# Towards chiral polaritons

Denis G. Baranov  Center for Photonics and 2D Materials, Moscow Institute of
Physics and Technology, Dolgoprudny 141700, Russia<EMAIL_ADDRESS>Christian Schäfer  MC2 Department, Chalmers University of Technology, Sweden
<EMAIL_ADDRESS>Maxim V. Gorkunov  Shubnikov Institute of
Crystallography, FSRC “Crystallography and Photonics” Russian Academy of
Sciences, Moscow 119333, Russia<EMAIL_ADDRESS>
###### Abstract

Coupling between light and material excitations underlies a wide range of
optical phenomena. Polaritons are eigenstates of a coupled system with
hybridized wave function. Owing to their hybrid composition, polaritons
exhibit at the same time properties typical for photonic and electronic
excitations, thus offering new ways for controlling electronic transport and
even chemical kinetics. While most theoretical and experimental efforts have
been focused on polaritons with electric-dipole coupling between light and
matter, in chiral quantum emitters, electronic transitions are characterized
by simultaneously nonzero electric and magnetic dipole moments. Geometrical
chirality affects the optical properties of materials in a profound way and
enables phenomena that underlie our ability to discriminate enantiomers of
chiral molecules. Thus, it is natural to wonder what kinds of novel effects
chirality may enable in the realm of strong light-matter coupling. Right now,
this field located at the intersection of nanophotonics, quantum optics, and
chemistry is in its infancy. In this Perspective, we offer our view towards
chiral polaritons. We review basic physical concepts underlying chirality of
matter and electromagnetic field, discuss the main theoretical and
experimental challenges that need to be solved, and consider novel effects
that could be enabled by strong coupling between chiral light and matter.

###### keywords:

Strong coupling, polaritons, optical cavities, resonances, chirality,
handedness, ab initio QED

##  1 Introduction

A system of a few interacting oscillators, either of classical or quantum
nature, is a primitive mathematical model that describes the behavior of a
wide spectrum of mechanical, thermodynamic, electromagnetic and other kinds of
physical systems. Polaritons are eigenstates of the system when
electromagnetic fields and matter couple strongly, i.e., when the coupling
rate exceeds all the decay and decoherence rates in the system, and share a
mixed light-matter characteristic.1 In the optical domain, polaritonic states
are often realized by means of coupling an optical cavity mode with excitonic
or vibrational transitions in resonant media 2, 3, 4. Owing to this hybrid
composition, polaritons show properties indicative of massless photonic and
massive electronic/vibrational excitations at the same time 5, thus offering
new ways for controlling electronic transport and modifying chemical kinetics
6, 7, 8, 9, 10, 11, 12.

Most theoretical and experimental efforts in this field have been focused on
coupling optical cavities with linearly- or circularly-polarized electronic
transitions of various quantum emitters (QEs). This is justified by the
abundance of electronic and vibrational emitters whose interaction with light
is dominated by the electric-dipole term of the Hamiltonian, while higher-
order contributions including the magnetic-dipole term are often minute. To
put it another way, natural materials very weakly respond to magnetic fields
oscillating with optical frequencies 13, 14. Nevertheless, in certain specific
cases, a non-negligible magnetic transition dipole moment can be of pivotal
importance. _Chiral_ media 15, 16 represent a practically relevant example.

A three-dimensional rigid body, which cannot be aligned with its mirror image
by a series of rotations and translations, is referred to as chiral 17. Two
mirrored versions of a chiral object are called left and right enantiomers,
and usually qualitative differences between them are obvious. However, an
attempt to define a unique quantitative measure of the geometric chirality
easily leads to confusion: one can introduce different seemingly meaningful
parameters which may vanish for different configurations and remain unrelated
to the chiral physical phenomena 18. Much more fruitful is the practical
approach, when one assesses the chirality of an object by its interactions
with other chiral entities.

For example, acting on a chiral biological receptor, opposite enantiomers of
the same molecule cause different response, perceived as different odor or
taste 19. In pharmaceutics, the opposite enantiomer of a drug molecule can be
useless at best, but often it is insidiously toxic for the chiral human body
20, 21. The majority of molecules involved in biological processes, such as
neurotransmitters (norepinephrine, ephedrine, and others) are chiral. In this
regard, there is a great demand from the pharmaceutical industry to develop
effective methods of separating chiral enantiomers.

Chirality profoundly affects interactions of matter with electromagnetic
radiation. As soon as a medium is composed of chiral constituents (such as
molecules or meta-atoms), it acquires a magneto-electric response, which mixes
E and B, and H and D, correspondingly 15. The study of interaction between
chiral light and matter has uncovered many fundamental effects, but, so far,
has been limited to weak light-matter coupling, where the presence of the
photonic field can be treated perturbatively. Some recent studies make an
attempt to address the behavior of strongly coupled system where at least one
of the constituents (either light or the material) is chiral 22, 23, 24. All
this raises a question: which new phenomena should arise from the chirality of
a QE, or of a cavity, or of both, when they become strongly coupled?

In this Perspective, we offer a conceptual view on the notion of chiral
polaritonics. We review the essential physics underlying strong light-matter
coupling, the chirality of matter and electromagnetic fields. We discuss the
main theoretical and experimental challenges for chiral polaritonics and
provide a perspective on its potential and future development. The
interdisciplinary nature of the problem suggests that a joint effort of
researches from classical electromagnetism, quantum optics, material science,
and biology might be needed in order to establish chiral polaritonics as the
versatile framework it promises to be.

##  2 Strong light-matter coupling

Let us begin by briefly recalling theoretical apparatus describing the
interaction of an optical emitter, such as an atom, a molecule, a lattice
vibration, or an artificial harmonic meta-atom, with a single electromagnetic
mode of an optical cavity. There are several frameworks to model such an
interaction, and the choice of the particular model in each situation depends
on the context. As long as anharmonic effects related to the saturability of
two-level system(s) may be ignored, and as long as the coupling strength is
not too high compared to resonant energies of the photonic and material
oscillators, the light-matter coupling can be conceptually modeled using the
following coupled-mode Hamiltonian 2, 3, 4:

$H=\left(\begin{array}[]{cc}\omega_{cav}-i\gamma_{cav}&g\\\
g^{*}&\omega_{0}-i\gamma_{0}\end{array}\right),$ (1)

where $\omega_{0}-i\gamma_{0}$ and $\omega_{cav}-i\gamma_{cav}$ are the
complex-valued frequencies characterizing the material and photonic
eigenmodes, correspondingly 25, and $g$ is the coupling constant. For a single
emitter coupled to a cavity we have $g=\mathcal{E}_{vac}\mu$, where
$\mathcal{E}_{vac}$ is the vacuum electric field of the cavity mode, and $\mu$
is the transition dipole moment of the emitter. If $N$ identical emitters are
coupled to the same cavity mode, the collective coupling constant reads
$g=\sqrt{N}\mathcal{E}_{vac}\mu$ 4. The two eigenvalues of the Hamiltonian Eq.
1 read:

$\omega_{\pm}=\frac{\omega_{cav}+\omega_{0}}{2}-i\frac{\gamma_{cav}+\gamma_{0}}{2}\pm\sqrt{|g|^{2}+\frac{1}{4}(\omega_{cav}-\omega_{0}-i(\gamma_{cav}-\gamma_{0}))^{2}}.$
(2)

Within this model of light-matter interaction, two different regimes may be
realized.

