# Metasurface-based hybrid optical cavities for chiral sensing

Nico S. Baßler  Max Planck Institute for the Science of Light, D-91058
Erlangen, Germany  Department of Physics, Friedrich-Alexander-Universität
Erlangen-Nürnberg (FAU), D-91058 Erlangen, Germany  Andrea Aiello  Max Planck
Institute for the Science of Light, D-91058 Erlangen, Germany  Kai P. Schmidt
Department of Physics, Friedrich-Alexander-Universität Erlangen-Nürnberg
(FAU), D-91058 Erlangen, Germany  Claudiu Genes  Max Planck Institute for the
Science of Light, D-91058 Erlangen, Germany  Department of Physics, Friedrich-
Alexander-Universität Erlangen-Nürnberg (FAU), D-91058 Erlangen, Germany
Michael Reitz  Max Planck Institute for the Science of Light, D-91058
Erlangen, Germany  Department of Chemistry and Biochemistry, University of
California San Diego, La Jolla, California 92093, USA

###### Abstract

Quantum metasurfaces, i.e., two-dimensional subwavelength arrays of quantum
emitters, can be employed as mirrors towards the design of hybrid cavities,
where the optical response is given by the interplay of a cavity-confined
field and the surface modes supported by the arrays. We show that, under
external magnetic field control, stacked layers of quantum metasurfaces can
serve as helicity-preserving cavities. These structures exhibit ultranarrow
resonances and can enhance the intensity of the incoming field by orders of
magnitude, while simultaneously preserving the handedness of the field
circulating inside the resonator, as opposed to conventional cavities. The
rapid phase shift in the cavity transmission around the resonance can be
exploited for the sensitive detection of chiral scatterers passing through the
cavity. We discuss possible applications of these resonators as sensors for
the discrimination of chiral molecules.

###### pacs:

42.50.Nn, 42.50.Pq, 42.25.Ja

Conventional isotropic (e.g., metallic) mirrors reverse the handedness (or
helicity) of circularly polarized light by turning right-circularly polarized
(RCP) light into left-circularly polarized (LCP) light and vice versa Coles
and Andrews (2012); Barnett _et al._ (2012). This makes it impossible to
realize helicity-preserving (HP) cavities or even chiral cavities (i.e.,
cavities only supporting light modes of a certain handedness), with
conventional mirrors Plum and Zheludev (2015); Aiello (2022). There is however
a great current scientific and technological interest in the design of HP
mirrors and resonators Hentschel _et al._ (2017); Semnani _et al._ (2020);
Feis _et al._ (2020); Voronin _et al._ (2022), in particular for the
enhancement of so-called dichroic effects. Dichroism refers to the (typically
weak) differential absorption of circularly polarized light by chiral
scatterers such as molecular enantiomers Tang and Cohen (2010). Enhancing
dichroic effects with optical resonators can result in better sensitivities
for the discrimination of molecular enantiomers Scott _et al._ (2020);
Mohammadi _et al._ (2018); Genet (2022); Mauro _et al._ (2023), a desired task
for biochemical applications. In the strong light-matter coupling regime,
chiral cavities have furthermore been proposed to create novel light-dressed
states of matter by breaking the time-reversal symmetry in materials, leading
to the emerging field of chiral polaritonics Hübener _et al._ (2021); Schäfer
and Baranov (2023).
In this work we show that HP cavities can be implemented with quantum
metasurfaces employed as mirrors. These structures have emerged as platforms
for achieving strong and highly directional light-matter interactions and can
most prominently be realized with cold atoms trapped in optical lattices Rui
_et al._ (2020). They can exhibit close to perfect reflection of incoming
light Bettles _et al._ (2016); Shahmoon _et al._ (2017); Rui _et al._ (2020);
Ballantine and Ruostekoski (2020); Alaee _et al._ (2020); Ballantine and
Ruostekoski (2021) and have numerous other applications e.g., as platforms for
topological quantum optics Bettles _et al._ (2017); Perczel _et al._ (2017a,
b), nonlinear quantum optics Bettles _et al._ (2020); Parmee and Ruostekoski
(2021); Moreno-Cardoner _et al._ (2021); Rusconi _et al._ (2021); Srakaew _et
al._ (2023); Pedersen _et al._ (2023) or quantum information processing
Plankensteiner _et al._ (2015); Facchinetti _et al._ (2016); Manzoni _et al._
(2018); Grankin _et al._ (2018); Guimond _et al._ (2019); Bekenstein _et al._
(2020). The main ingredient of our approach is to manipulate the polarization
of the incoming light field via the orientation of the effective two-level
systems that make up the metasurfaces, which can for instance be tuned via an
external magnetic field.
More generally, this work falls within the scope of hybrid cavities, i.e., the
design of optical resonators going beyond the simple textbook picture of a
single electromagnetic mode confined between two non-reactive mirrors.
Instead, strongly dispersive optical elements such as photonic crystals or
plasmonic metasurfaces are used as reflectors Zhou _et al._ (2014); Cernotík
_et al._ (2019); Denning _et al._ (2019); Fitzgerald _et al._ (2021); Bin-Alam
_et al._ (2021) with the aim to surpass the performance of standard cavities,
implying a highly non-Markovian behavior of the cavity as characterized by
non-Lorentzian, typically Fano-type lineshapes Miroshnichenko _et al._ (2010).

Figure 1: Helicity-preserving metasurface cavity. (a) A HP quantum metasurface
cavity of length $\ell\approx n\lambda/2$ can be constructed with composite
mirrors of orthogonal dipole orientation (e.g., along $x$ and $y$), with an
appropriate relative phase separation of $\phi_{m}=k_{l}\ell_{m}=\pi/2+2\pi
n_{m}$. (b), (c) Absolute value squared of RS vectors
$\mathbf{G}_{\pm}(\mathbf{R})$ for a cavity length $\ell=12.505\lambda$,
$\ell_{m}=5\lambda/4$, for square lattices with lattice spacing $a=0.8\lambda$
illuminated by RCP light with input polarization vector
$\mathbf{E}_{\text{in}}^{(+)}=(1,\mathrm{i},0)^{\top}/\sqrt{2}$ close to the
cavity resonance. The plots are normalized to the input intensity
$I_{\text{in}}=|\mathbf{E}_{\text{in}}^{(+)}|^{2}$. The metasurfaces are
defined with a finite curvature radius, leading to Gaussian confinement of the
mode (beam waist $w_{0}=8\lambda$). (d) Cavity transmission $|t_{c}|^{2}$ as a
function of the laser detuning $\Delta$ for different cavity lengths $\ell$.
(e) Chiral sensing with HP metasurface cavities: Ideal chiral scatterers
passing through a cavity with the same handedness yield a phase shift in the
cavity transmission output which can for instance be measured by homodyne
detection (LO: local oscillator) and leads to a phase detection sequence as
schematically illustrated in the diagram.

HP mirror – Let us present an implementation procedure for a HP mirror using a
stacked system of quantum metasurfaces. We start by introducing the formalism
for a single metasurface Baßler _et al._ (2023). To this end, we consider a 2D
quasi-infinite quantum emitter array where the emitters are situated in the
$xy$ plane at positions $\mathbf{r}_{j}$. For simplicity, one may imagine a
square lattice, however most of the results derived in the following are
equally valid for other Bravais lattices and can furthermore also be extended
to non-Bravais lattices Perczel _et al._ (2017a); Baßler _et al._ (2023). The
layer is comprised of $\cal N$ emitters with internal electronic structure
described by a $J=0\to J=1$ transition at transition frequency $\omega_{0}$.
In the following, we will work in the Cartesian polarization basis. The
transition dipole operator for each emitter can be written as
$\mathbf{d}=\sum_{\nu}\mathbf{d}_{\nu}\sigma_{\nu}+\mathrm{h.c.}$ with
$\mathbf{d}_{\nu}=\bra{g}\mathbf{d}\ket{\nu}$ and
$\sigma_{\nu}=\ket{g}\bra{\nu}$ is the corresponding lowering operator for
each electronic transition ($\nu=x,y,z$). In addition, we consider a laser
drive entering from the left in the form of a plane wave with positive-
frequency amplitude
$\mathbf{E}_{\text{in}}^{(+)}=(E_{\text{in},x},E_{\text{in},y},0)^{\top}$ and
laser frequency $\omega_{l}=2\pi c/\lambda=ck_{l}$ where $\lambda$ and $k_{l}$
are the laser wavelength and wavenumber, respectively.

In a frame rotating at the laser frequency, the Hamiltonian describing the
dynamics of the emitter array is given by the sum of the free evolution and
the dipole-dipole interaction ($\hbar=1$)

$\displaystyle\mathcal{H}_{0}+\mathcal{H}_{\text{d-d}}\\!=-\Delta\sum_{j,\nu}\sigma_{j,\nu}^{\dagger}\sigma_{j,\nu}^{\phantom{\dagger}}\\!+\\!\sum_{j,j^{\prime},\nu,\nu^{\prime}}\Omega_{jj^{\prime}}^{\nu\nu^{\prime}}\sigma_{j,\nu}^{\dagger}\sigma_{j^{\prime},\nu^{\prime}}^{\phantom{\dagger}},$
(1)

with the laser detuning $\Delta=\omega_{l}-\omega_{0}$ and $\sigma_{j,\nu}$ is
the lowering operator for the $\nu$-transition within a particular emitter
$j$. Assuming normally incident illumination, the laser drive adds as
$\mathcal{H}_{l}=\sum_{j,\nu}(\eta_{\nu}^{\phantom{\dagger}}\sigma_{j,\nu}^{\dagger}+\mathrm{h.c.})$
with Rabi frequencies
$\eta_{\nu}^{\phantom{i}}=d_{\nu}^{\phantom{i}}E^{(+)}_{\mathrm{in},{\nu}}$
and $\eta_{z}=0$. In addition to the coherent processes, the collective loss
of excitations due to spontaneous emission is described by the Lindblad term

$\displaystyle\mathcal{L}[\rho]\\!=\\!\sum_{j,j^{\prime},\nu,\nu^{\prime}}\Gamma_{jj^{\prime}}^{\nu\nu^{\prime}}\left[\sigma_{j,\nu}^{\phantom{\dagger}}\rho\sigma_{j^{\prime},\nu^{\prime}}^{\dagger}\\!-\\!\frac{1}{2}\left\\{\sigma_{j,\nu}^{\dagger}\sigma^{\phantom{\dagger}}_{j^{\prime},\nu^{\prime}},\rho\right\\}\right],$
(2)

where the last term denotes an anticommutator and the diagonal elements
describe the independent spontaneous emission of the emitters
$\Gamma_{jj}^{\nu\nu^{\prime}}=\Gamma_{0}\delta_{\nu\nu^{\prime}}$ with
$\Gamma_{0}=\omega_{0}^{3}d^{2}/(3\pi\epsilon_{0}c^{3})$ (we assume the dipole
moments to be identical in the following $d_{\nu}\equiv d$). The rates
$\Omega_{jj^{\prime}}^{\nu\nu^{\prime}}$,
$\Gamma_{jj^{\prime}}^{\nu\nu^{\prime}}$ describe coherent/incoherent
scattering of photons between emitters $j$ and $j^{\prime}$ and between
transitions $\nu$ and $\nu^{\prime}$ and can be derived as real and imaginary
parts of the photonic Green’s tensor (see App. A) as

$\displaystyle\Omega_{jj^{\prime}}^{\nu\nu^{\prime}}-\mathrm{i}\frac{\Gamma_{jj^{\prime}}^{\nu\nu^{\prime}}}{2}=-\mu_{0}\omega_{0}^{2}\,\mathbf{d}_{\nu}^{*}\cdot\mathbf{G}(\mathbf{r}_{jj^{\prime}})\cdot\mathbf{d}_{\nu^{\prime}}^{\phantom{*}},$
(3)

expressed in terms of the vacuum permeability $\mu_{0}$ and depending on the
interparticle separation
$\mathbf{r}_{jj^{\prime}}=\mathbf{r}_{j^{\prime}}-\mathbf{r}_{j}$. The Green’s
tensor is defined such that the real part of the self-interaction at
$j=j^{\prime}$ vanishes. From the steady-state solution of the quantum master
equation
$\dot{\rho}=\mathrm{i}[\rho,\mathcal{H}_{0}\\!+\\!\mathcal{H}_{\text{d-d}}\\!+\\!\mathcal{H}_{l}]\\!+\\!\mathcal{L}[\rho]$,
the dipole amplitudes and thereby the transmitted and reflected fields can be
computed (see App. A). The transmission matrix of the metasurface connecting
the polarization components of the input field to the outgoing field expresses
as Baßler _et al._ (2023)

