense Fields” and DFG Grant No. SM 292/5-2. A. F. O. gratefully acknowledges grants supporting his research at ICFO: Agencia Estatal de Investigación (the R&D project CEX2019-000910-S, funded by MCIN/AEI/10.13039/501100011033, Plan Nacional FIDEUA PID2019-106901GB-I00, FPI), Fundació Privada Cellex, Fundació Mir-Puig, Generalitat de Catalunya (AGAUR Grant No. 2017 SGR 1341, CERCA program), and EU Horizon 2020 Marie Skłodowska-Curie grant agreement no. 101029393.

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Footnotes

† Our choice of axes, which differs from ref. 22–24, simplifies Table 1. Our choice of phase coincides with that in ref. 24 and differs from the one in ref. 22 and 23. Note that some of the plots of Fig. 2(a) in ref. 24 are labeled with the wrong phase.

‡ Here we use the word dichroic in analogy to how it is used in the circularly polarized case in PECD, where a change of π in the phase between the two perpendicular components of the field leads to a change of sign of b_1,0.

§ Explicit expressions for the molecular rotational invariants g_1,−1, g_2,−2, and g_3,−2 are given in Appendices B–D.

¶ Symmetries not immediately apparent from the geometric symmetries of the system.³⁷

|| Another sensible choice, which slightly reduces the number of non-zero b̃_l,m coefficients, is to take the ω field along the z axis in order to obtain b̃_2N,m = 0 for m ≠ 0 in the signal corresponding to N-photon absorption of the ω field.

** This is considerably simplified by ordering the scalar products in the orientation integral as and using Table III in ref. 42.

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