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#  Non-Hermitian synthetic lattices with light-matter coupling

Amir Rahmani  Institute of Physics Polish Academy of Sciences, Al. Lotników
32/46, 02-668 Warsaw, Poland  Mateusz Kędziora  Institute of Experimental
Physics, Faculty of Physics,
University of Warsaw, ul. Pasteura 5, PL-02-093 Warsaw, Poland  Andrzej Opala
Institute of Physics Polish Academy of Sciences, Al. Lotników 32/46, 02-668
Warsaw, Poland  Institute of Experimental Physics, Faculty of Physics,
University of Warsaw, ul. Pasteura 5, PL-02-093 Warsaw, Poland  Michał
Matuszewski  Institute of Physics Polish Academy of Sciences, Al. Lotników
32/46, 02-668 Warsaw, Poland

###### Abstract

We propose that light-matter coupling can be used to realize synthetic
lattices. In particular, we consider a one-dimensional chain of exciton-photon
sites to create a comb lattice that exhibits a transition from a flat band to
a finite mass dispersion by tuning site-dependent light-matter coupling.
Moreover, in a non-Hermitian system with gain and loss, the flat band phase is
much more robust and the transition is accompanied by the appearance of
exceptional points in the complex energy spectrum. We demonstrate that by
engineering the light-matter coupling in the synthetic lattice, one can
explore various phases in the lasing regime. Our proposal paves the way for
studying non-Hermitian systems in higher dimensions.

##  I Introduction

Lattice models are ubiquitous in physics, with applications ranging from
approximations of real physical systems to efficient tools in theoretical
research such as lattice gauge theory. In the context of quantum simulation,
synthetic lattices are highly beneficial as a method for using the internal
degree of freedoms [1] or external states [2] to investigate nontrivial
topology and higher dimensions [1, 3]. The formation of different lattice
configurations may be implemented by using spin[4], Rydberg states[5], orbital
angular momentum states[6, 7] and photonic frequency comb [8].

Light-matter coupling may provide an advantage in investigations of non-
Hermitian physics [9] due to flows of energy or particles to and from a
system. A key feature of non-Hermitian systems is the presence of complex
eigenvalues, which can lead to a variety of interesting phenomena such as non-
reciprocal transport [10], topological phases [11], and exceptional points
[12]. A possible non-Hermitian system is a lattice with gain and loss at its
sites [13, 14]. Here we explore how light-matter coupling can provide an
additional degree of freedom for creating a synthetic lattice. We study a
simple model of a one-dimensional lattice in which the light-matter coupling
can be manipulated. As such, we provide an example of a comb lattice. In one
dimension, the lattice can be obtained from a two-leg ladder lattice [15] by
removing one leg while keeping all other connections and couplings unchanged.
Such lattices have been explored in the context of random walks [16] and
modeling chemical compounds [17].

Light can couple to matter both weakly or strongly, in the latter case leading
to hybrid light-matter quasiparticles called polaritons [18, 19]. Lattices of
photonic nodes can be created by structuring the sample itself by deposition
of a patterned layer on top of a cavity, sample etching [20] or etch-and-
overgrowth[21] procedures, as well as using a spatial light modulator to
excite selected nodes[22]. The ease of creating arbitrary geometries and
potential landscapes in polariton systems opened the way for studying a
variety of Hamiltonian models, including Lieb lattices [23, 24, 25, 26, 27,
28, 29, 30, 31, 32], Kagome[21], and honeycomb [20, 33] lattices, as well as
one-dimensional [34, 35] and two-dimensional [36] topological systems. Owning
to the possibility to manipulate gain and loss [37, 38] polaritons are an
ideal system for studying non-Hermitian effects [39]. A growing interest in
such systems is directly related to the possibility of exploring the physics
of exceptional points [40, 12, 41, 42], parity-time symmetry [43] or skin
effect [44, 45, 10, 46].

In this work, we propose that light-matter coupling in the intermediate regime
between strong and weak coupling can be exploited to introduce a synthetic
lattice with nontrivial properties. The considered lattice, schematically
shown in Fig. 1, is synthetic in the sense that it can be engineered to
possess specific features such as lasing and flat bands. The fact that photon
and exciton modes in the same micropillar couple allows to describe the system
formally as a one-legged comb model. This allows investigation of the physics
of a Lieb (Stub) lattice, including the appearance of a flat band. We show
that site-dependent tuning of light-matter coupling strength allows the
observation of the transition from a flat band to a dispersive spectrum.

