arXiv:2203.00580v1 [physics.optics] 1 Mar 2022

Ultrafast chirality: the road to efficient chiral measurements

David Ayuso1,2,∗ Andres F. Ordonez1,3,† and Olga Smirnova1,4‡1Max-Born-Institut, 12489 Berlin, Germany

2Department of Physics, Imperial College London, SW7 2AZ London, United Kingdom3ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Barcelona, Spain and

4Technische Universitat Berlin, 10623 Berlin, Germany

Today we are witnessing the electric-dipole revolution in chiral measurements. Here we reflect onits lessons and outcomes, such as the perspective on chiral measurements using the complementaryprinciples of “chiral reagent” and “chiral observer”, the hierarchy of scalar, vectorial and tensorialobservables, the new properties of the chiro-optical response in the ultrafast and non-linear do-mains, and the geometrical magnetism associated with the chiral response in photoionization. Theelectric-dipole revolution is a landmark event. It has opened routes to extremely efficient enantio-discrimination with a family of new methods. These methods are governed by the same principlesbut work in vastly different regimes – from microwaves to optical light; they address all molecu-lar degrees of freedom – electronic, vibrational and rotational, and use flexible detection schemes,i.e. detecting photons or electrons, making them applicable to different chiral phases, from gases toliquids to amorphous solids. The electric-dipole revolution has also enabled enantio-sensitive manip-ulation of chiral molecules with light. This manipulation includes exciting and controlling ultrafasthelical currents in vibronic states of chiral molecules, enantio-sensitive control of populations in elec-tronic, vibronic and rotational molecular states, and opens the way to efficient enantio-separationand enantio-sensitive trapping of chiral molecules. The word “perspective” has two meanings: “out-look” and “point of view”. In this perspective article, we have tried to cover both meanings.

Chirality is a fundamental concept of geometrical ori-gin ubiquitous in nature, from elementary particles tomolecules to macroscopic objects. Chiral molecules arecharacterised by the spatial arrangement of their nuclei,which makes them non-superimposable on their mirrorimage. This property underlies their important func-tion in chemistry and biology: chiral structure facilitatesrecognition at the molecular level, which is a key compo-nent of metabolic reactions in biological organisms.

Most biologically relevant molecules are chiral, andmany appear in nature only in one of their two possi-ble forms [1]. Whereas the amino acids found in liv-ing beings are essentially left-handed, sugars are right-handed. While the origin of biological homochirality,i.e. the single-handedness of key biomolecules, is stilldebated [2–11], its importance could be related to thefact that this geometrically protected quantity – hand-edness – carries a single bit of information [10] – left orright, i.e. “true” or “false”. Then, to enable such in-formation processing in biological systems, the balancebetween left and right must be broken. The measure ofsuch imbalance, the enantiomeric excess, is maximizedin homochiral systems, making them ideal for chirality-based information processing.

Since handedness is a key element in molecular recogni-tion, it is both fundamentally interesting and importantto learn how it is encoded in interactions between chiralmolecules and light.

Many applications use chirality for characterizingmolecular structures via linear light-matter interaction.

∗ [email protected]† [email protected]‡ [email protected]

For example, the helical structures of DNA, RNA, andsome proteins are routinely characterized by absorptioncircular dichroism (CD) measurements. The results aretypically interpreted on a phenomenological basis, e.g.by comparing to benchmark structures. The physical un-derstanding of chirality at the molecular level, i.e. at thelevel of the relevant properties of electronic structure anddynamics, is challenging. Indeed, little is known aboutthe dynamics of chirality in terms of concepts such aselectronic ring currents [12, 13] and the fields they maygenerate inside molecules [14, 15], ultrafast charge migra-tion [16–25] or nonlinear electronic responses [26]. Yet,the dynamical response provides a different and indepen-dent access to the physical mechanisms underlying thechiral function.

While understanding chiral interactions and time-resolving chiral electronic or vibronic dynamics are muchdesired, most of the existing ultrafast methods are re-stricted to weak interactions with the magnetic com-ponent of the light field (e.g. [27–36]). In the IR-VUV range, such restriction severely limits their effi-ciency for medium-size molecules or chiral moieties, withuseful time-resolved signals often just above the noise[30–34]. A recent experiment [33] graphically demon-strates the challenges: the time-resolved CD signal isonly a few percent of the static CD and is on the or-der of the baseline stability of the setup. Developingultrafast and highly enantio-sensitive approaches, whichtrack electronic/vibronic dynamics without relying onmagnetic interactions, is an important challenge.

This challenge has been taken on by the electric-dipolerevolution in chiral measurements. It began decades ago[37, 38], but is steadily picking up the pace now, justas we are writing these words. It involves several ex-

arX

iv:2

203.

0058

0v1

[ph

ysic

s.op

tics]

1 M

ar 2

022

2

tremely efficient enantio-sensitive methods that rely onpurely electric-dipole interactions and address electronic[37–78], vibrational [51, 66] and rotational [79–81] de-grees of freedom in molecules. Importantly, the analysisof the already existing methods allows one to identifythe key common principles underlying all these schemesand formulate general requirements for experimental se-tups [82], which should help one to create new efficientenantio-sensitive approaches, tailored to the needs, tasksand means of a given laboratory.

Here we formulate the key principles underlying effi-cient and ultrafast chiral measurements operating in theelectric-dipole approximation. We also describe wherewe see future challenges and frontiers in developing suchmethods. Throughout the paper we discuss severalenantio-sensitive phenomena. Respective discussions canbe identified in the text by looking at the correspond-ing subtitles highlighted in bold (e.g. Circular Dichro-ism).

Our perspectives include not only an outlook for thefuture, but also a general view on the basic principles un-derlying various chiral measurements and on how theseprinciples can be used to develop new measurement ap-proaches and increase their efficiency. This is why we be-gin our analysis with a tutorial-style introduction into thebasic principles underlying chiral measurements. Thisanalysis is presented in Section 2 and focuses on the in-teraction of chiral molecules with light (or electromag-netic, EM, fields). We introduce the concepts of chiralreagent and chiral observer as two fundamental principlesthat allow one to construct and analyse chiral measure-ments and establish a generalized perspective on chiralmeasurements common to different observation regimes.

These two principles offer complementary approaches.One of them, the principle of chiral reagent, involvesenantio-sensitive interactions of electromagnetic fieldswith molecules. Another, the chiral observer, does not.We show how both approaches can be upgraded in theirefficiency, i.e. realized within the electric-dipole approx-imation (Sections 3 & 4), enabling extremely efficientenantio-sensitive signals from gas-phase molecules andcreating new opportunities for efficient probes of chiralliquids.

Sections 3, 4 and 5 use the concepts introduced in Sec-tion 2 to develop a framework for understanding and re-alizing chiral measurements. To give a brief idea of othertopics covered here, we list below some specific questionsthat we address and highlight the most important con-clusions stemming from the analysis of these questions.

1. What is the structure of generic chiral observables(Section 2) and how do they map onto the ultra-fast response of molecules to light (Section 4)? Weshall see that there exists a hierarchy of chiral ob-servables encompassing scalar, vectorial and tenso-rial quantities. That is, there are infinitely manyof them, allowing for significant flexibility in con-structing chiral measurements, an important assetfor future experiments.

2. Do we need light’s chirality for chiral measurements(Section 3)? We shall point out that light’s chiral-ity is not necessary: it can be substituted by thechirality of an experimental setup. The flexibilityin designing chiral experimental setups offers addi-tional flexibility for efficient enantio-discrimination.

3. Is there a chirality measure for experimental setups(Section 3)? We shall see that such chirality mea-sure does indeed exist. It can therefore serve as aguideline for optimizing the enantio-sensitivity ofexperimental setups.

4. Can light be chiral already in the electric-dipole ap-proximation? What are the chirality measures forlight (Section 4)? We shall see that the chiralitymeasures for matter, light, or experimental setupshave an identical structure, and that there are ar-rays of these measures with an identical hierarchy,from scalars to vectors to tensors of various ranks.In the outcome of an experiment, the chirality mea-sure of matter couples to the chirality measure oflight or to the chirality measure of the experimentalsetup (Section 5).

5. How does geometry affect physics: what is thephysical mechanism of a chiral response (Section6)? Geometrical properties in real space, such ashandedness, are defined by electronic and nuclearconfigurations in molecules or lattice configurationsin solids. In condensed matter physics it has beenrecently realized that the manner in which theseproperties map onto the geometrical properties ofthe Hilbert space is very important. This map-ping leads to the concepts of Berry phase [83] andBerry curvature, topological phases [84, 85], andthe general concept of geometrical magnetism insolids, providing a novel framework for understand-ing electronic response [86]. About 20% of all ma-terials known today have electronic properties dic-tated by the topology of their electronic wave func-tions [87], and quantified by the Berry curvature.Therefore, we expect that geometry and topologyshould also play a prominent role in the electronicresponse of chiral molecules and should lead to newenantio-sensitive phenomena [88], especially in thenon-linear response.

Our expectations are based on our recent finding ofthe similarity between the Berry curvature in solidsand a geometric ‘propensity’ field generated by elec-tronic dynamics in chiral molecules [88]. This fieldmaps the geometric properties of nuclear arrange-ments sensed by chiral electron currents onto chi-ral observables in photoionization. In particular, itcontrols the strength of the enantio-sensitive pho-toionization current orthogonal to the polarizationplane of the driving laser field, known as the pho-toelectron circular dichroism (PECD) in one pho-ton ionization [37–51]. This control is formally

3

similar to how the Berry curvature controls, e.g.,the circular photogalvanic current in 3D materialslacking inversion symmetry, or the anomalous cur-rent in 2D two-band topological materials. We ex-pect that PECD is but a tip of the iceberg: thegeometric propensity field (Section 8.3) providesa guiding principle for uncovering a new class ofenantio-sensitive vectorial observables [88], akin tothe anomalous electron velocity due to the Berrycurvature in solids.

6. How can one develop efficient schemes to controlchiral optical responses (Section 6)? Section 6 out-lines our perspectives on efficient control and ma-nipulation of chiral molecules with light, and chartsthe roads for new and highly enantio-sensitive ap-proaches.

FIG. 1. Light helix in space. Electric field vector of circularlypolarized light makes helix in space. The pitch of the helixis given by light wave-length. For infrared and visible lightthe helix is orders of magnitude larger than the size of themolecule. It leads to extremely weak enantio-sensitive signalsfor small molecules, chiral moieties, local chiral centers suchas asymmetric carbons.

I. PROBING CHIRALITY: INTRODUCINGTHE CONCEPTS OF CHIRAL OBSERVER AND

CHIRAL REAGENT

How do we find out if an object is chiral or not, andhow do we probe its handedness? In the macro-world,we can compare the object with its mirror image. Thisdetection principle uses the concept of “chiral observer”;the rigorous definition will follow.

Probing chirality can also involve interaction betweentwo chiral objects. One example is a proper handshake,which requires the two enantiomers – the two hands – tohave the same handedness. This detection principle usesa “chiral reagent”, which interacts differently with left-and right-handed objects.

In the micro-world, a chiral reagent can simply be an-other chiral molecule. The basic reason for the different

outcome of a chemical reaction involving chiral moleculesis the key difference in their shape.

Light can also act as a chiral reagent. It is well knownthat the electric-field vector of a circularly polarized wavedraws a basic chiral structure in space: a helix (Fig.1). Thus, circularly polarized light is a chiral photonicreagent, which interacts differently with left- and right-handed molecules. The most prominent example beingthat left- and right-handed molecules absorb differentamounts of circularly polarized light of a given handed-ness, a phenomenon known as photo-absorption circulardichroism (CD).

Absorption CD (e.g. [32, 33, 89]) is still the methodof choice (e.g. [32, 33, 89]) among all-optical methods.Unfortunately, the CD signal is very weak (three to fiveorders of magnitude smaller than light absorption at thesame frequency), as it scales with the ratio of the molec-ular size to the light’s wavelength (i.e. pitch of the helix).Optical activity (optical rotation) is a complementary ap-proach based on detecting the rotation of the polarizationof a linearly polarized pulse propagating through a chiralmedium. It has the same unfavourable scaling [27, 28],especially in the IR-UV range and for small and medium-size molecules. This makes time-resolved measurementsvery challenging, especially in optically thin media, un-less one uses X-ray light to match the pitch of the helix tothe molecular size [90, 91] or very intense fields [34, 92–96] to generate and record high harmonic spectra. Bothroutes can enhance chiral dichroism from 0.01% to a fewpercent level. There are also ingenious field configura-tions that rely on suppression of the electric-dipole inter-action (e.g. [97? ? ]) to maximise the contribution of”traditional” magnetic-dipole terms, emerging due to thelight helix. Further improvements require one to aban-don weak magnetic interactions and rely exclusively onthe electric-dipole transitions.

There are two roads that one can take (Fig.2). Thefirst option is to develop new approaches that do notrely on the chiral properties of light. Along this roadwe will meet a chiral observer. The second option isto replace the inefficient (non-local) chiral reagent – thehelix drawn by the light’s polarization vector in space –with an efficient (local) chiral reagent – a helix, or anyother chiral structure, drawn by light’s electric vectorin time. We shall see that these two roads will take usto fundamentally different observables. But let us firstrecall how one can characterize the handedness of a chiralobject.

A. Chiral observables

Let us consider left-chewing and right-chewing cows[98, 99] as prototypical carriers of chiral dynamics[? ],and define a number that characterizes its handedness.While the jaws of a cow make a rotating motion, thefood moves in the direction orthogonal to the plane ofthis rotation, see Fig. 3. The chiral dynamics emerging in

4

FIG. 2. Two roads to efficient enantio-discrimination: chi-ral reagent (approaches that substitute inefficient (non-local)light helix in space by efficient (local) light helix in time)and chiral observer (approaches that do not rely on the chiralproperties of light).

this process is associated with two directions. The first is

the direction in which food goes, which is a vector ~J . The

second is the pseudo-vector of the angular momentum ~Lassociated with the rotation of the jaw. An example of achiral observable, h, is a number incorporating these two

key directions: a scalar product of ~J and ~L,

h = ~J · ~L . (1)

As ~L is a pseudo-vector and ~J is a vector, h is a pseu-doscalar. Since a pseudovector can be generated by tak-

ing a vector product of two vectors, e.g. ~L = [~r × ~p], thechiral observable can also be given by a triple product ofthree vectors. Pseudo-scalars change sign upon mirror re-flection and are, therefore, ideally suited to discriminateopposite enantiomers.

FIG. 3. Left- and right-chewing (chiral) cows convert in-plane

rotation of the jaw, characterized by the pseudo-vector ~L,into out-of-plane food motion, characterized by the vector ~J .Reprinted from [99], with permission from Elsevier.

An important conclusion from the above example isthat a chiral system can convert in-plane rotation intoa linear motion orthogonal to this plane. We shall seethat this can lead to helical electron currents excited inmolecules by circualrly polarized electric field. It is this

coupling that “merges” the pseudovector ~L with the vec-

tor ~J into a chiral molecular pseudoscalar h. Often the

pseudovector component ~L is responsible for encodingthe in-plane rotation induced e.g. by circuarly polarized

electric field, while the vectorial component ~J can be usedto read out the enantio-sensitive signal, provided that thechiral reference frame (chiral observer) is defined in theexperimental set-up.

1. Hierarchy of scalar, vectorial and tensorial observables

Experiments always measure “clicks”, which arescalars. Then, how are pseudoscalars encoded in experi-mental observables? Let us consider the two oldest andmost common optical techniques of chiral discrimination,photo-absorption circular dichroism and optical rotation,from this perspective.

Absorption Circular Dichroism. As we have al-ready pointed out, circularly polarized light can serve asa chiral reagent (Fig. 4a). Such a “reaction” involvesabsorption of light, and it is different in the two mirror-reflected versions of a chiral molecule. Note that theintensity of absorption is a scalar. However, this scalaris, in fact, a product of two pseudoscalars: one associatedwith the molecule and another associated with the lightfield. Light’s pseudoscalar is given by the optical chirality[100], which is proportional to light’s helicity in the caseof circularly polarized fields. The molecular pseudoscalaris given by the scalar product of the electric-dipole andmagnetic-dipole vectors. Thus, we need a chiral reagent,in this case the chiral light, to “hide” a pseudo-scalar in-side a scalar. Chiral reagents allow us to measure scalarenantio-sensitive observables .

How does the situation change if we have a setup thatallows us to measure vectorial observables, such as thepolarization of light transmitted through a medium ofrandomly oriented chiral molecules? The canonical ex-ample of such a setup is the detection of chirality viaoptical rotation. Optical rotation, observed by Biot in1815, was the first experiment that revealed molecularchirality, but its deep lessons have become especially per-tinent now in the light of the electric-dipole revolution.Namely, optical rotation does not use chiral light – ituses linearly polarized light and detects the rotation ofits polarization.

Optical Activity (Optical Rotation). When aplane wave of linearly polarized light passes through achiral medium, its plane of polarization is rotated by anangle that has the same magnitude but opposite signsin opposite enantiomers. Clearly, the characterization ofthis effect requires (i) specifying light’s propagation di-

5

CPL- Chiral structure (helix)- Enantio-sensitive scalar observables

CPL (dipole approximation)- Achiral structure- It defines an oriented plane- Enantio-sensitive vectorial observables

Two-colour field- Achiral structure- It defines an unoriented plane- Enantio-sensitive tensorial observables

a b c

FIG. 4. a, Circularly polarized light (CPL) defines a (chiral)helix in space and thus it can lead to enantio-sensitive scalarobservables. b, In the electric-dipole approximation, wherethe helical structure of CPL is neglected, the in-plane rota-tion of the electric-field vector defines a pseudo-vector, whichcan couple to molecular pseudoscalars and lead to enantio-sensitive vectorial observables. c, The Lissajous figure of atwo-colour field with orthogonally polarized ω and 2ω fre-quency components does not define an oriented plane, andtherefore it does not provide a pseudo-vector. It can lead toenantio-sensitive tensorial observables.

rection k and (ii) the ability to distinguish positive from

negative rotations around the polar vector k. Since ro-tations are characterized by pseudovectors, this abilityrequires the definition of a unitary rotation pseudovec-

tor θ such that θ · k 6= 0. Introducing polarizer, rotatingin the plane orthogonal to the light propagation vectordefines such pseudovector, because the direction of po-larizer rotation defines an oriented plane.

When taken together, the vector k associated with

light and the pseudovector θ associated with rotation ofpolarizer in the laboratory reference frame form a chiral

setup. The handedness of this setup is given by θ · k.This handedness of the setup emerges naturally within

a formal description of the phenomenon. Indeed, such a

description yields an angle of rotation ~δ = δk, where δ ∝∑b

(~da,b · ~mb,a

)/ (ω2ba − ω2

)is a molecular pseudoscalar

with opposite signs for the opposite enantiomers, a andb denote the ground and excited states, respectively, the

sum is taken over all excited states, ~da,b and ~mb,a are thetransition matrix elements for the electric and magnetic

dipole operators ~d and ~m, respectively, ω is the light

frequency and ωab ≡ ωb−ωa. In writing ~δ vectorially, wehave explicitly indicated that the rotation of the plane

of polarization (measured by the polarizer θ) is defined

with respect to k.As for any vectorial measurement, what the experi-

ment reveals are its components with respect to some ref-erence frame. In our case, the relevant unitary (pseudo)

vector in our reference frame is θ so that the measure-ment yields the scalar quantity δθ = θ ·~δ ∝ δ(θ · k). Thatis, the rotation angle δθ measured in the chosen referenceframe is the product of two pseudoscalars, one associatedwith the molecule (δ) and the other associated with the

setup (θ · k). If either the handedness of the molecules orof the setup is reversed, δθ changes sign.