Figure 1: The basic picture of strong light-matter coupling. (a) Schematic
illustration of an electronic transition of a two-level atom coupled to a
single photonic mode of an optical cavity. (b) Eigenvalues of the coupled-mode
Hamiltonian, Eq. 2, as a function of the coupling constant $g$ for the zero-
detuning case, $\omega_{cav}=\omega_{0}$. (c) The eigenvalues spectrum plotted
as a function of the cavity frequency for a fixed coupling constant.

Fig. 1(b) shows the eigenvalues of Hamiltonian Eq. 1 as a function of the
coupling constant $g$ for the zero-detuning case, $\omega_{cav}=\omega_{0}$.
Weak coupling regime occupies the left part of this diagram, where
$g<|\gamma_{cav}-\gamma_{0}|/2$. In this regime the two eigenstates have equal
real parts of their eigenenergies (although different imaginary parts). An
excited QE undergoes exponential decay into the ground state at a rate
$\gamma$, accompanied by spontaneous emission of a photon in the environment.
The point $g=|\gamma_{cav}-\gamma_{0}|/2$ marks the transition to the strong
coupling regime. Once the coupling constant exceeds the threshold value, the
real parts of eigenenergies split into two branches separated by the so called
Rabi splitting. Correspondingly, wave functions of the two eigenstates become
equal 50% mixes of photonic and material components, and are referred to as
upper and lower polaritons. When plotted as a function of the cavity frequency
for a fixed coupling constant, the energy spectrum displays the well-known
picture of anti-crossing between two polaritonic modes, Figure 1(c).

When the coupling constant becomes comparable to the resonant energies of the
system ($g>0.1\omega$), the system enters the ultrastrong coupling (USC)
regime in which all eigenstates become noticeably renormalized by the
interaction26. At this point, the trivial coupled-mode Hamiltonian as well as
the widely used rotating-wave approximation yield unphysical eigenstates and
becomes inaccurate, thus the underlying theoretical foundation of QED has to
be consulted. A simple and widely used model was derived by Hopfield 27, 28.
In coupled structures where the spatial coherence of the polariton is at the
center – for example, in systems exhibiting polariton condensation 29, 5 –
other approaches might be used that capture the non-local and non-linear
properties of polaritons, such as the Gross-Pitaevskii equation.

The hybrid wave function of polaritons has far-reaching implications for the
microscopic properties of the coupled system. In particular, the possibility
of strong light-matter coupling to affect the rates of chemical reactions
(ones involving the coupled material as a product or as a reactant) has been
the subject of intense experimental 6, 7, 11, 9, 30, 31 and theoretical 8, 32,
10, 33, 34, 35, 36 investigation in the past decade. Thus, it is natural to
wonder whether bridging polaritons with chirality may break the symmetric
yield of chiral products and enable other practically valuable effects.

##  3 Field and matter chirality

###  3.1 Chirality of light

In the following subsections we briefly overview the concepts of chiral light
and chiral matter. Chirality of light is most easily approached through the
notion of _handedness_. Left-handed (LCP) and a right-handed (RCP) circularly
polarized plane waves represent primordial examples of chiral electromagnetic
field, Fig. 2(a). LCP (RCP) wave in a transparent dielectric can be defined as
a monochromatic plane wave with its magnetic field $\pi/2$ ahead (behind) the
electric field everywhere in space:

$Z\mathbf{H}_{LCP}(\mathbf{r})=-i\mathbf{E}_{LCP}(\mathbf{r}),\ \
Z\mathbf{H}_{RCP}(\mathbf{r})=+i\mathbf{E}_{RCP}(\mathbf{r}).$ (3)

where $Z=\sqrt{\mu\mu_{0}/\varepsilon\varepsilon_{0}}$ is the medium
impedance. Considering an RCP (LCP) wave frozen in time yields a helix twisted
clockwise (counter-clockwise) in the propagation direction, Fig. 2(a). The
handedness quality of a circularly polarized plane wave is preserved upon
arbitrary rotation of the field in space, and also upon time reversal.
However, this quality is flipped upon a mirror reflection, or an inversion.
Thus, the handedness can be used as a basis for constructing a time-even
pseudo-scalar field.

To that end, consider the helicity operator
$\overset{\leftrightarrow}{\mathbf{\Lambda}}=\frac{1}{k}\big{(}\begin{smallmatrix}\nabla\times&0\\\
0&\nabla\times\end{smallmatrix}\big{)}$ acting on the complex electric and
magnetic field vectors $\Psi=(\mathbf{E},Z\mathbf{H})^{T}$. One immediately
realizes that LCP and RCP waves are eigenstates of this operator with
eigenvalues $\lambda=+1$ and $\lambda=-1$, respectively 37, 38.
Correspondingly, any linear combination of LCP (RCP) plane waves is an
eigenvector of $\overset{\leftrightarrow}{\mathbf{\Lambda}}$. Calculating the
local expected value of the helicity operator
$\langle\Psi(\mathbf{r})\rvert\overset{\leftrightarrow}{\mathbf{\Lambda}}\rvert\Psi(\mathbf{r})\rangle$
and adjusting the prefactor, we obtain the chirality density – a local measure
of handedness of a non-homogeneous field – that was originally introduced by
Lipkin 39:

$C(\mathbf{r},\omega)=\frac{\varepsilon_{0}\omega}{2}\mathfrak{I}\mathrm{m}[\mathbf{E}\cdot\mathbf{B}^{*}].$
(4)

This quantity characterizes the local geometry of light polarization,
expressing the degree of “screwiness”of the electric and magnetic field lines
40. It can also be interpreted as the normalized difference between the number
of left-handed and right-handed photons in a monochromatic field. On a side
note, the presence of material dissipation strongly modifies the expression
for chirality density 41.

Figure 2: The basics of electromagnetic field chirality. (a) Illustration of
the spatial structure of and RCP and LCP plane wave in free space. The arrows
indicate spatial distribution of the instantaneous electric field vectors in
space. (b) Schematic illustration of expansion of an arbitrary non-paraxial
electromagnetic field into Riemann-Silberstein vectors.

To grasp how the chirality density is generally applicable to arbitrary
monochromatic electromagnetic field distributions, one notices that a solution
of Maxwell’s equations in a transparent medium can always be expanded into two
components with well-defined handedness, called Riemann-Silberstein vectors
42:

$\mathbf{G}_{\pm}(\mathbf{r})=\frac{\mathbf{E}\left(\mathbf{r}\right)\pm
iZ\mathbf{H}(\mathbf{r})}{\sqrt{2}}.$ (5)

Each of the two Riemann-Silberstein vectors is an eigenstate of the helicity
operator, meaning that it only contains fields of the corresponding
handedness. According to our handedness convention, $\mathbf{G}_{+}$
($\mathbf{G}_{-}$) represents the LH (RH) field component.