$\displaystyle\bm{\mathcal{T}}_{m}=\mathds{1}+\mathrm{i}\frac{\widetilde{\Gamma}(0)}{2d^{2}}\bm{\alpha}_{\text{red}},$
(4)

where $\widetilde{\Gamma}(0)$ is the effective decay rate at zero quasi-
momentum and $\bm{\alpha}_{\text{red}}$ is the 2D polarizability tensor of the
metasurface, relating the induced dipole moment to the incoming electric
field. However, in the limit of large external magnetic fields
$\mu\absolutevalue{\mathbf{B}}\gg\widetilde{\Gamma}(0)$ (magnetic moment
$\mu$), all dipole transitions orthogonal to the magnetic field direction
become very off-resonant and one may focus on the polarization component in
the direction of the magnetic field, thereby reducing the description to an
effective two-level model Baßler _et al._ (2023). The transmission amplitude
of the metasurface for a single component $E_{\text{in},\nu}$ is then simply
given by Bettles _et al._ (2016); Shahmoon _et al._ (2017)

$\displaystyle
t_{m}=1+\frac{\mathrm{i}\widetilde{\Gamma}(0)/2}{\widetilde{\Omega}(0)-\Delta-\mathrm{i}\widetilde{\Gamma}(0)/2},$
(5)

where $\widetilde{\Omega}(0)=\sum_{j}\Omega_{0j}^{\nu\nu}$,
$\widetilde{\Gamma}(0)=\sum_{j}\Gamma_{0j}^{\nu\nu}$ describe the dipole-
induced collective frequency shift and decay rate arising from the $\nu$
transition dipoles (for an arbitrary index $0$ on the array). The complex
transmission and reflection amplitudes are connected as $t_{m}=1+r_{m}$ while
$|t_{m}|^{2}+|r_{m}|^{2}=1$. Most notably, if the laser frequency matches the
collective metasurface resonance
$\omega_{l}=\omega_{\nu}+\widetilde{\Omega}(0)$, perfect reflection of
incoming light is obtained as $|r_{m}|^{2}=1$. In the following, for the sake
of clarity, we proceed with the simplified two-level description. Finally, to
obtain a HP mirror, we consider now two copies of quantum metasurfaces
separated by a distance $\ell_{m}$, one with dipoles pointing in $x$-direction
and one with dipoles pointing in $y$-direction with a path length difference
of $k_{l}\ell_{m}=\phi_{m}=2\pi n_{m}+\pi/2$ ($n_{m}\in\mathbb{N}_{0}$)
between the two polarizations. The combination of these two mirrors is a
helicity-perserving mirror. The path length difference rotates the
$y$-polarization components by $\pi$, thereby reversing the mirror operation
which does not conserve helicity for an ordinary mirror. A full transfer
matrix calculation showing this can be found in App. E.

HP cavity – A HP optical cavity can now be simply implemented by two HP
metasurface mirrors separated by a distance $\ell$ (see Fig. 1(a)). The two
layers making up the mirror consist of dipoles with perpendicular dipole
orientations, leading to vanishing interactions between the two cavities in
the far field. For simplicity, we thus continue the discussion for a single
cavity while keeping in mind that the actual setup consists of two
noninteracting copies. A full discussion for both polarization components can
be found in App. E. Solving the coupled-dipole equations and neglecting the
contributions from all evanescent terms, a simple expression for the total
transmitted field can be obtained as
$E^{(+)}(z>\ell)=t_{c}E^{(+)}_{\text{in}}\mathrm{e}^{\mathrm{i}k_{l}z}$ with
the cavity transmission coefficient (assuming $k_{0}\approx k_{l}$, for
derivation see App. C)

$\displaystyle
t_{c}=\frac{\left(\Delta-\widetilde{\Omega}(0)\right)^{2}}{\left(\Delta-\widetilde{\Omega}(0)+\mathrm{i}\frac{\widetilde{\Gamma}(0)}{2}\right)^{2}+\frac{\widetilde{\Gamma}(0)^{2}}{4}\mathrm{e}^{2\mathrm{i}k_{l}\ell}}.$
(6)

We remark that instead of solving the coupled-dipole equations for the two
arrays, the same result can be obtained from classical transfer matrix theory
Cernotík _et al._ (2019); Reitz _et al._ (2022); Pedersen _et al._ (2023)
where the transfer matrix of a single metasurface can be expressed in terms of
the mirror polarizability
$\zeta_{m}=-\mathrm{i}r_{m}/t_{m}=\widetilde{\Gamma}(0)/[2(\widetilde{\Omega}(0)-\Delta)]$
as

$\displaystyle\mathbf{T}_{m}=\begin{pmatrix}1+\mathrm{i}\zeta_{m}&\mathrm{i}\zeta_{m}\\\
-\mathrm{i}\zeta_{m}&1-\mathrm{i}\zeta_{m}\end{pmatrix}.$ (7)

The total transfer matrix is then simply obtained as
$\mathbf{T}=\mathbf{T}_{m}\mathbf{T}_{f}\mathbf{T}_{m}$ with the free space
propagation matrix
$\mathbf{T}_{f}=\mathrm{diag}(\mathrm{e}^{\mathrm{i}k_{l}\ell},\mathrm{e}^{-\mathrm{i}k_{l}\ell})$.
The condition that the transmission ought to equal unity at the cavity
resonance $|t_{c}|^{2}=1$, yields the following expression for the cavity
resonance

$\displaystyle\Delta-\widetilde{\Omega}(0)=-\frac{\widetilde{\Gamma}(0)}{2}\tan(k_{l}\ell).$
(8)

To demonstrate that the resulting cavity consisting of two HP mirrors indeed
conserves the helicity, we compute the Riemann-Silberstein (RS) vectors
Bialynicki-Birula and Bialynicka-Birula (2013)

$\mathbf{G}_{\pm}(\mathbf{R})=\frac{1}{\sqrt{2}}\left(\mathbf{E}(\mathbf{R})\pm\mathrm{i}\mathcal{Z}\mathbf{H}(\mathbf{R})\right),$
(9)

which describe the combined electromagnetic field of chiral polarization and
$\mathcal{Z}=(\epsilon_{0}c)^{-1}$ is the vacuum impedance. The absolute value
of these quantities is plotted in Figs. 1(b), (c), for RCP light entering the
cavity, confirming that the cavity preserves the helicity while also showing a
strong field enhancement. The difference in absolute value between the RS
vectors can be seen as a measure for the chirality density inside the cavity.
The cavity itself is however not chiral as any elliptical input polarization
is supported. The magnetic field $\mathbf{H}(\mathbf{R})$ is determined via
Maxwell’s equations from the excitations of the electric dipoles on the
metasurface as detailed in App. A.

The transmission profile around the cavity resonance is illustrated in Fig.
1(d) for different cavity lengths for a lattice spacing of $a=0.8\lambda$
where the collective dipole shift is close to zero, i.e.,
$\widetilde{\Omega}(0)\approx 0$. If the cavity length $\ell$ exactly matches
$n\lambda/2$ ($n\in\mathbb{N}$), the cavity resonance coincides with the
resonance of the individual arrays and no transmission is obtained as all the
light is reflected. If $\ell$ becomes slightly larger, a narrow transmission
window opens up as the mirrors and the cavity now possess different resonance
frequencies. Further increasing the cavity length leads to a strongly
asymmetric Fano-type profile with a larger linewidth and the cavity resonance
drifting towards infinity for $\ell\to(n+\frac{1}{2})\lambda/2$. Once the next
multiple of $\lambda/2$ is approached, the cavity linewidth becomes narrow
again and the cavity resonance shifts towards
$\omega_{\nu}+\widetilde{\Omega}(0)$ as $\tan(k_{l}\ell)\to 0$. The distance
between the zero and the maximum of the transmission can be used as a measure
for the cavity linewidth $\kappa=\widetilde{\Gamma}(0)|\tan(k_{l}\ell)|$. We
present a coupled-modes theory for the input-output description of cavities
made from quantum metasurface mirrors in App. G.

Figure 2: Chiral sensing. (a) Cavity phase $\varphi_{c}=\mathrm{arg}(t_{c})$
versus small perturbations of the cavity length $\delta\ell_{s}$ for different
cavity lengths $\ell$ (square lattice, $a=0.8\lambda$). For each $\ell$, the
resonance condition described by Eq. (8) is fulfilled. (b) Illustration of
phase uncertainty in phase space for a coherent state. From a geometric point
of view, it becomes clear that the phase uncertainty becomes smaller when the
amplitude of the field is larger. (c) Homodyne detection signal
$\expectationvalue{m_{-}(t)}$ for (ideal) right- and left-handed chiral
scatterers (RHS/LHS) passing through an RCP cavity with corresponding shot
noise shown in light blue and red, respectively. The dashed black line
indicates the entry of a particle into the cavity and the dotted red line
indicates its exit. In addition, a rotational average was performed as
detailed in App. H. We have used $\Delta_{s}=10\Gamma_{0}$,
$\gamma_{s}=\Gamma_{0}$ with an average of one photon within the time interval
$\Gamma_{0}^{-1}$, i.e., $F=\Gamma_{0}$, and an integration time of
$T\Gamma_{0}=2000$. The linewidth for the metasurfaces was assumed to be the
same as for the infinite square lattice at $a=0.8\lambda$. The detuning of the
incoming laser was chosen to be $\Delta=\Gamma_{0}/100$.

Chiral sensing – We now consider the scenario depicted in Fig. 1(e) where
chiral scatterers with radiative linewidth $\gamma_{s}$ are sent through the
cavity. We assume the resonance of the scatterer $\omega_{s}$ to be far-
detuned from the cavity resonance
$\absolutevalue{\Delta_{s}}\gg\absolutevalue{\Delta_{c}},\gamma_{s}$ with
$\Delta_{s/c}=\omega_{l}-\omega_{s/c}$. In this case, the effect of a
scatterer with the same helicity as the cavity is to increase the path length
of light passing through the cavity and thereby effectively shift the cavity
length by a small amount
$\delta\ell_{s}=-\arctan(\gamma_{s}/2\Delta_{s})/k_{l}$ (for derivation see
App. F), such that the total cavity length is now given by
$\ell+\delta\ell_{s}$. On the other hand, a scatterer with the opposite
helicity as the cavity mode does not cause a shift (assuming ideal chiral
scatterers, in reality both helicities will lead to differential shifts). Due
to the quick phase switch around the cavity resonance for cavity lengths close
to $n\lambda/2$, a small perturbation of the cavity length can lead to a
considerable phase shift of the cavity transmission (see Fig. 2(a)). This is
the central idea of the sensing scheme discussed in the following.

One can then proceed to compute the relative phase change in the cavity
transmission between lengths $\ell$ and $\ell+\delta\ell_{s}$ on the cavity
resonance (assuming $\ell\approx n\lambda/2$)

$\varphi=\arg\frac{t_{c}(\ell+\delta\ell_{s})}{t_{c}(\ell)}\Bigr{|}_{\text{res},\ell\approx
n\frac{\lambda}{2}}\approx\arctan(\frac{1}{\tan(k_{l}\delta\ell_{s})}),$ (10)

which reaches a value of $\pi[\theta(k_{l}\delta\ell_{s})-1/2]$ as
$\delta\ell_{s}\to 0$, implying a phase jump from $-\pi/2$ to $\pi/2$ around
the cavity resonance for lengths close to $n\lambda/2$ ($\theta(x)$ is the
Heaviside function). If the cavity length departs from $n\lambda/2$, the
cavity linewidth increases and the phase switch gets diminished, as
illustrated in Figs. 1(d) and 2(a).

An experimental setup to measure this phase is homodyne detection as
illustrated in Fig. 1(e) where the phase between a local oscillator (for
instance obtained from beam splitting the input field) is compared to the
phase of the output field. Suppose we consider a signal beam with which we
drive the cavity
$\alpha(t)=\sqrt{F}\exp(-\mathrm{i}\omega_{0}t+\mathrm{i}\theta)$ and the
local oscillator field
$\alpha_{L}(t)=\sqrt{F_{\mathrm{LO}}}\exp(-\mathrm{i}\omega_{0}t+\mathrm{i}\theta_{\mathrm{LO}})$
with intensities (number of photons per unit of time) $F$ and
$F_{\mathrm{LO}}$. Then considering homodyne detection for a Fabry-Pérot
cavity leads to an uncertainty in the measured phase for an integration time
$T$ of the measurement and a quantum efficiency $\eta_{Q}$ which is encoded in
the intensity difference $m_{-}$ with variance $(\Delta
m_{-})^{2}_{\text{res}}=\eta_{Q}TF_{\mathrm{LO}}$ and expectation value
(assuming the phase variation to happen on a timescale much slower than the
optical frequency)

$\left\langle
m_{-}(t,T)\right\rangle=2\eta_{Q}\sqrt{FF_{\mathrm{LO}}}\absolutevalue{t_{c}}\int_{t}^{t+T}\mathrm{d}{t^{\prime}}\sin\left(\varphi(t^{\prime})\right).$
(11)

Here, we have approximated that $F_{\mathrm{LO}}\gg F$ and have taken
$\theta+\theta_{\mathrm{LO}}=2\pi n$ which can be obtained by phase matching
the local oscillator and the signal beam. Since the cavity is a linear
element, the resulting phase uncertainty (see sketch in Fig. 2(b)) is
independent of any cavity properties and only depends on the properties of the
state of the incoming beam, which is assumed to be classical for this
calculation. This uncertainty could however be improved upon by choosing a
phase-squeezed input field instead of a coherent one.

