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Synthetic chiral light for efficient control of chiral light matter interaction

This paper originated from efforts towards a high harmonic generation (HHG) based enantiosen-sitive effect taking place in the electric-dipole approximation and from the realization that the key physical element necessary for enantiosensitivity at the level of total number of photons ab-sorbed or emitted in this approximation is the chirality of the electric field. This summary will be mostly restricted to the latter point, which is the one I contributed to.

First we point out the difference between measurements which rely on light’s chirality, as e.g.

circular dichroism (CD), and measurements which rely on chiral setups and occur in the absence of chiral light, as e.g. optical activity (OA) and photoelectron circular dichroism (PECD). As discussed in the introduction of this thesis and as originally formulated in Ref. [15] (see Sec. 2.3), chiral setups rely on vector observables. Such directionality of the observable is not required in effects which rely on the chirality of the light. The main point of this paper is that the chirality of light, which in the case of circularly polarized light relies on its spatial dependence, can instead

be entirely encoded in the temporal dependence of the electric field. That is, in the form of a chiral polarization, where the electric field traces a chiral trajectory in (Ex, Ey, Ez) space as it oscillates39. This type of chirality does not rely on the spatial dependence of the field and it is therefore local in space and hence, unlike circularly polarized light, it has an enantiosensitive effect on chiral matter even within the electric-dipole approximation. We call this type of light locally chiral light (LCL). We show that the handedness of the LCL can be characterized via the pseudoscalars that can be formed with its vector Fourier components, and its sign can be reversed by changing the relative phase between the different frequencies. As shown in the introduction [see Eq. ([41])], these field pseudoscalars emerge naturally in interference terms in the perturbative description of light-matter interaction and can be used to control the enantiosensitive response of chiral matter already at the interference between one-photon and two-photon transitions.

Although this type of chirality has not been recognized before, LCL was unknowingly present in the works of Gerbasi et al. [37] and Eibenberger et al. [41]. However, only now it is clear that LCL was the key physical element explaining the enantiosensitive population transfer in those works. On the one hand, Gerbasi et al. did not recognize that the relative phase between the frequencies (and therefore the handedness) would typically change rapidly across the interaction region, condemning experimental measurements to failure. On the other hand, Eibenberger et al. knew of this problem but their solution is severely limited to having one of the wavelengths considerable bigger than the interaction region. Here we show that a chiral electric field can be obtained with only two frequencies ω and 2ω, where ω is elliptically polarized and 2ω is linearly polarized perpendicular to the plane defined by theω ellipse. In addition, we devise an experimental setup consisting of two noncollinear beams that keeps the handedness of the field constant across space (i.e. it is globally chiral) without imposing restrictions on the frequencies.

Since the enantiosensitivity takes place within the electric-dipole approximation and the setup is not restricted in terms of frequencies, our work unlocks the full potential of light to control the enantiosensitive response of chiral matter at the level of angle-integrated signals, i.e. without relying on the directionality of the detection (chiral setup).

As an example of what can be achieved using the proposed setup this paper shows calculations of HHG where a given set of even harmonics display perfect destructive interference in one enantiomer, and perfect constructive interference in the opposite enantiomer. The behavior can be completely reversed by reversing the handedness of the field. This corresponds to perfect chiral discrimination, an accomplishment unparalleled in the field of chiro-optical techniques.

Because of the generality of our findings, we expect our work to have a strong impact on the field of chiral discrimination as a whole, and in particular to the investigation of ultrafast processes in chiral molecules. The experimental confirmation of our results is underway.

3 Conclusions and outlook

We have discussed a broad range of effects that take place within the electric-dipole approxima-tion, which are typically orders of magnitude stronger than optical activity and circular dichroism.

Several of these effects were introduced here.

39In fact we have already seen how one of these trajectories may look like when we considered PXCD (see inset of Fig. 1 in Ref. [62]), where it described the FID of the expected value of the electric dipole after the pump.

We have introduced elementary instances of chiral wave functions which exemplify three basic forms in which chirality can be encoded in a wave function. These chiral wave functions can be formed as superpositions of atomic states, which points the way towards laser-sculpted chirality of initially achiral systems. Using these chiral wave functions we found a concrete mechanism showing how the interplay of two propensity rules can lead to the formation of net photoelectron currents (forward-backward asymmetries) in the photoionization of aligned chiral systems.

We have developed an analytical formulation that reveals helical (more precisely torus-knot-like) electronic currents formed in isotropic chiral molecules after excitation with a circularly polarized electric field. This formulation was extended to describe how net photoelectron currents (forward-backward asymmetries) result from subsequent photoionization with linearly polarized and even unpolarized light.

