††thanks: These two authors contributed equally.††thanks: These two authors
contributed equally.

# Effective bath model for arrays of coupled non-Hermitian nanoresonators

Vincent Vinel  Zejian Li  Adrien Borne  Adrien Bensemhoun  Ivan Favero
Cristiano Ciuti  Giuseppe Leo<EMAIL_ADDRESS>Laboratoire Matériaux
et Phénomènes Quantiques, MPQ UMR 7162, Université de Paris, CNRS, 75013,
Paris, France

###### Abstract

Nanophotonics systems have recently been studied under the perspective of non-
Hermitian physics. Given their potential for wavefront control, nonlinear
optics and quantum optics, it is crucial to develop predictive tools to assist
their design. We present here a simple model relying on the coupling to an
effective bath consisting of a continuum of modes to describe systems of
coupled resonators, and test it on dielectric nanocylinder chains accessible
to experiments. The effective coupling constants, which depend non-trivially
on the distance between resonators, are extracted from numerical simulations
in the case of just two coupled elements. The model predicts successfully the
dispersive and reactive nature of modes for configurations with multiple
resonators, as validated by numerical solutions. It can be applied to larger
systems, which are hardly solvable with finite-element approaches.

##  I Introduction

Nanophotonics deals with light behavior at nanoscopic scale, and encompasses
plasmonics [1, 2], photonic crystals [3, 4], and metamaterial optics [5, 6, 7,
8, 9]. While light confinement is a central challenge for these domains [8, 9,
10, 11], conceiving or exploring low quality factor (Q) systems is not always
straightforward, as scattering or dissipative processes are omnipresent. In
the past decade, nanophotonics has greatly benefited from non-Hermitian
physics [12, 13], which corresponds to the study of either time-dependent
Schrödinger equations or time-independent Schrödinger equations with operators
that are not Hermitian [14, 15]. Those non-Hermitian terms describe
dissipative processes such as the interaction with the environment. Non-
Hermitian approaches have already been adopted in plasmonics [16, 12, 13] and
more recently their dielectric counterparts [17] have also drawn a growing
interest for conceiving novel miniaturized optical devices for wavefront
shaping [7, 18], harmonic generation [19, 8, 10] and quantum photonics [20].

In this work, we theoretically study one-dimensional (1D) chains of $N$
dissipative dielectric nanoresonators by using an analytical non-Hermitian
quantum model. Our approach is to feed the model with complex coupling
parameters obtained from a numerical simulation in the case $N=2$, and then
validate it with longer chains by comparison with brute-force calculations. It
thereby becomes a predictive tool, essential for overcoming numerical
limitations associated to the study and design of larger systems of interest.
This model includes both direct coupling between resonators and coupling
mediated by the electromagnetic continuum acting as a reservoir. Such
couplings are determined by fitting the numerical solution of Maxwell’s
equations in the case of two coupled resonators. The latter is obtained via
finite-element-method (FEM) simulations on our trial system, which consists of
chains of equidistant Aluminum Gallium Arsenide (AlGaAs) nanocylinders along
$x$ [Fig. 1(a)]. Their size determines the spatial and spectral properties of
their eigenmodes [21, 22]. The numerical integration domain is bounded by a
perfectly matched layer (PML) that suppresses spurious electromagnetic-field
reflection at the edges of the integration domain. We focus on the hybrid
modes stemming from from magnetic dipoles (MDs) of single nanocylinders (see
more details in supplementary material). Their radius and height are set at
300 nm and 400 nm respectively, to lift the spectral degeneracy between out-
of-plane (MDz) and in-plane (MDx and MDy) magnetic dipolar modes, the coupling
strength being dictated by the gap $d$ between the nanocylinders. Figure
1(b,c) displays $N=2$ chain spectra for hybrid MDx and MDz as a function of
the gap. A pair of bonding and antibonding modes is formed, corresponding to
two complex eigenfrequencies in phase opposition. In the following, we will
neglect the coupling between two MDs oriented along different directions,
since the field overlap between two non-collinear MDs is 3 orders of magnitude
smaller than the one between two collinear MDs. We identify two coupling
regimes: an exponential-like decay of the frequency splitting for small gaps;
and a pseudo-periodic variation of the complex eigenfrequencies for larger
gaps. Outside each leaky resonator, the electromagnetic field decreases much
slower than for high-Q systems like micropillars [23] or whispering gallery
mode resonators [24, 25, 26, 27]. When more than one resonator is involved,
the corresponding eigenfrequencies oscillate and result in non-zero energy
splitting. A good figure of merit to assess the range of this interaction is
the scattering cross-section of each resonator mode, which can be as large as
ten times the cylinder cross-section in the case of MD [21, 17]. The
eigenfrequency spectra of a pair of MD eigenmodes exhibit degeneracy points
which occur at different gap values for real and imaginary parts and depend on
the specific mode. This feature has already been reported recently [28] and
offers interesting perspectives for meta-optics design and band engineering.
Additionally, the coupling between MDs modifies the quality factor of those
modes. For a single AlGaAs nanocylinder of the same dimensions, the Q factor
of the MDz is 7 and that the MDx and MDy is 5.5: our simulations thus confirm
that even long-range coupling between nanoresonators can increase Q factor,
although light remains poorly confined in those structures, which justifies
the introduction of a non-Hermitian formalism.