Neither the linearly polarized light, nor the polarizerare chiral on their own. Separately, they are not suited todetect chirality, but together they form the chiral setup,or the chiral reference frame of the observer. Thus, vec-torial observables allow one to substitute a chiral reagentby a chiral observer, i.e. a chiral setup formed by anycombination of fields and detectors.

One does not have to stop at vectorial observables. Achiral observer can also detect tensorial observables ofrank higher than one. The hierarchy continues ad in-finitum as the tensor rank increases. Let us consideran example of a chiral setup, which allows one to mea-sure vectorial chiral observables using circularly polarizedlight, already in the electric-dipole approximation, andthen discuss the modifications of the experimental setup,which allows one to measure tensorial observables.

Importantly, circularly polarized light in the electric-dipole approximation is not chiral: the spatial helixdrawn by the rotating electric-field vector as the lightpropagates in space (Fig. 4a) is lost in the electric-dipoleapproximation. Geometrically, in this approximation,the rotation of the electric field vector only defines anoriented plane (Fig. 4b). The light supplies a pseudovec-tor, which defines plane’s orientation and is the vectorproduct of the two orthogonal components of the light’spolarization. To compose a chiral observer, one needs tocomplement this pseudovector with a vector orthogonalto the polarization plane. This vector can be supplied bythe detector axis.

When the light field can provide a pseudovector, suchas the vector product of its two orthogonal polarizationcomponents, the enantio-sensitive and dichroic signal willbe a vector collinear with the light pseudovector. Butwhat if the the light field does not define an orientedplane, and therefore it cannot provide a pseudovector?

A two-color field with orthogonal ω and 2ω polariza-tions is one example of such a field (Fig. 4c). Its Lis-sajous figure changes the direction of rotation twice perlaser period. Such a field does not define an orientedplane and does not provide a pseudovector. Yet, it canstill be employed for enantio-discrimination [70, 72, 73]using a chiral observer. In this case, the chiral observerhas to supply not one, but two detector axes. The firstone should be along the direction orthogonal to the po-larization plane; the second one should break the symme-try between the two counter-rotating parts of the figureeight (see Fig. 4c), i.e. should be directed along the po-

larization of the fundamental field ~Eω. Thus, the chiralobserver can employ two axes defined by the detector.These two axes define a quadrupolar detector, which al-lows one to correlate measurements in two detection di-rections. The quadrupolar detector identifies positive ornegative correlations between the observables measuredalong two orthogonal axes (see Section 2.3). Such corre-lation of different detection directions results in tensorialchiral observables [101].

6

The concept of chiral observer embodies a powerfulprinciple of detecting chirality. It allows one to detectchirality using a collection of two (or more) non-chiralobjects, which are able to collectively form a single chi-ral object – a chiral reference frame, which is used todetect a vectorial (or tensorial) observable. Of course,none of these objects interacts with a chiral molecule inan enantio-sensitive way, as a chiral reagent does.

The power of the chiral observer comes from the free-dom to compose this observer from any vectors availablein the experiment, and the general framework for design-ing such measurements. The measurement no longer hasto rely on light’s magnetic field to probe chiral molecules,eschewing the need to focus on weak non-electric-dipoleeffects and taking maximum advantage of available toolsand experimental specifics.

B. Two electric-dipole revolutions in chiralmeasurements

Revolutions often come in sequence. The first electric-dipole revolution in chiral measurements has “revolution-ized” the observer: non-chiral light, interacting with mat-ter in the electric-dipole approximation, in combinationwith the detection system, has become a very efficienttool for chiral discrimination [37–53, 55? –81]. Sincethe concept of chiral observer is completely general, itcan be applied in a variety of experiments. Such exper-iments can detect photo-electrons [37–73], or molecularfragments [102, 103], or photons, such as optical light[74, 75], microwave emission [79–81], etc. Below we shallbriefly outline several “chiral observer” methods devel-oped so far, from photoionization to microwave detectionto non-linear optics.

Yet, there is one thing that a chiral observer cannotdo – it cannot detect enantio-sensitive scalar observables.This is where the second electric dipole revolution comesinto play. It aims to revolutionize the chiral reagent, i.e.to create and use light [104–109] or a combination of elec-tric fields [110–112] that is chiral already in the electricdipole approximation. This means that the electric fieldvector of such light, at any given point in space, shoulddraw a chiral Lissajous figure during its oscillation pe-riod. We shall refer to it as “synthetic chiral light” [104]to distinguish it from standard chiral light, i.e. standardlight helix in space.

Since synthetic chiral light (or chiral combination ofelectric fields including static or microwave electric fields)possesses its own pseudoscalar, it can provide access toscalar observables such as enantio-sensitive populationsof molecular electronic, vibronic and rotational states.Enantio-sensitive control of populations opens a way toextremely efficient manipulation of chiral molecules suchas e.g. enantio-separation, enantio-sensitive trapping,and possibly even to enantio-sensitive cooling if (or when)the huge challenges in cooling large molecules are re-solved.

The pseudoscalar of synthetic chiral light does notinclude its magnetic-field component: it is constructedfrom at least three non-coplanar electric-field compo-nents. Thus, the electric-field vector of such light needsto be 3D, a property that may arise in non-collinearconfigurations [76, 77, 104–108], in tightly focused laserbeams [78, 109] or inside nano-photonic structures [113],thanks to the emergence of longitudinal components[114]. What’s more, it has to be multi-color, since thethree electric field vectors have to be distinguishable. Italso means that such light can sense the chirality of mat-ter only via non-linear interactions, as it has to encodethe three field components into the light-induced transi-tion.

Importantly, the pseudoscalar – the chirality measureh of synthetic chiral light – is local. It is defined inthe electric-dipole approximation and therefore at ev-ery point in space: h = h(~r). To control total enantio-sensitive absorption in the entire sample, synthetic chirallight should maintain its handedness globally, across thewhole interaction region. That is, its pseudoscalar shouldmaintain its sign from one spatial point to another, or atthe very least be non-zero on average in space. This canbe achieved in different ways, one of them is discussed inSection 4.

If the light field is not globally chiral, i.e. its pseu-doscalar h = h(~r) is zero on average,

∫h(~r)d~r = 0, higher

multipoles of light’s handedness will start to play a keyrole. For example, we shall see that the presence of thechirality dipole

∫~rh(~r)d~r 6= 0, which characterizes the

spatial distribution of local handedness in a locally chiralfield with

∫h(~r)d~r = 0, would lead to enantio-sensitive

emission direction of light generated from the molecularsample via non-linear processes such as even harmonicgeneration [105].

Looking broadly, sculpting multi-color light beams inthree dimensions and applying them to chiral moleculesappears as a natural next step for the rapidly developingfield of generating and using vector beams. This step iscompelled by the rich opportunities arising from the in-teraction of such synthetic chiral light with chiral matter.

II. THE FIRST ELECTRIC-DIPOLEREVOLUTION IN CHIRAL MEASUREMENTS:

EFFICIENT CHIRAL OBSERVER

The “chiral observer” is a concept of chiral measure-ment in which the detection relies not on the interactionof two chiral objects, but rather on the interaction of achiral object with two or more achiral objects arrangedin such a way that together they form a chiral exper-imental setup, i.e. the experimental setup becomes achiral structure. A chiral observer can detect enantio-sensitive vectorial [82] and tensorial [115] observables. Inthe simplest case of vectorial obsevables, the chiral setupprovides a reference frame, right or left, defined by threenon-coplanar vectors. The chiral measurement has to be

7

resolved in this reference frame. Tensorial chiral observ-ables couple to chiral tensorial detectors, correlating twoor more detection directions in an enantio-sensitive way.Enantio-sensitive scalar observables, such as the total in-tensity of a signal, cannot be produced or measured insuch a setup, since the interaction of two chiral objectsis not involved.

A chiral observer does not require chiral light, andtherefore the chiral optical enantio-sensitive response canbe induced via strong electric-dipole interactions, lead-ing to highly efficient enantio-sensitive signals. This isthe essence of the first electric-dipole revolution: strongenantio-sensitive response without chiral light. It hasbrought a family of new methods benefiting from highly-efficient enentio-sensitive signals based on detecting elec-trons or photons: PECD in one-photon [37–51] andmulti-photon regimes [52–65], PXECD and PXCD [66],three wave mixing [75, 116–119] and enantio-sensitive mi-crowave spectroscopy [80, 112]. The relevant vectorialobservables are the induced polarization or photoelec-tron current. Tensorial observables related to multipo-lar bound polarizations [120] or multipolar photoelectroncurrents have been predicted in the two-photon regime[70, 71] and measured in the multiphoton regime [73].

The concept of chiral observer provides a set of rulesto find new vectorial or tensorial enantio-sensitive observ-ables, and to design the experimental setup required fortheir observation. These rules may also allow us to adaptthe setup to the specifics of the experimental tools avail-able in each laboratory. Let us formulate these simplerules.

Rule 1: The enantio-sensitive vectorial signal excitedby the light field in a randomly oriented ensemble of chiralmolecules and detected by a chiral observer has the followinggeneral form:

〈~v〉 = g~L, (2)

where g is a molecular pseudoscalar and ~L is a field pseu-dovector.

The lowest-order field pseudovector that can be con-structed without the help of the magnetic component ofthe light field is the vector product of the two orthogo-nal, phase-delayed electric-field components. Such a vec-tor product arises naturally in elliptically polarized fields,maximizing in circularly polarized fields. The respectivemolecular pseudoscalar is a triple product of three vec-tors: the first two are the non-collinear dipole momentsresponsible for the coupling with each of the two fieldcomponents present in the light pseudoscalar. The thirdis a vector ~v representing the desired vectorial observable.

The two types of vectorial observables outlined so farrequire observation of light generated due to the induced

dipole ~dif , ~v = ~dif (e.g. PXCD, see below), or ejected

photoelectrons with momentum ~k, ~v = ~k (e.g. PECD,see below). Thus, the molecular pseudoscalar g includes

the triple product of dipoles [ ~dik × ~dki] · ~dif for detecting

light and a similar construct in the case of detecting pho-

toelectrons [ ~dik × ~d∗ki] · ~k. The two pseudoscalars differby a single vector - the one characterizing the type ofobservation.

To illustrate Rule 1, consider the phenomenon thatheralded the electric-dipole revolution, photoelectron cir-cular dichroism (PECD) in one-photon ionization by cir-cularly polarized light.

Photoelectron circular dichroism, PECD, waspredicted by Ritchie [37], Cherepkov [121, 122] and Powis[38] and first detected by Bowering et al [39]. It has nowbeen extended to the multiphoton [52, 53] and strong-field ionization regimes [65] and is being adopted for in-dustrial applications [123].

In PECD, the circularly polarized field

~E(t) = ~Eωe−iωt + c.c. (3)

ionizes an isotropic sample of randomly oriented chiralmolecules. The resulting photoelectron angular distribu-tion (averaged over all molecular orientations) is foundto be asymmetric with respect to the polarization planeof the light. This so-called forward-backward asymmetryamounts to the generation of a net photoelectron currentperpendicular to the polarization plane, which displaysboth enantio-sensitivity and circular dichroism.

The PECD current can be written as [82]

~j = g~L, (4)

where

g ≡ 1

6

∫dΩk(~d∗~k,i ×

~d~k,i) · ~k, (5)

∫dΩk indicates integration over all directions of the pho-

toelectron momentum ~k, ~d~k,i ≡ 〈~k|~d|i〉 is the transition

dipole matrix element between the ground state |i〉 and

the continuum state |~k〉, and

~L ≡ ~E∗ω × ~Eω. (6)

Using the expression linking photoionization dipoles for

left and right enantiomers ~dleft~k,i= −~dright

−~k,i, one can show

that g has opposite values for opposite enantiomers [82],

i.e. it is responsible for the enantio-sensitivity of ~j. Usingthe standard expression for the amplitude of elliptically

polarized field ~Eω = E0(x+ iσy)/√

2, we can obtain the

respective light’s pseudovector: ~L = i|E0|2σz. It is pro-portional to the photon spin σz, therefore it vanishes forlinearly polarized fields (σ = 0) and points in oppositedirections for opposite circular polarizations (σ = ±1).

Thus, ~L is responsible for the circular dichroism of ~j.Moreover, since the observable ~j is perpendicular to thepolarization plane, the corresponding detector must beable to distinguish between the two opposite directionsperpendicular to the polarization plane, i.e. it must de-fine a reference vector (e.g. z), on which the vector of

8

photoelectron current ~j can be projected. Otherwise, thedetector will not have the capacity of recording the circu-lar dichroism or the enantio-sensitivity encoded in ~j. In-deed, one can show that the coefficient b1,0 usually mea-

sured in PECD satisfies b1,0 ∝ jz = z ·~j = gS, which isa product of the molecular handedness encoded in g and

handedness of the chiral setup encoded in S ≡ z · ~L [82].Due to the purely electric-dipole nature of PECD, theenantio-sensitive and dichroic signal recorded in b1,0 ∝ jzreaches several tens of percent of the total photoioniza-tion signal [37–69].

Importantly, the vector product of the two photoion-ization dipoles is equal to zero for the “flat” (plane wave)continuum: the formation of an enantio-sensitive elec-tron current in the continuum upon photoionization froma stationary state requires electron scattering from themolecular potential. This scattering imparts angular mo-mentum on the electron, which, together with its linear

momentum ~k, characterizes chirality of the electron dy-namics in the continuum, i.e. the chirality of the pho-toelectron current. Eq. 5 points to the physical originof PECD and its connection to the concept of geometricfields (See Section 8.4).

Rule 2: A chiral observer can detect enantio-sensitivesignals in the electric-dipole approximation due to even-order non-linear optical processes, such as those describedby even-order electric susceptibilities [76, 77, 124] χ(2), χ(4),... .

It is easy to understand this rule. Here we deal with

excitation or detection of induced polarization ~P . FromRule 1, we already know that any vectorial enantio-

sensitive signal, such as ~P , must be proportional to apseudovector coming from the electromagnetic field. Letus look at all available pseudovectors.

In the first order with respect to the field, there is

only one pseudovector ~Hω – the magnetic component ofEM wave. In the second order, we have the first op-portunity to construct the field pseudovector from the

vector product of the electric fields,[~Eω1× ~E∗ω2

]. In the

third order, to construct a field pseudovector, we have

to use the magnetic field again, e.g. ~Hω1( ~Eω2 · ~E∗ω3).

Here we have used the pseudovector ~Hω1and multi-

plied it by a scalar ~Eω2· ~E∗ω3

. In the fourth order, wecan use the field pseudovector in the electric-dipole ap-

proximation[~Eω1 × ~E∗ω2

]and combine it with the scalar

~Eω3 · ~E∗ω4. The rest is clear: in even orders, the pseu-

dovector can be constructed from electric field vectors,but in odd orders there is no such an opportunity and wehave to use the magnetic field instead. Thus, non-linearoptical processes of even order can be enantio-sensitivein the electric-dipole approximation, while the enantio-sensitivity of odd order processes relies on magnetic in-teractions. The same conclusion can be obtained usingsimple symmetry arguments [76, 125].

To illustrate Rules 1 and 2, we shall consider several

FIG. 5. Photoelectron circular dichroism (left) vs photoex-itation circular dichroism (right). In PECD circularly polar-ized light induces chiral current in continuum, while in PXCD,it induces chiral current in bound states.

phenomena which arise in different fields of research, buthave identical mechanism of enantio-sensitive response.

Photo-excitation circular dichroism: helicalcurrents in bound molecular states. Consider firstthe phenomenon of photo-excitation circular dichroism(PXCD), introduced recently by Beaulieu et al [66]. Itshows that chiral photoelectron currents, which underliethe photoionization circular dichroism (PECD) discussedabove, can also be excited in bound states (Fig.5).

In PXCD [66, 82] a short pulse

~E(t) =

∫dω ~Eωe

−iωt (7)

interacts with an isotropic sample of chiral molecules andcoherently excites a pair of vibrational or electronic states|1〉 and |2〉 via one-photon transitions. Coherent excita-tion of a pair of states by a circularly polarized pulseshould lead to dynamics. What kind of dynamics? Re-call that chiral media can convert in-plane rotation (ex-cited by the circularly polarized field) into linear motionorthogonal to the plane. Thus, one would expect that,after the pulse, the tip of the vector describing the in-duced polarization in the randomly oriented molecularensemble traces a helical trajectory in time (Fig. 6 ). Incontrast to continuum states, in bound states this mo-tion will get reflected from the outer turning point of thebound trajectory, reversing the direction of the motionorthogonal to the polarization plane of the exciting pumppulse. Formally, one can indeed show that, upon averag-ing over random molecular orientations (denoted as 〈...〉),the dynamics in the excited states does lead to oscilla-

tions of the expectation value of the induced dipole 〈~d〉in the direction perpendicular to the polarization plane[66]. In the frequency domain, this dipole is:

〈~d〉(ω2,1) = g~L, (8)

where

g ≡ 1

6(~d1,0 × ~d2,0) · ~d2,1 (9)

9

FIG. 6. Helical current traced by the tip of the excitedPXCD dipole [66].

and

~L ≡ ~E∗(ω1,0)× ~E(ω2,0). (10)

That is, we observe oscillations at the difference fre-quency ω2,1 ≡ ω2 − ω1 in the direction determined by~L. The excited dipole is indeed a result of the conversionof the initial in-plane rotation, excited in the electronicor vibrational degrees of freedom by the circularly po-larized pump field, into motion orthogonal to this plane.The phase of these oscillations is determined not only

by ~L but also by the sign of the molecular pseudoscalarg. Thus, oscillations excited in opposite enantiomers willhave opposite phases. Observing an enantio-sensitive re-sponse requires detecting the phase of the oscillating sig-nal, or equivalently the direction of the respective vecto-rial observable.

While PXCD is similar to PECD, the conversion ofin-plane rotation into motion orthogonal to the planehappens in bound states and requires at least two ex-cited states (instead of one continuum state in PECD).Therefore, we are now dealing with the Fourier compo-nents of the field at two different frequencies. This as-pect opens more options for the pump pulse polariza-

tions which lead to a non-zero ~L. For example, onecan use an elliptically polarized broadband pulse, e.g.~Eω = Eω(x+ iσy)/

√2, which yields ~L = iE∗ω1,0

Eω2,0σz, or

two linearly polarized pulses at an angle with respect to

each other, e.g. ~Eω1,0= Eω1,0

x and ~Eω2,0= Eω2,0

y, which

yields ~L = E∗ω1,0Eω2,0 z.