Moreover, by analyzing the properties of the system in the more realistic non-
Hermitian regime with gain and loss included, we find that the difference in
loss rates between excitons and photons leads to a great enhancement of
robustness of the flat band phase, and the appearance of a new flat band in a
strongly non-Hermitian case. These effects are shown to persist in the
nonlinear regime of lasing, where the signatures of a flat band and associated
compact localized states [47] (CLS) can be observed in a state resulting from
a long-time evolution starting from a random initial condition.

Our results show that manipulating the light-matter coupling strength in
lattices opens the way for the experimental realization of a new kind of
synthetic lattices that allow to explore strongly non-Hermitian physics. Our
model can be easily extended to higher dimensions or a larger number of light
and matter states, such as orthogonal light polarizations or higher exciton
states [48, 49]. It can be implemented in other physical platforms such as
cavity quantum electrodynamics [50] and coupled atom-light systems [51]. We
believe that our proposal may lead to engineering highly dimensional extended
lattices, providing an ideal platform for simulators of complex non-Hermitian
models.

##  II Model

The lattice described by our model shown in Fig. 1, consists of $2N$
micropillars ($N$ is the cell index, with two micropillars per unit cell)
forming a one-dimensional chain with lattice constant $a$. Each site is
described with two components, where we label exciton components with
superscript $X$ and photonic components with superscript $C$. We assume that
there is finite and uniform coupling between the sites via photonic
components, while exciton components are not coupled with each other due to
the much larger exciton effective mass. Photon and exciton modes in the same
site are coupled with each other with the strength corresponding to Rabi
frequencies which vary from site to site. We consider a staggered distribution
of couplings denoted by $\Omega_{1}$ in odd-numbered sites and $\Omega_{2}$ in
even-numbered sites, where $\Omega_{2}\leq\Omega_{1}$. In particular, such a
position-dependent light-matter coupling can be realized in practice using
site-dependent external pumping, which induces saturation of Rabi coupling
[52]. Photon tunneling between photon modes localized in neighboring
micropillars is described with $J$. As result, the model describing the system
can be considered effectively two-dimensional with an additional two-site
exciton-photon degree of freedom.

Figure 1: Scheme of our model. (Top) At lattice site $j$ photonic
($\psi_{j}^{C}$) and excitonic ($\psi_{j}^{X}$) states are coupled. The
neighboring photonic sites are coupled to each other with coupling rate $J$. A
possible choice of a unit cell is marked by the green square. (Bottom) In
practice the model can be implemented in a lattice of coupled micropillars.

Our model equations can be presented in terms of four fields inside a unit
cell. Denoting each cell with index $n=1,2,3,\cdots N$, we have $2N$
micropillars, for which we introduce fields $\psi^{C}_{j},\psi^{X}_{j}$
corresponding to photon and exciton components at site $j$, where
$j=1,2,\dots,2N$. The mean-field evolution equations are

$\displaystyle i\partial_{t}\psi^{C}_{j}=$
$\displaystyle-i\gamma_{C}\psi^{C}_{j}+J(\psi^{C}_{j-1}+\psi^{C}_{j+1})+\Omega_{j}\psi^{X}_{j}\,,$
(1a) $\displaystyle i\partial_{t}\psi^{X}_{j}=$
$\displaystyle-i\gamma_{X}\psi^{X}_{j}+g\hbar^{-1}|\psi^{X}_{j}|^{2}\psi^{X}_{j}+\Omega_{j}\psi^{C}_{j}\,,$
(1b)

where $\Omega_{j}=\Omega_{1}$ for odd $j$ and $\Omega_{j}=\Omega_{2}$ for even
$j$, $\gamma_{C}(\gamma_{X})$ denotes the decay rate from the photon (exciton)
modes, and $g$ is the nonlinear coefficient. Here $g$ is a complex number
taking into account both exciton-exciton interaction in its real part and gain
saturation effect in its imaginary part [42]. In the case of periodic boundary
conditions we assume $\psi_{N+1}^{C}=\psi_{1}^{C}$. For simplicity, we
consider the case where there is no exciton-photon detuning at any site.

