In this paper we present a unified picture of several enantiosensitive effects taking place within the electric-dipole approximation. A very considerable fraction of the concepts and equations presented in the introduction of this thesis were originally developed in this paper. The aim of this paper is to show that there are several novel enantiosensitive effects occurring in physi-cal situations spanning a somewhat broad spectrum (nonlinear optics, microwave spectroscopy, photoionization) that have more in common than what the original theoretical descriptions devel-oped for each of them reveals. We provide a unified picture for five effects: photoelectron circular dichroism (PECD), enantiosensitive microwave spectroscopy (EMWS), photoexcitation circular dichroism (PXCD), photoexcitation-photoelectron circular dichroism (PXECD), and enantiosen-sitive sum-frequency generation (SFG), and derive new expressions for PECD, EMWS, PXCD, and PXECD, which have the same fundamental structure as that for SFG, and clearly reveal the similarities between them. With the exception of PXECD all expressions result from first-order perturbation theory, the PXECD expression results from second-order (PXECD) perturbation theory.

We begin by discussing the measurement process in enantiosensitive effects, where despite the chirality of the medium, the quantity that is directly measured is nevertheless a scalar (a click in a detector). This scalar results from the product of two pseudoscalar quantities. One encodes the molecular handedness and the other encodes the handedness of the probe. In the case of circular dichroism (CD), the prototypical chiro-optical effect, the handedness of the probe is given by the helicity of the circularly polarized light (CPL), which is in turn given by the projection of the spin of the photon on its propagation direction. Only after we know both the handedness of the probe and the result of the measurement can we obtain the molecular

handedness. For the effects of interest here the light is not chiral and therefore the directionality of the detection setup must be brought into the picture. In the case of PECD, where the output is a net photoelectron current pointing in one of the two directions perpendicular to the polarization plane, the detector must consist of two plates, one in each perpendicular direction. Although these plates are otherwise indistinguishable, we treat them asymmetrically because, in order to calculate the net photoelectron current, we must subtract the output of one of them (usually called backward) from the output of the other (usually called forward). This effectively defines a reference vector ˆvref that allows us to assign a sign to the photoelectron current. The spin of the photon projected on the reference vector is what defines the handedness of the setup used to probe the chiral medium. Crucially, the fact that we use such a directional detector follows from the fact that we are measuring a vector observable. All the electric-dipole effects discussed here rely on the measurement of such a vector observable. Furthermore, we show that a symmetry analysis of how a vector observable resulting from interaction between CPL and a chiral isotropic medium reveals that vector observables are automatically enantiosensitive and circularly dichroic.

Vector observables have the same magnitude but opposite directions when either the enantiomer or the polarization is swapped, and are invariant under simultaneous swapping of enantiomer and polarization. This can be extended to other polarization schemes such as e.g. two different frequencies linearly polarized at a certain angle. In that case a change ofπ in the phase of one of the frequencies plays the role of changing the rotation direction of CPL.

Two clarifications are due here. First, although a typical CD experiment may seem directional in the sense that the detector must be put behind the sample, the measurement of CD does not in principle require a directional detection, as one can simply surround the sample by a single spherical detector and count the fluorescence photons in all directions. Second, although in PECD the reference vector is usually taken to coincide with the propagation direction of the light, which leads to the deceitful name “forward-backward asymmetry”, the FBA is in reality defined with respect to how the detectors are labeled and is fully independent of the direction of propagation of the light. To convince yourself of this, note that PECD can in principle be achieved using a standing rotating wave, where the direction of propagation of the light is undefined.

Having established the global picture we proceed to the derivation of the expressions for each of the effects. In the case of PECD we begin by showing that b1 is simply the signed magnitude of the net photoelectron current. As far as we know, this simple but insightful fact has not been stated previously in the literature. Then, taking into account that, similarly to a bound wave function, the scattering wave function rotates rigidly with the molecule, we choose to keep the photoelectron momentum ⃗k fixed in the molecular frame throughout the derivation (see introduction of this thesis). This allows us to apply the orientation averaging techniques in Ref.

[14] (see also Appendix 2 of the present paper), which are well known in the nonlinear optics and in the CD communities but apparently not in the photoionization community. After a few relatively simple steps which we explain in detail in the paper we obtain Eq. (22) [Eq. (14) in the paper].

This equation is valid for any polarization and has the promised structure. In comparison,^{36} the
expression originally derived by Ritchie [18] is expressed, as usual for photoionization, in terms
of sums over several indices which result from a partial wave expansion of the scattering wave
function and a fair amount of angular momentum algebra that produces a product of several
3j-symbols. As pioneering as it is, it is a rather complicated expression that can very unlikely
be appreciated as having anything in common with that describing the enantiosensitive SFG [see

36A rather unfair comparison given that Ritchie derived this almost 50 years earlier!

Eq. (10)] found by Giordmaine [17] a decade earlier. The equivalence of our expression and Ritchie’s original expression is demonstrated in Appendix 5 of the paper.