The PXCD signal is an attractive new enantio-senstivemolecular observable. It depends on molecular properties(dipoles) and hence it is molecule specific. It relies oncoherence between the states and, hence, it tracks coher-ence, electronic or vibronic. For example, in the exper-iment [66] such vibronic coherence, excited in Rydberg

states of fenchone and camphor by a circularly polarizedfemtosecond pulse, gave rise to chiral vibronic currentslasting for 1.2 ps. Its evolution reflected significant vari-ations of molecular chirality when probed via photoion-ization (see PXECD below). Finally, since the emissionassociated with the PXCD dynamics is a parametric pro-cess, signals from all molecules in the ensemble will addcoherently to form macroscopic dipole. This opens an op-portunity to control enantio-sensitive response not onlyon a single molecule level, but also at the macroscopiclevel. We will consider such macroscopic control of chi-ral emitters later, developing the concept of light withstructured chirality in Section 5.

Enantio-sensitive microwave spectroscopy. Aphenomenon analogous to PXCD exists in the microwavedomain. Known as enantio-sensitive microwave spec-troscopy (EMWS), it occurs in the case of purely ro-tational transitions, was discovered by Patterson et al[79] and recently further developed in [111, 126–130].Indeed, as shown in Ref. [82], the rotational problemcan be formulated in a way mathematically similar toPXCD in electronic or vibrational states, with the onlydifference that the expression for the molecular pseu-doscalar includes a sum over all magnetic quantum num-bers Mi of the asymmetric rotor states in each energylevel i = 0, 1, 2:

g ≡ 1

6

∑M0,M1,M2

(~d1M1,0M0× ~d2M2,0M0

) · ~d2M2,1M1, (11)

This difference stems only from the different nature of av-eraging over molecular orientations in PXCD and EMWS[82]. In averaging over molecular orientations, the co-herence of different rotational states is not important inPXCD, because it does not require coherent excitation ofrotational states. One can view different molecular ori-entations as different molecules (of the same kind) andperform classical averaging over molecular orientations[82]. In the case of EMWS, coherent excitation of rota-tional states is directly involved and the analysis requiresquantum averaging [82].

Enantio-sensitive non-linear wave-mixing.PXCD and EMWS are not the only two relatives inthe family of enantio-sensitive phenomena which occurin the electric-dipole approximation and follow Rule1. Although PXCD and EMWS are resonant processes,and oscillations take place in the absence of the drivingfield, the pseudoscalar g can loosely be interpreted asa second order susceptibility χ(2). Indeed, PXCD andEMWS are closely related to the original predictionsof Giordmaine [74] for sum- and difference-frequencygeneration in chiral liquids. The expression for thesecond-order induced polarization at, e.g. the differencefrequency can be rewritten in the form dictated by ourRule 1:

~P (2)(ω2,1) = χ(2)~L, (12)

The non-linear susceptibility χ(2) is a sum over states |i〉

10

2ωω1

0

1

2b c SHG

3ωω1

ω2

0

1

2SFG

3ωω1

ω2

0

1

2DFG

3ω

ω1 ω2

a PXCD

3ωω1

ω2

EMWS

FIG. 7. Diagrams for various nonlinear enantio-sensitive or chiral-sensitive processes. Different col-ors mark different directions of light polarization(e.g. ”green” -x, ”red” - y, ”blue” - z). a Diagramsfor photo-excitation circular dichroism (PXCD), difference-frequency generation, and enantio-sensitive microwave spec-troscopy (EMWS); b Sum-frequency generation (SFG). Ifω1 = ω2, SFG can not be enantio-sensitive, because therespective spatial directions become non-distinguishable. Itmeans that second harmonic generation (SHG) is prohib-ited in randomly oriented chiral molecules in the lowest-orderperturbative regime [116]. c Second harmonic generationbecomes enantio-sensitive in elliptically polarized field ω inthe non-perturbative regime, because the respective diagramrecords the direction of field rotation via one up and one downarrow.

and |j〉 involving terms of the form (~di,0 × ~dj,0) · ~dj,i; ~Lis given by Eq. (10).

Fig 7 shows non-linear optics-type diagram for PXCD,EMWS, (Fig 7a) and difference frequency generation(DFG) (Fig 7b), which unifies all three processes. Thearrows directed up describe amplitudes of photon ab-sorption, while the arrows directed down describes con-jugated amplitudes corresponding to photon emission.These amplitudes are excited by the respective (conju-gated or not) components of the incident field: the con-jugated field is associated with the arrow directed down.

Further advances in enantio-sensitive perturbativenon-linear optics are described in the review by Fisheret al [75].

Similarly, in the context of highly non-linear inter-actions, symmetry arguments [76, 77, 125] show thatthe interaction between a periodic (but not necessar-ily monochromatic) electric field and an isotropic chiralmedium can lead to enantio-sensitive polarization at evenmultiples of the fundamental frequency ω of the field.

That is, while ~P (2n+1)ω is identical for both enantiomers,~P 2nω has opposite signs for the opposite enantiomers.Analogously to the perturbative case, this means thatthere are electric field configurations for which even-orderhigh harmonics are possible only if the medium is chiral[76, 77]. Since in the non-perturbative regime second har-monic generation can involve additional up-down arrowscorresponding to virtual emission/absorption of photons,Fig. 7 c, it can also become enantio-sensitive.

In all of these methods, the ability to distinguish be-tween opposite enantiomers requires access to the phaseof the induced polarization. Thus, intensity measure-ments of the even order polarizations can distinguish chi-ral media from achiral media or different chiral mediafrom each other, but cannot distinguish between oppo-site enantiomers.

In the next sub-section (see cHHGd), we shall see howenantio-sensitive information can be mapped on the po-larization ellipse of the nonlinear optical response.

Chiral high harmonic generation in the electric-dipole approximation (cHHGd). High harmonicgeneration (HHG) is an extremely nonlinear process thatconverts intense radiation, usually in the IR domain,into high-energy photons with frequencies that are high-integer multiples of the incident field frequency [131]. Itcan be understood semi-classically as a sequence of threesteps, starting with strong-field ionization [132]. In thesecond step, the laser electric field takes the liberatedelectron away from the core, driving its oscillations inthe continuum. Note that the laser field can also interactwith the ionic core, driving rich multi-electron dynamics[133]. The third step is the electron-core recombination,resulting in the emission of the harmonic light. A typi-cal HHG spectrum contains information about both thestructure of the atomic/molecular system and the ultra-fast dynamics between ionization and recombination.

The third step of HHG, radiative electron-core recom-bination, is the inverse of photoionization. Therefore,just like in the photoionization in PECD, the recombi-nation dipole responsible for HHG driven by an ellipti-cally polarized field in isotropic chiral media can developa component which is orthogonal to the polarization ofthe driving field, i.e. along the direction of light prop-agation, as show in Fig. 8. This component oscillatesout of phase in media of opposite handedness. How-ever, unlike the photoelectron current in PECD, this chi-ral dipole component escapes direct observation in stan-dard HHG measurements: the dipole oscillating alongthe light propagation direction radiates orthogonal to it.This means that in the electric-dipole approximation thechiral component propagates in the direction orthogonalto the direction of the driving field and thus cannot beobserved in the macroscopic HHG signal. Indeed, themacroscopic HHG signal requires coherent addition ofthe light emitted from different positions in the chiralmedium, which is only possible if the harmonic light co-propagates with the driver. Indeed, in the first chiralHHG experiments [34] and subsequent setups [92–94, 96]this enantio-sensitive dipole component could not be ob-served, and the enantio-sensitive response relied on theinteraction of chiral molecules with the magnetic compo-nent of the laser field.

The fact that the electric-field vector of the intenselaser field driving HHG is orthogonal to its propagationdirection stops us from imprinting the strong chiral re-sponse shown in Fig. 8 into the macroscopic HHG signal,severely limiting the potential of the HHG camera. Thisproblem can be overcome by creating light with “for-ward” ellipticity, or transverse spin [114], and using todrive chiral HHG.

Consider two laser beams that propagate non-collinearly at a small angle, both carrying the same fun-damental frequency and linearly polarized in the prop-agation plane. In the overlap region, the total electric

11

0

0.5

1

0.25

0.75

1.25

Time

(fs)

FIG. 8. Time-dependent polarization driven in randomly ori-ented propylene oxide by an elliptically polarized field withintensity 5 · 1013W/cm2, wavelength 1770nm and 5% of el-lipticity (see [77] for details of the calculations). The polar-ization of the driving field is depicted in pink. The ultra-fast polarization response is 3D, chiral, and enantio-sensitive:the in-plane (achiral) polarization components are identicalin left- and right-handed molecules, whereas the (chiral) out-of plane component is out of phase in opposite enantiomers(compare with the helix in Fig. 6).

field becomes elliptically polarized in the plane of prop-agation (x, y), with the minor ellipticity component inthe propagation direction (y-axis), see Fig. 9a. Now thechiral dipole component has adequate orientation to gen-erate harmonic light that co-propagates with the drivingfield. Thus, it can be mapped onto the macroscopic HHGsignal [77].

Non-collinear optical setups are not the only way ofcreating electric-field vector components along the prop-agation direction which lead to forward ellipticity. Suchlongitudinal components naturally arise when light isconfined in space, as it happens in a tightly focused laserbeam, see Fig. 9b.

The HHG signal driven by light with forward ellip-ticity has two orthogonally polarized components: the(standard) component in the propagation plane, which isnot sensitive to chirality, and the enantio-sensitive com-ponent orthogonal to it, Fig. 9b, which is out of phasein media of opposite handedness. One might think thatthe harmonic light would be elliptically polarized, butthis is not necessarily the case. Indeed, the Rule 2 dic-tates that the enantio-sensitive and non-enantio-sensitivecomponents of the induced polarization carry differentharmonic frequencies: the non-enantio-sensitive compo-nent carries odd harmonics, the enantio-sensitive com-ponent carries even harmonics, Fig. 9 c, (see Rule 1).Hence, for relatively long pulses, the non-collinear setupin Fig 9 generates non-overlapping even and odd harmon-ics, which are orthogonally polarized. Only the phase ofthe even harmonics records the medium’s handedness [77]as it follows from Rules 1, 2. But, what if the drivingpulse is only a few cycles long, and therefore has broadspectrum?

Nonlinear optical rotation. In chiral HHG drivenby a few-cycle laser pulse which has forward ellipticity,the chiral (even-order) and achiral (odd-order) harmonicscan spectrally overlap. Then the emitted harmonic lightbecomes elliptically polarized [78] in the overlap region.

FIG. 9. Driving chiral HHG driven using light with“forward” ellipticity. a Schematic representation of a non-collinear setup for creating a field with “forward” ellipticity;b A Gaussian beam acquires a strong longitudinal compo-nent upon tight focusing, also leading to “forward ellipticity”,which has opposite sign on the opposite sides of the laser beamaxis. c Achiral response corresponds to odd harmonics, chiralresponse corresponds to even harmonics and is out of phase intwo enantiomers. d Spectral overlap of odd-order and even-order nonlinear-optical response in a few cycle pulses leads toopposite rotation of the polarization ellipse in opposite enan-tiomers.

Since the chiral components of the nonlinear-optical re-sponse are out of phase in opposite enantiomers, boththe spin (ellipticity) and the rotation of the polarizationellipse of the nonlinear-optical response in the spectraloverlap region will be opposite in opposite enantiomers.

Fundamentally, using the ultra-broad spectrum of atightly focused few-cycle pulse [78] is akin to using atwo-colour non-collinear setup. Such a setup allows oneto tailor the polarization of the driver in two and threedimensions and maps the molecular handedness onto thepolarization properties of the emitted harmonic light.

A polarizer placed before the high harmonic detectorand rotated by an angle 0 < α < 90 with respect to thelinearly polarized driving field can convert the enantio-sensitive orientations of the harmonic polarization ellipseinto different signal intensities [78]. Just like the tradi-tional optical activity, its nonlinear analogue does notrequire chiral light (see Section 2.1.1). As a result, the

12

total number of photons that reach the detector does notdepend on the medium’s handedness. Note that the ro-tation of the polarization ellipse of the emitted light [78]is analogous to the rotation of the linear polarization instandard optical rotation. Indeed, in the non-linear opti-cal rotation [78], the chiral setup is formed by the pseu-dovector defining rotations of the polarization ellipse and

by the vector defining the propagation direction k of the

light. k is relevant in this electric-dipole effect not only

because of phase-matching conditions along k but also

because k is encoded in the pattern of forward ellipticitiesof the driving field, a phenomenon known as transversespin-momentum locking [114].

Rule 3. A chiral observer allows the detection ofenantio-sensitive tensorial observables, e.g. quadrupolarcurrents and quadrupolar polarizations.[? ]

Tensorial observables are particularly relevant in two-color fields, such as a laser beam carrying ω and 2ω fre-quencies, arranged so that enantio-sensitivity at the levelof vectorial observables is symmetry forbidden. The two-color aspect of these fields naturally brings perspectivesof coherent control into the discussion. Let us considersome examples.

Photoionization in two-color fields withoutspin. The light pseudovector in many electric-dipole-based methods is proportional to the fields’s spin. Inthe absence of spin, there can be no vectorial enantio-sensitive observables. For example, laser beams carryingω and 2ω frequencies, linearly polarized and orthogonalto each other, do not carry spin: their vector productchanges direction during the laser cycle and is equal tozero on average. Hence, they cannot excite vectorialenantio-sensitive observables in randomly oriented en-sembles. Yet, enantio-sensitive photoionization signalstriggered by such fields have been predicted [70, 71] anddetected [73].

The point here is that, in addition to vectorial ob-servables such as induced polarization (ESMW, PXCD,difference- and sum-frequency generation, cHHGd), pho-toelectron current (PECD, PEXCD), chiral measure-ments can also yield tensorial observables. The two-colorarrangement above takes advantage of such observables.

A simple way to introduce tensorial observables isto consider photoionization [101]. Indeed, any photo-electron angular distribution W (θ, φ) can be decomposedin spherical harmonics as

W (θ, φ) =∑l,m

bl,mYl,m(θ, φ), (13)

where the values of the coefficients bl,m for fixed l and−l ≤ m ≤ l are the entries of a spherical tensor of rank land dimension (2l+ 1) [134]. The determination of thesecoefficients is therefore equivalent to a tensorial measure-ment.

Since vectors are tensors of rank one, for l = 1 weobtain the vectorial observables we have already dis-cussed. In PECD, for example, where W (k, θ, φ) is the

photoelectron angular distribution at photoelectron en-ergy k2/2, the three coefficients b1,m describe the vectorof the photoelectron current; b1,±1 = 0 due to symme-try and b1,0 6= 0 is responsible for the so-called forward-backward asymmetry. In the bound case, e.g. in PXCD,one could employ W (r, θ, φ) to describe bound electrondensity. In this case the b1,m coefficients describe the ex-pectation value of the electric-dipole operator couplingthe two states excited by the pump pulse.

Just like the b1,m coefficients encode vectorial proper-ties, the b2,m coefficients describe the quadrupolar partof W (θ, φ). In the context of photoionization with cir-cularly polarized light, it is well known that these co-efficients are not enantio-sensitive. However, they be-come enantio-sensitive and correspond to excitation of anenantio-sensitive quadrupolar photoelectron current re-sulting from the interference between a two-ω-photon anda one-2ω-photon ionization pathway when the ω and 2ωfields are linearly polarized perpendicular to each other,as predicted in [70, 71]. These predictions have beensupported by the observation of similar enantio-sensitivemultipolar signals with l ≥ 2 in strong-field ionization offenchone and camphor [73]. The analysis of the symme-try properties of this electric field configuration revealsthat, together with an appropriately oriented quadrupo-lar detector, it yields a chiral setup (see Fig. 4c), and itshandedness emerges naturally in the expressions for theenantio-sensitive response [115].

Let us consider interference of the one-photon ioniza-tion pathway of the initial state |0〉 triggered by the 2ω-field with the two-photon ionization pathway via an in-termediate state |1〉, triggered by the ω field. The fieldsare defined as

~E(t) = ~Eωe−iωt + ~E2ωe

−2iωt + c.c., (14)

with ~Eω = Eωx and ~E2ω = E2ωe−iφz. One can show that

the coefficient describing an xy quadrupole photoioniza-tion yields is [115]

b2,−2 = A(1)∗A(2)gS + c.c., (15)

where tilde indicates that the expansion was over thereal spherical harmonics, the coefficients A(1) and A(2)

depend on the detunings and pulse envelopes,

g =

∫dΩk[k·(~d~k,0×~d~k,1)](k·~d1,0)+[k·(~d~k,0×~d1,0)](k·~d~k,1)

(16)is a complex-valued molecular pseudoscalar with a struc-ture analogous to that found in PECD, and

S = ( ~Eω · ~Eω)[ ~E∗2ω · (x× y)] (17)

is a pseudoscalar that encodes the handedness of the chi-ral setup. Note the emergence of two axes, x and y, inthe expression for S. They highlight the role of the de-tector (see Fig. 4c) in defining a reference frame thatdistinguishes directions for which the product xy is pos-itive from directions for which it is negative (see also

13

discussion related to Fig. 4). Note that S depends onthe two-color phase φ, which can be used as a control pa-rameter. The phase φ is analogous to the relative phase

between ~E∗(ω1,0) and ~E(ω2,0) in PXCD and to the rela-tive phase between the two perpendicular components ofa circularly polarized light in PECD.

Just like vectorial enantio-sensitive electron current inphotoionization (PECD) has its counterpart in boundstates (PXCD), the quadrupolar current found in pho-toionization [70, 71, 73] has a quadrupolar analogue inthe context of bound excitation. That is, the samefield configuration and the same interference schemetranslated to the context of bound excitation leads tothe emergence of an enantio-sensitive quadrupole [120].There is, however, an important difference from the in-duced bound-state dipole in PXCD: the intereferencehere arises from the two pathways, one-photon and two-photon, and no longer requires coherent population oftwo electronic or vibronic states: a single final boundelectronic state is sufficient. Consequently, the gener-ated quadrupole is permanent: it does not oscillate andis associated with uniaxial orientation of the (initiallyisotropic) molecular sample.

Note that, in all phenomena considered in this section,the total intensity of the nonlinear-optical or photoelec-tron signal is not enantio-sensitive because the drivingfields are not chiral. Achieving enantio-sensitivity in thetotal signal intensity requires an efficient chiral photonicreagent: synthetic chiral light.

III. THE SECOND ELECTRIC-DIPOLEREVOLUTION IN CHIRAL MEASUREMENTS:

EFFICIENT CHIRAL REAGENT

A chiral reagent provides access to fundamentally dif-ferent enantio-sensitive observables upon interaction withchiral matter: total enantio-sensitive intensity signals,not accessible by a chiral observer. Circularly polarizedlight, the standard chiral photonic reagent, owes its hand-edness to the (chiral) helix that the electric-field vectordraws in space (see Fig. 10a), which can either be left-or right-handed. Its chirality is, however, non-local –atany given point in space, the electric-field vector drawsa planar circle. A fundamentally different way of endow-ing light with chirality is to encode it in time, making thetrajectory that the tip of the electric field vector traces intime chiral [104]. In contrast to the (standard) handed-ness of circularly polarized light, this new type of chiralityis defined locally, at each point in space. It arises alreadyin the electric-dipole approximation.

Concept 1. Locally chiral light is chiral within theelectric-dipole approximation: the tip of the electric fieldvector draws a (three-dimensional) chiral Lissajous figure intime. The generation of this light requires three orthogonalpolarization components and, at least, two colours.