We have introduced a classification of chiral effects that distinguishes two groups. The first group relies on the chirality of the electromagnetic field and yields enantiosensitive scalar observables.

The second group relies on the chirality of the setup, can take place in the absence of chiral electromagnetic fields, and yields enantiosensitive vector observables.

We have derived new mathematical expressions for the description of several chiral effects taking place within the electric-dipole approximation and relying on chiral setups and vector observables.

These expressions are considerably simpler than previous ones, provide a deeper physical insight into each of these chiral effects, and reveal the intimate connection between all these chiral effects.

The expression we derived for photoelectron circular dichroism reveals that the net photoelectron current results from the ⃗k-averaged absorption circular dichroism encoded in the propensity field i ⃗dki×d⃗ki (d⃗ki is the transition electric-dipole). The propensity field turns out to have the same form as the Berry curvature in two-band solids. First, this reveals an unexpected connection to solid state physics and the property often referred to as geometrical magnetism (Berry curvature) which ultimately leads to topological phases of matter. Second, it also reveals the geometrical origin of photoelectron circular dichroism. Furthermore, the generalization of our PECD expression to aligned samples shows how alignment is equivalent to an effective stretching of the propensity field.

We introduced a new type of chirality of light which is fully encoded in the Lissajous figure traced by the electric field at a single point in space. The local character of this chirality means that it can interact with the molecular chirality within the electric-dipole approximation. We introduced chiral measures to quantify the chirality of this light and propose an experimental arrangement for its production. Furthermore, we show that it can be made to have the same chirality at every point across the focus, and have numerical evidence of its incredible potential to control the enantiosensitive response of chiral matter.

All of these advancements provide a rather broad range of directions that can be followed in the future. These include:

• First, the development of laser-sculpted chiral matter from achiral matter using locally-chiral light.

• Second, the application of locally-chiral light to nanostructured materials to explore the interplay between chirality-ordering in matter and in light.

• Third, the investigation of geometrical and topological features of the propensity field in chiral molecules and its connections to effects in condensed matter physics.

• Fourth, the analysis of second and higher rank tensor observables and their application to chiral discrimination.

• Fifth, the application of locally-chiral light to different intensity regimes and spectral re-gions.

• Sixth, the exploration of chiral distillation techniques using locally-chiral light.

• Seventh, the application of coherent control techniques with locally-chiral light setups.

• Eight, the investigation of ultrafast processes occurring in chiral molecules using locally-chiral light.

• Finally, it is important to explore other locally-chiral light setups.

This list suggests that the work described in this thesis has the potential to open a new and rich field of research based on the concepts introduced here, namely synthetic chiral matter, synthetic chiral light, chiral setups, and the connection between geometrical/topological effects and chiral effects.

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⃝c2019 American Physical Society

Propensity rules in photoelectron circular dichroism in chiral molecules I: Chiral hydrogen

Andres F. Ordonez1, 2, and Olga Smirnova1, 2,

1Max-Born-Institut, Berlin, Germany

2Technische Universität Berlin, Berlin, Germany

This is the accepted version of an article published in Physical Review A 99, 043416 (2019). The published version is available online at http: // dx. doi. org/ 10. 1103/ PhysRevA. 99. 043416

Photoelectron circular dichroism results from one-photon ionization of chiral molecules by circu-larly polarized light and manifests itself in forward-backward asymmetry of electron emission in the direction orthogonal to the light polarization plane. What is the physical mechanism underlying asymmetric electron ejection? How “which way” information builds up in a chiral molecule and maps into forward-backward asymmetry?

We introduce instances of bound chiral wave functions resulting from stationary superpositions of states in a hydrogen atom and use them to show that the chiral response in one-photon ionization of aligned molecular ensembles originates from two propensity rules: (i) Sensitivity of ionization to the sense of electron rotation in the polarization plane. (ii) Sensitivity of ionization to the direction of charge displacement or stationary current orthogonal to the polarization plane. In the companion paper [Phys. Rev. A 99, 043417 (2019)] we show how the ideas presented here are part of a broader picture valid for all chiral molecules and arbitrary degrees of molecular alignment.