As an alternative to describing the passage of particles through the cavity
with transfer matrix theory, the dipole theory can be extended to include the
presence of an additional chiral scatterer which can be represented by coupled
electric and magnetic dipoles (see App. H). These equations of motion are
simulated in Fig. 2(c) for right- and left-handed scatterers (RHS/LHS)
entering an RCP cavity, showing a clear distinction in the resulting signal.
Here, the classical shot noise of a coherent input field is used to estimate
the phase error for a homodyne detection. One can also see several aspects of
the cavity physics in this plot. First, the relaxation to the steady state
occurs very slowly owing to the very small decay rate of the hybrid cavity. We
can also observe that with sufficient detector integration the shot noise can
be overcome in order to detect a single scatterer.

Let us finally briefly discuss the applicability of the chiral sensing scheme
to the discrimination of molecular enantiomers. Chiral molecules are in
general not perfect chiral scatterers. This is manifested in the fact that
their circular dichroism $\text{CD}=(A_{+}-A_{-})/(A_{+}+A_{-})$, i.e., the
difference in absorbance between RCP and LCP light, is not unity but some
small finite value. This can have several reasons, but the most physical one
is that the magnetic dipole linewidth of an optical transition is usually much
weaker than the electric dipole linewidth. This implies that the assumption of
coupling to only a single polarization component is unrealistic. Instead, both
enantiomers will lead to a small differential shift in optical path length.
Aside from the absolute magnitude of this change in path length however, the
presented strategy retains generality and the applicabilty is a question of
detailed system parameters. More so, we claim that we have mapped the problem
of chirality sensing of entantiomers onto a controllable cavity optomechanical
setup.

Conclusions and Outlook \- We have shown that HP mirrors and cavities can be
created from stacked quantum metasurfaces with orthogonal dipole orientation.
We remark that our proposal could be analogously implemented with arrays of
classical dipoles such as plasmonic lattices in which case no external
magnetic field control would be needed as the polarization can be controlled
by the geometry of the individual plasmonic elements Choudhary _et al._
(2023). We then proposed to use these narrow-linewidth HP cavity modes for the
optical sensing of chiral scatterers by discussing how the phase of the output
field is modified by an off-resonant scatterer passing through the cavity. We
furthermore discussed the phase uncertainty for homodyne detection which can
be minimized by tuning the input intensity and the integration time of the
detector. We also discussed briefly the applicability to the discrimination of
molecular enantiomers. Future endeavors will see the extension of our
formalism to additionally describe possible derogating effects such as motion,
vacancies and nonlinearities of the quantum emitter array.
We remark that, in addition to the use proposed in this work, layered
metasurfaces can enable a host of other applications. For instance, stacking
many of these layers leads to Bragg-mirror physics which could be used to
tailor the frequency windows of optical elements based on quantum
metasurfaces. Also, tilting the metasurfaces with respect to each other gives
rise to moiré superlattices which are known to exhibit exotic optoelectronic
phenomena in solid-state platforms Sunku _et al._ (2018). Preliminary
calculations show however that, for normally incident light, the twisting
angle between the layers does not matter. Even more general polarization
structures of the cavity mode, such as Faraday cavities Hübener _et al._
(2021), might also be implementable.

Acknowledgments – We acknowledge fruitful discussions with L. Mauro and J.
Fregoni which led to the initial idea for this project. This work was
supported by the Max Planck Society and the Deutsche Forschungsgemeinschaft
(DFG, German Research Foundation) – Project-ID 429529648 – TRR 306 QuCoLiMa
(“Quantum Cooperativity of Light and Matter”).

## References

  * Coles and Andrews (2012) M. M. Coles and D. L. Andrews, “Chirality and angular momentum in optical radiation,” Phys. Rev. A 85, 063810 (2012).
  * Barnett _et al._ (2012) S. M. Barnett, R. P. Cameron, and A. M. Yao, “Duplex symmetry and its relation to the conservation of optical helicity,” Phys. Rev. A 86, 013845 (2012).
  * Plum and Zheludev (2015) E. Plum and N. I. Zheludev, “Chiral mirrors,” Applied Physics Letters 106 (2015).
  * Aiello (2022) A. Aiello, “Helicity, chirality, and spin of optical fields without vector potentials,” Phys. Rev. A 106, 043519 (2022).
  * Hentschel _et al._ (2017) M. Hentschel, M. Schäferling, X. Duan, H. Giessen, and N. Liu, “Chiral plasmonics,” Science Advances 3, e1602735 (2017).
  * Semnani _et al._ (2020) B. Semnani, J. Flannery, R. Al Maruf, and M. Bajcsy, “Spin-preserving chiral photonic crystal mirror,” Light: Science & Applications 9, 23 (2020).
  * Feis _et al._ (2020) J. Feis, D. Beutel, J. Köpfler, X. Garcia-Santiago, C. Rockstuhl, M. Wegener, and I. Fernandez-Corbaton, “Helicity-preserving optical cavity modes for enhanced sensing of chiral molecules,” Phys. Rev. Lett. 124, 033201 (2020).
  * Voronin _et al._ (2022) K. Voronin, A. S. Taradin, M. V. Gorkunov, and D. G. Baranov, “Single-handedness chiral optical cavities,” ACS Photonics 9, 2652–2659 (2022).
  * Tang and Cohen (2010) Y. Tang and A. E. Cohen, “Optical chirality and its interaction with matter,” Phys. Rev. Lett. 104, 163901 (2010).
  * Scott _et al._ (2020) P. Scott, X. Garcia-Santiago, D. Beutel, C. Rockstuhl, M. Wegener, and I. Fernandez-Corbaton, “On enhanced sensing of chiral molecules in optical cavities,” Applied Physics Reviews 7, 041413 (2020).
  * Mohammadi _et al._ (2018) E. Mohammadi, K. L. Tsakmakidis, A. N. Askarpour, P. Dehkhoda, A. Tavakoli, and H. Altug, “Nanophotonic platforms for enhanced chiral sensing,” ACS Photonics 5, 2669–2675 (2018).
  * Genet (2022) C. Genet, “Chiral light–chiral matter interactions: an optical force perspective,” ACS Photonics 9, 319–332 (2022).
  * Mauro _et al._ (2023) L. Mauro, J. Fregoni, J. Feist, and R. Avriller, “Chiral discrimination in helicity-preserving Fabry-Pérot cavities,” Phys. Rev. A 107, L021501 (2023).
  * Hübener _et al._ (2021) H. Hübener, U. De Giovannini, C. Schäfer, J. Andberger, M. Ruggenthaler, J. Faist, and A. Rubio, “Engineering quantum materials with chiral optical cavities,” Nature Materials 20, 438–442 (2021).
  * Schäfer and Baranov (2023) C. Schäfer and D. G. Baranov, “Chiral polaritonics: Analytical solutions, intuition, and use,” J. Phys. Chem. Lett. 14, 3777–3784 (2023).
  * Rui _et al._ (2020) J. Rui, D. Wei, A. Rubio-Abadal, S. Hollerith, J. Zeiher, D. M. Stamper-Kurn, C. Gross, and I. Bloch, “A subradiant optical mirror formed by a single structured atomic layer,” Nature 583, 369 (2020).
  * Bettles _et al._ (2016) R. J. Bettles, S. A. Gardiner, and C. S. Adams, “Enhanced optical cross section via collective coupling of atomic dipoles in a 2D array,” Phys. Rev. Lett. 116, 103602 (2016).
  * Shahmoon _et al._ (2017) E. Shahmoon, D. S. Wild, M. D. Lukin, and S. F. Yelin, “Cooperative resonances in light scattering from two-dimensional atomic arrays,” Phys. Rev. Lett. 118, 113601 (2017).
  * Ballantine and Ruostekoski (2020) K. E. Ballantine and J. Ruostekoski, “Optical magnetism and Huygens’ surfaces in arrays of atoms induced by cooperative responses,” Phys. Rev. Lett. 125, 143604 (2020).
  * Alaee _et al._ (2020) R. Alaee, B. Gurlek, M. Albooyeh, D. Martín-Cano, and V. Sandoghdar, “Quantum metamaterials with magnetic response at optical frequencies,” Phys. Rev. Lett. 125, 063601 (2020).
  * Ballantine and Ruostekoski (2021) K. E. Ballantine and J. Ruostekoski, “Cooperative optical wavefront engineering with atomic arrays,” Nanophotonics 10, 1901–1909 (2021).
  * Bettles _et al._ (2017) R. J. Bettles, J. Minář, C. S. Adams, I. Lesanovsky, and B. Olmos, “Topological properties of a dense atomic lattice gas,” Phys. Rev. A 96, 041603 (2017).
  * Perczel _et al._ (2017a) J. Perczel, J. Borregaard, D. E. Chang, H. Pichler, S. F. Yelin, P. Zoller, and M. D. Lukin, “Photonic band structure of two-dimensional atomic lattices,” Phys. Rev. A 96, 063801 (2017a).
  * Perczel _et al._ (2017b) J. Perczel, J. Borregaard, D. E. Chang, H. Pichler, S. F. Yelin, P. Zoller, and M. D. Lukin, “Topological quantum optics in two-dimensional atomic arrays,” Phys. Rev. Lett. 119, 023603 (2017b).
  * Bettles _et al._ (2020) R. J. Bettles, M. D. Lee, S. A. Gardiner, and J. Ruostekoski, “Quantum and nonlinear effects in light transmitted through planar atomic arrays,” Comm. Phys. 3, 141 (2020).
  * Parmee and Ruostekoski (2021) C. D. Parmee and J. Ruostekoski, “Bistable optical transmission through arrays of atoms in free space,” Phys. Rev. A 103, 033706 (2021).
  * Moreno-Cardoner _et al._ (2021) M. Moreno-Cardoner, D. Goncalves, and D. E. Chang, “Quantum nonlinear optics based on two-dimensional Rydberg atom arrays,” Phys. Rev. Lett. 127, 263602 (2021).
  * Rusconi _et al._ (2021) C. C. Rusconi, T. Shi, and J. I. Cirac, “Exploiting the photonic nonlinearity of free-space subwavelength arrays of atoms,” Phys. Rev. A 104, 033718 (2021).
  * Srakaew _et al._ (2023) K. Srakaew, P. Weckesser, S. Hollerith, D. Wei, D. Adler, I. Bloch, and J. Zeiher, “A subwavelength atomic array switched by a single Rydberg atom,” Nature Physics (2023).
  * Pedersen _et al._ (2023) S. P. Pedersen, L. Zhang, and T. Pohl, “Quantum nonlinear metasurfaces from dual arrays of ultracold atoms,” Phys. Rev. Res. 5, L012047 (2023).
  * Plankensteiner _et al._ (2015) D. Plankensteiner, L. Ostermann, H. Ritsch, and C. Genes, “Selective protected state preparation of coupled dissipative quantum emitters,” Sci. Rep. 5, 16231 (2015).
  * Facchinetti _et al._ (2016) G. Facchinetti, S. D. Jenkins, and J. Ruostekoski, “Storing light with subradiant correlations in arrays of atoms,” Phys. Rev. Lett. 117, 243601 (2016).
  * Manzoni _et al._ (2018) M. T. Manzoni, M. Moreno-Cardoner, A. Asenjo-Garcia, J. V. Porto, A. V. Gorshkov, and D. E. Chang, “Optimization of photon storage fidelity in ordered atomic arrays,” New J. Phys. 20, 083048 (2018).
  * Grankin _et al._ (2018) A. Grankin, P. O. Guimond, D. V. Vasilyev, B. Vermersch, and P. Zoller, “Free-space photonic quantum link and chiral quantum optics,” Phys. Rev. A 98, 043825 (2018).
  * Guimond _et al._ (2019) P.-O. Guimond, A. Grankin, D. V. Vasilyev, B. Vermersch, and P. Zoller, “Subradiant Bell states in distant atomic arrays,” Phys. Rev. Lett. 122, 093601 (2019).
  * Bekenstein _et al._ (2020) R. Bekenstein, I. Pikovski, H. Pichler, E. Shahmoon, S. F. Yelin, and M. D. Lukin, “Quantum metasurfaces with atom arrays,” Nature Physics 16, 676–681 (2020).
  * Zhou _et al._ (2014) W. Zhou, D. Zhao, Y.-C. Shuai, H. Yang, S. Chuwongin, A. Chadha, J.-H. Seo, K. X. Wang, V. Liu, Z. Ma, and S. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quantum. Electron. 38, 1–74 (2014).
  * Cernotík _et al._ (2019) O. Cernotík, A. Dantan, and C. Genes, “Cavity quantum electrodynamics with frequency-dependent reflectors,” Phys. Rev. Lett. 122, 243601 (2019).
  * Denning _et al._ (2019) E. V. Denning, J. Iles-Smith, and J. Mork, “Quantum light-matter interaction and controlled phonon scattering in a photonic Fano cavity,” Phys. Rev. B 100, 214306 (2019).
  * Fitzgerald _et al._ (2021) J. M. Fitzgerald, S. K. Manjeshwar, W. Wieczorek, and P. Tassin, “Cavity optomechanics with photonic bound states in the continuum,” Phys. Rev. Res. 3, 013131 (2021).
  * Bin-Alam _et al._ (2021) M. S. Bin-Alam, O. Reshef, Y. Mamchur, M. Z. Alam, G. Carlow, J. Upham, B. T. Sullivan, J.-M. Ménard, M. J. Huttunen, R. W. Boyd, and K. Dolgaleva, “Ultra-high-Q resonances in plasmonic metasurfaces,” Nat. Commun. 12, 974 (2021).
  * Miroshnichenko _et al._ (2010) A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, “Fano resonances in nanoscale structures,” Rev. Mod. Phys. 82, 2257 (2010).
  * Baßler _et al._ (2023) N. S. Baßler, M. Reitz, K. P. Schmidt, and C. Genes, “Linear optical elements based on cooperative subwavelength emitter arrays,” Opt. Express 31, 6003–6026 (2023).
  * Reitz _et al._ (2022) M. Reitz, C. Sommer, and C. Genes, “Cooperative quantum phenomena in light-matter platforms,” PRX Quantum 3, 010201 (2022).
  * Bialynicki-Birula and Bialynicka-Birula (2013) I. Bialynicki-Birula and Z. Bialynicka-Birula, “The role of the Riemann–Silberstein vector in classical and quantum theories of electromagnetism,” J. Phys. A Math. 46, 053001 (2013).
  * Choudhary _et al._ (2023) S. Choudhary, R. W. Boyd, and J. E. Sipe, “Dark and bright modes, and their coherent control in dipolar metasurface bilayers,” Phys. Rev. A 107, 023521 (2023).
  * Sunku _et al._ (2018) S. S. Sunku, G. X. Ni, B. Y. Jiang, H. Yoo, A. Sternbach, A. S. McLeod, T. Stauber, L. Xiong, T. Taniguchi, K. Watanabe, P. Kim, M. M. Fogler, and D. N. Basov, “Photonic crystals for nano-light in moire graphene superlattices,” Science 362, 1153–1156 (2018).
  * Novotny and Hecht (2006) L. Novotny and B. Hecht, _Principles of Nano-Optics_ (Cambridge University Press, 2006).
  * Viviescas and Hackenbroich (2003) C. Viviescas and G. Hackenbroich, “Field quantization for open optical cavities,” Phys. Rev. A 67, 013805 (2003).
  * Craig and Thirunamachandran (1998) D. P. Craig and T. Thirunamachandran, _Molecular quantum electrodynamics: an introduction to radiation-molecule interactions_ (Courier Corporation, 1998).
  * Loudon (2000) R. Loudon, _The quantum theory of light_ (OUP Oxford, 2000).