An example of a locally chiral field [104] is shown inFig. 10b. The combination of a fundamental field, which

k k

x

yz

xy

z

ω

2ωω

2ω

a b

c d

FIG. 10. The concept of synthetic chiral light. a, Elec-tric field vector (red arrow) of circularly polarized light drawshelix as light propagates in space. b Synthetic chiral light islocally chiral, because the tip of its electric field vector (redarrow) draws chiral Lissajous figure at every point in space.Colour indicates temporal evolution of the trajectory, drawnby the tip of the electric field vector.

is elliptically polarized in the propagation (xy) plane,and an orthogonally polarized second harmonic gener-ates chiral Lissajous figure. Indeed, if we reflect field’strajectory, e.g. through the xy plane, the elliptically po-larized ω-field remains the same, but the z-polarized 2ωcomponent flips sign. These mirror-reflected Lissajousfigures cannot be superimposed by any rotation and/ortranslation.

Such locally chiral light can drive strongly enantio-sensitive optical signals in isotropic chiral matter viapurely electric-dipole interactions. Control over the tem-poral structure of the light field should enable efficientcontrol over the enantio-sensitive response of chiral mat-ter [104]. For the field presented in Figs. 10b, three keyparameters enable such control (for a given total inten-sity): the ellipticity of the fundamental field, the ampli-tude of the second harmonic, and the two-colour phasedelay.

As we know from Section 2, a chiral “object”, such aslocally chiral light, must have at least one pseudoscalarwhich characterizes its handedness. What is the chiralitymeasure (pseudoscalar) of this locally chiral light?Concept 2. Chiral correlation functions character-ize the local handedness of synthetic chiral light by record-ing the correlated interplay between the different frequencycomponents of the light wave, which encodes the handed-ness of the Lissajus figure. Chiral correlation functions char-acterize the strength of non-linear enantio-sensitive light-matter interaction in the electric-dipole approximation.

To characterize the handedness of light’s Lissajous fig-ure, one can take three snapshots of the electric-field vec-

tor ~F (t) at three successive instants of time t1, t2 andt3, and construct a triple product of these three vectors

14

~F (t1) · [~F (t2) × ~F (t3)]. If such a product is non-zero, itmeans that between t1 and t3 the tip of the electric fieldvector traced a chiral trajectory in the space of Fx, Fy, Fz.To make sure that not only a section, but the entire Lis-sajous curve is chiral, one can average the triple productover time:

H(3)(τ1, τ2) =

∫ T

0

dt ~F (t) · [~F (t+τ1)× ~F (t+τ2)] , (18)

with H(3)(τ1, τ2) being the third-, and the lowest-orderchiral field correlation function [? ]. The use of chiralcorrelation functions in the frequency domain, evaluatedat the field’s frequencies, is often more convenient thanthe direct application of time-domain expressions. First,it removes the arbitrary choice of τ1, τ2, etc. Second, itprovides a clear connection with the multi-photon pro-cesses that record the field’s handedness and its interac-tion with chiral matter. In the frequency domain, thechiral correlation function h(3) is simply a triple productinvolving three frequency components of the laser field,ω1, ω2, and −(ω1 + ω2) [104]:

h(3)(−ω3, ω1, ω2) = ~F ∗ω3· [~Fω1

× ~Fω2]. (19)

This triple product is non-zero if these three frequencycomponents are non-coplanar.

Rule 4. The scalar enantio-sensitive response of chiralmatter to locally chiral light is nonlinear and results fromthe interference between a chiral even-order process and anachiral odd-order process. Thus, it involves an odd numberof photons, leading to chiral light correlation functions ofodd order.

For example, h(3) characterizes the interference of twopathways leading to absorption in a 3-level system withchiral states 0, 1 and 2. The first (achiral) pathway is as-sociated with the linear response at frequency ω3, whichis not sensitive to chirality, and leads to polarization atω3 = ω1+ω2 along z. The second (chiral) pathway corre-sponds to sum-frequency generation, a second-order pro-cess that records the molecular handedness: the mediumabsorbs one ω1 photon and one ω2 photon from the fieldcomponents polarized in the x− y plane, generating po-larization at frequency ω3 along z. Sum-frequency gener-ation is symmetry-forbidden in a randomly oriented en-semble of achiral molecules [74]. That is, this secondpathway is unique to chiral media, and the induced po-larization is out of phase in media of opposite handed-ness. The two pathways interfere, making absorptionand emission at frequency ω3 strongly enantio-sensitive(Fig. 11). In a randomly oriented ensemble of chiralmolecules, the enantio-sensitive contribution to absorp-tion is =[χ(2)h(3)]. The physical meaning of h(3) is clearfrom Table I, comparing standard absorption CD andnon-linear absorption CD in the electric-dipole approx-imation: h(3) plays the role of optical chirality, whichcharacterizes the strength of enantio-sensitive absorptionin the linear response [100]. Likewise, in the non-linear

regime, the molecular pseudoscalar formed by the tripleproduct of the three relevant dipoles replaces the one typ-ical for the linear response – the scalar product betweenthe electric and magnetic dipoles.

3ωω1

ω2

0

1

2

2ωω1

0

1

2ba

FIG. 11. Enantio-sensitivity in absorption. Absorptionoccurs in a three-level system driven by fields with frequenciesω1, ω2, and ω3 polarized along x, y, and z, respectively. Thelack of inversion symmetry in a chiral molecule allows fordipole couplings between all states. The second-order (two-photon induced) polarization at ω3 = ω1 + ω2 is generatedalong z in randomly oriented chiral media.

Synthetic chiral light with two colors For a two-colour field, such as the one in Fig. 10c, h(3) = 0 simplybecause the field does not contain three frequencies. Itmeans that nonlinear 3-photon processes driven by thisfield are not enantio-sensitive, but it does not necessarilymean that the field is achiral. If the field is locally chiral,its handedness can be recorded in higher-order processes,which are characterized and controlled by higher-ordercorrelation functions. They involve additional dot prod-ucts of the electric field vectors, evaluated at differenttimes. The next-order chiral correlation function is

H(5)(τ1, τ2, τ3, τ4) =

∫ T

0

dt~F (t) · [~F (t+ τ1)× ~F (t+ τ2)]

· [~F (t+ τ3) · ~F (t+ τ4)] (20)

and, in general,

H(n)(τ1, τ2, ..., τn−1) =

∫ T

0

dt~F (t) · [~F (t+ τ1)× ~F (t+ τ2)]

... [~F (t+ τn−2) · ~F (t+ τn−1)](21)

where n is an odd number. For the field in Fig. 10b, thelowest-order non-zero chiral correlation function is H(5).Therefore, the lowest order enantio-sensitive response ofisotropic chiral matter to this light is of the fifth order.In the frequency domain [104]

h(5)(−2ω,−ω, ω, ω, ω) = ~F ∗2ω · [~F ∗ω × ~Fω][~Fω · ~Fω] (22)

describes and quantifies the lowest-order enantio-sensitive response of isotropic chiral media to this light.Here, again, the enantio-sensitivity arises from the in-terference of two pathways. The first, achiral, pathwayis associated with the linear response at frequency 2ω,

15

TABLE I. Absorption CD for two types of chiral light.

Type of light Absorption CD Molecular psudoscalar Light pseudoscalarNatural light

(helix in space) =χemOC [100] in resonance χem ∝ [~df,i · ~mf,i] OC ∝ [ ~E∗ω · ~Bω]Synthetic light

(helix in time) =χ(2)h(3) in resonance χ(2) ∝ [~d2,0 · (~d2,1 × ~d1,0)] [135] h(3) ∝ ~E∗(ω2,0) · [ ~E(ω2,1)× ~E(ω1,0)] [104]

which leads to induced polarization at 2ω along z. Inthe second, chiral, pathway, the medium absorbs three ωphotons from the major field component and emits oneω photon into the minor ellipticity component, also gen-erating polarization at frequency 2ω along z (see [104]),orthogonal to the polarization plane of the ω-field. Thecombination of up– and down-arrows records the direc-tion of rotation of the driving field. The second pathwayexists only in chiral media, and the induced polarizationis out of phase in media of opposite handedness. In-terference between these two pathways enables enantio-sensitive absorption and emission at the frequency 2ωand the possibility of achieving enantio-sensitive popula-tions of excited electronic states. The enantio-sensitivecontributions to these observables can be written as aproduct of two pseudoscalars [104]: h(5) characterizingthe field’s handedness and a molecular pseudoscalar in-volving first- and fourth-order susceptibilities.

Locally chiral light vs globally chiral light Whilelocally chiral light and the concept of local chirality havebeen introduced [104] very recently, the second electric-dipole revolution started almost two decades ago. Usingquantum control strategies, Gerbasi, Brumer, Saphiroand co-workers proposed a two-step optical scheme forenantio-purification of randomly oriented mixtures of op-posite enantiomers, which works in the electric-dipole ap-proximation [110]. In the first step, a combination ofthree laser pulses with mutually orthogonal linear po-larizations was used to selectively excite one of the twoenantiomers to a selected vibrational state. In the sec-ond step, the photo-excited molecules were forced to fliphandedness by a sequence of two linearly polarized pulses.Their simulations predicted 95% of enantio-purity whenstarting from a racemic mixture of dimethylallene [110].By controlling the relative phases between the laser fieldsin the first step, they were able to control whether theleft-handed molecules were turned into right-handed orvice versa. This is probably the earliest example of ap-plication of locally chiral light to enantio-manipulationof molecules. The relative phases between the colorsfully control the shape of light’s Lissajous figure and itshandedness, controlling the outcome of the interferencein Fig11.

Yet, there is an important caveat to this scheme. Lo-cally chiral fields carrying three orthogonally polarizedcolours can be realized in the overlap region of two(or more) laser beams that propagate in different di-rections. However, the phase delay between the non-collinear beams, i.e. the relative time at which theirwavefronts reach a specific point in space, is space-

dependent. As a result, the handendess of the generatedlocally chiral field changes periodically in space, and sodoes the enantio-sensitive response of chiral matter.

For example, for the laser parameters proposed in[110], considering cross-propagating beams, the field’shandedness would change in space with periodicities onthe order of a few micrometers. The application ofsuch a field to a racemic mixture of isotropically dis-tributed left- and right-handed molecules would create anon-homogeneous distribution of left- and right-handedmolecules, which would be periodically distributed inspace. This structured distribution would be, on average,still racemic, unless using extremely tight laser focusingor thin media.

This problem is alleviated in the case of the longerwavelengths associated with the microwave radiation,which leads to significantly wider spatial regions wherethe field maintains its local handedness, on the order of afew tens of centimeters. Eibenberger, Doyle and Patter-son pioneered enantio-selective population of rotationalstates using phase-controlled microwave fields [112] to-gether with Schnell and co-workers [136], who demon-strated an alternative implementation.

Unfortunately, if one tries to apply an equivalentscheme to achieve enantio-sensitive populations of elec-tronic states, one has to face the above problem: thefield’s local handedness changes rapidly in space, destroy-ing the enantio-sensitivity in the total (global) integratedresponse of the macroscopic medium. To translate thehuge enantio-sensitivity enabled by locally chiral fieldsto the macroscopic response of the medium, at the levelof total signal intensities, the field also needs to be glob-ally chiral.Concept 3. A field is globally chiral if and only if its

global (macroscopic) structure is chiral.

While this definition is somewhat redundant when ap-plied to standard circularly polarized light, which is ei-ther left- or right-handed everywhere in space, it be-comes particularly relevant in the characterization of themacroscopic response of isotropic chiral matter to locallychiral light, whose handedness can be spatially struc-tured. Such synthetic chiral light is globally chiral ifits handedness, characterized by the nth-order chiral cor-relation function, survives integration in space, i.e. if∫h(n)(~r)d~r 6= 0.In particular, if the field’s handedness is maintained

in space, i.e. if the phase of h(n) remains constant, onecan achieve the highest possible degree of control overthe enantio-sensitive response of chiral matter: quench it

16

in one enantiomer while maximizing it in its mirror twin[104].

The locally chiral field in Fig. 10 can be created ina way that maintains the same handedness globally inspace [104] using a non-collinear setup, see Fig. 12. Itcontains two laser beams that propagate at a small an-gle, with each beam carrying two cross-polarized phase-locked colors: the fundamental and its second harmonic.By controlling the ω,2ω phase delays in the two beams,one achieves full control over the field’s local handed-ness globally in space. This field enables complete dis-crimination between left- and right-handed randomly ori-ented chiral molecules via high harmonic generation spec-troscopy [104].

k2k1

F2ωF1

ω

F2ω2F1

ω2

ω

2ω

a b

α=5º

30 32 34 36 38 40 42 44Harmonic number

CR

(%

)In

tens

ity

0

100

200

0.5

1

R

L

CR=2IL−IRIL+IR−

xy

z

FIG. 12. Locally and globally chiral light. a, Syntheticchiral light that is locally and globally chiral can be createdwith two non-collinear beams carrying cross polarized ω and2ω colours [104]. In the overlap region, the total ω field iselliptical in the xy plane, the 2ω field is z-polarized, generatingthe chiral Lissajous curve in the inset. b, Even harmonicintensity emitted by randomly oriented left- and right-handedpropylene oxide and the chiral response, see [104] for details.The field’s chirality, and thus the enantio-sensitive responseof the medium, is fully controlled by the ω,2ω phase delays inthe two beams.

IV. NEW ENANTIO-SENSITIVEOBSERVABLES VIA STRUCTURING LIGHT’S

CHIRALITY

Synthetic chiral light that is locally and globally chi-ral makes an extremely efficient chiral photonic reagent[104]. Yet, the first dipole revolution taught us thatwe do not need to rely on the (global) handedness oflight to detect the chirality of matter efficiently. Canwe apply these lessons to synthetic chiral light? Canwe measure strongly enantio-sensitive signals using lightthat is locally chiral (h(n)(~r) 6= 0), but globally achiral(∫h(n)(~r)d~r = 0)? The answer is “yes”. Applying the

concepts from the first dipole revolution leads to newenantio-sensitive observables, which arise upon structur-ing light’s local handedness.

Concept 4. Chirality-structured light is light whosehandedness is non-trivially structured in space.

The possibility of structuring the local properties of

light in space [137], including both its intensity and phase[138], creates unique opportunities for imaging [139] andmanipulating [140] properties of matter. Likewise, struc-turing light’s chirality [100, 141–144] could open new ef-ficient routes for enantio-sensitive imaging and controlof chiral matter. With structuring of light performedlocally, the control extends to the level of individualmolecules. One example of the new type of structuredlocally chiral light is the chirality polarized light [105].

a b- + - + - +

-+-+ -+

unpolarized

charge polarized

unpolarized

chirality polarized

L R R RL L

L R L R L R

FIG. 13. Polarization of charge versus polarization ofchirality. a, 1D arrangement of charged units that is: (i)neutral and unpolarized, and (ii) neutral and polarized. b,1D arrangement of chiral units that is: (i) achiral (racemic)and unpolarized, and (ii) achiral and polarized.

The concept of polarization of chirality applies to bothlight and matter and is somewhat analogous to polariza-tion of charge. A periodic distribution of alternating pos-itive (+q) and negative (−q) charges in one dimension isunpolarized if the particles are uniformly distributed andpolarized if this distribution is periodically modified (Fig.13a). Likewise, a periodic distribution of chiral units ofalternating handedness can have polarization of chiral-ity if the units are not uniformly distributed, see Fig.

13b. Here, we find dipoles of chirality ~dc = h~r0, where~r0 is the vector connecting two nearby chiral units andh = hR = −hL is the handedness of one chiral unit. Notethat, regardless the value of ~dc, the medium is racemicand achiral, just like the medium of alternating negativeand positive charges is neutral.

This concept can immediately be applied to the syn-thetic chiral light using the same non-collinear setup asin Fig. 12a. Now, however, we can control the ω, 2ωdelays in each beam so that the field’s handedness, char-acterized by its fifth-order chiral correlation function, isnot maintained globally in space as in Ref. [104] but gen-erates the “dimers” of alternating handedness: dipoles ofchirality, see Fig. 14a, polarized along x.

This periodic, racemic space-time structure will inter-act differently with chiral matter of opposite handednessby generating a periodic near-field pattern of the enantio-sensitive nonlinear-optical local response. The mappingof this near-field nonlinear-optical response into the far-field image will be sensitive to the chirality dipole of light.

Fig. 14b shows the harmonic intensity emitted fromrandomly oriented fenchone as a function of the emis-sion angle. The total (angle-integrated) signal is thesame for left- and right-handed molecules, as the over-all field is achiral. However, the emission direction ishighly enantio-sensitive: while the left-handed moleculesemit light to the left (towards negative angles), the right-

17

a

x ( m)

|h(5

) | (ar

b. u

.)

L R LR L R LR L R LR

Divergence (degrees)

-4 40-8 80

0.5

1

0 2-2

Inte

nsity

0

0.4

0.8 b H12(R)(L)

FIG. 14. Chirality polarized light can be created usingthe setup of Fig. 12a, but adjusting the ω,2ω phase delaysso the field’s handedness, characterized by its 5th-order chiralcorrelation function h(5), is not maintained globally in space,in contrast to [104]. Here, it creates a periodic structure ofdipoles of chirality. a, h(5), its phase (i.e. the field’s handed-ness) is encoded in the colours. The arrows indicate the direc-tion polarization of chirality, which is imprinted in the non-linear response of chiral matter. b, 12th-harmonic emissionfrom randomly oriented fenchone, see [105] for λ = 1030nm,

F(1)ω = F

(2)ω = 0.015a.u. and F2ω = Fω/10.

handed molecules emit light to the right (positive angles).By looking into a specific emission direction, a chiral

observer will allow us to distinguish between left- andright-handed molecules. Note that neither the light field,nor the detector are chiral on their own. However, byselecting a specific emission direction in the far field, themeasurement setup, created by the (achiral) light fieldand the detector, becomes chiral.

V. TOWARDS EFFICIENT CONTROL,IMAGING AND MANIPULATION OF CHIRAL

MOLECULES WITH LIGHT

We now outline some of the new opportunities offeredby substituting concepts and tools of enantio-sensitivemolecular imaging and control, which use linear responseand require the magnetic field component of the lightwave, by concepts and tools that rely on non-linear re-sponse and do not require the magnetic field component.In terms of imaging, in this section we address the op-portunities for photon-based spectroscopies. The nextsection will focus on new opportunities arising in photo-electron spectroscopy.

A. Enantio-sensitive and molecular specificspectroscopy with synthetic chiral light

In this subsection we describe the fundamental originsof enantio– and molecular sensitivity in the interaction ofsynthetic chiral light with chiral matter. Possible appli-cations of these ideas to enantio-separation are discussedin the subsequent subsection.

1. Enantio-sensitive molecular markers

The enantio-sensitive response of chiral matter to syn-thetic chiral light relies on the coherent interplay betweenthe two contributions to light-induced polarization, achi-ral and chiral, the latter having opposite phase in me-dia of opposite handedness. One can control the ampli-tude and the relative phase of these two contributionsby controlling the handedness of the light field, so thatthey interfere constructively in one enantiomer and de-structively in its mirror twin, maximizing the enantio-sensitivity.