It should be noted that, in the most general sense, a combination of
electromagnetic fields is chiral, if its mirror image cannot be aligned with
itself by a series of rotations and translations. Clearly, the local density
(4), being perfect for characterizing plane waves, cannot exhaustively cover
the structured light chirality. There are plenty of solutions of Maxwell’s
equations, such as chiral vortex beams, having $C=0$ everywhere in space, yet
being geometrically chiral.43. It is unclear at the moment whether this
distribution chirality, also dubbed as Kelvin’s chirality, can be quantified
by an appropriate measure, but various ways of its conversion from and into
the local chirality density in tight focuses of light beams are being
intensively explored.44, 45. When it comes to chiral light-matter
interactions, however, it is the chirality density (4) that still plays the
major role. Peculiar effects determined by the distribution chirality, though
being more subtle, promise intriguing new possibilities along with the surge
of interest in structured light in general.

###  3.2 Chirality of matter

Not only electromagnetic field, but rigid geometrical bodies as well can be
characterized with chirality. A geometrical body in three-dimensional space is
said to be chiral, if it cannot be superimposed upon its mirror image by
rotations and translations, Fig. 3(a). Geometrical chirality has fundamental
implications on the electromagnetic response of matter. Any medium composed of
chiral microscopic elements, such as molecules or meta-atoms, acquires a
magneto-electric response, wherein electric displacement $\bf D$ becomes
linked to $\mathbf{H}$, and magnetic induction $\bf B$ becomes linked to
$\mathbf{E}$. For a reciprocal bi-isotropic material the constitutive
relations take the form 15:

$\left(\begin{array}[]{c}\mathbf{D}\\\
\mathbf{B}\end{array}\right)=\left(\begin{array}[]{cc}\varepsilon_{0}\varepsilon(\omega)&+i\kappa(\omega)/c\\\
-i\kappa(\omega)/c&\mu_{0}\mu(\omega)\end{array}\right)\left(\begin{array}[]{c}\mathbf{E}\\\
\mathbf{H}\end{array}\right),$ (6)

where $\varepsilon$ and $\mu$ are the scalar relative permittivity and
permeability, correspondingly, and
$\kappa=\kappa^{\prime}+i\kappa^{\prime\prime}$ is a complex dimensionless
pseudoscalar _Pasteur parameter_ of the coupling between the electric and
magnetic fields. Generally, any of these quantities can be a tensor
(pseudotensor), in which case the medium is called bi-anisotropic. In the case
of a bi-isotropic medium with a pseudoscalar Pasteur parameter, its real part
$\kappa^{\prime}$ introduces a difference in phase velocities of RCP and LCP
plane waves, resulting in the optical rotation (OR) of linear polarization
direction. The imaginary part $\kappa^{\prime\prime}$ gives rise to different
absorption of RCP and LCP waves, which amounts to the circular dichroism (CD).

Figure 3: The basics of matter chirality. (a) Visual representation of a
chiral object $S$ and its mirrored enantiomer $S^{\prime}$. The two objects
are related by a mirror transformation and cannot be mapped into one another
by a series of rotation and translations. (b) Configurational and
constitutional chirality. Configurational chirality involves identical
constituent elements arranged in a geometrically chiral shape. Constitutional
chirality involves non-identical elements arranged in an otherwise non-chiral
shape. (c) Examples of chiral meta-atoms: metallic helices, twisted split-ring
resonators, and DNA origami scaffolds. (d) Schematic illustration of a chiral
source of electromagnetic field involving a pair of parallel electric and
magnetic dipole transitions. Panel (b) adapted with permission from refs. 46.
Parts of panel (c) adapted with permission from refs. 47 and 48.

Prominent examples of naturally chiral compounds are different inorganic
crystals such as quartz or cinnabar. Much more common is the chirality on the
molecular scale of organic substances. Although the corresponding effects of
bi-isotropy described by Eq. (6) as well as of a more sophisticated bi-
anisotropy remain also typically weak, the valuable information encoded in OR
and CD spectra motivates persistent improvement of the corresponding
techniques of optical chirality diagnostics 49.

Profound chiral optical phenomena occur when the molecular chirality is given
an opportunity to govern a bottom-up self-assembly of chiral superstructures.
For a particular example, one can consider chiral nematic liquid crystals –
cholesterics. Here the structural chirality of constituent organic molecules
induces smooth twisting of the otherwise homogeneous nematic. Upon appropriate
preparation and alignment, cholesterics form stable helical configurations
with variable pitch 50.

Besides the variety of naturally occurring organic and inorganic substances,
chirality can be engineered in nanostructures at the mesoscopic level 51, 52,
53. Truly fascinating prospects of boosting the chiral optics to its
fundamental limits have opened with the advent of metamaterial concept, as the
most sophisticated top-down nanotechnology approaches have been applied to
produce designer meta-structures for versatile optical functionality 54.
Typically, their chirality stems from that of constituting elements – _meta-
atoms_. The mirror symmetry of the latter can be broken in different ways. A
meta-atom chirality may originate purely from the geometry of its
construction, in which case, it is referred to as _configurational_ chirality,
Fig. 3(b). Alternatively, it may be determined by the mutual arrangement of
specific different parts of an otherwise achiral structure, in which case it
is referred to as _constitutional_ chirality. Examples of artificial chiral
nanostructures include metallic helices 55, 56, chiral arrangements of non-
chiral meta-atoms 57, and numerous bio-inspired chiral plasmonic
nanostructures build around DNA scaffolds 52, 58, 59, Fig. 3(d). It is truly
remarkable how almost infinitesimally small bio-molecules can impart chiral
optical response to otherwise achiral plasmonic structures. The latter can
amplify and effectively transfer subtle chiral fingerprints of bio-molecules
from the ultraviolet to the more convenient visible range 60. Moreover,
proteins can trigger the formation of chiral plasmonic complexes which then
directly visualize the structural molecular chirality by the visible plasmonic
CD 61.

Metasurfaces – periodic arrangements of chiral meta-atoms in planar arrays –
have proven to perform extremely strong chiral light-matter interactions
within the optical paths smaller and even much smaller than the wavelength 62.
Moreover, the palette of available chiral optical phenomena greatly exceeds
the capabilities of natural materials: a chiral metasurface is characterized
by a set of scattering matrix elements $r_{\mu\nu}$ and $t_{\mu\nu}$
describing transmission and reflection of light from polarization state $\nu$
into polarization state $\mu$. Depending on the metasurface design, some of
these coefficients can vanish, while others become dominant. The metasurface
point symmetry group plays the key role in the analysis of all such
possibilities.

Chiral metasurfaces possessing a three-fold or higher rotational symmetry
around a vertical axis scatter normally incident light in a relatively simple
fashion 63, 64. The only relevant chiral characteristics of such metasurfaces
are the transmission CD, which describes the difference in transmittances of
RCP and LCP waves:

$\textrm{CD}=\frac{|t_{RR}|^{2}-|t_{LL}|^{2}}{|t_{RR}|^{2}+|t_{LL}|^{2}},$ (7)

and OR, which describes the difference in phase delays. Qualitatively, such
metasurfaces are similar to slabs of natural chiral Pasteur media 65. The
quantitative differences, however, are immense: the CD and OR delivered by
metasurfaces can take any physically meaningful value including the extreme
maxima and minima (${\rm CD}=\pm 1$ and ${\rm OR}=\pm 90^{\circ}$) after the
light travels only fractions of its wavelength 66. Recent transition to all-
dielectric platforms has enabled creating strongly chiral and transparent
samples 67, 68, 69. Moreover, precisely engineering quasi-bound states in the
continuum 70 it is possible to realize the maximum chirality combining full
transparency to waves of one circular polarization with full absorption of
waves of the opposite circular polarization 71.