##  Appendix A Dyadic Green’s function

The free-space photonic Green’s tensor is given by (evaluated at the resonance
of the emitters $\omega_{0}=ck_{0}$)

$\displaystyle\mathbf{G}(\mathbf{R})=\left(\mathds{1}+\frac{1}{k_{0}^{2}}\nabla\otimes\nabla\right)\frac{\mathrm{e}^{\mathrm{i}k_{0}R}}{4\pi
R}-\frac{\mathds{1}}{3k_{0}^{2}}\delta(\mathbf{R}),$ (12)

where $\otimes$ denotes the dyadic product, $R=|\mathbf{R}|$ and the last term
removes the divergence on the self-interaction terms at $\mathbf{R}=0$. This
can be expressed more explicitly as

$\displaystyle\mathbf{G}(\mathbf{R})=\frac{\mathrm{e}^{\mathrm{i}k_{0}R}}{4\pi
k_{0}^{2}}\left[\left(\frac{k_{0}^{2}}{R}+\frac{\mathrm{i}k_{0}}{R^{2}}-\frac{1}{R^{3}}\right)\mathds{1}+\left(-\frac{k_{0}^{2}}{R}-\frac{3\mathrm{i}k_{0}}{R^{2}}+\frac{3}{R^{3}}\right)\frac{\mathbf{R}\otimes\mathbf{R}}{R^{2}}\right]-\frac{\mathds{1}}{3k_{0}^{2}}\delta(\mathbf{R}),$
(13)

where $\mathds{1}$ is the $3\times 3$ identity matrix and
$\mathbf{R}=(x,y,z)^{\top}$. For the scattering problem considered here, a
Fourier decomposition of the Green’s tensor with respect to the in-plane wave
vector components $\mathbf{q}=(q_{x},q_{y})^{\top}$ is useful. Decomposing
also $\mathbf{R}=(\mathbf{r}_{\parallel},z)^{\top}$ into lattice plane and
out-of-plane coordinate, one can make use of the Weyl expansion Novotny and
Hecht (2006)

$\displaystyle\frac{\mathrm{e}^{\mathrm{i}k_{0}R}}{R}=\frac{\mathrm{i}}{2\pi}\int\differential\mathbf{q}\,\frac{1}{q_{z}}\mathrm{e}^{\mathrm{i}\mathbf{q}\cdot\mathbf{r}_{\parallel}}\mathrm{e}^{\mathrm{i}q_{z}|z|},$
(14)

with $q_{z}=\sqrt{k_{0}^{2}-q^{2}}$. The Green’s tensor (including the
divergent self-interaction) can now be written as

$\displaystyle\mathbf{G}(\mathbf{R})=\frac{\mathrm{i}}{8\pi^{2}k_{0}^{2}}\int\differential\mathbf{q}\,\frac{k_{0}^{2}\mathds{1}-\bar{\mathbf{q}}\otimes\bar{\mathbf{q}}}{q_{z}}\mathrm{e}^{\mathrm{i}\mathbf{q}\cdot\mathbf{r}_{\parallel}}\mathrm{e}^{\mathrm{i}q_{z}|z|},$
(15)

where
$\bar{\mathbf{q}}=(q_{x},q_{y},\text{sgn}(z)\sqrt{k_{0}^{2}-q^{2}})^{\top}$.

From Maxwell’s equations for monochromatic fields, one can find the magnetic
Green’s tensor (including the self-interaction) as

$\mathbf{G}_{M}(\mathbf{R})=\frac{\mathrm{i}}{\omega_{0}}\curl\mathbf{G}(\mathbf{R})=\frac{\mathrm{i}}{4\pi\omega_{0}}\curl\left(\mathds{1}\frac{\mathrm{e}^{\mathrm{i}k_{0}R}}{R}\right)=\frac{(k_{0}R+\mathrm{i})\mathrm{e}^{\mathrm{i}k_{0}R}}{4\pi\omega_{0}R^{3}}\begin{pmatrix}0&-z&y\\\
z&0&-x\\\ -y&x&0\end{pmatrix}.$ (16)

which can be used to determine the magnetic field emitted from an electric
dipole. Here, the curl of a matrix $\mathbf{A}$ is defined (using Einstein sum
convention) as $(\curl\mathbf{A})_{il}=\varepsilon_{ijk}\partial_{j}A_{kl}$
with $\varepsilon_{ijk}$ the Levi-Civita symbol.

##  Appendix B Fourier space treatment of bilayer system: Intra- and
interlayer interactions

As mentioned in the main text, the HP cavity setup can be decomposed into two
cavities consisting of two metasurfaces each. We will thus restrict our
discussion here to bilayers. The bilayer system forming the cavity can be
treated analytically by a Fourier transform of the dipole operators on both
left ($L$) and right ($R$) sublattice. We define the Fourier transformation
for the Pauli operators (considering a single polarization degree of freedom)
as

$\displaystyle\sigma_{\mathbf{q}}^{L/R}=\sum_{j\in
L/R}\sigma_{j}\mathrm{e}^{-\mathrm{i}\mathbf{q}\cdot\mathbf{r}_{j}},\qquad\sigma_{j}^{L/R}=\frac{1}{\mathcal{N}}\sum_{\mathbf{q}}\sigma_{\mathbf{q}}^{L/R}\mathrm{e}^{\mathrm{i}\mathbf{q}\cdot\mathbf{r}_{j}^{L/R}}.$
(17a)

The effective non-Hermitian Hamiltonian describing the free propagation of a
single excitation on the two independent arrays may then be expressed in
Fourier space as

$\displaystyle\widetilde{\mathcal{H}}_{\text{eff}}=\sum_{\mathbf{q}}\left(-\Delta+\widetilde{\Omega}(\mathbf{q})-\mathrm{i}\frac{\widetilde{\Gamma}(\mathbf{q})}{2}\right)\left(\sigma_{\mathbf{q}}^{R,\dagger}\sigma_{\mathbf{q}}^{R}+\sigma_{\mathbf{q}}^{L,\dagger}\sigma_{\mathbf{q}}^{L}\right),$
(18)

where $\widetilde{\Omega}(\mathbf{q})$, $\widetilde{\Gamma}(\mathbf{q})$
denote the in-plane frequency shifts and decay rates, respectively (definition
see below in Eq. (21)). Importantly however, in addition to Eq. (18), one is
left with a term describing the interlayer interaction between the arrays
which is discussed in more detail below. The laser drive under general oblique
incidence with respect to the array plane is included as

$\displaystyle\mathcal{H}_{l}=\sum_{j\in
L}\left(\eta^{L}\mathrm{e}^{\mathrm{i}\mathbf{k}_{\parallel}\cdot\mathbf{r}_{j}}\sigma_{j}^{\dagger}+\mathrm{h.c.}\right)+\sum_{j\in
R}\left(\eta^{R}\mathrm{e}^{\mathrm{i}\mathbf{k}_{\parallel}\cdot\mathbf{r}_{j}}\sigma_{j}^{\dagger}+\mathrm{h.c.}\right),$
(19)

where $\mathbf{k}_{\parallel}$ is the wave vector of the laser parallel to the
array plane, $k_{z}=\sqrt{k_{l}^{2}-k_{\parallel}^{2}}$ and the Rabi drive for
the two arrays differs by a phase factor due to the free propagation of the
incoming field along the $z$-direction
$\eta^{R}=\eta^{L}\mathrm{e}^{\mathrm{i}k_{z}\ell}$. The linearized equations
of motion for the expectation values of the dipole amplitudes
$\beta_{\mathbf{q}}^{L/R}=\expectationvalue{\sigma_{\mathbf{q}}^{L/R}}$ can be
expressed in the weak excitation limit
$\expectationvalue{\sigma_{j}^{z}}\approx-1$ in Fourier space as

$\displaystyle\derivative{\beta^{L}_{\mathbf{q}}}{t}$
$\displaystyle=\mathrm{i}(\Delta-\mathcal{M}_{\mathbf{q}})\beta^{L}_{\mathbf{q}}-\mathrm{i}\mathcal{M}^{LR}_{\mathbf{q}}\beta^{R}_{\mathbf{q}}-\mathrm{i}\mathcal{N}\delta_{\mathbf{q},\mathbf{k}_{\parallel}}\eta^{L},$
(20a) $\displaystyle\derivative{\beta^{R}_{\mathbf{q}}}{t}$
$\displaystyle=\mathrm{i}(\Delta-\mathcal{M}_{\mathbf{q}})\beta^{R}_{\mathbf{q}}-\mathrm{i}\mathcal{M}^{LR}_{\mathbf{q}}\beta^{L}_{\mathbf{q}}-\mathrm{i}\mathcal{N}\delta_{\mathbf{q},\mathbf{k}_{\parallel}}\eta^{R},$
(20b)

from which we can see that an incident laser only probes the surface modes
corresponding to $\mathbf{q}=\mathbf{k}_{\parallel}$. The term
$\mathcal{M}_{\mathbf{q}}=\widetilde{\Omega}(\mathbf{q})-\mathrm{i}\widetilde{\Gamma}(\mathbf{q})/2$
describing the collective frequency shifts and decay rates of an individual
array and is given by a sum over all in-plane interactions (for an arbitrary
index $0$ on one of the two arrays)