The next step is to develop spectroscopy, which wouldprovide access to molecule-specific information recordedin such interference, such as the relative amplitude andphase between the chiral and the achiral non-linear re-sponses. The latter should be naturally sensitive to thetype of the molecule and its conformation [106]. As aresult, the handedness of the light field that maximizesenantio-sensitivity of the optical response should also bemolecule-specific, opening new opportunities for efficientmolecular recognition and enabling the design of molec-ular markers: molecule-specific “fingerprints” of chiralmolecules, based on the relative amplitude and phase ofchiral and achiral non-linear responses.

For example, the intensity of the enantio-sensitiveemission at frequency 2ω driven by the locally chiral field(Fig. 10) is

I = Iach + Ich + a cos(φM + φω,2ω) (23)

where Iach and Ich are the intensities associated withthe achiral and chiral pathways, respectively, φM =arg(χ(4) − χ(1)) is the molecular phase that depends onthe first- and fourth-order susceptibilities, φω,2ω is the 2-

colour phase delay, and a ∝ σMσL|χ(1)||χ(4)||h(5)|, whereσM = ±1 depends on the molecular handedness andσL = ±1 keeps track of the sign of ellipticity of the lightfield. Experimentally, one can vary φω,2ω together withthe amplitude of the 2ω-field component to find the laserparameters that maximize chiral dichroism. These opti-mal parameters should be molecule-specific and may en-able efficient molecular recognition. Note that the phaseφω,2ω and the amplitude of the 2ω field component con-trols the light pseudoscalar, which couples to the respec-tive molecular pseudoscalar in the observables. Sinceboth the light and molecular pseudoscalars are differ-ent in different non-linearity orders [104], corresponding

18

to different frequencies of the emitted light, frequency-resolved optimal light parameters offer an additional di-mension (i.e. frequency) to this type of enantio-sensitive,molecule-specific spectroscopy.

2. Ultrafast optical rotation: multi-dimensional nonlinearspectroscopy with sub-cycle-controlled optical waveforms.

The sensitivity of an enantio-sensitive non-linear re-sponse to the two-colour phase (see Eq. 23) points tonew opportunities in non-linear spectroscopy with sub-cycle-controlled optical waveforms.

One example are few-cycle pulses, where the tem-poral structure of the electric field vector E(t) =E0a(t) cos(ωt + φCEP) depends on the carrier-envelopephase (CEP) φCEP. The sensitivity of the electronic dy-namics to the instantaneous value of the oscillating elec-tric field leads to strong CEP dependence of the nonlinearresponse [145–148]. For our purposes, the CEP acts asan additional spectroscopic parameter somewhat equiv-alent to the two-colour phase delay in long laser pulses,which carry two well-defined frequency components suchas ω and 2ω. In fact, changing the relative phase betweenthese two colors shapes individual oscillations of the totallaser field on the sub-cycle scale. Along this route, onecan use intense, linearly polarized light pulses to drivethe nonlinear analogue of optical rotation, now driven bypurely electric-dipole interactions [78]. Importantly, suchpulses should be confined both in space and in time.

Confinement in space can be achieved by tightly focus-ing the beam into a medium of randomly oriented chiralmolecules, as shown in Fig, 9b. Thanks to tight focusing,the field acquires ellipticity in the direction of light prop-agation, i.e. “forward ellipticity”. The chiral mediumconverts this forward ellipticity into the enantio-sensitiveresponse orthogonal to the propagation plane (the planeof the Figure.)

Confinement in time, arising in nearly single-cyclepulses with controlled CEP, ensures that the pulse hasultra-broad spectral bandwidth. The latter enables theinterference of the odd-order achiral response in the po-larization plane and the even-order chiral response or-thogonal to it.

As a result, the generated nonlinear polarization be-comes elliptically polarized and enantio-sensitive: the el-lipticity and the rotation angle of the non-linear opticalresponse will have opposite signs in media of oppositehandedness, signifying the non-linear optical activity (ro-tation) [78]. The rotation angle is controlled by the CEPof the laser pulse. In contrast to conventional opticalactivity, this nonlinear effect is driven by purely electric-dipole interactions and leads to giant rotation angles (andellipticities) already at the single-molecule level, enablingthe possibility of highly efficient chiral discrimination inoptically thin media. Since the CEP of the pulse con-trols the enantio-sensitive response in this non-linear op-tical activity, the value of the CEP that maximizes the

enantio-sensitive response of a chiral molecule may con-stitute a molecular marker. The enantio-sensitive sig-nal plotted vs frequency and the CEP phase presents atwo-dimensional set of data, which may be perceived asa “molecular QR-code”, sensitive to both the moleculeand possibly its conformer. While the uniqueness ofsuch measure is yet to be proven, the respective inves-tigation is interesting in its own right, as it may resultin a new interesting path in ultrafast non-linear spec-troscopy. Adding additional spectroscopic parameters toa few-cycle laser pulse, such as a frequency-dependentphase delay [149], may increase the dimensionality andconsequently the sensitivity of this approach, extendingopportunities for ultrafast enantio-sensitive imaging andcontrol in the electric-dipole approximation.

B. Enantio-sensitive manipulation.

Locally and globally chiral electric fields create op-portunities for excitation of only one of the two enan-tiomers of a chiral molecule to rotational, vibrational orelectronic states and open routes to achieving efficientenantio-manipulation. Enantio-sensitive excitations torotational states have already been demonstrated usingmicrowave fields with three orthogonally polarized com-ponents [112, 136, 150–152] including opportunities forits optimal control [151, 152].

A recent experiment [130] demonstrated an alterna-tive path to enantio-sensitive control over rotational ex-citations. This experiment realises the concept of chiralobserver and combines the optical centrifuge [153–155]with Coulomb explosion imaging. The chiral experimen-tal setup consists of an optical centrifuge, i.e. a linearlypolarised laser field rotating (with acceleration) at theslow, rotational, time scale about the propagation direc-tion. The additional axis is provided by the Coulomb ex-plosion imaging detector. The respective molecular pseu-doscalar should include the projection of the total angu-lar momentum (transferred from the field to the moleculeand recorded in its rotational excitation) on the detectoraxis, i.e. on the direction of the Coulomb explosion.

Alternatively, one can also project the total angularmomentum supplied by the optical centrifuge on a dif-ferent axis, e.g. the direction of an additional externalstatic field. This setup has recently been proposed byYachmenev et al [111]. The combination of strong in-frared fields forming the optical centrifuge and static elec-tric field polarized along the centrifuge rotational axis isan example of locally and globally chiral electric field ar-rangement.

Synthetic chiral light in the optical domain enablescontrol over the electronic degrees of freedom, and thusthe possibility of exciting a selected molecular enan-tiomer to a desired electronic or vibronic state. Enantio-sensitive coherent control over such chiral electronicclouds opens additional routes for enantio-selective ma-nipulation, since the properties of the photo-excited

19

molecules, e.g. their dipole moments or polarizabilities,can be substantially different from those in the groundstate.

Synthetic chiral light may also allow one to measurestrongly enantio-sensitive photo-absorption signals, withthe ambitious goal of enhancing the enantio-sensitivityof standard photo-absorption circular dichroism by sev-eral orders of magnitude. Along this route, introducingsynthetic chiral light to attosecond transient absorptionspectroscopy (ATAS) may allow one to induce and mea-sure strongly enantio-sensitive dynamics with attosecondtime resolution using photon-based measurements.

C. Enantio-separation.

Intense laser beams allow one to manipulate, accel-erate, decelerate and trap particles [156–158]. Enantio-sensitive optical potentials, which could be applied forselective manipulation, trapping and sorting of chiralparticles with specific handedness have recently beendemonstrated in linear light-matter interaction regime[141, 159–166]. A promising route to efficient enantio-separation was proposed by Cameron and co-workers[144, 167], who suggested creating gratings of chiral lightof alternating handedness, which could then send oppo-site molecular enantiomers in opposite directions. Theyfound strongly enantio-sensitive deflection angles in chi-ral molecules with strong magnetic dipoles, such as he-licene, demonstrating the feasibility of the method. How-ever, the proposed optical setups are chiral only beyondthe electric-dipole approximation, and thus the sensitiv-ity of this method to smaller molecules and localized chi-ral structures such as due to an asymmetric carbon, couldbe limited by the weakness of the non-electric-dipole in-teractions. Synthetic chiral light may allow one to over-come this limitation, enhancing these opportunities evenfurther.

VI. SYNTHETIC CHIRAL MATTER:IMPRINTING CHIRALITY ON ACHIRAL

MATTER

Synthetic chiral light can also be used to endow achi-ral matter with chiral properties. Achiral matter excitedwith such locally chiral light becomes “synthetic chiralmatter”. That is, one can aim to imprint chirality onatoms [168], achiral molecules [127], and solids, and readit out on ultrafast scales. In molecules, Owens et al.[127] showed that a chiral arrangement of a DC field andan optical centrifuge [153, 154] produces PH3 moleculesthat rotate around a P–H bond in a direction determinedby the centrifuge, with the P–H bond oriented along theDC field. Formally, in this case the chirality of the elec-tric field is imprinted on the molecular rotational states,which can be detected using e.g. ESMW spectroscopy.

In the case of electronic states, relatively simple su-perpositions of angular momentum eigenstates suffice toform chiral electronic wave functions in atomic hydro-gen [168], i.e. synthetic chiral atoms, which are al-ready oriented in space. These chiral atoms can alsodisplay PECD upon ionization with circularly polarizedfield [108, 168–170]. The experimental demonstration ofPECD on chiral atoms will require efficient excitationschemes. Such schemes have recently been proposed inthe first theoretical works on using locally chiral lightimprinting chirality on atomic ensembles [108, 170], sup-porting our general expectation that non-perturbative in-teraction of locally chiral light with atoms will in generallead to chiral superpositions of states.

Interesting opportunities may arise for imprinting chi-rality on atoms in optical lattices and on electrons insolids. Recently, the spatial symmetry of the light’s Lis-sajous figure has been used to imprint topological prop-erties onto a trivial two-dimensional hexagonal material[171]. Analogously, the longitudinal components emerg-ing in tightly focused pulses can be used to make locallychiral light, which can break the symmetry of a cubic lat-tice, by introducing “forward-backward” asymmetry inthe field-modified hopping coefficients and thus turningthe cubic lattice into a chiral object. Chiral crystals havebeen shown to possess interesting topological properties[172]. Therefore, imprinting chirality on lattices mightoffer a way to induce light-driven ultrafast topology.

Looking broadly, the outcome of the interaction be-tween such synthetic chiral matter and either ‘natural’matter or light (chiral or not) has not been explored andraises many interesting questions. How and to what ex-tent will the chirality of synthetic chiral matter influencethe outcome of its interaction with ‘natural’ chiral mat-ter or light? Is synthetic chiral matter useful for enantio-selective chemical synthesis? Can we design efficient cat-alysts for asymmetric synthesis based on synthetic chiralmatter? Is it possible to implement ultra sensitive chi-rality detectors based on chiral Rydberg atoms? Howwill many-body effects influence the dynamics of chiralRydberg atoms trapped in optical lattices? And can weperform quantum simulations of phenomena unique tochiral crystals [173] using chiral Rydberg atoms in opti-cal lattices?

VII. OPPORTUNITIES FOR IMAGING ANDCONTROL OF CHIRAL MOLECULES VIA

PHOTOIONIZATION

Photoelectrons are extremely sensitive structuralprobes of matter in gas and condensed phase. Inmolecules, electron scattering on the nuclei during pho-toionization provides information about their spatial ar-rangements. This information is recorded in the key pho-toionization observable: the angular and energy resolvedphotoelectron distribution; angle-resolved photo-electronspectroscopy can be viewed as “diffraction from within.”

20

PECD and PXCD are two important milestones inunderstanding chiral electronic and vibronic response ofmolecules to light. PECD in photoionization from astationary state is generated by chiral continuum cur-rents, while PXCD in photoexcitation is generated by chi-ral bound currents. Probing PXCD via photoionizationwith a circularly polarized field would inevitably “mix”these two types of chiral dynamics and such time-resolvedprobe could be understood in terms of “synchronisation”of chiral bound and continuum currents. Controlling theinterplay of pump and probe pulses can be interpreted interms of controlling this synchronisation. Thus, formu-lating standard photoionization observables in terms ofthe currents and fields they generate [101] may provide ahelpful physical picture and a language for communicat-ing the observations.

Since standard photoionization observables are definedin momentum space, the continuum electron currentsand the fields they generate should also be defined inmomentum space [101]. Such perspective is customary

in condensed matter physics, where one analyzes the ~k-dependent fields generated by electrons photo-excitedfrom a valence to a conduction band. Electron scatter-ing on the lattice sites may lead to swirling electron tra-jectories. These, in turn, generate a geometric magneticfield, known as Berry curvature. The geometric magneticfield in solids plays an important role in understandingthe electronic response, especially in materials with non-trivial topological properties.

Can the perspective offered by analysing the electronicresponse in molecular photoionization via a geometricmagnetic field generated by swirling electron photoion-ization current be fruitful, especially for chiral molecules?Could it allow us to uncover new enantio-sensitive ob-servables in photoionization? What can we learn fromthe topology of field lines? How does this field reflect in-formation about molecular dynamics prior to ionization?Interestingly, the standard photoionization observables

formulated in terms of local, i.e. ~k-dependent currentsand the fields they generate, provide some answers tothese questions (see Section 8.4 below).

As an extremely sensitive probe of molecular chiral-ity, PECD, has already given rise to a family of time-resolved methods which use various pump-probe setupsand detect angularly resolved photoelectron spectra. Allthese methods involve two- or higher order multiphotonprocesses. An important player in such setups is thepolarization of pump and probe pulses. For example,two-photon ionization of a chiral molecule can be per-formed in many different ways: linearly polarized pump- circularly polarized probe (TD PECD [64]), circularlypolarized pump - linearly polarized probe (PXECD [66]),circularly polarized pump - circularly polarized probe co-rotating (coherent control of PECD [72, 101]) or counter-rotating to each other.

Interestingly, chiral properties can also be probed us-ing linearly polarized pump and probe pulses, but con-ditions apply. What are these conditions? What is the

difference in photoionization observables detected usingthese different schemes?

In this section, we first provide the perspective on suchtwo-photon measurements, which demonstrates that eachof these schemes addresses different molecular properties,because it relies on different molecular pseudoscalars andtherefore exposes a different and independent aspect ofmolecular chirality. Next, we discuss the physical picturebehind these schemes and perspectives for imaging andcontrol of chiral molecular dynamics inspired by theseschemes. Finally, we discuss the geometric magnetic fieldand the physical origin of PECD using the language ofcurrents and fields generated by electrons during pho-toionization. At this point we would also address theX-ray community, which identifies imaging of molecularcurrents as an important milestone.

A. Chiral molecular fingerprints in the two-photonangular resolved photoionization

In these subsection we are dealing with methods ofchiral detection relying on the principle of the chiral ob-server, just like the PECD and PXCD methods consid-ered in Section 3. However, there is an important differ-ence. While PECD and PXCD are one-photon processesand involve two light field vectors that form a single light

pseudovector ~L (see Eq. 2), PEXCD is a two-photonprocess. In two-photon and, in general, multiphotonprocesses more than two light vectors may be relevant.

Therefore, different pseudovectors ~L1, ~L2, . . . can beformed. This gives rise to a hierarchy of chiral measures(see Section VIII). Analogously, more than three molec-ular vectors may be relevant, and thus several differentmolecular pseudoscalars g1, g2, . . . can be formed. Theavailability of multiple light pseudovectors and multiplemolecular pseudoscalars is reflected in a generalization ofEq. (2) according to

~v =∑i

gi~Li . (24)

That is, instead of a single product between a molecu-lar pseudoscalar and a light pseudovector, the enantio-sensitive vectorial observable ~v is given by a sum of sev-eral such products, involving all possible combinations ofall available vectors. Rule 7 in Section VIII provides themost general expression and points to its formal origins.

The array of different molecular pseudoscalars simplyreflects the fact that the description of a complex chiralobject requires the specification of more than one pseu-doscalar. For example, while a simple helix has just onehelicity, a compound helix where a loosely wound helix isformed from a tightly wound helix requires the specifica-tion of two independent helicities: one for the loose helixand one for the tight helix (see Fig. 15). In this sense,the coupling between molecular pseudoscalars and lightpseudovectors in Eq. (24) reveals how each of the ‘helic-ities’ of a chiral molecule (a rather complex ‘compound

21

helix’) couples to the light field. (Similar situation with”nested” chiral structures can occur due to the presenceof several chiral centers with different handedness in achiral molecule).

FIG. 15. An object displaying compound chirality in theform of two independent handedness: a helix made of a moretightly bound helix.

Two photon pump-probe process are a perfect example

illustrating Eq. (24). Suppose that the pump ~E1 inducesa transition from the ground state |g〉 into a bound ex-

cited state |j〉 and the probe ~E2 induces a transition from

the state |j〉 into the continuum |~k〉. For pulses and timedelays short compared to the rotational dynamics, thephotoelectron current is given by [82]

~J =

6∑i=0

gi~Li, (25)

where the characteristic molecular pseudoscalars gi are:

g0 =1

30

∣∣∣~dj,g∣∣∣2 ∫ dΩk

[(~d∗~k,j ×

~d~k,j

)· ~k], (26)

g1 =1

30

∫dΩk

[(~d∗~k,j ×

~d~k,j

)· ~dg,j

] (~dj,g · ~k

), (27)

g2 =1

30

∫dΩk

[(~dg,j × ~d∗~k,j

)· ~k] (

~dj,g · ~d~k,j), (28)

g3 =1

30

∫dΩk

[(~di,j × ~d~k,j

)· ~k] (

~dj,g · ~d∗~k,j), (29)

with

g4 = g∗1 , g5 = g∗2 , g6 = g∗3 . (30)

Importantly, these pseudoscalaras couple to differentlight pseudovectors:

~L0 =∣∣∣ ~E1

∣∣∣2 ( ~E∗2 × ~E2

), ~L1 =

[(~E∗2 × ~E2

)· ~E∗1

]~E1,

(31)

~L2 =(~E1 · ~E2

)(~E∗1 × ~E∗2

), ~L3 =

(~E1 · ~E∗2

)(~E∗1 × ~E2

),

(32)

~L4 = ~L∗1, ~L5 = ~L∗2, ~L6 = ~L∗3, (33)

which define constrains on the observation of each molec-ular pseudoscalar gi in terms of polarization of pump andprobe pulses.

For example, the g0~L0 term is associated with thePECD from the excited state |j〉 and requires only an el-liptically polarized probe field. The pseudovector fomed

by photoionization vectors (~d∗~k,j×~d~k,j) is important in its

own right, since it can be understood as a geometric mag-

netic field ~B(~k) associated with photoionization of chiralmolecules [168] (see subsection VII D). Note that g0 and

g1 encode different components of ~B(~k): g0 encodes theradial field component and g1 encodes the field compo-nent along the direction of the bound dipole connectingthe ground and excited states.

Accessing g1~L1 requires a different and more sophis-ticated arrangement of pump and probe polarizations.