Figure 1:  AlGaAs nanocylinder ensembles and numerical simulations for short
chains. (a): schematic of the system. The gap denotes edge-to-edge distance
between nanocylinders. (b) and (c): numerical simulation for $N=2$: real part
of the eigenfrequency $\omega$ as a function of gap for MDx and MDz
respectively, with colorbar providing the quality factor
$Q=\Re(\omega)/2\Im(\omega)$. Inset: 3D plot of the electric field norm in a
nanocylinder. (d) case $N=3$ for MDx. Electric field norm inside nanocylinders
for each mode for a gap of $0.75~{}\rm{$\mathrm{\SIUnitSymbolMicro m}$}$.
Normalized powers inside the resonators for each mode are 1: (0.27, 0.47,
0.26); 2: (0.50, 0, 0.50); 3: (0.22, 0.56, 0.22).

##  II Non-Hermitian quantum formalism

To describe the dynamics of an ensemble of leaky resonators, let us consider
an Hamiltonian consisting of a tight-binding term for the nanoresonators
coupled to an effective bath. The non-Hermitian character is obtained by
tracing out the bath degrees of freedom. Our basic idea is to extract the
effective bath parameters from exact FEM results for two coupled resonators,
and then use the formalism to predict the modes for an arbitrary number and
arrangement of nanoresonators. In the rotating-wave approximation, the
Hamiltonian reads (with $\hbar=1$):

$\displaystyle\hat{H}=$
$\displaystyle\hat{H}_{\rm{sys}}+\hat{H}_{\rm{bath}}+\hat{H}_{\rm{int}},$ (1)
$\displaystyle\hat{H}_{\rm{sys}}=$
$\displaystyle\sum_{j}\omega_{j}\hat{a}_{j}^{\dagger}\hat{a}_{j}+\sum_{j,j^{\prime}}J(d_{jj^{\prime}})(\hat{a}_{j}^{\dagger}\hat{a}_{j^{\prime}}+\mathrm{h.c.}),$
(2) $\displaystyle\hat{H}_{\rm{bath}}=$
$\displaystyle\int\mathrm{d}\eta~{}\omega_{\eta}\hat{\alpha}_{\eta}^{\dagger}\hat{\alpha}_{\eta},$
(3) $\displaystyle\hat{H}_{\rm{int}}=$
$\displaystyle\sum_{j}\int\mathrm{d}\eta~{}\mathrm{i}(g_{j\eta}\hat{a}_{j}^{\dagger}\hat{\alpha}_{\eta}-g_{j\eta}^{*}\hat{\alpha}_{\eta}^{\dagger}\hat{a}_{j}).$
(4)