$\displaystyle\mathcal{M}_{\mathbf{q}}=\sum_{j\in\text{in-
plane}}\left(\Omega_{0j}-\mathrm{i}\frac{\Gamma_{0j}}{2}\right)\mathrm{e}^{-\mathrm{i}\mathbf{q}\cdot\mathbf{r}_{j}}=\widetilde{\Omega}(\mathbf{q})-\mathrm{i}\frac{\widetilde{\Gamma}(\mathbf{q})}{2}=-\frac{3}{2}\Gamma_{0}\lambda_{0}\widetilde{G}(\mathbf{q};0),$
(21)

which can be expressed in terms of the Fourier transform of the Green’s tensor
(single polarization component) with respect to its first two arguments
$\widetilde{G}(\mathbf{q};z)=\sum_{j}\mathrm{e}^{-\mathrm{i}\mathbf{q}\cdot\mathbf{r}_{j}}G(\mathbf{R})$.
The term $\mathcal{M}^{LR}_{\mathbf{q}}$, which describes the interaction
between the two arrays, involves a summation over all coherent and dissipative
out-of-plane interactions with the emitters of the opposing lattice

$\displaystyle\mathcal{M}_{\mathbf{q}}^{LR}=\sum_{j\in\text{out-of-
plane}}\left(\Omega_{0j}-\mathrm{i}\frac{\Gamma_{0j}}{2}\right)\mathrm{e}^{-\mathrm{i}\mathbf{q}\cdot\mathbf{r}_{j}}=-\frac{3}{2}\Gamma_{0}\lambda_{0}\widetilde{G}(\mathbf{q};\ell),$
(22)

which simply corresponds to evaluating the 2D lattice transform of the Green’s
tensor at $z=\ell$. In steady state, the equations of motion for the
coherences (20) become

$\displaystyle\left[\Delta-\mathcal{M}_{\mathbf{q}}-\frac{\left(\mathcal{M}_{\mathbf{q}}^{LR}\right)^{2}}{\Delta-\mathcal{M}_{\mathbf{q}}}\right]\beta^{L}_{\mathbf{q}}$
$\displaystyle=\mathcal{N}\delta_{\mathbf{q},\mathbf{k}_{\parallel}}\left(\eta^{L}+\frac{\mathcal{M}_{\mathbf{q}}^{LR}\eta_{\mathbf{q}}^{R}}{\Delta-\mathcal{M}_{\mathbf{q}}}\right),$
(23a)
$\displaystyle\left[\Delta-\mathcal{M}_{\mathbf{q}}-\frac{\left(\mathcal{M}_{\mathbf{q}}^{LR}\right)^{2}}{\Delta-\mathcal{M}_{\mathbf{q}}}\right]\beta^{R}_{\mathbf{q}}$
$\displaystyle=\mathcal{N}\delta_{\mathbf{q},\mathbf{k}_{\parallel}}\left(\eta^{R}+\frac{\mathcal{M}_{\mathbf{q}}^{LR}\eta_{\mathbf{q}}^{L}}{\Delta-\mathcal{M}_{\mathbf{q}}}\right).$
(23b)

By making use of Poisson’s summation formula, the Fourier transform of the
Green’s tensor can be turned into a sum over all vectors $\mathbf{g}$ of the
reciprocal lattice $\Lambda^{*}$ and can be approximated in the far field
$z\gg\lambda$ as

$\displaystyle\widetilde{G}(\mathbf{q};z)=\frac{\mathrm{i}}{2\mathcal{A}}\sum_{\mathbf{g}\in\Lambda^{*}}\frac{\mathrm{e}^{\mathrm{i}\sqrt{k_{0}^{2}-(\mathbf{q}+\mathbf{g})^{2}}|z|}}{\sqrt{k_{0}^{2}-(\mathbf{q}+\mathbf{g})^{2}}}\approx\frac{\mathrm{i}}{2\mathcal{A}}\frac{\mathrm{e}^{\mathrm{i}q_{z}|z|}}{q_{z}},$
(24)

where we have only kept the $\mathbf{g}=0$ contribution and took all other
contributions to be evanescent as it is the case for subwavelength lattices.
The interaction between the arrays is then simply described by the plane wave
term

$\displaystyle\mathcal{M}_{\mathbf{q}}^{LR}=-\mathrm{i}\frac{\widetilde{\Gamma}(\mathbf{q})}{2}\mathrm{e}^{\mathrm{i}q_{z}\ell},$
(25)

with the amplitude of the interaction is governed by the effective in-plane
decay rate
$\widetilde{\Gamma}(\mathbf{q})=3\Gamma_{0}/(4\pi)(\lambda_{0}^{2}/\mathcal{A})(k_{0}/q_{z})$.
This leads to the following expressions for the dipole amplitudes on left and
right layer:

$\displaystyle\beta^{L}_{\mathbf{q}}=\mathcal{N}\delta_{\mathbf{q},\mathbf{k}_{\parallel}}\eta^{L}\left[\frac{\Delta-\mathcal{M}_{\mathbf{q}}-\mathrm{i}\frac{\widetilde{\Gamma}(\mathbf{q})}{2}\mathrm{e}^{2\mathrm{i}q_{z}\ell}}{(\Delta-\mathcal{M}_{\mathbf{q}})^{2}+\frac{\widetilde{\Gamma}(\mathbf{q})^{2}}{4}\mathrm{e}^{2\mathrm{i}q_{z}\ell}}\right],\qquad\beta^{R}_{\mathbf{q}}=\mathcal{N}\delta_{\mathbf{q},\mathbf{k}_{\parallel}}\eta^{L}\left[\frac{\left(\Delta-\mathcal{M}_{\mathbf{q}}-\mathrm{i}\frac{\widetilde{\Gamma}(\mathbf{q})}{2}\right)\mathrm{e}^{\mathrm{i}q_{z}\ell}}{(\Delta-\mathcal{M}_{\mathbf{q}})^{2}+\frac{\widetilde{\Gamma}(\mathbf{q})^{2}}{4}\mathrm{e}^{2\mathrm{i}q_{z}\ell}}\right].$
(26)

##  Appendix C Transmission, intracavity field, and resonance condition

The total electric field can be computed as the sum of the incident field and
the dipole-scattered field
$E^{(+)}(\mathbf{R})=E_{\text{in}}^{(+)}\mathrm{e}^{\mathrm{i}kz}+E_{\text{dip}}^{(+)}(\mathbf{R})$.
The dipole field is given by the sum over all individually emitted fields and
can be expressed as a sum over all wave vectors as (only considering the field
along the $z$-direction)

$E^{(+)}_{\text{dip}}(z)=-\frac{3\pi\Gamma_{0}}{k_{0}d}\sum_{j}G(z\hat{e}_{z}-\mathbf{r}_{j})\beta_{j}=-\frac{3\pi\Gamma_{0}}{k_{0}d\mathcal{N}}\sum_{\mathbf{q}}\left[\widetilde{G}^{L}(\mathbf{q};z)\beta_{\mathbf{q}}^{L}+\widetilde{G}^{R}(\mathbf{q};z)\beta_{\mathbf{q}}^{R}\right],$
(27)

where $\widetilde{G}^{L}$, $\widetilde{G}^{R}$ denote the Fourier transforms
of the Green’s tensor with respect to the left and right sublattice. This
leads to the following expressions for the transmitted and intracavity field
(neglecting all contributions from evanescent terms)

$\displaystyle E^{(+)}(z>\ell)$
$\displaystyle=\left[E_{\text{in}}^{(+)}\mathrm{e}^{\mathrm{i}k_{z}z}-\mathrm{i}\frac{\widetilde{\Gamma}(\mathbf{k}_{\parallel})}{2d}\left(\beta^{L}_{\mathbf{k}_{\parallel}}+\mathrm{e}^{-\mathrm{i}q_{z}\ell}\beta^{R}_{\mathbf{k}_{\parallel}}\right)\mathrm{e}^{\mathrm{i}q_{z}z}\right]\approx
t_{c}E_{\text{in}}^{(+)}\mathrm{e}^{\mathrm{i}k_{z}z},$ (28a) $\displaystyle
E^{(+)}(0<z<\ell)$
$\displaystyle=\left[E_{\text{in}}^{(+)}\mathrm{e}^{\mathrm{i}k_{z}z}-\mathrm{i}\frac{\widetilde{\Gamma}({\mathbf{k}_{\parallel}})}{2d}\left(\beta^{L}_{\mathbf{k}_{\parallel}}\mathrm{e}^{\mathrm{i}q_{z}z}+\mathrm{e}^{\mathrm{i}q_{z}(\ell-z)}\beta^{R}_{\mathbf{k}_{\parallel}}\right)\right],$
(28b)

Figure 3: Plots of intracavity field intensity for a bilayer system where the
resonance condition for $\Delta$ is always satisfied. The $z$-coordinate is
normalized to the cavity length so that the different cases can be compared.
The plots are generated by considering the infinite square lattice with
$a=0.8\lambda$ in Fourier space. Eq. (24) was evaluated until the reciprocal
lattice sum converged so that evanescent contributions to the interlayer
interaction and cavity field are also included.

where the cavity transmission coefficient reads

$\displaystyle
t_{c}=\frac{\left[\Delta-\mathcal{M}_{\mathbf{k}_{\parallel}}-\mathrm{i}\frac{\widetilde{\Gamma}({\mathbf{k}_{\parallel}})}{2}\right]^{2}}{(\Delta-\mathcal{M}_{\mathbf{k}_{\parallel}})^{2}+\frac{\widetilde{\Gamma}({\mathbf{k}_{\parallel}})^{2}}{4}\mathrm{e}^{2\mathrm{i}k_{z}\ell}},$
(29)

and we have assumed $q_{z}\approx k_{z}$. The condition that the cavity
transmission should equal unity at the cavity resonance $|t_{c}|^{2}=1$ yields
a quadratic equation for the collective resonance
$\Delta-\widetilde{\Omega}(\mathbf{k}_{\parallel})$ with a single solution
given by

$\displaystyle\Delta-\widetilde{\Omega}(\mathbf{k}_{\parallel})=-\frac{\widetilde{\Gamma}({\mathbf{k}_{\parallel}})}{2}\frac{\sin(2k_{z}\ell)}{1+\cos(2k_{z}\ell)}=-\frac{\widetilde{\Gamma}({\mathbf{k}_{\parallel}})}{2}\tan(k_{z}\ell),$
(30)

implying that the resonance frequency of the cavity is given by
$\omega_{c}=\omega_{0}+\widetilde{\Omega}(\mathbf{k}_{\parallel})-\widetilde{\Gamma}({\mathbf{k}_{\parallel}})\tan(k_{z}\ell)/2$.
The results presented in the manuscript correspond to
$\mathbf{k}_{\parallel}=0$, i.e., perpendicular illumination of the cavity. In
Fig. 3 we plot the intensity of the intracavity field at different resonances
including the contribution of the evanescent terms, leading to a field
distribution which is maximal at the mirror surfaces and exhibiting a sine-
like behavior in between.

The results can be easily generalized to several polarization components
(e.g., in the Cartesian polarization basis) by considering the vector of
dipole amplitudes
$\bm{\beta}_{\mathbf{q}}=(\beta_{\mathbf{q},x},\beta_{\mathbf{q},y},\beta_{\mathbf{q},z})^{\top}$
with $\beta_{\mathbf{q},\nu}=\expectationvalue{\sigma_{\mathbf{q},\nu}}$ and
by taking into account the full Green’s tensor Eq. (15).

##  Appendix D Comparison of Fourier space solution with finite-size real
space simulation

The analytical results derived above are valid for infinite lattices. In Figs.
4(a)-(c), we compare the result of Eq. (28a) with the transmission profile
obtained by a real space simulation of the coupled-dipole equations for a
finite number of emitters on the layers, for different cavity lengths. For a
finite lattice, also other modes than $\mathbf{q}=0$ are excited by the
incoming laser. The scattering contributions from these modes lead to a
transmission larger than unity and vanish for larger array sizes, converging
towards the analytical solution. While the results presented below correspond
to plane wave illumination, for a Gaussian beam with beam width covering only
a few lattice sites, the infinite lattice case is approached more quickly as
boundary effects are avoided.