Indeed, ~L1 is non-zero only when ~E2 is elliptically po-

larized and in addition ~E1 has a component perpendic-

ular to the plane defined by ~E2. In contrast, ~L2 and~L3 are non-zero even if both the pump and the probeare linearly polarized, as long as they are neither par-allel nor perpendicular to each other, suggesting a con-nection to some recent works using such arrangement ofpump and probe pulses to excite rotational states of chi-ral molecule and induce enantio-sensitive molecular ori-entation [126, 128, 174]. Note also that while ~L2 vanishes

for co-rotating circularly polarized pump and probe, ~L3

vanishes for counter-rotating pump and probe. Thesesimple rules together with the [Eqs. (25)-(33)] specifyingthe coupling between molecular pseudoscalars and fieldpseudovectors can be combined to determine the valuesof each molecular pseudoscalar gi.

In the case where the final state |~k〉 can be reachedvia two different intermediate states, additional molecu-lar pseudoscalars and field pseudovectors (see Ref. [82])resulting from the two-path interference contribute to thegeneration of the photoelectron current. Importantly,they also can be expressed [82] in compact form similar toEqs. (26)-(33). Unlike the direct terms, these terms os-cillate with the time delay between pump and probe andrecord the dynamics excited by the pump. One exampleof such dynamics are helical currents excited in boundstates by ultrashort circualry polarized pulses (PXCD).Below we provide a perspective on their detection.

B. Nonlinear photoionization probes of molecularchirality

In the previous section we gave the formal descriptionof the two-photon pump-probe setups for probing chiralmolecular structure and dynamics. Here we discuss theunderlying physical picture describing two-photon pho-toionization as an interplay of chiral bound and contin-uum currents. This interplay can be disentangled in PX-ECD, which induces photoionization using linearly polar-ized light to decrease the influence of continuum currents.

22

Probing helical currents in bound states (PX-ECD) via photoionization. Helical currents excitedin chiral molecules by short circularly polarized pulses(PXCD, see Section 3) can be probed by photoionization.Since induced polarization oscillates out of phase in op-posite enantiomers, the respective electron currents flowin opposite directions. Photoionization by linearly po-larized short pulse can reveal this direction of the boundcurrent: the liberated electron should typically continueto move in the same direction as it was moving in thebound states. Thus, two photoelectron detectors placedalong the propagation direction (spin) of the pump pulsewill show the forward-backward asymmetry: more for-ward electrons in one enantiomer and more backwardelectrons in the opposite enantiomer.

Interference of two photoionization signals originatingfrom the two excited PXCD states (see Fig. 5) recordsthe coherence between these states and also its time evo-lution as a function of the delay between the exciting(pump) and photoionizing (probe) pulses. Since bothpulses are short, interference of photoionization signalscoming from the two intermediate states excited by thepump may cover a considerable range of photoelectronenergies. In this range, one can observe that the pho-toelectron current oscillates with the time-delay betweenthe two pulses and has opposite directions for oppositeenantiomers, or for opposite spins of the pump pulse.That is, thanks to the molecular chirality, the spin of thepump photon can be read out from the photoelectron an-gular distribution even though the ionizing step is carriedout with linearly polarized light.

Indeed, one can show [66] that the photoelectron cur-rent is proportional to two terms:

JPXECDz (k) = σ[(~d01 × ~d02) · ~Dr

12(k)] sin(∆E21τ)−σ[(~d01 × ~d02) · ~Di

12(k)] cos(∆E21τ),(34)

where JPXECDx (k) = JPXECD

y (k) = 0, z and σ are thepropagation direction and spin of the pump pulse, re-spectively, τ is the pump–probe delay, k is the photoelec-tron momentum, ∆E21 is the energy difference between

the excited states, ~d01 and ~d02 are the transition dipoles

from the ground state to the excited states, ~D12(k) =~Dr12(k) + i ~Di

12(k) is a complex Raman-type photoioniza-tion vector that connects excited bound states via thecommon continuum. Since ~D12(k) encodes coherence

between the excited states, it plays the role of ~d12 inEqs. (8)-(10) for PXCD. The two triple products in Eq.(34) are the molecular pseudoscalars characterizing theenantio-sensitive signal in PEXCD. The photoionization

vector ~D12 in Eq. (34) is given by

~D12(~k) ≡ −4(~D1 · ~D∗2

)~k +

(~D∗2 · ~k

)~D1 +

(~D1 · ~k

)~D∗2 .(35)

Note that this general expression shows that every

available vector ( ~D1, ~D∗2 ,~k) can be used to “complete”

the triple product in Eq. (34). Physically, the appear-ance of the second and the third term in Eq. (35) is

due to partial alignment of molecules by the pump pulse.To validate this statement it is sufficient to consider anisotropic probe pulse, which cannot be sensitive to theinitial alignment by the pump. One can show (see SI ofRef. [66]) that only the first term in Eq. (35) survives inthis case.

How accurately can the continuum current imagethe current in the bound states? The difficulties hereare similar to those encountered in the so-called to-mographic imaging of molecular orbitals in High Har-monic Spectroscopy [175]. Namely, the bound-continuummapping is only exact in the case of the plane wavecontinuum: in this case, the total photoelectron cur-

rent ~JPXECDtot , integrated over all photoelectron energies

~JPXECDtot ≡

∫~JPXECDPW (k)dk is indeed proportional to the

bound current ~JPXECDtot ≡ ~JPXCD, since one can show

that −(1/2)∫~Di,PW12 (k)dk ≡ ∆E12

~d12. However, thestructure of the continuum in chiral molecules is a lotmore complex than simple plane waves. Perhaps, theconnection between the bound and continuum currentscan be established by introducing an additional unknown

function f(k), such that −(1/2)∫~Di,PW12 (k)f(k)dk ≡

∆E12~d12. This function will then have to be recon-

structed together with the bound current, e.g. iteratively,starting with f(k) = 1 for the plane wave continuum,similar to the efforts in tomographic reconstruction ofmolecular orbitals in High Harmonic Spectroscopy [176].

The first PEXCD images [66] were recorded via ex-citation of Rydberg bands in fenchone and camphoremolecules using a circularly polarized femtosecond pumppulse carried at 201 nm (with 80 meV 1/e2 bandwidth)and probing it using a time-delayed, linearly polarizedprobe pulse carried at 405 nm (with 85 meV at 1/e2

bandwidth). Although the results demonstrate excita-tion of a chiral vibronic wave-packet in these molecules,detailed information about the specific nature of thesedynamics requires further analysis. Such analysis couldprovide much desired insight into chiral molecular dy-namics at femtosecond time-scales and presents one ofthe exciting future opportunities for this field.

Two-color coherent control. Another interestingaspect of chiral molecular dynamics may result from theinterplay of bound and continuum chiral currents. Bothcurrents are present if both pump and probe pulses arecircularly polarized. Indeed, the photon spin carried bythe pump can be transferred to the current excited inbound states. At the same time, the photon spin car-ried by the probe pulse can lead to chiral continuumcurrents. This combination of pump and probe polariza-tions has been recently explored by Goetz et al [72] andused for two-photon coherent control of the chiral photo-electron current associated with the coefficient b1,0 andmultipolar currents associated with the coefficient b3,0.Goetz et al [72] achieved very significant enhancement ofenantio-sensitivity of photoionization observables by op-timizing the arrival of each frequency in the pump andprobe pulses. Since the scheme involves the absorption of

23

two circularly polarized photons, the optimization couldhave been related to achieving the best synchronizationbetween the bound and continuum currents. Whetherthe control has been associated with such synchronisa-tion remains to be seen, but exploring and exploiting theinterplay of bound and continuum currents for enhanc-ing chiral photoelectron signal may present an interestingfuture direction.

Time-dependent PECD implies exciting molecularvibronic dynamics by a linearly polarized pump pulseand probing it with a circularly polarised pulse [64]. Aswe have seen from previous examples, excitation with acircularly polarized pump imprints the photon spin ontothe bound states dynamics and excites helical currentsin chiral molecules. At first glance one may think thata linearly polarized pump cannot excite chiral dynam-ics, because the spin is required to define the “helic-ity” of the excited bound current. Indeed, since a lin-early polarized pump is a superposition of two counter-rotating circular pulses, we should expect the excitationof two PXCD currents of opposite handedness in a givenenantiomer. However, the probe – a circularly polarizedpulse, will break this symmetry between left and righthelical PXCD currents. Indeed, two co-rotating photonsand two counter-rotating photons produce different pho-toionization signals, because they correspond to differentmolecular pseudoscalars [see e.g. Eqs. (28) and (29) forg2 and g3 in case of a single intermediate state. Eq.(35) of ref. [82] generalizes this result for two intermedi-ate states]. Thus, the time-dependent PECD is a probehighlighting the synchronisation of bound and continuumchiral currents pondered above. It presents a differentialmeasure encoding two chiral currents of opposite hand-edness with different amplitudes. The results of time-dependent PECD experiments could be re-interpreted asthe images of such interplay [64].

C. Enantio-separation via photoionization

The possibility of selectively exciting only one of thetwo enantiomers of a chiral molecule to an electronic stateopens several routes for enantio-separation that we chartbelow.

One option is to use synthetic chiral light [104](See section III) to selectively excite one of the two enan-tiomers of a chiral molecule to a desired state, tuning thefrequencies of the light field in (possibly multi-photon)resonance with a specific electronic or vibronic transition.Next, the photo-excited molecules could be photoionizedwith a second laser pulse, yielding enantio-selective ion-ization. These molecular ions with well-defined handed-ness could then be extracted with a static field.

Another option is to use achiral light without pho-ton spin to achieve enantio-sensitive uniaxial orientationof chiral molecules on the electronic time-scale [120].

It is often assumed that molecular orientation canonly occur on rotational time-scales. However, in chi-

ral molecules, this does not have to be the case. Ouranalysis [120] in the perturbative regime predicts thatphase-locked, orthogonally polarized fields with frequen-cies ω and 3ω can induce field-free permanent electronicdipoles in initially isotropic samples of chiral moleculesvia resonant electronic excitation. The dipole’s orienta-tion is enantio-sensitive and it is controlled by the rel-ative phase between ω and 3ω fields, which determinesthe sub-cycle direction of rotation of the total electricfield. In contrast to the photo-excited circular dichroism(PXCD) [66], here not only the excited electron but alsothe molecule correlated to the excitation acquires orien-tation.

This effect is fundamentally multi-photon. In the fre-quency domain, the interference between the two path-ways, 3 × ~ω vs. 1 × 3~ω, is sensitive to the molecularorientation and handedness. This leads to orientation-dependent excitation and thus uniaxial orientation of theexcited molecules, on the electronic excitation time scale.The orientation is perpendicular to the polarization planeand is reflected in the emergence of a field-free permanentdipole.

This fundamental phenomenon points to interestingopportunities for creating enantio-sensitive permanentdipoles via orientation-dependent excitation of Rydbergstates upon resonance-enhanced multiphoton ionization(REMPI). The non-perturbative ω and 3ω fields shouldalso be explored, since in such regime hitting resonancesdoes not necessarily require carefully tuning the light fre-quency to specific molecular transitions. Indeed, in thenon-perturbative regime one can take advantage of light-induced energy shifts of excited states. These are knownas Freeman resonances [177] and are virtually inevitable

at intensities I ∼ 1013 W/cm2

and above. They will alsolead to orientation of molecular ions after orientation-selective resonantly enhanced multi-photon ionization.By selectively depleting randomly oriented neutrals, pref-erential orientation in the neutral ensemble is also cre-ated. After that a static field can be used to spatiallyseparate opposite enantiomers.

D. Geometric magnetism in chiral molecules

The excitation of enantio-sensitive photoelectron cur-rents (PECD) in the electric-dipole approximation can belinked to the concept of the geometrical magnetism intro-duced by M. Berry [83]. One of its manifestations is theBerry curvature in solids, which enables a class of phe-nomena in condensed matter systems including anoma-lous electron velocity, the Hall effect, and related topo-logical phenomena [86]. A geometric magnetic field alsoappears in photoionizaton of chiral molecules by circu-larly polarized fields. This field arises due to “curly” or“twisted” polarization in vibronic states or due to “curly”or “twisted” currents. The “twist” originates from thechiral arrangements of the nuclei and does not vanishupon averaging over the random molecular orientations.

24

The geometric magnetic field arising in chiral moleculesunderlies several classes of chiral photoionization observ-ables. It is related to the so-called propensity field thatwe have introduced recently [178].

The propensity field in photoionization involvesthe vector product of two conjugated photoionizationdipoles, [178]

~B(~k) = i[~d~k,g × ~d∗~k,g] = i[~p~k,g × ~p

∗~k,g

]

(Ek − Eg)2. (36)

where ~d~k,g and ~p~k,g = i(Ek − Eg)~d~k,g are transition

dipoles in the length and velocity gauges, respectively,and Ek and Eg are the energies of the photoelectron andof the ground state, respectively.

As usual for photoionization observables, the propen-

sity field ~B(~k) is a function of the photoelectron momen-

tum ~k. ~B(~k) quantifies the absorption circular dichroism

(CD) for a specific state |~k〉. The direction of ~B(~k) in-dicates a preferred direction in the molecular frame: if acircularly polarized light propagates along this direction,it maximizes the CD for a transition from the ground

state into a specific final state |~k〉. Loosely speaking, the

direction of ~B(~k) defines the axis in the molecular framealong which the rotational symmetry of the molecule is

broken to the highest extent, for a given final state ~k.

The magnitude of ~B(~k) is proportional to the corre-sponding CD, i.e. it is proportional to the difference

between the populations of the state |~k〉 obtained withleft and right circularly polarized light propagating along~B(~k). Indeed, if we denote the direction of ~B(~k) by

eB(~k) ≡ ~B(~k)/|B(~k)|, we obtain [88, 101, 178]

~B(~k) · eB = |~d+~k,g|2 − |~d−~k,g|

2, eB ≡~B(~k)

|B(~k)|, (37)

where ~d±~k,gare the photoionization dipoles for ionization

by left or right circularly polarized fields propagating

along the direction specified by ~B(~k). Thus, the vector

field ~B(~k) provides the molecule-specific ~k-resolved mapof maximal possible photoionization CD.

One can show that the propensity field is analogous tothe Berry curvature in two band solids, [88, 178] which isresponsible for the circular photogalvanic effect in chiralsolids [179] in the same way as the propensity field isresponsible for PECD in gas phase chiral molecules. Thechiral geometric magnetic field introduced in Ref. [88]provides a generalization of the propensity field.

Geometric field in molecular photoionization[88] The propensity field is related to photoionizationfrom a single state. In a more general case of pho-toionization from a superposition of two excited states|j〉 + e−iφij |i〉 (φij = ωijt, with ωij ≡ ωi − ωj being thetransition frequency between the states), the propensity

field encodes the coherence between the excited states:

~Bij(~k, φij) = −1

2i[~d∗~ki ×

~d~kj

]eiφij + c.c.

≡ ~Qij(~k) cosφij + ~Pij(~k) sinφij , (38)

where we have introduced the displacement ~Qij(~k) and

current ~Pij(~k) quadratures:

~Qij(~k) ≡ −<i[~d∗~ki ×

~d~kj

], (39)

~Pij(~k) ≡ =i[~d∗~ki ×

~d~kj

]. (40)

Eq. (38) describes the geometric field oscillating at thefrequency ωij . For i = j = g, φij = 0 and Eq. (38)reduces to Eq. (36). For any number of states Eqs. (36)and (38) can be generalized as

~B(~k, t) =1

2

∑i,j

i[~d~kj × ~d∗~ki

]eiφij . (41)

We call ~B(~k, t) in Eq. (41) the geometric field in molec-ular photoionization.

Applying inversion (~r → −~r, ~k → −~k) to reversemolecular handedness, we find that the displacement andcurrent quadratures in left- (S) and right-handed (R)

molecules are connected via ~Q(S)ij (~k) = ~Q

(R)ij (−~k) and

~P(S)ij (~k) = ~P

(R)ij (−~k). The ~Q(~k) quadratures are related

to the lack of rotational symmetry of electron density

[around the direction of ~Q(~k)]. In turn, the ~P (~k) quadra-tures are related to the lack of rotational asymmetry com-ing from the circulating bound state currents (around

the direction of ~P (~k)). The physical origin of the ge-ometric magnetic field are “twisted” polarizations and“curly” currents. Notably, the currents induced by thelight pulses generate the geometric field, which does notvanish in the molecular frame.

The geometric propensity field reflects the geometryof the molecular photoionization dipoles and gives rise tothree classes of enantio-sensitive observables, relying onvarious quadratures of the geometric field [88]. The newenantio-sensitive observables of Class I have been com-pletely overlooked so far. Class I observables can only ap-pear if the current in molecular bound states was excitedprior to photoionization. Thus, Class I observables canserve as messengers of charge-directed reactivity: chem-ical reactivity driven by ultrafast chiral electron dynam-ics. The first member of Class I observables is molecu-lar orientation circular dichroism in photoionization [88].Observables of Class II and III include the PECD (andtime-dependent PECD) current and an infinite array ofits multipolar versions. Most of these observables havenot been studied so far.

Concept 5. The geometric field [88] is a molecular frameproperty unique to every molecule. It underlies vectorial and

25

tensorial enantio-sensitive observables in photoionization, inthe electric-dipole approximation. Its flux in photoelectronmomentum space quantifies the PEXCD, its integral over allphotoelectron momenta quantifies chiral current excited inbound states, its multipole moments characterize multipolarcurrents, which can be excited by light fields without netspin. The geometric field in photoionization is similar tothe Berry curvature in solids.

Future directions. The geometric field so far servedfor us as a heuristic principle for discovering and classify-ing new enantio-sensitive observables. Future directionscan include identification of new members of Classes I-III,establishing the connection between the topology of thegeometric field and the topology of the respective molec-ular bound and continuum states, and the applicationof enantio-sensitive molecular orientation [88], which oc-curs in neural molecules and molecular ions, for enantio-separation and ultrafast molecular imaging. Other possi-bilities include exploiting the analogy between chiral ef-fects in photoionization and a broad class of topologicalphenomena in solids, aiming to create observables whichencode both chiral and topological properties of matter[180], such as the quantized circular dichroism.

VIII. THE HIERARCHY OF CHIRALMEASURES

This section concludes the paper by offering a unifiedview on chiral measurements and chiral observables asa cornerstone of such measurements, ultimately definingtheir efficiency.

In chiral measurements performed with electromag-netic fields we usually deal with the following objects: achiral molecule and either a chiral light (or a combinationof electric fields), or a chiral setup. An enantio-sensitivemeasurement couples the pseudoscalars of the two ob-jects: the chiral molecule and the chiral light and/or elec-tric fields (the “chiral reagent” type of the interaction),or the chiral molecule and the chiral setup (the “chiralobserver” type of the interaction). Remarkably, not onlythe pseudoscalars of all these chiral objects have verysimilar structure, but they also form a very similar hier-archy associated with the increasing order of non-linearinteractions.

A graphical example of such structural similarity ofchiral measures emerges from the comparison of the laserand the setup pseudoscalars. Table II shows that in everyorder of non-linearity the structure of these two pseu-doscalars is the same, only the specific vectors are dif-ferent. Namely, the setup pseudoscalar always containsvectors associated with the detector axes, which “substi-tute” one or more light vectors of the light pseudoscalar.