To calculate the spectrum of the system, we consider the eigenvalue equation
$\tilde{H}\tilde{\Psi}=E(k)\tilde{\Psi}$ at $g=0$, where $\tilde{\Psi}$ is a
plane wave solution with momentum $k$. We obtain

$\displaystyle\tilde{H}=\begin{pmatrix}-i\hbar\gamma_{C}&\hbar\Omega_{1}&\hbar
J(1+e^{-ika})&0\\\ \hbar\Omega_{1}&-i\hbar\gamma_{X}&0&0\\\ \hbar
J(1+e^{ika})&0&-i\hbar\gamma_{C}&\hbar\Omega_{2}\\\
0&0&\hbar\Omega_{2}&-i\hbar\gamma_{X}\end{pmatrix}\,.$ (2)

In the following, we will analyze in detail the properties of the system in
both linear and nonlinear regimes.

Figure 2: Examples of spectra and eigenstates in the linear Hermitian case.
(a)-(b) Energy spectra in momentum space for an infinite system. In (a) we
assume $\Omega_{2}=0$ (Stub lattice). In this case there are two degenerate
flat bands with $E=0$, one corresponding to the Lieb lattice flat band, and
another corresponding to isolated exciton sites $\psi^{X}_{2n}$. In (b) we
have $\Omega_{2}=0.4~{}\mathrm{ps}^{-1}$, and the flat bands split into two
dispersive bands. Panel (c) shows eigenenergy spectrum in function of
$\Omega_{2}$ in a finite system with $N=30$. (d),(e) Examples of density
distributions for eigenstates corresponding to the flat band in (a) and
dispersive band in (b), respectively. Corresponding points are marked with a
red star and a circle in (a) and (b). Color scale is in arbitrary units. Other
parameters are $\Omega_{1}=1~{}\mathrm{ps}^{-1}$, $J=0.8~{}\mathrm{ps}^{-1}$.

##  III Hermitian case

For the sake of clarity, we start our study with the linear Hermitian case
$\gamma_{C}=\gamma_{X}=0$ and $g=0$. When $\Omega_{2}=0$, even exciton nodes
$\psi^{X}_{2n}$ are completely isolated from all other nodes. The remaining
photonic nodes and odd exciton nodes form the so-called one-dimensional Lieb
lattice (also called Stub lattice) model [24, 53, 54]. In this case one can
find the dispersion equation analytically,
$E(k)=0,\pm\hbar\sqrt{\Omega_{1}^{2}+2J^{2}(1+\cos(ka))}$. The model exhibits
a flat band with $E=0$ and infinite mass separated by gaps from two dispersive
bands, see Fig. 2(a). Flat bands possess a number of intriguing physical
phenomena, including compact localized states, sensitivity to perturbations
and disorder, strongly correlated phases, and topological states [47]. Note
that in addition to the Lieb lattice flat band, the full model (1) has a
trivial flat band that corresponds to isolated exciton sites.

When the $\Omega_{2}$ parameter is nonzero, the two degenerate flat bands at
$E=0$ split into two dispersive bands, as shown in Fig. 2(b) and Fig. 2(c).
Hence, in the Hermitian case, flat bands are present only in the limit of
$\Omega_{2}=0$. On the other hand, in the symmetric case,
$\Omega_{1}=\Omega_{2}$, the two bands with positive energy and the two bands
with negative energy coalesce with each other, marking a transition to the
uniform one-legged ladder model. In the intermediate regime $0<\Omega_{2}<1$,
the two middle bands that emerged from the flat bands preserve some properties
of the flat band eigenstates. In particular, the structure of the eigenstates
in the middle bands resembles the CLS of the Lieb lattice, as in Fig. 2(d). In
contrast, the eigenstates of the dispersive top and bottom bands resemble
standard bulk states, see Fig. 2(e).

Figure 3: Linear non-Hermitian case. In (a) and (b) we show the real and
imaginary parts of eigenvalues in function of quasimomentum for the same
parameters as in Fig. 2(b), but with nonzero decay terms
$\gamma_{C}=2.2~{}\mathrm{ps}^{-1}$ and $\gamma_{X}=0.1~{}\mathrm{ps}^{-1}$.
In contrast to the Hermitian case, flat bands are present in the system for a
non-zero coupling $\Omega_{2}=0.4~{}\mathrm{ps}^{-1}$. Exceptional points
(EPs) are marked with a black open circle. In (c) and (d) we show real parts
of eigenvalues in function of $\Omega_{2}$ for the fixed
$\Omega_{1}=1~{}\mathrm{ps}^{-1}$. Qualitatively different spectra are shown
in (d) for a smaller decay rate $\gamma_{C}=0.54~{}\mathrm{ps}^{-1}$. The
eigenstates corresponding to the points marked in panels (a) and (b) are shown
in panels (e)-(g).