Figure 4: Comparison between analytical expression of transfer matrix/Fourier
space transmission lineshapes (Eq. (29), dotted black curve) with real space
simulation (solid curves) for square lattices with $a=0.8\lambda$ for
different cavity lengths (a) $\ell=5.52\lambda$, (b) $\ell=5.55\lambda$, and
(c) $\ell=5.7\lambda$. The number of emitters per metasurface is given by
$\mathcal{N}$. For the numerical simulations, a small array curvature with
radius $R\approx 3.5\cdot 10^{8}\lambda$ was implemented since the finite size
of the array would otherwise lead to instabilities due to transverse
waveguiding modes Guimond _et al._ (2019). Simulations for other curvature
radii show similar results.

##  Appendix E Transfer matrix for HP mirror

In this section, we show that the proposed setup indeed preserves the helicity
by making use of the transfer matrix method. We start by constructing the
transfer matrix for a single HP mirror consisting of two metasurface layers
with dipoles oriented along $x$ and $y$. From this, the cavity properties can
then be determined in a straightforward manner. Consider the setup of mirrors
with corresponding transfer matrices of

$\mathbf{T}_{m}=\begin{pmatrix}1+\mathrm{i}\zeta_{m}&\mathrm{i}\zeta_{m}\\\
-\mathrm{i}\zeta_{m}&1-\mathrm{i}\zeta_{m}\end{pmatrix},$ (31)

where $\zeta_{m}$ is the polarizability of the metasurface. The first mirror
only affects the $x$ polarization and the second only the $y$ polarization
component. Between them the light propagates freely according to

$\mathbf{T}_{f}=\begin{pmatrix}\mathrm{e}^{\mathrm{i}k\ell_{m}}&0\\\
0&\mathrm{e}^{-\mathrm{i}k\ell_{m}}\end{pmatrix},$ (32)

for both polarization components. In the basis
$(E^{\leftarrow}_{x},E^{\rightarrow}_{x},E^{\leftarrow}_{y},E_{y}^{\rightarrow})^{\top}$,
where the subscript dennotes the polarization and the superscript the
propagation direction of the electric field, the total transfer matrix can be
expressed in block notation as

$\mathbf{T}_{\text{HP}}=\begin{pmatrix}\mathbf{T}_{m}&0\\\
0&\mathds{1}\end{pmatrix}\begin{pmatrix}\mathbf{T}_{f}&0\\\
0&\mathbf{T}_{f}\end{pmatrix}\begin{pmatrix}\mathds{1}&0\\\
0&\mathbf{T}_{m}\end{pmatrix}.$ (33)

To see now that this matrix represents a HP mirror for $\ell_{m}=\lambda/4$,
we must perform a basis transformation which rotates
$(E^{\leftarrow}_{x},E^{\rightarrow}_{x},E^{\leftarrow}_{y},E_{y}^{\rightarrow})^{\top}$
into
$(E^{\leftarrow}_{+},E^{\rightarrow}_{+},E^{\leftarrow}_{-},E_{-}^{\rightarrow})^{\top}$.
This is given by

$\mathbf{U}=\frac{1}{\sqrt{2}}\begin{pmatrix}1&0&1&0\\\ 0&1&0&1\\\
-\mathrm{i}&0&\mathrm{i}&0\\\ 0&\mathrm{i}&0&-\mathrm{i}\end{pmatrix}.$ (34)

Applying this basis transformation to the transfer matrix yields

$\mathbf{U}^{\dagger}\mathbf{T}_{\text{HP}}\mathbf{U}=\begin{pmatrix}1+\mathrm{i}\zeta_{m}&\mathrm{i}\zeta_{m}&0&0\\\
-\mathrm{i}\zeta_{m}&1-\mathrm{i}\zeta_{m}&0&0\\\
0&0&1+\mathrm{i}\zeta_{m}&\mathrm{i}\zeta_{m}\\\
0&0&-\mathrm{i}\zeta_{m}&1-\mathrm{i}\zeta_{m}\end{pmatrix}\begin{pmatrix}\mathrm{i}&0&0&0\\\
0&-\mathrm{i}&0&0\\\ 0&0&\mathrm{i}&0\\\ 0&0&0&-\mathrm{i}\end{pmatrix},$ (35)

which is indeed block-diagonal in the circular basis with identical subblocks
for the two polarization components. The diagonal matrix on the right-hand
side accounts for the free propagation. In total, it can thus be formulated
that the resulting optical element describes a “helicity-preserving mirror
plus free propagation of $\lambda/4$”.

##  Appendix F Cavity phase shift due to presence of chiral scatterer

For further discussions, we focus on one of the equivalent subblocks in Eq.
(35). A chiral molecule can be interpreted as a scatterer that induces
different path lengths for the two polarization components $\delta
l_{s}^{\pm}$.

The transfer matrix for a single scatterer expresses as

$\displaystyle\mathbf{T}_{s}=\begin{pmatrix}1+\mathrm{i}\zeta_{s}&\mathrm{i}\zeta_{s}\\\
-\mathrm{i}\zeta_{s}&1-\mathrm{i}\zeta_{s}\end{pmatrix},$ (36)

where the polarizability of a single scatterer with linewidth $\gamma_{s}$ and
detuning from the laser frequency $\Delta_{s}=\omega_{l}-\omega_{s}$ is given
by $\zeta_{s}=-(\gamma_{s}/2)/(\mathrm{i}\gamma_{s}/2+\Delta_{s})$. In the
limit $\absolutevalue{\Delta_{s}}\gg\gamma_{s}$, this can be understood as an
effective path length added to the free space propagation inside the cavity
due to the approximation

$\displaystyle\mathbf{T}_{s}\approx\begin{pmatrix}\mathrm{e}^{\mathrm{i}k_{l}\delta\ell_{s}}&0\\\
0&\mathrm{e}^{-\mathrm{i}k_{l}\delta\ell_{s}}\end{pmatrix},$ (37)

with the effective length change of the cavity caused by the scatterer

$\displaystyle\delta\ell_{s}=-\frac{\arctan(\gamma_{s}/2\Delta_{s})}{k_{l}}.$
(38)

The cavity transmission coefficient is obtained from the last entry of the
total transfer matrix $t_{c}=1/T_{22}$ and reads

$\displaystyle
t_{c}(\ell+\delta\ell_{s})=\frac{\left(\Delta-\widetilde{\Omega}(0)\right)^{2}}{\left(\Delta-\widetilde{\Omega}(0)+\mathrm{i}\frac{\widetilde{\Gamma}(0)}{2}\right)^{2}+\frac{\widetilde{\Gamma}(0)^{2}}{4}\mathrm{e}^{2\mathrm{i}k_{l}(\ell+\delta\ell_{s})}},$
(39)

and the relative phase shift due to the presence of the scatterer in the
cavity can be expressed as (on the cavity resonance
$\Delta-\widetilde{\Omega}(0)=-\frac{\widetilde{\Gamma}(0)}{2}\tan(k_{l}\ell)$)

$\displaystyle\varphi$
$\displaystyle=\arg\left[\frac{t_{c}(\ell+\delta\ell_{s})}{t_{c}(\ell)}\right]_{\text{res}}=-\arccot(\cot(2k_{l}\delta\ell_{s})-\csc(2k_{l}\delta\ell_{s})\sec(2k_{l}\ell)^{2})$
(40)
$\displaystyle\approx\arctan(\cot(2k_{l}\delta\ell_{s}))-\frac{k_{l}^{2}\bar{\ell}^{2}}{2}\cot(k_{l}\bar{\ell}),$

where the last approximation holds for $\ell\approx n\lambda/2$ and
$\bar{\ell}=\ell-\lfloor 2\ell\rfloor/2$ is the relative distance from this
half-integer factor of the wavelength.

Enantiomers can now be differentiated by illuminating the cavity with
circularly polarized light. Within the polarization subblocks, the two
enantiomers will lead to different effective path lengths
$\delta\ell_{s}^{\pm}$ and hence to a different signal in the homodyne
measurement.

##  Appendix G Coupled-modes theory

$c_{\lambda}(\omega,t)$ | bath operator of mode $\lambda$ |
---|---|---
$b_{\lambda}$ | system oscillator mode $\lambda$ |
$\omega_{\lambda}$ | frequency of oscillator mode $\lambda$ |
$b_{\text{in},\lambda}$ | input operator of port $\lambda$ | $b_{\text{in},\lambda}(t)=\lim_{t_{1}\to\infty}\frac{1}{2\pi}\int\mathrm{d}{\omega}\,\mathrm{e}^{-\mathrm{i}\omega(t-t_{1})}c_{\lambda}(\omega,t_{1})$
$b_{\text{out},\lambda}$ | output operator of port $\lambda$ | $b_{\text{out},\lambda}(t)=\lim_{t_{0}\to-\infty}\frac{1}{2\pi}\int\mathrm{d}{\omega}\,\mathrm{e}^{-\mathrm{i}\omega(t-t_{0})}c_{\lambda}(\omega,t_{0})$
$W_{\lambda,\lambda^{\prime}}$ | coupling between oscillator mode $\lambda$ and input operator $\lambda^{\prime}$ |
$t_{c}^{\text{TM}}$ | transmission function from transfer matrix theory |
$t_{c}^{\text{CM}}$ | transmission function from coupled-modes theory |
$\kappa$ | cavity mode linewidth |
$\omega_{c}$ | cavity mode frequency |
Table 1: Definitions for section S7.

The results discussed so far describe the classical light propagation through
a hybrid cavity formed by quantum metasurfaces. For many applications (such as
sensing discussed below), one is however interested in a quantum input-output
description which would also allow to describe non-classical effects such as
e.g., the photon statistics of the transmitted field.

Figure 5: Comparison between cavity transmission function $t_{c}^{\text{TM}}$
obtained from transfer matrix theory/coupled-dipole formalism (dashed curves)
and the cavity transmission function $t_{c}^{\text{CM}}$ of the coupled-modes
theory (solid curves). Here we chose $\widetilde{\Omega}(0)=0$ as the origin.
The real part of the transmission functions is shown in red and the imaginary
part in blue. From left to right, we show the comparison for decreasing cavity
lengths (a) $\ell=5.1\lambda$, (b) $\ell=5.05\lambda$, and (c)
$\ell=5.01\lambda$, which corresponds to increasing reflectivity of the
metasurface at the resonance. The transmission is shown within the frequency
window $[\omega_{c}-\kappa,\omega_{c}+\kappa]$. The vertical line at
$\Delta=0$ indicates the metasurface resonance and the other vertical line
indicates the cavity resonance.

To this end, we assume a phenomenological coupled-modes model (as introduced
in Ref. Cernotík _et al._ (2019)) where the cavity mode and surface modes of
the metasurface are modeled by harmonic oscillators coupled to common vacuum
modes. We follow the notation in Ref. Viviescas and Hackenbroich (2003) which
assumes equations of motion
$\dot{b}_{\lambda}=-\mathrm{i}\omega_{\lambda}b_{\lambda}-\mathrm{i}\sum_{\lambda^{\prime}}W_{\lambda,\lambda^{\prime}}b_{\text{in},\lambda}$
for the oscillator modes $b_{\lambda}$ and input noise operators
$b_{\text{in},\lambda}$ (see table of definitions 1 above). Then, the input-
output relations may be formulated in compact matrix-vector form in frequency
domain as

$\mathbf{b}_{\text{out}}(\omega)=\mathbf{S}(\omega)\mathbf{b}_{\text{in}}(\omega),$
(41)

where the scattering matrix assumes the following form in the Markovian limit

$\displaystyle\mathbf{S}(\omega)=\mathds{1}-2\pi\mathrm{i}\mathbf{W}^{\dagger}\mathbf{D}^{-1}(\omega)\mathbf{W},$
(42)

with

$\displaystyle[\mathbf{D}(\omega)]_{\lambda\lambda^{\prime}}=(\omega-\omega_{\lambda})\delta_{\lambda\lambda^{\prime}}+\mathrm{i}\pi[\mathbf{W}\mathbf{W}^{\dagger}]_{\lambda\lambda^{\prime}}.$
(43)

The transmission can then be found as matrix elements of the scattering matrix
$t_{c}^{\text{CM}}=S_{12}$.