Combining chirality with broken rotational symmetry allows obtaining even more
sophisticated polarization responses, which expands the multitude of chiral
metasurface functionalities further 72, 73, 74, 75. Carefully engineered
arrays of silicon helices 76 or silicon bars of different height 77, 78
achieve lossless maximum chirality, when transparency to waves of one circular
polarization is accompanied by a full reflection of the other.

CD, being an optical signature of geometric chirality, should not be confused
with similar effects inherent to achiral objects. For example, metasurfaces
having _at most_ two-fold rotational symmetry around a vertical axis can
exhibit unequal RCP-to-LCP and LCP-to-RCP transmission described by the so-
called circular conversion dichroism:
$\textrm{CCD}=(|t_{RL}|^{2}-|t_{LR}|^{2})/(|t_{RL}|^{2}+|t_{LR}|^{2})$.
Although the co-polarized transmittances are bound to be equal for achiral
structures, such metasurfaces may exhibit unequal total power transmission
upon illumination with RCP and LCP waves:
${\left|t_{RR}\right|}^{2}+{\left|t_{LR}\right|}^{2}\neq{\left|t_{LL}\right|}^{2}+{\left|t_{RL}\right|}^{2}$
79. Violating high-order rotational symmetry is crucial for designing
handedness-preserving mirrors 80, 81, which are an essential building block
for constructing chiral polaritons.

One can also introduce the notion of a chiral source of electromagnetic field.
An elementary chiral source is a combination of parallel electric $\mathbf{p}$
and magnetic $\mathbf{m}$ point dipoles with a $\pm\pi/2$ phase difference,
Fig. 3(c):

$\mathbf{m}=-i\xi c\mathbf{p},$ (8)

where $\xi$ is a real-valued parameter. $\xi=\pm 1$ defines an ideal LH (’+1’)
and RH (’-1’) chiral source, respectively, that emits only fields of the
corresponding handedness 82, 38.

This notion of chiral field sources naturally translates to quantum emitters
(QEs). A chiral QE is one, whose transition between the ground and excited
levels is quantified by a combination of collinear electric and magnetic
dipole moments satisfying Eq. 8 83. In realistic QEs whose transitions do
possess electric and magnetic dipole moments at the same time, they are
usually not aligned ideally as required by Eq. 8, but possess a perpendicular
component of the magnetic dipole moment $\mathbf{m}\bot\mathbf{p}$. This
component can be associated with the so called Omega-type bi-anisotropic
response of the medium 15.

Knowledge of the transition dipole moments of a particular QE allows one to
describe an ensemble of those in terms of macroscopic permittivity and Pasteur
parameter, as in Eq. 6. Although often assumed non-dispersive, correct
frequency dispersion of Pasteur parameter (as well as permittivity) will be
crucial for description of strong coupling between quantum emitters and
electromagnetic field. As long as a single electronic transition of the QE is
involved, the permittivity and the Pasteur parameter of a macroscopic ensemble
of such EQs with volume density $\rho$ can be described by the Lorentz model
84:

$\displaystyle\varepsilon(\omega)=\varepsilon_{\infty}+f\frac{\omega_{P}^{2}}{\omega_{0}^{2}-\omega^{2}-i\gamma\omega},\
\
\kappa(\omega)=g\frac{\omega_{P}^{2}}{\omega_{0}^{2}-\omega^{2}-i\gamma\omega},$
(9)

where $\omega_{P}^{2}=\rho e^{2}/(3\varepsilon_{0}m)$ is the plasma frequency
of the transition ($1/3$ accounts for isotropic orientation of QEs), $e$ and
$m$ are the electron mass and charge, respectively,
$f=2m\omega_{0}|\mathbf{p}|^{2}/(e^{2}\hbar)$ and
$g=2m\omega\mathfrak{I}\mathrm{m}[\mathbf{p}\cdot\mathbf{m}^{*}]/(e^{2}\hbar
c)$ are the electric and magneto-electric oscillator strengths of the chiral
transition, $\omega_{0}$ and $\gamma$ are the transition frequency and
linewidth, respectively.

Accounting for light-matter interactions determined by further higher-order
multipoles of molecular transitions predictably expands the palette of chiral
optical effects especially arising when structured light beams are involved.
Although it is hard to access the higher-order multipoles by plane light
waves, tightly focused optical vortices with strongly inhomogeneous field
distributions provide a propper new set of tools. Intriguing effects are
observed, for example, as chiral molecular electric quadrupoles asymmetrically
interact with optical vortices of different handedness giving rise to the so
called helical dichroism.85

###  3.3 Chirality versus spin of light

In quantum electrodynamics, the fundamental importance of chiral light forms
becomes obvious already on the very basic level of analysis of intrinsic
symmetry of free space photons 86. For a particle with mass, such symmetry can
be conveniently analyzed in its rest frame. As rest frames are generally
nonexistent for photons, one has to consider the space transformations
preserving the photon propagation direction. Circularly polarized plane waves
are natural eigenstates of such coordinate transforms. The absence of the rest
frame, on the other hand, does not allow for a proper introduction of a photon
_spin_ , which otherwise appears as an intrinsic angular momentum of a resting
particle with mass. Considering photon parity with respect to the coordinate
inversion, however, one recognizes the necessity to introduce a close analog
of spin describing the intrinsic angular momentum of a photon – its spin
angular momentum (SAM).

In terms of the classical electrodynamics, SAM of an electromagnetic field can
be treated as a continuous field variable. From this standpoint, the SAM
density is a vector quantity, which for a monochromatic electromagnetic field
$\\{\mathbf{E}(\mathbf{r}),\mathbf{H}(\mathbf{r})\\}$ in free space reads: 87:

$\mathbf{s}=\frac{1}{4\omega}\mathfrak{I}\mathrm{m}\left[\varepsilon_{0}{\mathbf{E}^{*}}\times\mathbf{E}+\mu_{0}{\mathbf{H}^{*}}\times\mathbf{H}\right].$
(10)

A trivial situation wherein an electromagnetic field possesses SAM density is
presented by an RCP or LCP plane wave. In this case, the SAM density is
parallel or anti-parallel to the wave vector depending on the handedness of
the wave.

Another far less obvious situation when an electromagnetic field features a
spin is presented by evanescent plane waves. Consider a monochromatic
transverse-magnetic evanescent wave propagating in free space along the $x$
axis, and attenuating exponentially in the $z$ direction:
$\mathbf{H}\propto\widehat{\mathbf{y}}e^{ik_{x}x-\kappa_{z}z}$,
$\mathbf{E}\propto\left(-i\kappa_{z}\widehat{\mathbf{x}}+k_{x}\widehat{\mathbf{z}}\right)e^{ik_{x}x-\kappa_{z}z}$,
where the wave vector components satisfy
$k_{x}^{2}-\kappa_{z}^{2}=(\omega/c)^{2}$. After simple algebra we find that
such a wave also carries a non-zero SAM density component, which is
perpendicular to the propagation and the attenuation direction 87:

$\mathbf{s}_{T}\propto\mathfrak{R}\mathrm{e}\
\mathbf{k}\times\mathfrak{I}\mathrm{m}\ \mathbf{k}.$ (11)

This component of SAM arises due to the elliptical polarization of the
electric field in the propagation plane, Fig. 4(a), and is known as the
transverse spin angular momentum 87, 88.