We then move on to a general description of the PXCD effect introduced in the previous paper [62].

In this case, we begin by pointing out that in order to form a molecular pseudoscalar that encodes
the molecular handedness it is necessary to have at least three molecular vectors (equivalently
one vector and one pseudovector). Any enantiosensitive effect must somehow probe these vectors
in such a way that they end up forming a triple product. In the case of photoionization these
vectors are the photoelectron momentum and the real and imaginary parts of the transition
electric-dipole. For bound transitions, where there is no scattering direction and the transition
electric-dipoles can be chosen real, one can instead rely on the three transition electric-dipole
vectors in a three-level system^{37}. We derive an expression for the expected value of the
electric-dipole which results from coherent excitation of the two excited states for arbitrary polarizations.

The expression derived in the previous paper for PXCD is a particular case of the expression in this paper. This expression shows that the interference term, which is second order in the field amplitudes, is enantiosensitive and describes an oscillation in a direction perpendicular to the polarization vectors defined by the incident fields at the frequency corresponding to the energy difference between the two excited states. In essence this is the free induction decay (FID) version of the enantiosensitive difference-frequency generation (DFG) predicted by Giordmaine [17]. Nevertheless, besides occurring in the absence of the exciting fields, the fact that one can use a broad band pulse centered at a single frequency to populate both excited states coherently requires a rather relaxed interpretation of what DFG means.

We also show that the DFG variant of the EMWS results obtained by Patterson et al. in Ref.

[35] for rotational transitions can be written in virtually the same way as the PXCD signal [cf.

Eqs. (17) and (22) in the paper^{38}]. In other words, PXCD and EMWS are essentially the
electronic/vibrational and rotational expressions of the same underlying phenomenon. While in
hindsight this might seem evident, the noteworthy aspect of this is that, while for PXCD the
molecular orientation is treated as a parameter of the wave function, in EMWS the orientation
itself is the main degree of freedom of the wave function. The equations required to achieve
this equivalence are derived in Appendix 3 of this paper. Furthermore, the expressions derived
using the usual machinery for rotational states [43], which were specifically derived to account
for Patterson’s et al. experiment, hardly reveal this connection.

The expression presented in the introduction [Eq. (16)] for the enantiosensitive effect obtained by Patterson et al. [33] with the static electric field is also derived in this paper. Furthermore we show that it works not only for rotational but also for electronic and vibrational states [cf. Eq.

(28) in the paper].

Finally, we derive the expression for the PECD net photoelectron current in pump probe setups with two intermediate states for arbitrary polarizations of pump and probe [see Eqs. (30)-(35) in the paper]. The equations for PXECD derived in the previous paper are a particular case of the equations in this paper. Unlike the previous expressions where there is a single product between a molecular pseudoscalar and a field pseudovector, in these expressions we obtain a sum of such products. This happens because this photoionization scheme probes five molecular

37An enantiosensitive effect is achieved with only two levels in CD because there the transition magnetic-dipole pseudovector is taken into account.

38There is a typo in Eq. (22). The M superscripts on the transition dipoles should be replaced by L superscripts.

vectors (four transition dipoles and the photoelectron momentum) and there are ten ways to form different pseudoscalars with five vectors. In fact, we have recently learned that only six of this ten pseudoscalars are really independent from each other [see Eq. (24) in Ref. [14]]. Nevertheless, the expressions in this paper are written in a particularly practical form that connects a single molecular pseudoscalar with a single field pseudovector [see Eq. (31) in Ref. [14]]. Even for six instead of ten different pseudoscalars, our expressions reveal how rapidly the complexity of the enantiosensitive response increases by adding a second photon. This complexity makes Eqs. (30)-(35) very valuable because they show which combinations of the pump and probe polarizations address which molecular pseudoscalars. In Eqs. (33)-(35) we classify these terms according to their polarization selection rules in three main groups. The first group is nonzero only for chiral fields and has not yet been tested experimentally. The second group is nonzero for circular or elliptical fields. The third group contains the remaining contributions. In particular, the third group shows that two-photon PECD can occur even when the pump and probe pulses are both linearly polarized, provided the fields are neither parallel nor orthogonal to each other [see Eq.

(35)]. This third group has not been tested experimentally yet. Importantly, the contributions to the interference term do not depend on the relative phase between the two pulses and therefore the pump and probe do not need to be phase-locked. Nevertheless, it must be kept in mind that we ignore any nuclear motion between pump and pulse, and therefore the expressions are valid only for short enough pump-probe time delays.

In summary, besides recognizing the importance of measuring vectors (as opposed to scalars) that allow the use of chiral setups in the absence of chiral light, we have achieved a clear common theoretical framework enantiosensitive effects occurring within the electric-dipole approximation across the fields of nonlinear optics, photoionization, and microwave spectroscopy. A further generalization of the reasoning presented in this work towards the topic of tensor observables [see e.g. Eq. (23)] is in the process of submission.