##  IV Linear non-Hermitian case

We analyze the effect of nonzero decay $\gamma_{C},$ $\gamma_{X}$ on the
spectra and eigenstates of the system, keeping the interactions $g=0$. The
results are presented in Fig. 3. It should be noted that adding a uniform
decay rate $\gamma_{C}=\gamma_{X}>0$ would lead only to an addition of a
constant imaginary part to the eigenvalues of the system, with no effect on
the real part of the spectrum or the eigenfunctions. However, in real systems
the decay rate of photons $\gamma_{C}$ is typically much higher than the decay
rate for excitons $\gamma_{X}$. We find that this leads to a dramatic change
in the spectra. An Example is shown in Figs. 3(a) and 3(b), where real and
imaginary parts of eigenenergies are shown for the same parameters as in the
Hermitian case of Fig. 2(b), but with $\gamma_{C}=2.2~{}\mathrm{ps}^{-1}$ and
$\gamma_{X}=0.1~{}\mathrm{ps}^{-1}$. In contrast to the hermitian case, we
find that despite nonzero $\Omega_{2}$, the system spectrum contains two flat
bands in the entire Brillouin zone, while the dispersive bands develop small
regions of flat dispersion at the two ends of the Brillouin zone. The
transition from dispersive dependence at low quasimomentum to purely imaginary
dependence at high quasimomenta is marked by the occurrence of exceptional
points. We note that previously a purely imaginary dispersion was predicted to
occur in the Bogoliubov excitation spectrum of a polariton condensate coupled
to a reservoir [55]. However, it occurrs at low momenta while the mini-flat
bands that appear in our model are present at high quasimomenta and the model
does not assume condensation or include interactions with an uncondensed
reservoir.

The real part of energy eigenvalues is presented as a function of $\Omega_{2}$
in Figs. 3(c) and 3(d) for two values of the photon decay rate $\gamma_{C}$.
Upon changing the decay rate, exceptional points may emerge in the spectrum.
While in the case of a high decay rate there are no energy gaps in the real
part of the energy, in the low decay rate case the gaps are open at low values
of $\Omega_{2}$ and the system spectrum is more similar to the Hermitian case
of Fig. 2(c). Nevertheless, even in this case flat bands survive to a nonzero
value of $\Omega_{2}$, in contrast to the Hermitian limit. This shows that
including dissipation to the model, which occurs naturally in photonic
systems, makes flat bands much more robust. Moreover, increasing $\gamma_{C}$
may induce exceptional points. At low decay rates, we have three distinct
bands (including a flat band). This is shown in panel (d) for low values of
$\Omega_{2}$. But by increasing the decay rate, the band gaps start
decreasing, while exceptional points emerge at the gap closing. An example of
a gapless spectrum is shown in panel (c). In general, increasing decay rate
leads to coalescence of eigenvalues which gives rise to exceptional points and
the disappearance of the gap. At larger values of the decay rate, exceptional
points are present for any value of $\Omega_{2}$, and so the real part has no
gap in panel (c). However, for a lower value of the decay rate the exceptional
points may appear only at certain range of $\Omega_{2}$. For this reason, the
spectrum in panel (d) displays band gaps.

Finally, in Figs. 3(e)-(g) we show three examples of density distributions of
eigenstates corresponding to different bands. The eigenstate of the flat band
shown in Fig. 3(e) preserves the characteristic structure of the CLS of the
Hermitian model, with all odd photonic sites being empty. In contrast, the
state from the flat mini-band at high quasimomentum in Fig. 3(f) is
characterized by a complementary pattern, with all even photonic sites being
empty. An eigenstate from the dispersive band in the center of the Brillouin
zone depicted in Fig. 3(g) shows no particular structure of photonic states,
which are distributed more or less uniformly, with only a small admixture of
excitonic states.