Figure 4: Relation between chirality and spin. (a) Illustration of the spatial
distribution of the electric and magnetic fields in a TM-polarized evanescent
wave propagating along the $x$ axis and exponentially decaying along the $z$
axis. Elliptical polarization of the electric field produces a transverse
component of the spin angular momentum perpendicular to the linear momentum of
the wave (b) Spin-momentum locking in evanescent waves. An elliptically
polarized electric dipole source placed near a waveguiding surface features
directional coupling to guided waves due to the spin-orbit coupling.

This property of evanescent fields leads to a fundamental connection between
light’s SAM density and linear momentum near planar surfaces and enables
unidirectional excitation of waveguide modes in various nanophotonic platforms
89, 88, 90. Consider a TM-polarized bound mode supported by an interface (such
as a surface wave propagating along a metallic surface, or a waveguide mode of
a dielectric film). An electric dipole with an appropriate elliptical
polarization placed near a waveguiding surface will produce an electromagnetic
field that will couple preferentially to those guided modes, whose transverse
SAM density matches that of the circular-polarized source, Fig. 4(b). An
analogous spin-momentum locking arises for TE-polarized bound modes 91.

It is crucial to emphasize the similarities and the differences between the
SAM density and the chirality of light. SAM density is a time-odd vector,
while chirality density is a time-even pseudo-scalar. To put this into
perspective, one may consider a circularly-polarized homogeneous wave. The SAM
density in such a wave is collinear with the momentum, and time reversal
leaves its handedness unchanged, but flips the SAM density. A lot of excellent
works dealing with this mechanism of unidirectional excitation have used terms
”chiral coupling”, ”chiral emission”, and alike (see, e.g., refs. 92, 90, 93,
94, 95), while the studied systems had little to do with true chirality of
light. Chirality involves chiral sources represented by a combination of
collinear electric and magnetic dipoles, as opposed to circularly polarized
electric dipole engaged in the effect of spin-orbit coupling.

##  4 Constructing chiral polaritons

Now we are in a position to discuss the notion of a chiral polariton, by which
we are going to refer to the coupled state of an interacting light-matter
system, where both the optical mode _and_ the material excitation are chiral.
Maxwell’s equations in free space admit solutions in the form of chiral
photons, so one could already examine chiral polaritons emerging in this
situation. However, from the practical standpoint, we are usually interested
in strong interaction with confined optical modes. Thus, it would seem natural
to load an ordinary resonator – such as a Fabry-Pérot cavity mode – with
chiral matter and expect excitation of chiral polaritons upon excitation of
such a system with a circularly polarized electromagnetic field. This kind of
system has been realized in ref. 22 by coupling chiral metallic meta-atoms to
the normal-incidence resonance of a Fabry-Pérot cavity. Similar setup was
realized in ref. 23 by coupling chiral organic emitters to a non-chiral
surface plasmon mode of a metallic film.

The simple intuition outlined above faces a fundamental obstacle. To exemplify
the problem, consider a perfect electric mirror illuminated at normal
incidence by an LCP or an RCP plane wave
$\mathbf{E}_{inc}\left(\mathbf{r}\right)\propto{\left(1,\pm
i\lambda,0\right)}^{T}e^{ikz}$, $\lambda=\pm 1$. The field continuity at the
surface of the mirror requires that the reflected wave is described by the
same complex polarization ${\left(1,\pm i\lambda,0\right)}^{T}$. Thus,
reflection of the circularly polarized wave at a normal by a metallic mirror
preserves its SAM density, but reverses its wave vector, which in the end
flips the handedness of the reflected wave. This handedness flipping of the
wave travelling between two mirrors creates a standing wave with zero
chirality density, Fig. 5(a). As a result, the quasinormal modes (QNMs) of a
Fabry-Pérot cavity at normal incidence do not possess any handedness. This
property of circularly polarized standing waves imposes severe constraints on
the behavior of polaritonic systems. In particular, this helicity flipping
ultimately limits the magnitude of CD that can be observed with a chiral film
strongly coupled to an ordinary Fabry-Pérot cavity despite the resonant
enhancement of the electromagnetic field inside the cavity 22.

The problem of handedness flipping poses a quest for optical cavities
supporting QNMs of well-defined handedness, whose performance is schematically
illustrated in Fig. 5(a) 96. A natural question then arises: what design rules
does one need to follow in order to engineer a cavity that preserves the
handedness of electromagnetic field (upon scattering)? This turns out to be a
more fundamental issue which we address below.

Figure 5: Handedness-preserving optical cavities. (a) The difference between
an ordinary Fabry-Pérot cavity, and a handedness-preserving planar cavity. An
ordinary Fabry-Pérot resonator supports standing waves with linear
polarization. A hypothetical handedness-preserving cavity, on the contrary,
supports a chiral standing wave with a linear polarization at every point in
space. (b) Sketch of the handedness-preserving cavity composed of two
dielectric photonic crystal mirrors. The color shows the optical chirality
density of the resulting standing wave. (c) Left: single-handedness chiral
cavity realized by stacking two dielectric low-symmetry mirrors. Right:
simulated electric field enhancement spectra. (d) A compact maximally chiral
optical cavity represented by a metallic helix producing a resonant response
only to one incident handedness and being nearly transparent to the other
handedness. Panel (b) adapted with permission from ref. 97. Parts of panel (c)
adapted from ref. 98. Panel (d) adapted with permission from ref. 99.

###  4.1 Handedness-preserving cavities

Handedness of light may be converted in different ways upon interaction with
scatterers, even non-chiral ones. As we showed above, any field distribution
satisfying Maxwell’s equations can be expanded into two components of well
defined handedness $\mathbf{G}_{\pm}$. Imagine a scenario when an incident
field ${\mathbf{E}}_{inc}(\mathbf{r})$ of a certain handedness (for example, a
purely LH field) interacts with an arbitrary scatterer, such as a metasurface
or a compact nanoparticle. For this field only the $\mathbf{G}_{+}$ component
is different from zero. The scattered field, on the other hand, would
generally contain both handedness components:
$\mathbf{G}^{scat}_{+}\left(\mathbf{r}\right)\neq
0,\mathbf{G}^{scat}_{-}\left(\mathbf{r}\right)\neq 0$.

Some cavities, such as plasmonic and dielectric nanoantennas, exhibit spatial
regions of enhanced chirality density even upon excitation with linearly
polarized light 100, 101, 75, but QNMs of these systems still do not possess a
well-defined pure handedness. In this context, the concept of duality 102
plays an important role. Using the framework of Riemann-Silberstein vectors
one can show that a piece-wise homogeneous and isotropic system preserves the
handedness of any incident electromagnetic field at a frequency $\omega$ if
and only if all the materials have a constant impedance across all media
constituting the system, including the environment 102:

$\frac{\varepsilon\left(\mathbf{r},\omega\right)}{\mu(\mathbf{r},\omega)}=\frac{\varepsilon_{env}\left(\mathbf{r},\omega\right)}{\mu_{env}(\mathbf{r},\omega)}=\rm
const\ \ \forall\mathbf{r}.$ (12)

Structures that satisfy this criterion are referred to as dual.
Correspondingly, if the structure is surrounded by free space, Eq. 12 requires
equal permittivity and permeability of the resonator,
$\varepsilon(\mathbf{r},\omega)=\mu(\mathbf{r},\omega)$.