$\hat{H}_{\rm{sys}}$ is the bare resonators’ Hamiltonian, where each
nanoresonator has one mode with annihilation operator $\hat{a}_{j}$ and
frequency $\omega_{j}$, $j$ and $j^{\prime}$ denoting different resonators. In
the following, we will consider equally spaced identical resonators
($\omega_{j}=\omega_{0}$, $d_{jj^{\prime}}=d$). $J(d)$ describes the coherent
coupling between two resonators via evanescent field, and is essentially
determined by the distance $d$ between them for a given set of resonators. The
second term $\hat{H}_{\rm{bath}}$ describes the continuum of radiation modes
represented by the annihilation operators $\hat{\alpha}_{\eta}$ indexed by
$\eta$, and $\hat{H}_{\rm{int}}$ is the interaction Hamiltonian between the
system and the bath, where $g_{j\eta}$ is the coupling between the $j^{th}$
nanoresonator and the radiation mode $\hat{\alpha}_{\eta}$. The operators
$\hat{a}_{j}$ and $\hat{\alpha}_{\eta}$ obey bosonic commutation relations,
i.e. $[\hat{a}_{j},\hat{a}^{\dagger}_{j^{\prime}}]=\delta_{jj^{\prime}}$ and
$[\hat{\alpha}_{\eta},\hat{\alpha}^{\dagger}_{\eta^{\prime}}]=\delta(\eta-\eta^{\prime})$.
All commutators between $\hat{a}_{j}$ or $\hat{a}^{\dagger}_{j}$ and
$\hat{\alpha}_{\eta}$ or $\hat{\alpha}^{\dagger}_{\eta}$ are taken to be zero.
In the Heisenberg picture, one can derive the quantum Langevin equation [29]
(see Supplementary Material for the derivation) for $\hat{a}_{j}$:

$\displaystyle\dfrac{\mathrm{d}\hat{a}_{j}}{\mathrm{d}t}=-\mathrm{i}\left[\omega_{0}\hat{a}_{j}+J(d)\sum_{j^{\prime}}\hat{a}_{j^{\prime}}\right]-\sum_{k}\int_{-\infty}^{\infty}\mathrm{d}t^{\prime}\Gamma_{jk}(t-t^{\prime})\hat{a}_{k}(t^{\prime})+\hat{F}_{j}(t),$
(5)

where we have defined the damping kernel
$\Gamma_{jk}(\tau)=\Theta(\tau)\int\mathrm{d}\eta~{}g_{j\eta}g_{k\eta}^{*}e^{-\mathrm{i}\omega_{\eta}\tau}$
and the Langevin force
$\hat{F}_{j}(t)=\int\mathrm{d}\eta~{}g_{j\eta}e^{-\mathrm{i}\omega_{\eta}(t-t_{0})}\hat{\alpha}_{\eta}(t_{0})$
where $t_{0}\rightarrow-\infty$.

After applying Fourier transform, adopting the convention
$\tilde{A}(\omega)=\int_{-\infty}^{\infty}\mathrm{d}te^{\mathrm{i}\omega
t}A(t)$, to Eq. (5), we obtain the equations in the frequency domain:

$\displaystyle\omega\tilde{a}_{j}(\omega)=\omega_{0}\tilde{a}_{j}(\omega)+J(d)\sum_{j^{\prime}}\tilde{a}_{j^{\prime}}(\omega)-\mathrm{i}\sum_{k}\tilde{\Gamma}_{jk}(\omega)\tilde{a}_{k}(\omega)+\mathrm{i}\tilde{F}_{j}(\omega),$
(6)

which can be cast into the following matrix form

$\mathcal{M}(\omega,d)\begin{bmatrix}\tilde{a}_{1}(\omega)\\\ \vdots\\\
\tilde{a}_{N}(\omega)\end{bmatrix}+\mathrm{i}\begin{bmatrix}\tilde{F}_{1}(\omega)\\\
\vdots\\\ \tilde{F}_{N}(\omega)\end{bmatrix}=0,$ (7)

with eigenvalues $\lambda^{(i)}$ that can be obtained by diagonalization. We
can then solve for the resonant frequencies

$\Omega_{\text{res}}=\left\\{\omega_{i}^{\star}\in\mathbb{R}~{}|~{}\exists
i,~{}\omega_{i}^{\star}=\operatorname*{arg\,min}_{\omega}|\lambda^{(i)}(\omega)|^{2}\right\\},$
(8)

which, by definition, give local maxima of the amplitude of the frequency
response, and the corresponding damping