I. INTRODUCTION

Photoelectron circular dichroism (PECD) [1–4] her-alded the “dipole revolution” in chiral discrimination:

chiral discrimination without using chiral light. PECD belongs to a family of methods exciting rotational [5–

8], electronic, and vibronic [9,10] chiral dynamics with-out relying on relatively weak interactions with magnetic fields. In all these methods the chiral response arises already in the electric-dipole approximation and is sig-nificantly higher than in conventional techniques, such as e.g. absorption circular dichroism or optical rotation, known since the XIX century (see e.g. [11]). The connec-tion between these electric-dipole-approximaconnec-tion-based methods is analyzed in [12]. The key feature that dis-tinguishes them from standard techniques is that chiral discrimination relies on a chiral observer - the chiral refer-ence frame defined by the electric field vectors and detec-tor axis [12]. In PECD, ionization with circularly polar-ized light of a non-racemic mixture of randomly-oriented chiral molecules results in a forward-backward asymme-try (FBA) in the photoelectron angular distribution and is a very sensitive probe of photoionization dynamics and of molecular structure and conformation [13,14]. PECD yields a chiral response as high as few tens of percent of the total signal and the method is quickly expanding from the realm of fundamental research to innovative applica-tions, becoming a new tool in analytical chemistry [15–

17]. PECD is studied extensively both experimentally [4, 18–44] and theoretically [2, 3, 12, 45–57] and was re-cently pioneered in the multiphoton [58–67], pump-probe [68], and strong-field ionization regimes [69,70].

ordonez@mbi-berlin.de

smirnova@mbi-berlin.de

In this work we focus on the physical mechanisms un-derlying the chiral response in one-photon ionization at the level of electrons and introduce “elementary chiral in-stances” - chiral electronic wave functions of the hydrogen atom.

In molecules, with the exception of the ground elec-tronic state, the chiral configuration of the nuclei is not a prerequisite for obtaining a chiral electronic wave func-tion. Thus, one may consider using a laser field to im-print chirality on the electronic wave function of an achi-ral nuclear configuration. The ability to create a chiachi-ral electronic wave function in an atom via a chiral laser field [71] implies the possibility of creating perfectly ori-ented (and even stationary) ensembles ofsynthetic chiral molecules (atoms with chiral electronic wave functions) with a well defined handedness in a time-resolved fashion from an initially isotropic ensemble of atoms. Such time-resolved chiral control may open new possibilities in the fields of enantiomeric recognition and enrichment if the ensemble of synthetic chiral atoms is made to interact with actual chiral molecules. From a more fundamental point of view, the elementary chiral instances could be excited in atoms arranged in a lattice of arbitrary sym-metry to explore an interplay of electronic chirality and lattice symmetry possibly leading to interesting synthetic chiral phases of matter.

Here our goal is to understand how molecular proper-ties such as the probability density and the probability current give rise to PECD and how they affect the sign of the FBA in the one-photon ionization regime. In a forthcoming publication we will use the hydrogenic chiral wave functions to extend this study into the strong-field regime. As a first step towards our goal, we consider the case of photoionization from a bound chiral state into an achiral Coulomb continuum, and restrict the analysis to aligned samples.

As can be seen in Fig. 2 of [12] and in Figs. 3 and 5 of the companion paper [1], within the electric-dipole approximation, the photoelectron angular distribution of isotropic or aligned samples can display a FBA only if the sample is chiral. This is in contrast with other dichroic effects observed in oriented or aligned achiral systems (see e.g. [72,73]).

An isotropic continuum such as that of the hydrogen atom cannot yield a FBA in an isotropically oriented en-semble (see [45] and Appendix VII A), because in this case the continuum is not able to keep track of the molec-ular orientations and therefore the information about the chirality of the bound state is completely washed out by the isotropic orientation averaging. However, this does not rule out the emergence of the FBA in an aligned en-semble, where only a restricted set of orientations comes into play. Therefore, the fact that we use an isotropic continuum shall not affect our discussion on the origins of PECD in any way beyond what is already obvious, namely, that the FBA we discuss relies entirely on the chirality of the bound state and that it vanishes if we include all possible molecular orientations.

In Sec. II we introduce the chiral hydrogenic states.

In Sec. III we use the chiral hydrogenic states to fo-cus on physical mechanisms underlying PECD in aligned molecules. In Sec. IV we discuss effects on the FBA that result from increasing the complexity of the initial state. In the companion paper [1] we show that optical propensity rules also underlie the emergence of the chiral response in photoionization in the general case of arbi-trary chiral molecules and arbiarbi-trary degree of molecular alignment, and we also expose the link between the chiral response in aligned and unaligned molecular ensembles.

SectionVconcludes this paper.