The first row of Table II compares linear chiral mea-sures involving the light and the setup pseudoscalars re-spectively. This comparison shows that in PECD the

light pseudovector [ ~E∗ω × ~Eω] substitutes the light pseu-

dovector ~B∗ω in CD, while the detector axis z substitutes

the light vector ~Eω.The second-order phenomena involving the light and

setup pseudoscalars are shown in the second row of thetable. Here non-linear CD(2) and PXCD/ESMW use thesame light pseudovector, which is given by the cross prod-uct of the electric fields at two different frequencies, butthe third vector necessary to form the desired triple prod-uct has a very different nature. Indeed, the light vectorcontributing to the light pseudoscalar in CD(2) is substi-tuted by the detector axis z in PXCD/ESMW. The samehappens in the fourth order of nonlinearity: the setuppseudoscalar pertinent for tensorial observables such as,e.g., the quadrupole current, substitutes the two laservectors employed in CD(4) by the detector axes. Thepresence of the two detector axes reflects the tensorialnature of the required detection scheme.

Thus, the overall structure of any chiral measure con-tributing to experimental observables is encoded in pseu-doscalar expressions formed by dot and cross productsbetween appropriate vectors. In particular, the chiralmeasures in the electric dipole approximation in TableII involve a triple product of three vectors in the lowestorder and are subsequently complemented by one scalarproduct of two vectors per each subsequent order of non-linearity. For example, the two-photon pump-probe pho-toionization of randomly oriented chiral molecules leadsto the appearance of one additional (with respect toPECD) scalar product of light fields in the setup pseu-doscalar (see the last row of Table II).

Table II reveals not only the common overall struc-ture of pseudoscalars but also the flexibility in addressingvarious molecular properties for distinguishing oppositeenantiomers. The interchangeability of molecular, lightand setup vectors is a great asset for chiral experiments,provided that the vectors are chosen wisely. Dependingon the type of observation, enantio-sensitive response ofthe same molecular sample can have different strengthand different requirements to light pseudovector.

For example, let us compare molecular pseudoscalarsfor the linear CD and the non-linear absorption circu-lar dichroism CD(2) [104] in Table II. (Note that pseu-doscalars of CD(2) also describe the three level enantio-sensitive population transfer [112, 136]). We see that inthe latter case, instead of relying on the molecular mag-netic transition dipole, one can rely on the cross productof two electric transition dipoles; instead of relying onthe magnetic field one can rely on the cross product ofthe electric field at two different frequencies.

One also has freedom in choosing the setup vectors.They can be constructed not only by introducing detec-tors for electrons, as done in PECD, but also by usingadditional electronic or vibrational/rotational degrees offreedom introduced via molecular alignment or coinci-dence detection involving other electrons, fragments, etc.

Formally, different dot and cross products inlight/molecular/setup pseudoscalars appear as a result ofthe orientation averaging procedure. Indeed, the isotropyof the molecular sample is the reason why these expres-

26

TABLE II. Hierarchy of chiral measures.

Phenomenon Molecular pseudoscalar Light/setup pseudoscalar

Linear CD [~df,i · ~mf,i] [100] [ ~E∗ω · ~Bω] [100]

PECD∫

dΩk[~k · (~d∗~k,i × ~d~k,i)] [82] [z · ( ~E∗ω × ~Eω)] [82]

Non-linear CD(2) [~d2,0 · (~d2,1 × ~d1,0)] or χ(2) ∝ Σm,n[~dn,0 · (~dn,m × ~dm,0)]Fn,m [74, 117] ~E∗(ω2,0) · [ ~E(ω2,1)× ~E(ω1,0)] [104]

PXCD/ESMW [~d2,0 · (~d2,1 × ~d1,0)] [66, 82] [z · ( ~E∗ω × ~Eω)] [82]

Non-linear CD(4) (~d0,1 · ~d1,2)[~d0,2 · (~d2,4 × ~d4,3)] or χ(4) ( ~Eω · ~Eω)[ ~E∗2ω · ( ~E∗ω × ~Eω)] [104]

Quadrup. currents∫

dΩk(k · ~d1,0)[k · (~d~k,0 × ~d~k,1)] + (k · ~d~k,1)[k · (~d~k,0 × ~d1,0)] [115] ( ~Eω · ~Eω)[ ~E∗2ω · (x× y)] [115]

Permanent quadrup. [(Q2,2~d2,1) · (~d1,0 × ~d2,0)] + [(Q2,2

~d1,0) · (~d2,1 × ~d2,0)] [120] ( ~Eω · x)[ ~E∗2ω · ( ~Eω × y)] [120]

PECD(2) (circ. [ ~E(ω1,0) · ~E∗(ωk,1)]

pump, circ. probe)∫

dΩk(~d1,0 · ~d∗~k,1)[~k · (~d1,0 × ~d~k,1)] [82] z · [ ~E∗(ω1,0)× ~E(ωk,1)] [82]

TD-PECD (linear [ ~E∗(ω2,0) · ~E(ω1,0)]

pump, circ. probe)∫

dΩk(~d2,0 · ~d1,0)[~k · (~d∗~k,2 × ~d~k,1)] [82] z · [ ~E∗(ωk,2)× ~E(ωk,1)] [82]

PEXCD (circ. [ ~E∗(ωk,2) · ~E(ωk,1)]

pump, linear probe)∫

dΩk(~d∗~k,2 · ~d~k,1)[~k · (~d2,0 × ~d1,0)] [82] z · [ ~E∗(ω2,0)× ~E(ω1,0)] [82]

sions do not depend on the relative orientations betweenmolecular and setup vectors (e.g. through the terms of

the form ~df,i · ~Eω), but instead only on the relative ori-entations between either molecular vectors among them-

selves (e.g. ~df,i · ~mf,i), or the setup vectors among them-

selves (e.g. ~Eω · ~B∗ω).

In general, if e.g. the process involves several photons,more vectors become available and a simple Rule 1 re-quires generalization.

Rule 7 enantio-sensitive observables pertinent to ran-domly oriented molecular ensemble can be written in thegeneral form [101, 181]:

v =∑i,j

giMijSj , (42)

where the gi’s and the Sj ’s are molecular and setup (orlight) pseudoscalars, respectively, and the Mij ’s are cou-pling constants.

The gi’s result from different possible contractions ofLevi-Civita and Kronecker delta tensors with moleculartensors, which then lead to the dot and cross productsdiscussed above. The same applies to the Sj ’s but usingthe setup (or light) instead of molecular tensors. In thesimplest cases there is a single possible contraction sothat the sum over i and j reduces to a single term. Thisis precisely what occurs in the case of Rule 1 in Sec.II: the measured click, a scalar v, corresponding to theprojection of the respective vectorial observable vif ontothe detector axis z, converts the laser pseudovector L intothe setup pseudoscalar S by projecting L on the detectoraxis: S = Lz. The resulting expression v = vif z = gS isindeed the simplest case of Rule 7.

IX. CONCLUSIONS

Our perspective on ultrafast chirality presented hereoffers a unifying framework for understanding and quan-tifying enantio-sensitive phenomena in interaction of chi-ral molecules with electromagnetic fields. This frame-work is applicable to all frequency regimes, from mi-crowaves to infrared, to visible, to X-rays and underliesspectroscopic tools probing electronic, vibronic or rota-tional states of chiral molecules and detecting photons orphoto-electrons. We have described the new concepts ofsynthetic and locally chiral light, polarization of chirality,and chiral geometric field. These concepts lead to newapplications in ultrafast optic, molecular photoionizationand nano-photonics. The perspective on synthetic chi-ral light in confined environments and its applications innanophotonics will be discussed separately.

ACKNOWLEDGEMENTS

Discussions with Prof. M. Ivanov, Prof. A. Stein-berg, Prof. M. Stockman and Dr. I. Nowitzki wereextremely important at different stages of this workand are gratefully acknowledged. A.F.O. and O.S.gratefully acknowledge the MEDEA Project, which hasreceived funding from the European Union’s Horizon2020 Research and Innovation Programme under theMarie Sk lodowska-Curie Grant Agreement No. 641789.A.F.O. and O.S. gratefully acknowledge support fromthe DFG SPP 1840 “Quantum Dynamics in TailoredIntense Fields” and DFG Grant No. SM 292/5-2.A.F.O. gratefully acknowledges grants supporting his re-search at ICFO: Agencia Estatal de Investigacion (theR&D project CEX2019-000910-S, funded by MCIN/AEI/10.13039/501100011033, Plan Nacional FIDEUAPID2019-106901GB-I00, FPI), Fundacio Privada Cellex,Fundacio Mir-Puig, Generalitat de Catalunya (AGAURGrant No. 2017 SGR 1341, CERCA program), and EUHorizon 2020 Marie Sk lodowska-Curie grant agreementNo 101029393.

27

[1] L. G. Wade, Chirality in Drug Research, Prentice Hall,2003.

[2] W. A. Bonner, Origins of life and evolution of the bio-sphere, 1991, 21, 59–111.

[3] J. Cohen, Science, 1995, 267, 1265–1266.[4] R. M. Hazen, T. R. Filley and G. A. Goodfriend, Pro-

ceedings of the National Academy of Sciences, 2001, 98,5487–5490.

[5] D. G. Blackmond, Proceedings of the National Academyof Sciences of the United States of America, 2004, 101,5732–5736.

[6] R. Breslow and Z.-L. Cheng, Proceedings of the NationalAcademy of Sciences, 2009, 106, 9144–9146.

[7] I. Weissbuch and M. Lahav, Chemical Reviews, 2011,111, 3236–3267.

[8] J. E. Hein and D. G. Blackmond, Accounts of ChemicalResearch, 2012, 45, 2045–2054.

[9] J. E. Hein, D. Gherase and D. G. Blackmond, in Chem-ical and Physical Models for the Emergence of Biolog-ical Homochirality, ed. P. Cintas, Springer Berlin Hei-delberg, Berlin, Heidelberg, 2013, pp. 83–108.

[10] A. Brewer and A. P. Davis, Nature Chemistry, 2014, 6,569.

[11] F. Jafarpour, T. Biancalani and N. Goldenfeld, Phys.Rev. Lett., 2015, 115, 158101.

[12] I. Barth and J. Manz, Angewandte Chemie Interna-tional Edition, 2006, 45, 2962–2965.

[13] T. Bredtmann and J. Manz, Angewandte Chemie Inter-national Edition, 2011, 50, 12652–12654.

[14] I. Barth and J. Manz, Phys. Rev. A, 2007, 75, 012510.[15] K.-J. Yuan and A. D. Bandrauk, Phys. Rev. A, 2013,

88, 013417.[16] L. Cederbaum and J. Zobeley, Chemical Physics Letters,

1999, 307, 205 – 210.[17] J. Breidbach and L. S. Cederbaum, The Journal of

Chemical Physics, 2003, 118, 3983–3996.[18] A. I. Kuleff and L. S. Cederbaum, Journal of Physics

B: Atomic, Molecular and Optical Physics, 2014, 47,124002.

[19] S. Lunnemann, A. I. Kuleff and L. S. Cederbaum,Chemical Physics Letters, 2008, 450, 232 – 235.

[20] F. Remacle, R. D. Levine, E. W. Schlag andR. Weinkauf, The Journal of Physical Chemistry A,1999, 103, 10149–10158.

[21] F. Remacle and R. D. Levine, Proceedings of the Na-tional Academy of Sciences, 2006, 103, 6793–6798.

[22] F. Calegari, D. Ayuso, A. Trabattoni, L. Belshaw,S. De Camillis, S. Anumula, F. Frassetto, L. Poletto,A. Palacios, P. Decleva, J. B. Greenwood, F. Martınand M. Nisoli, Science, 2014, 346, 336–339.

[23] F. Calegari, G. Sansone, S. Stagira, C. Vozzi andM. Nisoli, Journal of Physics B: Atomic, Molecular andOptical Physics, 2016, 49, 062001.

[24] M. Nisoli, P. Decleva, F. Calegari, A. Palacios andF. Martın, Chemical Reviews, 2017, 117, 10760–10825.

[25] D. Ayuso, A. Palacios, P. Decleva and F. Martın, Phys.Chem. Chem. Phys., 2017, 19, 19767–19776.

[26] S. Mukamel, Principles of nonlinear optical spec-troscopy, Oxford University Press, New York, 1995.

[27] J. Meyer-Ilse, D. Akimov and B. Dietzek, Laser & Pho-tonics Reviews, 2013, 7, 495–505.

[28] S. Ghosh, G. Herink, A. Perri, F. Preda, C. Manzoni,D. Polli and G. Cerullo, ACS photonics, 2021, 8, 2234–2242.

[29] J. R. Rouxel, A. Rajabi and S. Mukamel, Journal ofChemical Theory and Computation, 2020, 16, 5784–5791.

[30] L. Ye, J. R. Rouxel, S. Asban, B. Rosner andS. Mukamel, Journal of Chemical Theory and Compu-tation, 2019, 15, 4180–4186.

[31] J. R. Rouxel, Y. Zhang and S. Mukamel, Chem. Sci.,2019, 10, 898–908.

[32] H. Rhee, Y.-G. June, J.-S. Lee, K.-K. Lee, J.-H. Ha,Z. H. Kim, S.-J. Jeon and M. Cho, Nature, 2009, 458,310–313.

[33] M. Oppermann, J. Spekowius, B. Bauer, R. Pfister,M. Chergui and J. Helbing, The Journal of PhysicalChemistry Letters, 2019, 10, 2700–2705.

[34] R. Cireasa, A. E. Boguslavskiy, B. Pons, M. C. H. Wong,D. Descamps, S. Petit, H. Ruf, N. Thire, A. Ferre,J. Suarez, J. Higuet, B. E. Schmidt, A. F. Alharbi,F. Legare, V. Blanchet, B. Fabre, S. Patchkovskii,O. Smirnova, Y. Mairesse and V. R. Bhardwaj, NaturePhysics, 2015, 11, 654–658.

[35] M. Cho, Nature Physics, 2015, 11, 621–622.[36] D. Baykusheva, D. Zindel, V. Svoboda, E. Bommeli,

M. Ochsner, A. Tehlar and H. J. Worner, Proceedingsof the National Academy of Sciences, 2019, 116, 23923–23929.

[37] B. Ritchie, Phys. Rev. A, 1976, 13, 1411–1415.[38] I. Powis, The Journal of Chemical Physics, 2000, 112,

301–310.[39] N. Bowering, T. Lischke, B. Schmidtke, N. Muller,

T. Khalil and U. Heinzmann, Phys. Rev. Lett., 2001,86, 1187–1190.

[40] T. Lischke, N. Bowering, B. Schmidtke, N. Muller,T. Khalil and U. Heinzmann, Phys. Rev. A, 2004, 70,022507.

[41] S. Turchini, N. Zema, G. Contini, G. Alberti, M. Alagia,S. Stranges, G. Fronzoni, M. Stener, P. Decleva andT. Prosperi, Phys. Rev. A, 2004, 70, 014502.

[42] M. Stener, G. Fronzoni, D. D. Tommaso and P. Decleva,The Journal of Chemical Physics, 2004, 120, 3284–3296.

[43] S. Stranges, S. Turchini, M. Alagia, G. Alberti, G. Con-tini, P. Decleva, G. Fronzoni, M. Stener, N. Zema andT. Prosperi, The Journal of Chemical Physics, 2005,122, 244303.

[44] D. Di Tommaso, M. Stener, G. Fronzoni and P. Decleva,ChemPhysChem, 2006, 7, 924–934.

[45] S. Turchini, D. Catone, G. Contini, N. Zema, S. Irrera,M. Stener, D. Di Tommaso, P. Decleva and T. Prosperi,ChemPhysChem, 2009, 10, 1839–1846.

[46] L. Nahon, G. A. Garcia and I. Powis, Journal of Elec-tron Spectroscopy and Related Phenomena, 2015, 204,322 – 334.

[47] A. Ferre, C. Handschin, M. Dumergue, F. Burgy,A. Comby, D. Descamps, B. Fabre, G. A. Garcia,R. Geneaux, L. Merceron, E. Mevel, L. Nahon, S. Petit,B. Pons, D. Staedter, S. Weber, T. Ruchon, V. Blanchetand Y. Mairesse, Nature Photonics, 2015, 9, 93–98.

28

[48] S. Turchini, Journal of Physics: Condensed Matter,2017, 29, 503001.

[49] K. Fehre, N. M. Novikovskiy, S. Grundmann, G. Ka-stirke, S. Eckart, F. Trinter, J. Rist, A. Hartung, D. Tra-bert, C. Janke, G. Nalin, M. Pitzer, S. Zeller, F. Wie-gandt, M. Weller, M. Kircher, M. Hofmann, L. P. H.Schmidt, A. Knie, A. Hans, L. B. Ltaief, A. Ehres-mann, R. Berger, H. Fukuzawa, K. Ueda, H. Schmidt-Bocking, J. B. Williams, T. Jahnke, R. Dorner, M. S.Schoffler and P. V. Demekhin, Phys. Rev. Lett., 2021,127, 103201.

[50] P. Kruger and K.-M. Weitzel, Angewandte Chemie In-ternational Edition, 2021, 60, 17861–17865.

[51] G. A. Garcia, L. Nahon, S. Daly and I. Powis, NatureCommunications, 2013, 4, 2132.

[52] C. Lux, M. Wollenhaupt, T. Bolze, Q. Liang, J. Kohler,C. Sarpe and T. Baumert, Angewandte Chemie Inter-national Edition, 2012, 51, 5001–5005.

[53] C. S. Lehmann, N. B. Ram, I. Powis and M. H. M.Janssen, The Journal of Chemical Physics, 2013, 139,234307.

[54] C. S. Lehmann, N. B. Ram, I. Powis and M. H. M.Janssen, The Journal of Chemical Physics, 2013, 139,234307.

[55] M. M. Rafiee Fanood, I. Powis and M. H. M. Janssen,The Journal of Physical Chemistry A, 2014, 118,11541–11546.

[56] M. M. R. Fanood, N. B. Ram, C. S. Lehmann, I. Powisand M. H. M. Janssen, Nature Communications, 2015,6, 7511.

[57] C. Lux, M. Wollenhaupt, C. Sarpe and T. Baumert,ChemPhysChem, 2015, 16, 115–137.

[58] C. Lux, A. Senftleben, C. Sarpe, M. Wollenhaupt andT. Baumert, Journal of Physics B: Atomic, Molecularand Optical Physics, 2015, 49, 02LT01.

[59] S. Beaulieu, A. Comby, B. Fabre, D. Descamps,A. Ferre, G. Garcia, R. Geneaux, F. Legare, L. Na-hon, S. Petit, T. Ruchon, B. Pons, V. Blanchet andY. Mairesse, Faraday Discuss., 2016, 194, 325–348.

[60] A. Kastner, C. Lux, T. Ring, S. Zullighoven, C. Sarpe,A. Senftleben and T. Baumert, ChemPhysChem, 2016,17, 1119–1122.

[61] J. Miles, D. Fernandes, A. Young, C. Bond, S. Crane,O. Ghafur, D. Townsend, J. Sa and J. Greenwood, An-alytica Chimica Acta, 2017, 984, 134–139.

[62] A. Kastner, T. Ring, H. Braun, A. Senftleben andT. Baumert, ChemPhysChem, 2019, 20, 1416–1419.