$\gamma_{i}^{\star}=-\Im[\lambda^{(i)}(\omega_{i}^{\star})].$ (9)

For a compact notation, we can assign the complex frequency to the resonant
mode
$\omega^{\text{res}}_{i}=\omega^{\star}_{i}-\mathrm{i}\gamma_{i}^{\star}$. In
the case $N=2$, with identical resonators, the matrix can be explicitly
written as :

$\mathcal{M}(\omega,d)=\begin{bmatrix}\omega_{0}-\omega-\mathrm{i}\tilde{\Gamma}_{\rm{diag}}(\omega,d)&J(d)-\mathrm{i}\tilde{\Gamma}_{\rm{off}}(\omega,d)\\\
J(d)-\mathrm{i}\tilde{\Gamma}_{\rm{off}}(\omega,d)&\omega_{0}-\omega-\mathrm{i}\tilde{\Gamma}_{\rm{diag}}(\omega,d)\end{bmatrix}$
(10)

where we have further assumed
$\tilde{\Gamma}_{11}=\tilde{\Gamma}_{22}:=\tilde{\Gamma}_{\rm{diag}}$ and
$\tilde{\Gamma}_{12}=\tilde{\Gamma}_{21}:=\tilde{\Gamma}_{\text{off}}$ by
symmetry.

The coupling functions $J$, $\tilde{\Gamma}_{\rm{diag}}$ and
$\tilde{\Gamma}_{\rm{off}}$ can be fitted from the simulation results
presented in Fig. 1. To simplify the treatment, we expand them to first order
in $\omega$, which allows us to determine a set of possible reservoir
functions from the simulation of the $N=2$ system (see supplementary material
for a detailed derivation). For an $N$-resonator chain, the matrix in Eq. (7)
can be written as:

$\mathcal{M}_{N}(\omega,d)=(\omega-\omega_{0})I(N)+J(d)Sec(N)-\mathrm{i}\big{(}\tilde{\Gamma}_{jk}(\omega,d)\big{)}_{\\{j,k\\}\in\llbracket
1;N\rrbracket^{2}}$ (11)

with $I(N)$ being the $N\times N$ identity matrix, and $Sec(N)$ the $N\times
N$ secondary diagonal matrix.

##  III Results and discussion

The same model is used to predict the modes in the presence of three
resonators [Fig. 2(a) solid blue line] and four resonators [Fig. 2(c) solid
blue line], using only what we learned from the $N=2$ case behavior as a
function of the gap. The main features of the fully numerical simulations are
captured by our model: exponential-like decay, pseudo periodicity, and
degeneracy points of both imaginary and real parts of the eigenfrequencies.
All this confirms that a relatively simple analytical non-Hermitian formalism
can predict the physics of non-trivial nanophotonic systems.

Figure 2:  Comparison between the analytical model and the FEM simulations.
(a) Case $N=3$: real part (top panel) and imaginary part (bottom panel) of the
eigenfrequency $\omega$ as a function of gap for MDx. (b) Probability density
map in the case $N=3$, depending if next-nearest neighbor coupling is
implemented (right, NNN) or not (left, NN). Modes are labelled according to
Fig. 1(d). (c) Case $N=4$: real part (top panel) and imaginary part (bottom
panel) of the eigenfrequency $\omega$ as a function of gap for MDx.

Strikingly, degeneracy points for real parts on one hand, and for imaginary
parts on the other hand, arise for the same gaps in the cases $N=2$, $3$ and
$4$. The fact that they are independent of $N$ indicates that they only depend
on single resonator’s characteristics. The numerical spectra of the eigenmodes
in the cases $N=3$ and $4$ are fairly described by our formalism (see the
relative error in the Fig. S4 of the supplementary material). Furthermore,
following the correspondence between the probability density and the power
stored inside each resonator, the eigenmodes solution of the non-unitary
dynamics in the case $N=3$ [Fig. 2(b)] displays more symmetric results than
the FEM simulations [Fig. 1(d)]. Therefore, in order to refine our model, we
explore coupling to the next-nearest neighbor.