As one can see, duality is a very stringent requirement that calls for
magnetic materials, $\mu\neq 1$. Since they are very rare in the visible
range, this motivates the search for non-magnetic handedness-preserving
structures. Feis _et al._ 97 demonstrated a handedness-preserving cavity
formed by a pair of periodic dielectric mirrors, Fig. 5(b). Each mirror
operates by virtue of first-order diffraction into large transverse momentum
($k_{\parallel}$) modes. Thanks to the presence of an inversion center, the
system itself is achiral, meaning it supports both the LH and the RH QNMs at
the same time. Voronin et al. pushed this idea further and studied a _single-
handedness_ chiral optical cavity 98. By utilizing low-symmetry dielectric
mirrors adopted from ref. 81, they designed a Fabry-Pérot cavity that supports
a QNM of a well-defined handedness, and does not support QNMs of the opposite
handedness in the relevant wavelength range, Fig. 5(c).

In addition to planar infinitely extended resonator, there has been a progress
in designing compact nanostructures with chiral optical resonances. Garcia et
al. experimentally demonstrated what is called a maximally chiral object 38 of
a finite size operating in the IR 99. Their structure is a helix-shaped
metallic nanoresonator, Fig. 5(d), that is nearly transparent for incident
light of a certain handedness, while excitation with the opposite handedness
produces resonant extinction. Any maximally chiral object is necessarily dual
38, implying that the excited QNMs should have a well defined handedness. The
interior of the resulting nanocavity can be loaded with an excitonic material
similarly to ref. 103, which studied theoretically a similar plasmonic
nanostructure interfaced with an achiral excitonic material.

###  4.2 A theoretical model for chiral polaritonics

The availability of a handedness-preserving cavity is only one part of the
story, next we have to understand how to utilize it. Exploring the
possibilities of chiral polaritonics is no small endeavour and will demand
ample work on both theoretical and experimental side. Intuition, often
generated by simple models as illustrated in this section, serves as an
important catalyst. We refer the interested reader to Ref. 104 for a detailed
derivation.

The starting point for all theoretical descriptions is the non-relativistic
minimal-coupling Hamiltonian. Since molecules are usually small in relation to
the wavelength of the confined modes it is beneficial to apply the Power-
Zienau-Wooley transformation and retain only the electric and magnetic dipolar
terms as well as electric quadrupoles 105, 106. The lowest-order Hamiltonian
for chiral interaction takes therefore the form

$\displaystyle\hat{H}$
$\displaystyle=\hat{H}_{M}+\hat{H}_{L}-\frac{1}{\varepsilon_{0}}\sum_{n}^{N}\hat{\boldsymbol{\mu}}_{n}\cdot\hat{\mathbf{D}}_{\perp}(\mathbf{r}_{n})-\sum_{n}^{N}\hat{\textbf{m}}_{n}\cdot\hat{\mathbf{B}}(\mathbf{r}_{n})+\hat{H}_{LMc}$
(13)

with the electric and magnetic transition-dipole moments
$\hat{\boldsymbol{\mu}}_{n},~{}\hat{\textbf{m}}_{n}$ related by the transition
elements $\textbf{m}_{n}^{01}=-ic\xi\boldsymbol{\mu}_{n}^{01}$, displacement
$\hat{\mathbf{D}}_{\perp}(\mathbf{r}_{n})$ and magnetic
$\hat{\mathbf{B}}(\mathbf{r}_{n})$ field. The corrections collected in
$\hat{H}_{LMc}$ are essential for gauge-invariance, realistic descriptions and
large interaction strength 104. Importantly, the dipolar moments are defined
with respect to the center of mass of each individual molecule. Following Ref.
104, we can obtain an intuitive solution for the interaction between N
strongly simplified chiral emitters and the standing chiral field. In a
simplified form, the analytic solution

$\displaystyle\omega_{\pm}\approx\frac{1}{\sqrt{2}}\sqrt{\omega_{cav}^{2}+\omega_{0}^{2}+8\xi\lambda
g^{2}\pm\sqrt{[\omega_{cav}^{2}-\omega_{0}^{2}]^{2}+16g^{2}(\omega_{cav}+\omega_{0}\xi\lambda)(\omega_{cav}\xi\lambda+\omega_{0})}}$
(14)

provides an intuitive way to understand the interplay of the handedness of
field $\lambda$ and molecules $\xi$. The polaritonic eigenvalues
$E_{\pm}=\hbar\omega_{\pm}+E_{vac}$, where
$E_{vac}=\hbar(\omega_{+}+\omega_{-})/2$ is the ground state energy of the
coupled chiral system, will exhibit an avoided crossing $\propto
g(1+\lambda\xi)$. Fig. 6 illustrates this handedness depending hybridization
that gives rise to a plethora of possible applications.

Figure 6: Spectrum of polaritonic eigenvalues $\omega_{\pm}$ of the chiral
Hopfield model with a left-handed cavity mode as a function of the cavity
frequency $\omega_{cav}$ in relation to the matter excitation $\omega_{m}$ and
chiral factor $\xi$. The eigenvalues are calculated with typical values for
dye molecules (see Ref. 104 for details).

The handedness preserving mirrors are therefore able to lift the degeneracy
between same ($\lambda\xi>0$) and opposite ($\lambda\xi<0$) combinations – we
obtain non-intrusive control over the enantiomers. Furthermore, this extends
also to the correlated ground-state of the combined system. While it sounds
like a dream come true, it is important to realize that the degree of
chirality is often weak and the energetic difference between the enantiomers
therefore small. Nevertheless, chiral polaritionics suggests a plethora of
possible applications, some will demand better engineered fields, others might
work out-of-the-box. In the following, we will discuss some of those
applications, debating short-term challenges and long-term prospects.
Furthermore, we briefly discuss which theoretical approaches might be suited
and how existing techniques can be extended.

##  5 Prospects and Challenges in Chiral Polaritonics

Chirality is a long standing and widely impactful concept in natural science.
If we add sugar to our coffee, enjoy a good meal, or are in the need of
medication - chirality has a defining influence at every point of our life and
even the existence of life, as we know it, itself. Understanding and
controlling the chiral state of a substance is thus pivotal. Early work in
chiral catalysis was awarded in 2001 with the Nobel price in chemistry 107. In
the following, we will illustrate a selected set of promising directions,
discuss associated challenges and highlight recent development in the
theoretical description of chiral polaritonics.