Now, we make a physical assumption for the form of this coupling matrix, see
for instance Cernotík _et al._ (2019)

$\mathbf{W}=\begin{pmatrix}\sqrt{\frac{\kappa}{4\pi}}&\sqrt{\frac{\kappa}{4\pi}}\\\
\sqrt{\frac{\widetilde{\Gamma}(0)}{2\pi}}&0\\\
0&\sqrt{\frac{\widetilde{\Gamma}(0)}{2\pi}}\end{pmatrix}.$ (44)

By this choice we identify $b_{1}$ as the cavity mode and $b_{2}$ and $b_{3}$
as the surface modes of the metasurface. Thus, we also choose
$\omega_{2}=\omega_{3}=\omega_{0}+\widetilde{\Omega}(0)$. This satisfies the
condition that the cavity transmission ought to be zero at the metasurface
resonance $t_{c}^{\text{CM}}(\omega_{0}+\widetilde{\Omega}(0))=0$. The other
condition that needs to be satisfied is that the cavity transmission becomes
unity at the cavity resonance $|t_{c}^{\text{CM}}(\omega_{c})|^{2}=1$. It
turns out that any choice $\omega_{1}=\omega_{c}+\delta$ and
$\kappa=\left(\frac{\widetilde{\Gamma}(0)}{\omega_{c}}+4\frac{\omega_{c}}{\widetilde{\Gamma}(0)}\right)\delta$
will satisfy this condition. The natural choice $\delta=0$ will however not
lead to a physical solution. Checking against the transfer matrix theory, we
find that the choice $\delta=\omega_{c}$ gives the correct linewidth.

The comparison between the transfer matrix results and the result for the
coupled-modes theory is shown in Fig. 5 for different cavity lengths $\ell$.
We can see that for $\ell\approx n\lambda/2$, the coupled-modes theory
perfectly fits the transmission around the resonance. For larger deviations
from this condition, the transmission functions show differences. This can be
interpreted as the contributions of the transmission that occur via the free
space modes instead of via the cavity-confined mode for a bad cavity (the
transmission of a metasurface becomes unity for large detuning).

##  Appendix H Chiral response of far-detuned molecular scatterer in HP
metasurface cavity

$\omega_{s}$ | transition frequency of molecule |
---|---|---
$\mathbf{d}_{s}$ | electric dipole moment of molecule |
$\bm{\mu}_{s}$ | magnetic dipole moment of molecule |
$\ell_{s}$ | position of molecule along the $z$ axis |
$\beta_{s}$ | coherence of molecular transition | $\beta_{s}=\expectationvalue{\ket{e_{s}}\bra{g_{s}}}$
$\mathbf{d}_{i}$ | dipole moment of $i$th metasurface in stack |
$\beta_{i}$ | coherence of $i$th metasurface in stack |
$\ell_{i}$ | position of $i$th metasurface in stack |
$\Delta_{s}$ | detuning of molecule | $\omega_{l}-\omega_{s}$
$\gamma_{s}$ | total molecule linewidth | $\gamma_{s}=\gamma^{m}_{s}+\gamma^{e}_{s}$
$\gamma^{m}_{s}$ | molecule magnetic linewidth | $\gamma^{m}_{s}=\frac{k_{0}^{3}}{3\pi\mu_{0}}\mu_{s}^{2}$
$\gamma^{e}_{s}$ | molecule electric linewidth | $\gamma^{e}_{s}=\frac{k_{0}^{3}}{3\pi\epsilon_{0}}d_{s}^{2}$
$\hat{e}_{\pm}$ | circular basis vectors | $\hat{e}_{\pm}=(\hat{e}_{x}\pm\mathrm{i}\hat{e}_{y})/\sqrt{2}$
$\xi$ | polarization of incoming light (RCP/LCP) | $\xi=\pm$
$\expectationvalue{\cdot}_{\text{rot}}$ | rotational average | see Ref. Craig and Thirunamachandran (1998)
$\ell_{m}$ | HP mirror metasurface spacing |
$\beta^{x}_{L}$ | coherence of left metasurface with dipole moment in $x$-direction |
$\beta^{y}_{L}$ | coherence of left metasurface with dipole moment in $y$-direction |
$\beta^{x}_{R}$ | coherence of right metasurface with dipole moment in $x$-direction |
$\beta^{y}_{R}$ | coherence of right metasurface with dipole moment in $y$-direction |
$\beta^{\xi}_{L}$ | circular superposition of left metasurfaces | $\beta^{\xi}_{L}=\beta^{x}_{L}+\xi\beta^{y}_{L}$
$\beta^{\xi}_{R}$ | circular superposition of right metasurfaces | $\beta^{\xi}_{R}=\beta^{x}_{R}+\xi\beta^{y}_{R}$
$\hat{v}$ | a hat indicates that the vector is normalized (unit vector) | $\hat{v}=\mathbf{v}/\norm{\mathbf{v}}$
$\mathcal{M}_{\mathbf{q}}$ | interaction between $\mathbf{q}$-mode of a metasurface and magnetic moment of molecule in far field |
$\mathbf{E}_{\text{in}}$ | incoming electric field |
$\mathbf{B}_{\text{in}}$ | incoming magnetic field |
$d$ | magnitude of dipole moments of HP mirror metasurfaces | $d=\norm{\mathbf{d}_{i}}$
$\widetilde{\Gamma}$ | linewidth of metasurfaces | $\widetilde{\Gamma}=\widetilde{\Gamma}(0)=\frac{k_{0}^{3}}{3\pi\epsilon_{0}}d^{2}$
Table 2: Definitions for section S8.

Consider now that we place a chiral molecular scatterer with electric moment
$\mathbf{d}_{s}$ and magnetic moment $\bm{\mu}_{s}$ inside the cavity. We now
assume that both these moments correspond to the same transition, implying
cross-terms in the polarizability for the electric and magnetic components,
i.e., an electric field will also excite the magnetic dipole moment.

In order to continue, we first need to consider the magnetic far field due to
an array of electric dipoles due to the cross term in the Green’s function
given in Eq. (16) (the Green’s function for the electric field generated by a
magnetic dipole is the same but multiplied by minus one). The procedure to
calculate the far field in the subwavelength regime is essentially the same as
for the purely electric Green’s function. To see this, consider that both
Green’s functions can be written as a differential operator acting on the
scalar Green’s function. As such,

$\mathcal{M}_{\mathbf{q}}\approx-\mathrm{i}\frac{k_{0}^{3}}{3\pi\sqrt{\epsilon_{0}\mu_{0}}}\frac{\bm{\mu}_{s}^{*}\cdot(\hat{e}_{z}\times\mathbf{d})}{2}\mathrm{sgn}(z)\mathrm{e}^{\mathrm{i}q_{z}\absolutevalue{z}}$
(45)

gives the interaction due to light propagating from the metasurface to the
molecule located at position $z$. Here, $\mathbf{d}$ indicates the dipole
moment of the metasurface (also see table of definitions 2). Now, the
interaction between $N$ metasurface layers with dipole moments
$\mathbf{d}_{i}$ and a single molecule can be written as (denoting for
simplicity $\widetilde{\Gamma}(0)\equiv\widetilde{\Gamma}$ and omitting the
coupling of the molecule to other surface modes of the array with
$\mathbf{q}\neq 0$)

$\displaystyle\derivative{\beta_{i}}{t}$
$\displaystyle=(\mathrm{i}\Delta-\widetilde{\Gamma}/2)\beta_{i}-\frac{k_{0}^{3}}{3\pi\epsilon_{0}}\sum_{j\neq
i}\mathbf{d}_{i}^{*}\cdot\mathbf{d}_{j}\mathrm{e}^{\mathrm{i}k_{z}\absolutevalue{\ell_{i}-\ell_{j}}}\beta_{j}-\mathrm{i}\mathbf{d}_{i}^{*}\cdot\mathbf{E}_{\text{in}}\mathrm{e}^{\mathrm{i}k_{l}\ell_{i}}{}$
$\displaystyle-\frac{k_{0}^{3}}{3\pi\sqrt{\epsilon_{0}\mu_{0}}}\text{sgn}(\ell_{i}-\ell_{s})\mathbf{d}_{i}^{*}\cdot(\hat{e}_{z}\crossproduct\bm{\mu}_{s})\mathrm{e}^{\mathrm{i}k_{z}\absolutevalue{\ell_{i}-\ell_{s}}}\beta_{m}-\frac{k_{0}^{3}}{3\pi\epsilon_{0}}\mathbf{d}_{i}^{*}\cdot\mathbf{d}_{m}\mathrm{e}^{\mathrm{i}k_{z}\absolutevalue{\ell_{i}-\ell_{s}}}\beta_{j},$
(46a) $\displaystyle\derivative{\beta_{s}}{t}$
$\displaystyle=(\mathrm{i}\Delta_{s}-\gamma_{s}/2)\beta_{s}-\frac{k_{0}^{3}}{3\pi\epsilon_{0}}\sum_{i}\mathbf{d}_{s}^{*}\cdot\mathbf{d}_{i}\mathrm{e}^{\mathrm{i}k_{z}\absolutevalue{\ell_{i}-\ell_{j}}}\beta_{i}-\mathrm{i}\mathbf{d}_{s}^{*}\cdot\mathbf{E}_{\text{in}}\mathrm{e}^{\mathrm{i}k_{l}\ell_{s}}-\mathrm{i}\bm{\mu}_{i}^{*}\cdot\mathbf{B}_{\text{in}}\mathrm{e}^{\mathrm{i}k_{l}\ell_{s}}{}$
$\displaystyle-\frac{k_{0}^{3}}{3\pi\sqrt{\epsilon_{0}\mu_{0}}}\sum_{i}\text{sgn}(\ell_{i}-\ell_{s})\bm{\mu}_{s}^{*}\cdot(\hat{e}_{z}\crossproduct\mathbf{d}_{i})\mathrm{e}^{\mathrm{i}k_{z}\absolutevalue{\ell_{i}-\ell_{s}}}\beta_{i},$
(46b)

where $\beta_{i}$ are the coherences of the metasurfaces and $\beta_{s}$ is
the coherence of the molecular transition. $\ell_{i}$ and $\ell_{s}$ denote
the positions of the metasurfaces and the molecular scatterer along the $z$
axis. This neglects the coupling between the molecule and all other surface
modes with different momenta. This can be taken into account by renormalizing
the decay rate
$\gamma_{s}=\frac{k_{0}^{3}}{3\pi\epsilon_{0}}d_{s}^{2}+\frac{k_{0}^{3}}{3\pi\mu_{0}}\mu_{s}^{2}$
which will however only affect the total transmission which is of no
particular interest to us.

For the proposed setup of four metasurface layers with dipole moments
$(d\hat{e}_{x},d\hat{e}_{y},d\hat{e}_{x},d\hat{e}_{y})$ at positions
$(0,\ell_{m},\ell,\ell+\ell_{m})$ with $\ell_{m}=\lambda/4$, it is thus clear
that the superpositions $\beta^{x}_{L}+\xi\beta^{y}_{L}$ and
$\beta^{x}_{R}+\xi\beta^{y}_{R}$ couple to a specific polarization component.
For left or right-handed illumination, we thus consider the equations of
motion only in this subspace

$\displaystyle\derivative{\beta^{\xi}_{L}}{t}$
$\displaystyle=(\mathrm{i}\Delta-\widetilde{\Gamma}/2)\beta^{\xi}_{L}-\frac{\widetilde{\Gamma}}{2}\mathrm{e}^{\mathrm{i}k_{z}\ell}\beta_{R}^{\xi}-\frac{\sqrt{\widetilde{\Gamma}}}{2}\mathrm{e}^{\mathrm{i}k_{z}\ell_{s}}\hat{e}_{-\xi}\cdot\left[\sqrt{\gamma^{e}_{s}}\hat{d}_{s}+\mathrm{i}\xi\sqrt{\gamma^{m}_{s}}\hat{\mu}_{s}\right]\beta_{s}-\mathrm{i}\sqrt{\widetilde{\Gamma}}E_{\text{in}},$
(47a) $\displaystyle\derivative{\beta^{\xi}_{R}}{t}$
$\displaystyle=(\mathrm{i}\Delta-\widetilde{\Gamma}/2)\beta^{\xi}_{R}-\frac{\widetilde{\Gamma}}{2}\mathrm{e}^{\mathrm{i}k_{z}\ell}\beta^{\xi}_{L}-\frac{\sqrt{\widetilde{\Gamma}}}{2}\mathrm{e}^{\mathrm{i}k_{z}(\ell-\ell_{a})}\hat{e}_{\xi}\cdot\left[\sqrt{\gamma^{e}_{s}}\hat{d}_{s}+\mathrm{i}\xi\sqrt{\gamma^{m}_{s}}\hat{\mu}_{m}\right]\beta_{s}-\mathrm{i}\sqrt{\widetilde{\Gamma}}E_{\text{in}}\mathrm{e}^{\mathrm{i}k_{z}\ell},$
(47b) $\displaystyle\derivative{\beta_{s}}{t}$
$\displaystyle=[\mathrm{i}\Delta_{s}-(\gamma^{m}_{s}+\gamma^{e}_{s})/2]\beta_{s}-\frac{\sqrt{\widetilde{\Gamma}}}{2}\mathrm{e}^{\mathrm{i}k_{z}\ell_{s}}\left[\sqrt{\gamma^{e}_{s}}\hat{d}_{s}^{*}-\mathrm{i}\xi\sqrt{\gamma^{m}_{s}}\hat{\mu}_{s}^{*}\right]\cdot\hat{e}_{-\xi}\beta^{\xi}_{L}$
$\displaystyle-\frac{\sqrt{\widetilde{\Gamma}}}{2}\mathrm{e}^{\mathrm{i}k_{z}(\ell-\ell_{s})}\left[\sqrt{\gamma^{e}_{s}}\hat{d}_{s}^{*}-\mathrm{i}\xi\sqrt{\gamma^{m}_{s}}\hat{\mu}_{s}^{*}\right]\cdot\hat{e}_{\xi}\beta_{R}^{\xi}.$
(47c)