It is important to note that the coupling between two resonators $j$ and $j+2$
should be different whether a resonator $j+1$ is present or not. This implies
that next-nearest-neighbor coupling cannot be extracted from the $N=2$ case,
but from the $N=3$ at least. Therefore, the knowledge of long-range coupling
in a chain of a $N$ nanoresonators depends on the knowledge of the
corresponding $N-1$ chain. However, the resolution of such a model would prove
tedious, with had-to-extract coupling constants through iterative calculation
processes, resulting in numerical challenges to predict modes of longer
chains. From the information extracted in the case $N=2$, we performed an
analytical calculation of the $N=3$ (resp. N=4) chain with simplified next-
nearest-neighbor coupling [Fig. 2(a) and (b)] (resp. (c)) to clarify whether
it could improve our model. For this purpose, we introduce
$J(2r+2d)+\mathrm{i}\tilde{\Gamma}_{\rm{off}}(2r+2d)$, the coupling between
two nanoresonators separated by $2r+2d$ , where r is the radius of the
nanocylinder. Conservative and dissipative coupling constants, $J(2r+2d)$ and
$\tilde{\Gamma}_{\rm{off}}(2r+2d)$, were extracted from Maxwell’s equations
solutions in the case $N=2$. This simplified next-nearest-neighbor coupling
improves the agreement of probability densities [Fig. 2(b)] and spectra [Fig.
2(a) and (c)] with FEM simulations (see the relative error in the Fig. S5 of
the supplementary material). Such a refinement confirms that those modes
involve long-range interaction between coupled resonators.

From Fig. 2, it also appears that the agreement between the analytical non-
Hermitian model and the numerical Maxwell’s equations resolution is more
significant for larger gaps. This can be ascribed to the hypothesis of linear
dependence of the coupling constants on $\omega$, which enabled us to fit the
complex eigenfrequencies. For a narrower gap between nanocylinders, the field
overlap grows stronger, which implies a wider frequency splitting of the
eigenmodes [30, 31], as well as a deformation of near-field both outside and
inside the resonators. In this regime, the coupling can no longer be expressed
as a perturbation, increasing the deviation of our analytical model from
brute-force calculations. Finally, we computed spectra for 1D equidistant
chains with different number of sites $N$ (Fig. 3). For a chain of $N$
resonators, $N$ hybrid modes are expected for a given MD. While analytical
non-Hermitian calculations derive dispersion from eigenmodes of the $N$-site
chains, numerical simulations are quickly limited by the size of the
calculation space, which is defined by the PML size and the meshing of the
system. Indeed, for $N=6$, FEM simulations require about 100 GBytes of RAM,
which meets the upper limit of our local calculation resources. Additionally,
hybridization of the eigenmodes with the PML modes prevents us to identify
them properly. This is less prominent for MDy and MDz hybrid modes, whose
symmetries differ more from spherical PML than MDx modes. This observation
underpins the necessity of finding alternative means to model and design
coupled nanophotonic systems of larger size. In opposition, analytical
resolution of the tight-binding non-Hermitian problem is possible for large
$N$, as illustrated in Fig. 3(b) for $N=200$, where we assumed no frequency
dependence in the coupling constants for this specific calculation. In this
case, the eigenfrequencies of the 1D chain tend to form a continuum of modes,
whose spectral localization depends on the coupling constants of the system.

Figure 3: Comparison between numerical simulations and analytical model for
$d=750~{}$nm. (a) Real part (top panel) and imaginary part (bottom panel) of
the eigenfrequencies $\omega$ vs. the number N of resonators in the chain.
Blue dashes: spectra calculated with the Hamiltonian model; grey crosses: FEM
simulations. (b) Complex-plane representation of the analytically calculated
eigenfrequencies for chains of $N=9$ (black triangles) and $N=200$ (red
crosses).