###  5.1 Anticipated novel phenomena

#### Controlling spontaneous emission:

In the weak coupling regime, single-handedness cavities could enable
differential spontaneous emission rates for left and right-handed enantiomers
of chiral sources. Yoo and Park introduced the notion of chiral Purcell factor
back in 2015 108; however, their description can be questioned because the
authors assumed that LH and RH enantiomers of a given chiral source would show
unequal spontaneous emission rates in free space, while in fact they must be
equal (the empty environment of the QE is symmetric with respect to
inversion). Voronin et al. studied the radiation of chiral point sources
coupled to their single-handedness cavities and observed a strong asymmetry in
the emission intensities, suggesting that the fluctuating electromagnetic
field within a single-handedness cavity acquires a strong chiral character 98.
Such an observation follows intuitively from the handedness-dependent mode
density in a chiral environment that gives rise to handedness-dependent
Purcell factors. Thus, electronically excited states of opposite enantiomers
of a chiral molecule would acquire different lifetimes, Fig. 7(b), which could
be interesting in the context of selective de-excitation of chiral molecules
83. Recent studies demonstrate intrinsic asymmetry of photoluminescence from
chiral excitons hosted in J-aggregates 109; single-handedness cavities may
further enhance this emission asymmetry.

#### Red-shifting chiral transitions:

Either symmetric two-handedness or asymmetric single-handedness chiral
cavities can be utilized for engineering of chiral polaritons by loading them
with resonant Pasteur media, which could have promising implications for
chiral sensing. Indeed, chiral electronic transitions of the majority of bio-
molecules are located in the UV range, where spectroscopic measurements are
challenging. The two polaritons of a strongly coupled system are expected to
experience an anti-crossing, wherein the lower polariton shifts towards the
visible range, Fig. 7(a), where measuring CD and characterizing the
enantiomeric state of a racemic would be much easier. Recent theoretical
efforts are beginning to explore this strategy. For example, Feis et al. and
Beutel et al. studied the response of a symmetric handedness-preserving cavity
loaded with a non-resonant Pasteur medium and shown enhancement of CD 97, 110,
while Mauro et al. examined the behavior of a single-handedness cavity loaded
with _resonant_ chiral medium 111.

Figure 7: Potential implications of chiral strong coupling. (a) Selective de-
excitation of chiral molecules. Two opposite enantiomers of a chiral molecule
coupled with an asymmetric single-handedness optical cavity. Asymmetric
coupling strength leads to unequal spontaneous decay rates from the excited
states of two enantiomers. (b) Red-shifting chiral transitions by strong
coupling. A single chiral electronic transition of a molecule located in the
UV range. Strongly coupling this molecule with a photonic mode causes of the
the lower polariton (LP) to red-shift towards the visible range while the
upper polariton (UP) is shifted deeper into the UV domain. (c) Steering the
equilibrium of an ensemble of racemic mixture by ultra-strong coupling of
chiral emitters to a single-handedness optical cavity.

#### Leveraging chiral vacua:

The chiral nature of vacuum fluctuations in single-handedness cavities might
enable unusual vacuum energy landscapes. A single harmonic oscillator
possesses a vacuum (a.k.a. zero-point) energy of $U_{vac}=\hbar\omega/2$. In a
polaritonic system made up of two interacting harmonic excitations the vacuum
energy therefore is $U_{vac}=\hbar(\omega_{+}+\omega_{-})/2$, where
$\omega_{\pm}$ are the two polaritonic transition energies.112, 113 Such
changes in the correlated ground state will differ for each enantiomer and
increase with the molecular concentration as $\sqrt{N}$.104, 114 This hints at
a possibility that chiral molecules will experience different ground-state
energies inside a single-handedness cavity depending on their enantiomeric
state, thus steering the molecular structure to a different equilibrium.
Irrespective the exciting possibilities, a few obstacles remain. First,
molecular chirality is small, a challenge and opportunity to be revisited in
the following. Second, simplified few-mode theories are well suited to
describe spectral effects but can be expected to provide only a qualitative
trend for the correlated ground state energy that would be predicted following
Casimir. 115, 116

#### Enhancing the Weak Chirality:

The parameter $\xi$ characterizing the degree of matter chirality is extremely
weak for realistic molecules ($\xi\sim 10^{-4}...10^{-6}$, ref 84), which
translates into a minute asymmetry of the polaritonic energy spectra between
right and left enantiomers. One possibility to address this issue is to design
a chiral cavity that would compensate for this smallness. The discriminating
effect in an ensemble of emitters coupled to a single chiral mode scales with
$\sqrt{N}g\lambda\xi$, where $\lambda$ characterizes the relationship between
the electric and magnetic fields of the chiral optical mode. This suggests
that chiral molecules with $|\xi|\ll 1$ would produce a substantial energy
asymmetry if they are coupled to a chiral optical mode with $\lambda\gg 1$.
This is not possible in purely transverse chiral standing waves, where the
electric and magnetic field are related via $Z\mathbf{H}=\pm i\mathbf{E}$.
However, in subwavelength cavities, where the field has a substantial
longitudinal component, it does no longer hold . Thus, the issue could be
addressed by designing a compact chiral nanocavity, as in ref. 99, whose
quasi-normal mode is dominated by the longitudinal magnetic field,
$|Z\mathbf{H}|\gg|\mathbf{E}|$, and maintains a chiral character.

#### Exploring autocatalysis and homochirality:

Why do sugar (sucrose) and many other substances exist in nature in such high
excess compared to their chiral images? What leads to this large imbalance
between enantiomers? This simple question remains unanswered, although it
marks a defining problem for the development of life.117 While still debated,
it is conjectured that an initial unbalance between the enantiomers was
amplified by autocatalytic reactions of chemical and/or biological nature 118.
The origin of this unbalance remains however unknown, and speculations range
from a pure chance due to statistical fluctuations to the parity symmetry
breaking in weak interactions 119. The problem is obvious, without a way to
carefully adjust minute changes in the environmental conditions, it is
challenging to observe and manipulate a possible mechanism.

Chiral polaritonics might provide here a unique possibility. The optical
fields inside the chirality-preserving cavity undergo vacuum fluctuations that
break the chiral symmetry of free space104. How strong the electromagnetic
environment varies inside the cavity from the free-space conditions can be
well controlled by changing size, shape and quality of the optical mirrors. It
has been demonstrated that optical cavities can be used to control anything
from single atoms 120, over chemical reactions 121, up to biomolecules and
bacteria 122. Such a sensitive tool could now be used to fine-tune the degree
to which the electromagnetic environment prefers a given handedness. Control
over the chiral environment allows then control over the initial conditions of
homochirality. Furthermore, as the chiral polaritonic interaction depends on
the number of collectively coupled emitters with the corresponding handedness,
this effect is inherently self-catalysing. While the details of this approach
remain to be investigated, chiral polaritonics could serve as a novel tool to
explore homochirality.

###  5.2 Development of Theoretical Descriptions

Simplified analytical models are mandatory to build intuition – the foundation
for innovative developments and starting point for sophisticated descriptions.
However, a faithful description of experimental reality, which provides in the
long run the necessary microscopic understanding, will inevitably require
sophisticated microscopic theory. How complex such a microscopic description
has to be will depend heavily on the system at hand and the relevant
observable, as sketched in Fig. 8. The value of the classical approximation
for light should not be underestimated. Providing Maxwell’s equations with an
adequate polarizability will often suffice to obtain accurate far-field
spectra. As an example, consider the typical Hopfield system with a set of
quantum harmonic oscillators describing molecular dipoles and a quantized
photonic mode. In the linear regime, its spectrum is identical to the purely
classical solution obtain from Maxwell’s equation 123, 124. Clearly, Maxwell’s
equation does not provide access to the microscopic dynamic governed by
quantum mechanics – the moment we intend to control the material we are bound
to consider its full complexity. However, it is often sufficient to combine
quantum matter with classical light.