II. HYDROGENIC CHIRAL WAVE FUNCTIONS We will describe three types of hydrogenic chiral wave functions. The first type (p-type) is of the form

⃓⃓χ±p⟩︁

= 1

√2(|3p±1⟩+|3d±1⟩), (1) where|nlm⟩denotes a hydrogenic state with principal quantum numbern, angular momentuml, and magnetic quantum numberm. χ+p(⃗r)is shown in Fig. 1. The su-perposition of states with even and odd values oflbreaks the inversion symmetry and leads to a wave function po-larized (hence the subscriptp) along thez axis, which is indicated by an arrow pointing down in Fig. 1. m=±1 implies a probability current in the azimuthal direction and is indicated by a circular arrow in Fig. 1. The com-bination of these two features results in a chiral wave function, as is evident from its compound symbol. The sign of mdetermines theenantiomer and, as usual, the twoenantiomers are related to each other through a

re-Figure 1. Top: contour map ofχ+p (⃗r)[Eq. (1)] on they= 0 plane, where it only takes real values. Dashed (solid) lines indicate negative (zero or positive) contours. Bottom: iso-surface ⃓⃓χ+p(⃗r)⃓⃓ = 0.01 a.u. colored according to the phase.

The chiral symbol on the upper left corner indicates the po-larization of the density (vertical arrow) and the probability current in the azimuthal direction (curved arrow).

flection; in this case, across thex= 0 plane, as follows from the symmetry of spherical harmonics1.

The second type (c-type) is given by

⃓⃓χ±c⟩︁

= 1

√2(|3p±1⟩+ i|3d±1⟩), (2) which differs from |χp⟩ only in the imaginary coeffi-cient in front of|3d±1⟩. At first sight, since⟨⃗r|3p±1⟩and

1We could have also defined opposite enantiomers through an in-version, and in this case instead of changingmwe would change the relative sign between|3p1and|3d1⟩. Both definitions of the opposite enantiomer are equivalent and are related to each other via a rotation.

Figure 2. (a) Cut of χ+c (⃗r)[Eq. (2)] on they = 0plane. The black lines indicate the contours of⃓⃓χ+c (⃗r)⃓⃓while the colors indicate its phase. The white arrows indicate the direction of the component of the probability current in the y = 0plane.

(b)Isosurfaces⃓⃓χ+c (⃗r)⃓⃓= 0.011 a.u.and(c) ⃓⃓χ+c (⃗r)⃓⃓= 0.005 a.u.colored according to the phase. (d) Trajectory followed by an element of the probability fluid⃓⃓χ+c (⃗r)⃓⃓2. The rotation around thez axis is counterclockwise. The radial distance in this specific trajectory varies between 1 and 18.5 a.u.

⟨⃗r|3d±1⟩are complex functions, one would not expect im-portant differences between p and c states, however, as shown in Fig. 2, thep andcstates are qualitatively dif-ferent. We can see that instead of the polarization along z, there is probability current circulating around a nodal circle of radius 6 a.u. in the z = 0 plane, as indicated by the two circular arrows in Fig. 2 (a). Analogously to thepstates, where the polarization of the probability density is determined by the relative sign between|3p±1⟩ and |3d±1⟩, in the c states the direction of the proba-bility current is determined by the relative sign between

|3p±1⟩ and i|3d±1⟩. This out-of-plane (polar) current combined with the in-plane (azimuthal) current due to m=±1leads to a chiral probability current (hence thec subscript), visualized in Fig. 2(d) via the trajectory fol-lowed by an element of the probability fluid|χ+c|2. This single trajectory (also known as a streamline in the con-text of fluids) clearly shows how, although pure helical motion of the electron is not compatible with a bound state, helical motion can still take place in a bound state via opposite helicities in theinnerandouterregions2. As can be inferred from the cut ofχ+c (⃗r)in they= 0plane [Fig. 2 (a)], trajectories passing far from the nodal cir-cle, like that shown in Fig. 2 (d), circulate faster in the azimuthal direction than around the nodal circle while those close to the nodal circle have the opposite behavior and look like the wire in a toroidal solenoid. Interestingly, a probability current with the same topology was found in Ref. [74] when analyzing the effect of the (chiral) weak interaction on the hydrogenic state2p1/2.

So far we have only considered wave functions with achiral probability densities whose chirality relies on

non-2 We will say that a point is in the inner/outer region if thez component of its probability current is positive/negative.

zero probability currents. The helical phase structure of χ±c (⃗r) [see Figs. 2 (b) and (c)] suggests that we can construct a wave functionχ±ρ (⃗r)with chiral probability density (hence the subscriptρ) by taking the real part of χ±c (⃗r), i.e.