[63] S. T. Ranecky, G. B. Park, P. C. Samartzis, I. C. Gian-nakidis, D. Schwarzer, A. Senftleben, T. Baumert andT. Schafer, Phys. Chem. Chem. Phys., 2022, 24, 2758–2761.

[64] A. Comby, S. Beaulieu, M. Boggio-Pasqua,D. Descamps, F. Legare, L. Nahon, S. Petit, B. Pons,B. Fabre, Y. Mairesse and V. Blanchet, The Journal ofPhysical Chemistry Letters, 2016, 7, 4514–4519.

[65] S. Beaulieu, A. Ferre, R. Geneaux, R. Canonge,D. Descamps, B. Fabre, N. Fedorov, F. Legare, S. Petit,T. Ruchon, V. Blanchet, Y. Mairesse and B. Pons, NewJournal of Physics, 2016, 18, 102002.

[66] S. Beaulieu, A. Comby, D. Descamps, B. Fabre, G. A.Garcia, R. Geneaux, A. G. Harvey, F. Legare, Z. Masın,L. Nahon, A. F. Ordonez, S. Petit, B. Pons, Y. Mairesse,O. Smirnova and V. Blanchet, Nature Physics, 2018, 14,484–489.

[67] I. Powis, in Photoelectron Circular Dichroism in ChiralMolecules, John Wiley & Sons, Ltd, 2008, ch. 5, pp.267–329.

[68] M. H. M. Janssen and I. Powis, Phys. Chem. Chem.Phys., 2014, 16, 856–871.

[69] R. Hadidi, D. K. Bozanic, G. A. Garcia and L. Nahon,Advances in Physics: X, 2018, 3, 1477530.

[70] P. V. Demekhin, A. N. Artemyev, A. Kastner andT. Baumert, Phys. Rev. Lett., 2018, 121, 253201.

[71] P. V. Demekhin, Physical Review A, 2019, 99, 063406.[72] R. E. Goetz, C. P. Koch and L. Greenman, Phys. Rev.

Lett., 2019, 122, 013204.[73] S. Rozen, A. Comby, E. Bloch, S. Beauvarlet,

D. Descamps, B. Fabre, S. Petit, V. Blanchet, B. Pons,N. Dudovich and Y. Mairesse, Phys. Rev. X, 2019, 9,031004.

[74] J. A. Giordmaine, Physical Review, 1965, 138, A1599–A1606.

[75] P. Fischer and F. Hache, Chirality, 2005, 17, 421–437.[76] O. Neufeld, D. Ayuso, P. Decleva, M. Y. Ivanov,

O. Smirnova and O. Cohen, Phys. Rev. X, 2019, 9,031002.

[77] D. Ayuso, A. F. Ordonez, P. Decleva, M. Ivanov andO. Smirnova, Opt. Express, 2022, 30, 4659–4667.

[78] D. Ayuso, A. F. Ordonez, M. Ivanov and O. Smirnova,Optica, 2021, 8, 1243–1246.

[79] D. Patterson and J. M. Doyle, Physical Review Letters,2013, 111, 023008.

[80] D. Patterson, M. Schnell and J. M. Doyle, Nature, 2013,497, 475–477.

[81] K. K. Lehmann, in Frontiers and Advances in MolecularSpectroscopy, ed. J. Laane, Elsevier, 2018, pp. 713–743.

[82] A. F. Ordonez and O. Smirnova, Physical Review A,2018, 98, 063428.

[83] M. V. Berry, Proceedings of the Royal Society of London.A. Mathematical and Physical Sciences, 1984, 392, 45–57.

[84] C. L. Kane and E. J. Mele, Physical review letters, 2005,95, 226801.

[85] B. A. Bernevig, T. L. Hughes and S.-C. Zhang, science,2006, 314, 1757–1761.

[86] R. Resta, Reviews of Modern Physics, 1994, 66, 899–915.

[87] J. G. C. Felser, Proceedings of Nobel symposium on chi-ral matter, 2022.

[88] A. F. Ordonez, D. Ayuso, P. Decleva and O. Smirnova,arXiv:2106.14264 [physics], 2021.

[89] N. Berova, P. L. Polavarapu, K. Nakanishi and R. W.Woody, Comprehensive Chiroptical Spectroscopy, Wiley,2013.

[90] Y. Zhang, J. R. Rouxel, J. Autschbach, N. Govind andS. Mukamel, Chem. Sci., 2017, 8, 5969–5978.

[91] J. R. Rouxel, M. Kowalewski and S. Mukamel, Struc-tural Dynamics, 2017, 4, 044006.

[92] O. Smirnova, Y. Mairesse and S. Patchkovskii, Journalof Physics B: Atomic, Molecular and Optical Physics,2015, 48, 234005.

[93] D. Ayuso, P. Decleva, S. Patchkovskii and O. Smirnova,Journal of Physics B: Atomic, Molecular and OpticalPhysics, 2018, 51, 06LT01.

[94] D. Ayuso, P. Decleva, S. Patchkovskii and O. Smirnova,Journal of Physics B: Atomic, Molecular and OpticalPhysics, 2018, 51, 124002.

29

[95] Y. Harada, E. Haraguchi, K. Kaneshima andT. Sekikawa, Phys. Rev. A, 2018, 98, 021401.

[96] D. Baykusheva and H. J. Worner, Phys. Rev. X, 2018,8, 031060.

[97] K. A. Forbes and D. L. Andrews, Journal of Physics:Photonics, 2021, 3, 022007.

[98] P. Jordan and R. d. L. Kronig, Nature, 1927, 120, 807–807.

[99] T. Bungo, Y. Nakano, K. Okano, M. Hayashida,H. Kawagoe, H. Furusawa, K. Yasukochi, T. Matsuishi,K. Izumi, M. Shimojo, M. Furuse and Y. Masuda, Ap-plied Animal Behaviour Science, 1999, 64, 227–232.

[100] Y. Tang and A. E. Cohen, Phys. Rev. Lett., 2010, 104,163901.

[101] A. F. Ordonez and O. Smirnova, arXiv:2009.03660[physics], 2020.

[102] M. Pitzer, M. Kunitski, A. S. Johnson, T. Jahnke,H. Sann, F. Sturm, L. P. H. Schmidt, H. Schmidt-Bocking, R. Dorner, J. Stohner, J. Kiedrowski,M. Reggelin, S. Marquardt, A. Schießer, R. Berger andM. S. Schoffler, Science, 2013, 341, 1096–1100.

[103] M. Pitzer, Journal of Physics B: Atomic, Molecular andOptical Physics, 2017, 50, 153001.

[104] D. Ayuso, O. Neufeld, A. F. Ordonez, P. Decleva,G. Lerner, O. Cohen, M. Ivanov and O. Smirnova, Na-ture Photonics, 2019, 13, 866–871.

[105] D. Ayuso, A. F. Ordonez, P. Decleva, M. Ivanov andO. Smirnova, Nature Communications, 2021, 12, 3951.

[106] D. Ayuso, New opportunities for ultrafast and highlyenantio-sensitive imaging and control of chiral nu-clear dynamics: towards enantio-selective attochem-istry, 2021.

[107] N. Mayer, M. Ivanov and O. Smirnova, Imprinting chi-rality on atoms using synthetic chiral light, 2021.

[108] G. P. Katsoulis, Z. Dube, P. Corkum, A. Staudte andA. Emmanouilidou, arXiv:2112.02670 [physics], 2021.

[109] M. Khokhlova, E. Pisanty, S. Patchkovskii, O. Smirnovaand M. Ivanov, Enantiosensitive steering of free-induction decay, 2021.

[110] D. Gerbasi, P. Brumer, I. Thanopulos, P. Kral andM. Shapiro, The Journal of Chemical Physics, 2004,120, 11557–11563.

[111] A. Yachmenev, J. Onvlee, E. Zak, A. Owens andJ. Kupper, Phys. Rev. Lett., 2019, 123, 243202.

[112] S. Eibenberger, J. Doyle and D. Patterson, Phys. Rev.Lett., 2017, 118, 123002.

[113] P. Lodahl, S. Mahmoodian, S. Stobbe, A. Rauschenbeu-tel, P. Schneeweiss, J. Volz, H. Pichler and P. Zoller,Nature, 2017, 541, 473–480.

[114] K. Y. Bliokh and F. Nori, Physics Reports, 2015, 592,1–38.

[115] A. F. Ordonez and O. Smirnova, arXiv:2009.03655[physics], 2020.

[116] P. Fischer, K. Beckwitt, F. W. Wise and A. C. Albrecht,Chemical Physics Letters, 2002, 352, 463–468.

[117] P. Fischer, A. D. Buckingham and A. C. Albrecht, Phys-ical Review A, 2001, 64, 053816.

[118] P. Fischer, A. D. Buckingham, K. Beckwitt, D. S.Wiersma and F. W. Wise, Physical Review Letters,2003, 91, 173901.

[119] M. A. Belkin, T. A. Kulakov, K.-H. Ernst, L. Yan andY. R. Shen, Physical Review Letters, 2000, 85, 4474–4477.

[120] A. F. Ordonez and O. Smirnova, in Molecular Beamsin Physics and Chemistry: From Otto Stern’s Pioneer-ing Exploits to Present-Day Feats, ed. B. Friedrich andH. Schmidt-Bocking, Springer International Publishing,2021, pp. 335–352.

[121] N. A. Cherepkov, Physics Letters, 1972, 40A, 119–121.[122] N. A. Cherepkov, Journal of Physics B: Atomic and

Molecular Physics, 1981, 14, L623.[123] M. H. J. I. Powis, Current Trends in Mass Spectrometry,

2017, 5, 16.[124] R. Boyd, Nonlinear Optics, Elsevier, 3rd edn., 2008.[125] O. Neufeld, D. Podolsky and O. Cohen, Nature Com-

munications, 2019, 10, 405.[126] A. Yachmenev and S. N. Yurchenko, Physical Review

Letters, 2016, 117, 033001.[127] A. Owens, A. Yachmenev, S. N. Yurchenko and

J. Kupper, Physical review letters, 2018, 121, 193201.[128] E. Gershnabel and I. S. Averbukh, Phys. Rev. Lett.,

2018, 120, 083204.[129] I. Tutunnikov, J. Floß, E. Gershnabel, P. Brumer, I. S.

Averbukh, A. A. Milner and V. Milner, Physical ReviewA, 2020, 101, 021403.

[130] A. A. Milner, J. A. M. Fordyce, I. MacPhail-Bartley,W. Wasserman, V. Milner, I. Tutunnikov and I. S. Aver-bukh, Phys. Rev. Lett., 2019, 122, 223201.

[131] F. Krausz and M. Ivanov, Rev. Mod. Phys., 2009, 81,163–234.

[132] P. B. Corkum, Physical Review Letters, 1993, 71, 1994.[133] O. Smirnova and M. Ivanov, in Multielectron High Har-

monic Generation: Simple Man on a Complex Plane,John Wiley & Sons, Ltd, 2014, ch. 7, pp. 201–256.

[134] D. M. Brink and G. R. Satchler, Angular Momentum,Clarendon Press, Oxford, 2nd edn., 1968.

[135] P. Fischer, D. S. Wiersma, R. Righini, B. Champagneand A. D. Buckingham, Phys. Rev. Lett., 2000, 85,4253–4256.

[136] C. Perez, A. L. Steber, S. R. Domingos, A. Krin,D. Schmitz and M. Schnell, Angewandte Chemie Inter-national Edition, 2017, 56, 12512–12517.

[137] H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R.Dennis, D. L. Andrews, M. Mansuripur, C. Denz,C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Mar-rucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser,N. P. Bigelow, C. Rosales-Guzman, A. Belmonte, J. P.Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stil-goe, J. Romero, A. G. White, R. Fickler, A. E. Willner,G. Xie, B. McMorran and A. M. Weiner, Journal ofOptics, 2016, 19, 013001.

[138] E. Pisanty, G. J. Machado, V. Vicuna-Hernandez,A. Picon, A. Celi, J. P. Torres and M. Lewenstein, Na-ture Photonics, 2019, 13, 569–574.

[139] S. W. Hell, Angewandte Chemie International Edition,2015, 54, 8054–8066.

[140] M. Padgett and R. Bowman, Nature Photonics, 2011,5, 343–348.

[141] F. Patti, R. Saija, P. Denti, G. Pellegrini, P. Biagioni,M. A. Iatı and O. M. Marago, Scientific Reports, 2019,9, 29.

[142] M. Li, S. Yan, Y. Zhang, P. Zhang and B. Yao, J. Opt.Soc. Am. B, 2019, 36, 2099–2105.

[143] D. S. Bradshaw and D. L. Andrews, Opt. Lett., 2015,40, 677–680.

[144] R. P. Cameron, A. M. Yao and S. M. Barnett, The Jour-nal of Physical Chemistry A, 2014, 118, 3472–3478.

30

[145] G. G. Paulus, F. Grasbon, H. Walther, P. Villoresi,M. Nisoli, S. Stagira, E. Priori and S. De Silvestri, Na-ture, 2001, 414, 182–184.

[146] A. Baltuska, T. Udem, M. Uiberacker, M. Hentschel,E. Goulielmakis, C. Gohle, R. Holzwarth, V. S.Yakovlev, A. Scrinzi, T. W. Hansch and F. Krausz, Na-ture, 2003, 421, 611–615.

[147] A. Schiffrin, T. Paasch-Colberg, N. Karpowicz,V. Apalkov, D. Gerster, S. Muhlbrandt, M. Korb-man, J. Reichert, M. Schultze, S. Holzner, J. V. Barth,R. Kienberger, R. Ernstorfer, V. S. Yakovlev, M. I.Stockman and F. Krausz, Nature, 2013, 493, 70–74.

[148] T. T. Luu, M. Garg, S. Y. Kruchinin, A. Moulet, M. T.Hassan and E. Goulielmakis, Nature, 2015, 521, 498–502.

[149] G. L. Yudin, A. D. Bandrauk and P. B. Corkum, Phys.Rev. Lett., 2006, 96, 063002.

[150] V. A. Shubert, D. Schmitz, C. Perez, C. Medcraft,A. Krin, S. R. Domingos, D. Patterson and M. Schnell,The Journal of Physical Chemistry Letters, 2016, 7,341–350.

[151] M. Leibscher, E. Pozzoli, C. Perez, M. Schnell, M. Siga-lotti, U. Boscain and C. P. Koch, arXiv preprintarXiv:2010.09296, 2020.

[152] M. Leibscher, T. F. Giesen and C. P. Koch, The Journalof chemical physics, 2019, 151, 014302.

[153] J. Karczmarek, J. Wright, P. Corkum and M. Ivanov,Phys. Rev. Lett., 1999, 82, 3420–3423.

[154] D. M. Villeneuve, S. A. Aseyev, P. Dietrich, M. Spanner,M. Y. Ivanov and P. B. Corkum, Phys. Rev. Lett., 2000,85, 542–545.

[155] L. Yuan, S. W. Teitelbaum, A. Robinson and A. S.Mullin, Proceedings of the National Academy of Sci-ences, 2011, 108, 6872–6877.

[156] H. L. Bethlem, G. Berden and G. Meijer, Phys. Rev.Lett., 1999, 83, 1558–1561.

[157] S. Y. T. van de Meerakker, H. L. Bethlem and G. Meijer,Nature Physics, 2008, 4, 595–602.

[158] C. Maher-McWilliams, P. Douglas and P. F. Barker,Nature Photonics, 2012, 6, 386–390.

[159] R. J. Hernandez, A. Mazzulla, A. Pane, K. Volke-Sepulveda and G. Cipparrone, Lab Chip, 2013, 13, 459–467.

[160] A. Canaguier-Durand, J. A. Hutchison, C. Genet andT. W. Ebbesen, New Journal of Physics, 2013, 15,123037.

[161] G. Tkachenko and E. Brasselet, Nature Communica-tions, 2014, 5, 3577.

[162] G. Tkachenko and E. Brasselet, Nature Communica-tions, 2014, 5, 4491.

[163] A. Hayat, J. P. B. Mueller and F. Capasso, Proceedingsof the National Academy of Sciences, 2015, 112, 13190–

13194.[164] I. D. Rukhlenko, N. V. Tepliakov, A. S. Baimuratov,

S. A. Andronaki, Y. K. Gun’ko, A. V. Baranov andA. V. Fedorov, Scientific Reports, 2016, 6, 36884.

[165] R. Ali, F. A. Pinheiro, R. S. Dutra, F. S. S. Rosa andP. A. Maia Neto, Nanoscale, 2020, 12, 5031–5037.

[166] G. Pellegrini, M. Finazzi, M. Celebrano, L. Duo, M. A.Iatı, O. M. Marago and P. Biagioni, The Journal ofPhysical Chemistry C, 2019, 123, 28336–28342.

[167] R. P. Cameron, S. M. Barnett and A. M. Yao, NewJournal of Physics, 2014, 16, 013020.

[168] A. F. Ordonez and O. Smirnova, Physical Review A,2019, 99, 043416.

[169] A. F. Ordonez and O. Smirnova, Physical ChemistryChemical Physics, 2022.

[170] N. Mayer, M. Ivanov and O. Smirnova,arXiv:2112.02658 [physics], 2021.

[171] A. Jimenez-Galan, R. E. F. Silva, O. Smirnova andM. Ivanov, Nature Photonics, 2020, 14, 728–732.

[172] G. Chang, B. J. Wieder, F. Schindler, D. S. Sanchez,I. Belopolski, S.-M. Huang, B. Singh, D. Wu, T.-R.Chang, T. Neupert, S.-Y. Xu, H. Lin and M. Z. Hasan,Nature Materials, 2018, 17, 978.

[173] D. S. Sanchez, I. Belopolski, T. A. Cochran, X. Xu, J.-X. Yin, G. Chang, W. Xie, K. Manna, V. Suß, C.-Y.Huang, N. Alidoust, D. Multer, S. S. Zhang, N. Shu-miya, X. Wang, G.-Q. Wang, T.-R. Chang, C. Felser,S.-Y. Xu, S. Jia, H. Lin and M. Z. Hasan, Nature, 2019,567, 500–505.

[174] I. Tutunnikov, J. Floß, E. Gershnabel, P. Brumer andI. S. Averbukh, Phys. Rev. A, 2019, 100, 043406.

[175] J. Itatani, J. Levesque, D. Zeidler, H. Niikura, H. Pepin,J. C. Kieffer, P. B. Corkum and D. M. Villeneuve, Na-ture, 2004, 432, 867–871.

[176] C. Vozzi, M. Negro, F. Calegari, G. Sansone, M. Nisoli,S. De Silvestri and S. Stagira, Nature Physics, 2011, 7,822–826.

[177] R. R. Freeman, P. H. Bucksbaum, H. Milchberg,S. Darack, D. Schumacher and M. E. Geusic, PhysicalReview Letters, 1987, 59, 1092–1095.

[178] A. F. Ordonez and O. Smirnova, Physical Review A,2019, 99, 043417.

[179] F. de Juan, A. G. Grushin, T. Morimoto and J. E.Moore, Nature Communications, 2017, 8, 15995.

[180] K. Schwennicke and J. Yuen-Zhou, arXiv preprintarXiv:2105.05469, 2021.

[181] D. L. Andrews and T. Thirunamachandran, The Jour-nal of Chemical Physics, 1977, 67, 5026–5033.