Figure 8: Summary of classical and quantum theoretical methods for description
of strong chiral light-matter interaction. Left: description/modelling of
chiral systems using Maxwell’s equations in terms of macroscopic constitutive
parameters gives access to spectroscopic far-field quantities, such as cross-
sections. Middle: semi-classical description using combined treatment with
Maxwell’s equations for the field and Schrödinger equation for matter gives
access to, e.g., population dynamics of the matter encoded by its density
matrix with feedback on the optical fields. Right: full quantum description of
chiral light-matter interaction gives access to quantum correlated features,
such as ground-state changes and combined statistics.

The complexity of quantum matter imposes a necessity for efficient
descriptions such as time-dependent density-functional theory125 which can be
combined with a self-consistent propagation of classical light 126, 127, 128,
129. This approach remains accurate in predicting the eigenstates of the
combined system even in the case of strong interaction with optically confined
modes130. Naturally, the major limitation of this approach is that quantum
fluctuations of the electromagnetic field are not considered. While mixed
quantum-classical approaches are partially able to compensate for the
statistical nature of quantum fluctuations 131, 132, 133, 134, much remains to
be desired. Changes in the correlated ground-state are for instance entirely
caused by quantum fluctuations.

Following the canonical quantization procedure 135, 106, the photonic fields
are represented as a sum over many quantum harmonic oscillators. This vastly
enlarges the already excessive size of the material Hilbert space and renders
a direct solution of the associated Schrödinger equation impossible for
anything remotely realistic. Typical coping strategies assume, for instance, a
strong simplification of the material.136, 137, 104 In general, however, this
approach is best suited to collect intuition and extensions to a more faithful
representation of the material become important when working with e.g.
chemical reactions. The complexity of light and matter is dominated by the
complexity of the material. For weak light-matter interaction, the associated
Fockspace can be efficiently truncated. Ultra-strong interactions require
effective photon-free and non-perturbative approaches138, 139.

If the complexity of the material represents the major limitation, the
solution should be to adjust existing many-body techniques to include
quantized light-matter coupling. Two such approaches are quantum-
electrodynamical density-functional theory (QEDFT)140, 141, 142, 138 and QED
coupled-cluster (QED-CC)143, 144, 145, 146. Their QED adapted form inherits
the fundamental strength and weaknesses of their respective quantum mechanical
counterparts. While this novel field of ab initio QED147 is still in its
infancy, already the initial development provides access to real-space
resolved Lamb-shifts148, modification of chemical reactivity 34, and
intermolecular interactions 149. Chiral polaritonics calls now for an
intensified development in two directions.

#### Beyond the electric dipole:

While many optical effects, such as Mie-scattering, imply matter to be of the
size of the optical wavelength, the molecular constituents of the material are
of much smaller size and the electric dipole commonly dominates the
interaction. This convenient feature led to a clear dominance of the electric
dipole approximation for the interaction between quantized fields with matter
– to a point that the implications of this approximation had been partially
forgotten150, 151. Now, within the realm of chiral strong-coupling, it will be
required to remedy this development. It seems natural to extend the
description simply to the next order in the multipolar expansion135 and the
multipolar framework is known to perform well in the single mode limit of
strong coupling151. In addition, full minimal coupling descriptions could
complement this process, providing an efficient sanity check via the
restrictions enforced by gauge invariance. The latter is only intact for the
full Hilbert space. Furthermore, a resolution of the photonic momentum might
question the single-mode approximation, bringing the field in closer contact
to Casimir physics. Riso et al. have recently taken first steps in this
direction by minimally coupling a single cavity mode to chiral molecules
described by coupled-cluster theory114. An alternative direction is based on
extensions of (current-)density functional theory140, 138, 152.

#### Scaling up:

The most elegant or rigorous theory is useless if unable to predict and
understand the problem at hand. Thus, we are facing the question: Given a
satisfying theoretical description of a chiral molecule, how do we account for
the chiral environment comprising chiral cavities and nanoplasmonic particles?
The recent debate around the theoretical description of ground-state
polaritonic chemistry121, 153, 154, 155 illustrates the limitation of
oversimplified models. While ab initio QEDFT calculations34 have shown
qualitative agreement with experiments, much remains to be understood and the
simplified description of the photonic system to be scrutinized. Collective
light-matter interaction and complex environments represent now the next
challenge127. Conceptually, this is closely related to other multi-scale
problems, such as impurities in condensed matter156 or active centers in
biological structures157, 158. While parallelization strategies can account,
under severe approximations, for a few hundred of molecules159, mesoscopic
ensembles and realistic macroscopic environments will only become available
when efficient embedding strategies are employed127, 160, 161, 162, 163, 164,
165. Lastly, plasmonic structures will take a prominent role for chirality166,
especially due to their capability to enhance the inherently weak chirality
(see Sec. 5.1). Charge migration and the microscopic surface structure renders
simple eigenmode quantization strategies questionable if our chiral molecule
of interest is in close proximity to the nanoparticle. Density-functional
theory presents a reliable choice where first connections to optical cavities
are already available126. The explicit simulation of sizeable plasmonic
structure is, however, too costly. Mixed descriptions along the lines of
quantum hydrodynamics167 or the embedding of quantum-dynamics into classical
permittivity for larger fractions of the system127, 168 could help to
alleviate some of the computational weight necessary to describe realistic
plasmonic systems.

##  6 Conclusions

To conclude, we have presented our vision of a potentially new area of
nanophotonics - chiral polaritonics. Guided by the notion of chirality of
electromagnetic field and matter, we established the general requirements to
resonant optical systems in which chiral polaritonic eigenstates may emerge.
Recent theoretical efforts discussed in this Perspective already indicate that
chiral polaritonic systems may feature non-trivial optical phenomena, where
the interplay of light and matter chirality is of key importance for
determining the eigenfunctions of the system, as well as its response to
external electromagnetic fields.

We have speculated on the potential of a collection of novel effects that can
be enabled by strong coupling between chiral light and matter. Established
directions for the utilization of chirality involve plasmonic or biological
nanostructures166, the here proposed usage of chiral polaritonics adds further
versatility. The addition of ’light’, being the fundamental reason for
dispersive/van-der-Waals interactions, is more than a mere hurdle – it
provides a new opportunity for the description and control of chirality.
Finally, development of existing and new theoretical tools might be necessary
in order to accurately describe various observables of a strongly coupled
chiral system, such as far-field response functions, molecular populations,
and quantum correlations. Due to the multifaceted nature of the problem, a
coordinated effort of researchers with backgrounds in classical
electromagnetism, quantum optics, and material science will be needed in order
to succeed in the hunt for chiral polaritons.

The work was supported by the Russian Science Foundation (21-72-00051). The
work of M.V.G. was supported by the Ministry of Science and Higher Education
of the Russian Federation within the State assignment of FSRC “Crystallography
and Photonics” RAS. C.S. acknowledges funding by the Swedish Research Council
(VR) through Grant No. 2016-06059. D.G.B. acknowledges support from BASIS
Foundation (grant 22-1-3-2-1).

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