⃓⃓χ±ρ⟩︁

= 1

√2(︁⃓⃓χ±c⟩︁

+ c.c.)︁ (3)

= 1

2(|3p±1⟩+ i|3d±1⟩ − |3p1⟩+ i|3d1⟩)

= 1

√2[∓ |3px⟩+|3dyz⟩].

It turns out that this wave function is not chiral. Nev-ertheless, increasing the l values by one results in the wave function we are looking for3. The third type (ρ-type) of chiral wave function is given by

⃓⃓

⃓χ±ρ(421)⟩︂

= 1

√2

(︂⃓⃓⃓χ±c(421)⟩︂

+ c.c.)︂

(4)

= 1

2(|4d±1⟩+ i|4f±1⟩ − |4d1⟩+ i|4f1⟩)

= 1

√2

(︁∓ |4dxz⟩+⃓⃓4fyz2⟩︁)︁

3It is also possible to obtain a chiralρstate without increasing the value oflby replacing thecstate in Eq. (3) by a superposition of thep[Eq. (1)] andc[Eq. (2)] states. However, the resulting state is less symmetric and does not provide any more insight than the one obtained in Eq. (4) so we decided to skip it in favor of clarity.

Figure 3. Isosurfaces χ+ρ(421)(⃗r) = ±0.001 a.u. (left) and χ+ρ(421)(⃗r) = ±0.008 a.u. (right) [Eq. (4)] viewed along the x(top) andz (bottom) axes.

Figure 4. Top: cut of χ+c(421)(⃗r) [Eq. (6)] on the y = 0 plane. The black lines indicate the contours of |χ+c(421)(⃗r)| while the colors indicate its phase. The white arrows indicate the direction of the component of the probability current in they= 0plane. Bottom: isosurfaces|χ+c(421)(⃗r)|= 0.001 a.u.

(left) and |χ+c(421)(⃗r)|= 0.004 a.u.(right) colored according to the phase.

and is shown in Fig. 3 for m = 1. In Eq. (4) we introduced the notation

⃓⃓

⃓χ±p(nl|m|)⟩︂

≡ 1

√2(|n, l,± |m|⟩+|n, l+ 1,± |m|⟩) (5)

⃓⃓

⃓χ±c(nl|m|)⟩︂

≡ 1

√2(|n, l,± |m|⟩+ i|n, l+ 1,± |m|⟩) (6)

⃓⃓

⃓χ±ρ(nl|m|)⟩︂

≡ 1

√2

(︂⃓⃓⃓χ±c,(nl|m|)⟩︂

+ c.c.)︂

, l≥2, (7) which includes straightforward modifications to the simplest cases in Eqs. (1), (2), and (4) that we have already considered. Figure 4 shows χ+c(421)(⃗r), which was used in Eq. (4), and Figs. 5 and 6 the m = 2 variations χ+c(422)(⃗r) and χ+ρ(422)(⃗r) = (︁⟨⃗r|4dx2y2⟩ − ⟨⃗r|4fxyz⟩)︁

/√

2 4, which will be used for the analysis of PECD in the next subsection. As can be seen in Figs. 3 and 6, like the c states, the ρ states also have helical structures of opposite handedness in the inner andouter regions.

Theρstates are particularly meaningful because they mimic the electronic ground state of an actual chiral molecule in the sense that unlike thep and thecstates, their chirality is completely encoded in the probability density and does not rely on probability currents. The decomposition ofρstates into cstates is the chiral ana-logue of the decomposition of a standing wave into two waves traveling in opposite directions, and, as we shall see in the next subsection, it will provide the corresponding advantages.

Finally, note that according to Barron’s definition of true and false chirality [75], the p states display false chi-rality because a time reversal yields the opposite enan-tiomer, while the c and ρ states display true chirality because a time-reversal yields the same enantiomer.

III. THE SIGN OF THE

FORWARD-BACKWARD ASYMMETRY IN ALIGNED CHIRAL HYDROGEN

Now we consider photoionization from the chiral bound states just introduced via circularly polarized light. For this, we require the scattering wave function ψ(−)

k . In the case of hydrogen, this wave function is known an-alytically [76]. ψk()(⃗r) has cylindrical symmetry with respect to⃗kand is shown in Fig. 7 fork= 0.3 a.u. in a

4Interestingly, when plotted as in Fig. 6, the statesχ±ρ(l+1,l,l)(⃗r) form a topological structure known as torus link with linking number±l.