Here $\xi=\pm$ indicates the incoming polarization and we have split the decay
rate of the molecular transition into a magnetic and electric part
$\gamma_{s}=\gamma^{m}_{s}+\gamma^{e}_{s}$. The vectors $\hat{e}_{\pm}$ are
the circular basis vectors. Assuming steady state for $\beta_{s}$ in the
dispersive regime $\Delta_{s}\gg\gamma_{s}$, one obtains two terms for the
left and right side. One term that renormalizes the coupling between the
metasurface layers and another term which renormalizes the metasurface
linewidth and frequency. The second term is of the form

$-\mathrm{i}\frac{\widetilde{\Gamma}}{4\Delta_{s}}\mathrm{e}^{2\mathrm{i}k_{z}\ell_{s}}\hat{e}_{-\xi}\cdot\left[\sqrt{\gamma^{e}_{s}}\hat{d}_{m}+\mathrm{i}\xi\sqrt{\gamma^{m}_{s}}\hat{\mu}_{s}\right]\left[\sqrt{\gamma^{e}_{s}}\hat{d}_{s}^{*}-\mathrm{i}\xi\sqrt{\gamma^{m}_{s}}\hat{\mu}_{s}^{*}\right]\cdot\hat{e}_{-\xi}$
(48)

for the left and

$-\mathrm{i}\frac{\widetilde{\Gamma}}{4\Delta_{s}}\mathrm{e}^{2\mathrm{i}k_{z}(\ell-\ell_{s})}\hat{e}_{\xi}\cdot\left[\sqrt{\gamma^{e}_{s}}\hat{d}_{s}+\mathrm{i}\xi\sqrt{\gamma^{m}_{s}}\hat{\mu}_{s}\right]\left[\sqrt{\gamma^{e}_{s}}\hat{d}_{s}^{*}-\mathrm{i}\xi\sqrt{\gamma^{m}_{s}}\hat{\mu}_{s}^{*}\right]\cdot\hat{e}_{\xi}$
(49)

for the right.

It is well known that a rotational average of the terms in brackets yields the
same result for both terms

$\expectationvalue{\hat{e}_{\pm\xi}\cdot\left[\sqrt{\gamma^{e}_{s}}\hat{d}_{s}+\mathrm{i}\xi\sqrt{\gamma^{m}_{s}}\hat{\mu}_{s}\right]\left[\sqrt{\gamma^{e}_{s}}\hat{d}_{s}^{*}-\mathrm{i}\xi\sqrt{\gamma^{m}_{s}}\hat{\mu}_{s}^{*}\right]\cdot\hat{e}_{\pm\xi}}_{\text{rot}}=\left[\gamma_{s}+2\xi\sqrt{\gamma^{m}_{s}\gamma_{s}^{e}}\imaginary(\hat{d}_{s}^{*}\cdot\hat{\mu}_{s})\right],$
(50)

where
$\mathrm{i}\sqrt{\gamma^{m}_{s}\gamma_{s}^{e}}\imaginary(\hat{d}_{s}^{*}\cdot\hat{\mu}_{s})$
is usually identified as the rotary strength of the molecular transition which
is nonzero for chiral molecules and zero for achiral molecules. The limit of a
perfect chiral scatterer for $\hat{d}_{s}\in\mathbb{R}$ and
$\hat{\mu}_{s}\in\mathrm{i}\mathbb{R}$ is when the two are aligned and have
the same magnitude so that
$\sqrt{\gamma^{m}_{s}\gamma_{s}^{e}}\imaginary(\hat{d}_{s}^{*}\cdot\hat{\mu}_{s})=\pm\gamma_{s}$.
This implies that the scatterer only couples to one of the polarization
components.

##  Appendix I Derivation of phase uncertainty for homodyne detection

$M_{-}(t,T)$ | intensity difference operator |
---|---|---
$m_{-}$ | detected intensity difference |
$F$ | intensity of signal beam ($\\#$photons/time) | unit [Hz]
$F_{\mathrm{LO}}$ | intensity of LO beam ($\\#$photons/time) | unit [Hz]
$T$ | detector integration time |
$\eta_{Q}$ | quantum efficiency of detector | $0\leq\eta_{Q}\leq 1$
$\theta$ | phase of signal beam |
$\theta_{\mathrm{LO}}$ | phase of LO |
Table 3: Definitions for section S9.

We consider balanced homodyne detection and we follow closely section 6.11 of
Loudon (2000). First, we define the intensity difference operator

$\displaystyle
M_{-}(t,T)=\mathrm{i}\int_{t}^{t+T}\textrm{d}t^{\prime}\Bigl{[}t_{c}^{*}a^{\dagger}(t^{\prime})a_{\mathrm{LO}}(t^{\prime})-t_{c}a^{\dagger}_{\mathrm{LO}}(t^{\prime})a(t^{\prime})\Bigr{]},$
(51)

where $T$ is the detector integration time ($T$ and $t$ are not to be confused
with the cavity transmission function $t_{c}$), $a(t)$ is given by

$\displaystyle
a(t)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\textrm{d}\omega\,a(\omega)\mathrm{e}^{-\mathrm{i}\omega
t}.$ (52)

and $a_{\mathrm{LO}}(t)$ is the annihilation operator of the local oscillator
field, still defined, mutatis mutandis, by Eq. (52). The signal and the LO are
prepared in the _monochromatic_ coherent states $\ket{\alpha}$ and
$\ket{\alpha_{\mathrm{LO}}}$, respectively, such that

$\displaystyle a(t)\ket{\alpha}=\alpha(t)\ket{\alpha},\qquad\text{and}\qquad
a_{\mathrm{LO}}(t)\ket{\alpha_{\mathrm{LO}}}=\alpha_{\mathrm{LO}}(t)\ket{\alpha_{\mathrm{LO}}},$
(53)

where

$\displaystyle\alpha(t)=\sqrt{F}\exp(-\mathrm{i}\omega_{0}t+\mathrm{i}\theta),\qquad\text{and}\qquad\alpha_{\mathrm{LO}}(t)=\sqrt{F_{\mathrm{LO}}}\exp(-\mathrm{i}\omega_{0}t+\mathrm{i}\theta_{\mathrm{LO}}).$
(54)

By definition $F$ and $F_{\mathrm{LO}}$ are the number of photons per unit of
time of the signal and the LO beams, respectively. In a typical experimental
setup $F_{\mathrm{LO}}\gg F$.

The detected intensity difference $m_{-}$ is calculated as

$\displaystyle m_{-}=\eta_{Q}\left\langle M_{-}(t,T)\right\rangle,$ (55)

where $\eta_{Q}$ is the quantum efficiency of the detectors and here and
hereafter $\langle
O\rangle=\langle\alpha,\alpha_{\mathrm{LO}}|O|\alpha,\alpha_{\mathrm{LO}}\rangle$.
A straightforward calculation gives

$\displaystyle\left\langle
M_{-}(t,T)\right\rangle=2\sqrt{FF_{\mathrm{LO}}}\absolutevalue{t_{c}}\int_{t}^{t+T}\mathrm{d}{t^{\prime}}\sin\left(\theta-\theta_{\mathrm{LO}}+\varphi(t^{\prime})\right),$
(56)

where we have defined $t_{c}=\absolutevalue{t_{c}}\exp(\mathrm{i}\varphi(t))$.
Since we consider the case of a particle passing through the cavity, we keep
the phase time-dependent but assume the transmission is not affected
significantly by the passage.

The difference photocount variance $(\Delta m_{-})^{2}$ is given by

$\displaystyle(\Delta
m_{-})^{2}=\eta_{Q}^{2}\left\\{\left\langle\left[M_{-}(t,T)\right]^{2}\right\rangle-\left\langle
M_{-}(t,T)\right\rangle^{2}\right\\}+\eta_{Q}(1-\eta_{Q})\int_{t}^{t+T}\textrm{d}t^{\prime}\left\langle
a_{\mathrm{LO}}^{\dagger}(t^{\prime})a_{\mathrm{LO}}(t^{\prime})-\absolutevalue{t_{c}}^{2}a^{\dagger}(t^{\prime})a(t^{\prime})\right\rangle,$
(57)

where

$\displaystyle\left\langle\left[M_{-}(t,T)\right]^{2}\right\rangle=\absolutevalue{t_{c}}^{2}\left[T\left(F+F_{\mathrm{LO}}\right)+4FF_{\mathrm{LO}}\left(\int_{t}^{t+T}\mathrm{d}{t^{\prime}}\sin\left(\theta-\theta_{\mathrm{LO}}+\varphi(t^{\prime})\right)\right)^{2}\right].$
(58)

Gathering all the pieces together, we eventually find

$\displaystyle(\Delta
m_{-})^{2}=\eta_{Q}T\Bigl{[}\eta_{Q}\absolutevalue{t_{c}}^{2}\left(F+F_{\mathrm{LO}}\right)+\left(1-\eta_{Q}\right)\left(\absolutevalue{t_{c}}^{2}F+F_{\mathrm{LO}}\right)\Bigr{]}.$
(59)

Small changes in the transmission function of the Fabry-Pérot cavity due to
the passage of the particle produce small changes in the mean photocounts at
the two detectors in the homodyne setup. The limiting resolution is determined
by the intrinsic uncertainty in the homodyne detection, quantified by the
difference photocount variance $(\Delta m_{-})^{2}$. The uncertainty
$\Delta\varphi$ in the transmission function caused by the homodyne
uncertainty is obtained from

$\displaystyle\Delta\varphi=\frac{\sqrt{(\Delta
m_{-})^{2}}}{\displaystyle\absolutevalue{\frac{\textrm{d}m_{-}}{\textrm{d}\varphi}}}.$
(60)

From Eqs. (55) and (56) we have

$\displaystyle\absolutevalue{\frac{\textrm{d}m_{-}}{\textrm{d}\varphi}}=2\eta_{Q}T\sqrt{FF_{\mathrm{LO}}}\absolutevalue{t_{c}}\absolutevalue{\int_{t}^{t+T}\mathrm{d}{t^{\prime}}\cos\left(\theta-\theta_{\mathrm{LO}}+\varphi(t^{\prime})\right)}.$
(61)

For a signal and a local oscillator matched with one of the cavity resonances
$\omega_{m}=2\pi(m-1/2)/{t_{\text{RT}}}$ we have $\absolutevalue{t_{c}}=1$ and
$\theta-\theta_{\mathrm{LO}}+\varphi(t)=\delta\varphi(t)$, so that

$\displaystyle(\Delta
m_{-})^{2}_{\text{res}}=\eta_{Q}T\left(F+F_{\mathrm{LO}}\right),\qquad\text{and}\qquad\absolutevalue{\frac{\textrm{d}m_{-}}{\textrm{d}\varphi}}_{\text{res}}=2\eta_{Q}T\sqrt{FF_{\mathrm{LO}}}\absolutevalue{\int_{t}^{t+T}\mathrm{d}{t^{\prime}}\cos\left(\delta\varphi(t^{\prime})\right)}.$
(62)

Substituting Eq. (62) into (60) we obtain

$\displaystyle\Delta\varphi=\frac{\sqrt{F+F_{\mathrm{LO}}}}{2\sqrt{\eta_{Q}T}\sqrt{FF_{\mathrm{LO}}}\absolutevalue{\int_{t}^{t+T}\mathrm{d}{t^{\prime}}\cos\left(\delta\varphi(t^{\prime})\right)}}\approx\frac{1}{2(\eta_{Q}TF)^{1/2}\absolutevalue{\int_{t}^{t+T}\mathrm{d}{t^{\prime}}\cos\left(\delta\varphi(t^{\prime})\right)}},$
(63)

where the last approximate equality holds when $F_{\mathrm{LO}}\gg F$.
Similarly, in this regime

$\frac{\sqrt{(\Delta
m_{-})^{2}_{\text{res}}}}{\sqrt{F_{\mathrm{LO}}}}=\sqrt{\eta_{Q}T}.$ (64)