##  IV Conclusion

Overall, the analytical non-Hermitian quantum formalism used here offers
adequate means to compute modal information on 1D chains of $N$ resonators,
and it can be naturally extended to nonlinear quantum optics [32]. By
increasing the length of the chain, one could study the transition between a
discrete ensemble of nanoresonators and a photonic crystal. Considering the
wide variety of geometries and properties accessible via nanostructured
dielectric materials, we envisage that such nanophotonic systems could be a
promising model for implementing more complex interactions in 1D, 2D or 3D
metasystems. They could also constitute toy systems to study for example
topological edge states [33, 34]. A direct application for such a predictive
model could be the improvement of Q factor optimization of leaky
nanoresonators through non-Hermitian coupling.

###### Acknowledgements.

This work has been supported by the ANR grants NANOPAIR (ANR-18-CE92-0043) and
NOMOS (ANR-18-CE24-0026). The authors would like to thank Carlo Gigli his
contribution to the numerical simulations platform. Data underlying the
results presented in this paper are not publicly available at this time but
may be obtained from the authors upon reasonable request. The authors declare
no conflicts of interest.

## References

  * Chu  _et al._ [2019] A. Chu, C. Gréboval, N. Goubet, B. Martinez, C. Livache, J. Qu, P. Rastogi, F. A. Bresciani, Y. Prado, S. Suffit, S. Ithurria, G. Vincent, and E. Lhuillier, ACS Photonics 6, 2553 (2019), https://doi.org/10.1021/acsphotonics.9b01015 .
  * Teperik and Degiron [2012] T. V. Teperik and A. Degiron, Phys. Rev. Lett. 108, 147401 (2012).
  * Kosaka  _et al._ [1998] H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, Phys. Rev. B 58, R10096 (1998).
  * Edrington  _et al._ [2001] A. C. Edrington, A. M. Urbas, P. DeRege, C. X. Chen, T. M. Swager, N. Hadjichristidis, M. Xenidou, L. J. Fetters, J. D. Joannopoulos, Y. Fink, and E. L. Thomas, Advanced Materials 13, 421 (2001).
  * Decker  _et al._ [2015] M. Decker, I. Staude, M. Falkner, J. Dominguez, D. N. Neshev, I. Brener, T. Pertsch, and Y. S. Kivshar, Advanced Optical Materials 3, 813 (2015).
  * Liu  _et al._ [2016] S. Liu, M. B. Sinclair, S. Saravi, G. A. Keeler, Y. Yang, J. Reno, G. M. Peake, F. Setzpfandt, I. Staude, T. Pertsch, and I. Brener, Nano Letters 16, 5426 (2016), https://doi.org/10.1021/acs.nanolett.6b01816 .
  * Gigli  _et al._ [2021] C. Gigli, G. Marino, A. Artioli, D. Rocco, C. D. Angelis, J. Claudon, J.-M. Gérard, and G. Leo, Optica 8, 269 (2021).
  * Gigli  _et al._ [2019] C. Gigli, G. Marino, A. Borne, P. Lalanne, and G. Leo, Frontiers in Physics 7, 221 (2019).
  * Koshelev and Kivshar [2021] K. Koshelev and Y. Kivshar, ACS Photonics 8, 102 (2021), https://doi.org/10.1021/acsphotonics.0c01315 .
  * Koshelev  _et al._ [2020] K. Koshelev, S. Kruk, E. Melik-Gaykazyan, J.-H. Choi, A. Bogdanov, H.-G. Park, and Y. Kivshar, Science 367, 288 (2020), https://science.sciencemag.org/content/367/6475/288.full.pdf .
  * Jin  _et al._ [2021] W. Jin, Q.-F. Yang, L. Chang, B. Shen, H. Wang, M. A. Leal, L. Wu, M. Gao, A. Feshali, M. Paniccia, K. J. Vahala, and J. E. Bowers, Nature Photonics 15, 346 (2021).
  * Lupu  _et al._ [2013] A. Lupu, H. Benisty, and A. Degiron, Opt. Express 21, 21651 (2013).
  * Cortes  _et al._ [2020] C. L. Cortes, M. Otten, and S. K. Gray, The Journal of Chemical Physics 152, 084105 (2020), https://doi.org/10.1063/1.5131762 .
  * Moiseyev [2011] N. Moiseyev,  _Non-Hermitian quantum mechanics_ (Cambridge University, 2011).
  * Bender [2007] C. M. Bender, Reports on Progress in Physics 70, 947 (2007).
  * Alaeian and Dionne [2014] H. Alaeian and J. A. Dionne, Phys. Rev. B 89, 075136 (2014).
  * Gigli  _et al._ [2020] C. Gigli, T. Wu, G. Marino, A. Borne, G. Leo, and P. Lalanne, ACS Photonics 7, 1197 (2020), https://doi.org/10.1021/acsphotonics.0c00014 .
  * Rocco  _et al._ [2020] D. Rocco, C. Gigli, L. Carletti, G. Marino, M. A. Vincenti, G. Leo, and C. De Angelis, IEEE Photonics Journal 12, 1 (2020).
  * Marino  _et al._ [2019a] G. Marino, C. Gigli, D. Rocco, A. Lemaître, I. Favero, C. De Angelis, and G. Leo, ACS Photonics 6, 1226 (2019a), https://doi.org/10.1021/acsphotonics.9b00110 .
  * Marino  _et al._ [2019b] G. Marino, A. S. Solntsev, L. Xu, V. F. Gili, L. Carletti, A. N. Poddubny, M. Rahmani, D. A. Smirnova, H. Chen, A. Lemaître, G. Zhang, A. V. Zayats, C. D. Angelis, G. Leo, A. A. Sukhorukov, and D. N. Neshev, Optica 6, 1416 (2019b).
  * Carletti  _et al._ [2015] L. Carletti, A. Locatelli, O. Stepanenko, G. Leo, and C. D. Angelis, Optics Express 23, 26544 (2015).
  * Gili  _et al._ [2016] V. F. Gili, L. Carletti, A. Locatelli, D. Rocco, M. Finazzi, L. Ghirardini, I. Favero, C. Gomez, A. Lemaître, M. Celebrano, C. D. Angelis, and G. Leo, Opt. Express 24, 15965 (2016).
  * St-Jean  _et al._ [2017] P. St-Jean, V. Goblot, E. Galopin, A. Lemaître, T. Ozawa, L. Le Gratiet, I. Sagnes, J. Bloch, and A. Amo, Nature Photonics 11, 651 (2017).
  * Armani  _et al._ [2003] D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, Nature 421, 925 (2003).
  * Baker  _et al._ [2014] C. Baker, W. Hease, D.-T. Nguyen, A. Andronico, S. Ducci, G. Leo, and I. Favero, Opt. Express 22, 14072 (2014).
  * Parrain  _et al._ [2015] D. Parrain, C. Baker, G. Wang, B. Guha, E. G. Santos, A. Lemaitre, P. Senellart, G. Leo, S. Ducci, and I. Favero, Opt. Express 23, 19656 (2015).
  * Roland  _et al._ [2020] I. Roland, A. Borne, M. Ravaro, R. D. Oliveira, S. Suffit, P. Filloux, A. Lemaître, I. Favero, and G. Leo, Opt. Lett. 45, 2878 (2020).
  * Pichugin and Sadreev [2019] K. N. Pichugin and A. F. Sadreev, Journal of Applied Physics 126, 093105 (2019), https://doi.org/10.1063/1.5094188 .
  * Ciuti and Carusotto [2006] C. Ciuti and I. Carusotto, Physical Review A 74, 033811 (2006).
  * Zhang  _et al._ [2012] F. Zhang, L. Kang, Q. Zhao, J. Zhou, and D. Lippens, New Journal of Physics 14, 033031 (2012).
  * Vial and Hao [2016] B. Vial and Y. Hao, Journal of Optics 18, 115004 (2016).
  * Li  _et al._ [2021] Z. Li, A. Soret, and C. Ciuti, Phys. Rev. A 103, 022616 (2021).
  * Smirnova  _et al._ [2020] D. Smirnova, D. Leykam, Y. Chong, and Y. Kivshar, Applied Physics Reviews 7, 021306 (2020), https://doi.org/10.1063/1.5142397 .
  * Kruk  _et al._ [2019] S. Kruk, A. Poddubny, D. Smirnova, L. Wang, A. Slobozhanyuk, A. Shorokhov, I. Kravchenko, B. Luther-Davies, and Y. Kivshar, Nature Nanotechnology 14, 126 (2019).