Figure 4:  (a) Real part of energy of excitations in a nonlinear steady state
in function of $\Omega_{2}$ for $N=30$. The system is in the weakly nonlinear
regime. Parameters are $\gamma_{C}=2.2~{}\mathrm{ps}^{-1}$ and
$\gamma_{X}=-0.1~{}\mathrm{ps}^{-1}$, other parameters as in Fig. 3. Red dots
show the energy of the linear excitation mode with zero imaginary part. All
other excitations have energies with negative imaginary parts. In panels
(b)-(d) we show the evolution of densities in photonic sites. In panel (b) we
used $\Omega_{2}=0.18~{}\mathrm{ps}^{-1}$, which corresponds to a non-zero
real eigenenergy, see panel (a). In panel (c) we used
$\Omega_{2}=0.46~{}\mathrm{ps}^{-1}$, for which the real part of the energy is
very near the zero-energy flat band. The resulting density shows signatures of
random localization. Panel (d) shows an example of dynamics in a weakly
dissipative case where we used $\gamma_{C}=0.54~{}\mathrm{ps}^{-1}$,
$\gamma_{X}=-0.12~{}\mathrm{ps}^{-1}$, and
$\Omega_{2}=0.85~{}\mathrm{ps}^{-1}$. Here the system reaches the steady state
after initial Rabi oscillations. We assume the nonlinear parameter $\hbar
g=0.005-0.005i~{}$$\mathrm{meV^{2}~{}ps}$ which consists of both density-
preserving interactions and density-dependent decay.

##  V Lasing

In the previous section we considered a lattice in the linear non-Hermitian
regime. A natural extension is a system in the presence of nonlinearity,
$g\neq 0$. Lasing corresponds to a steady state in the case of positive gain.
To achieve the steady state regime, we add a new feature to our dissipative
model, that is, we consider a balance between gain and nonlinear decay. In
this case particles escaping the lattice can be replenished by a pumping term.

This can be analyzed simply by assuming $\gamma_{X}<0$ (as the pump source,
eg. a nonresonant optical pump creating excitons) while $\gamma_{C}>0$ (photon
decay). In this regime, we need to assume a nonzero-complex nonlinear term
($g\neq 0$) in Eqs. (1) to reach a stable steady state. We keep the associated
parameters in the so called weak nonlinear regime when $|g||\psi_{1,n}|^{2}$
and $|g||\psi_{2,n}|^{2}$ are at least one order of magnitude lower than
$\hbar\gamma_{C}$. As an aside, one possible source of disorder that may
become important in our modeling is due to randomness in pumping, which is not
considered here, but the results remain unchanged with a minor percent of
disorder strength [56]. We provide examples of the dynamics in the steady-
state regime in Figure 4.

Additionally, we calculate the spectrum of small Bogoliubov excitations around
the steady state. As we are dealing with a homogeneous case, in the steady
state case we may consider a stationary solution in the form
$\psi_{j}^{SC,SX}e^{-iEt/\hbar}$. The phase factor holds information about the
eigenvalues of the model in real space. Indeed, by substituting the above
ansatz in Eq. (1), one can find a set of time-independent equations

$\displaystyle 0=$ $\displaystyle(-i\hbar\gamma_{C}-E)\psi^{SC}_{j}+\hbar
J(\psi^{SC}_{j-1}+\psi^{SC}_{j+1})+\hbar\Omega_{j}\psi^{SX}_{j}\,,$ (3a)
$\displaystyle 0=$
$\displaystyle(-i\hbar\gamma_{X}-E)\psi^{SX}_{j}+\hbar\Omega_{j}\psi^{SC}_{j}+g|\psi^{SX}_{j}|^{2}\psi^{SX}_{j}\,,$
(3b)

which can be solved to find the eigenvalues $E$. To this end, we solve model
equations (Eq. (1)) numerically by employing Runge-Kutta method, and use the
corresponding excitonic occupations in the steady state solution
$|\psi^{SX}|^{2}$ in the above equations.

In the steady state, we are interested in excitations for which the imaginary
part of the energy eigenvalue $\mathrm{Im}[E]$ is zero, since for
$\mathrm{Im}[E]>0$ the steady state is unstable, and for $\mathrm{Im}[E]<0$
the excitations decay in time. Excitation modes with $\mathrm{Im}[E]=0$ can be
considered as discrete analogs of Goldstone modes. We assume that the
parameter that can be manipulated externally is $\Omega_{j}$. As such, it
would be useful to consider the variations of energies with $\Omega_{2}$ in a
steady state.

Real parts of excitation eigenvalues are shown in Fig. 4(a). Here we assume
that the interactions in a chain of micropillars are effectively repulsive via
the effect of exciton-exciton interactions. Considering the case when particle
conserving interactions and gain saturation are of the same order of
magnitude, we choose $\hbar g=0.005-0.005i$ $\mathrm{meV^{2}~{}ps}$ as our
effective nonlinear parameter taking into account both conservative and
dissipative nonlinear processes. We note that in comparison to the linear case
in Fig. 3(c), at small $\Omega_{2}$ a new split band is present in Fig. 4(a).
This splitting from the zero energy band is a nonlinear effect, that is, the
nonlinear term induces a positive onsite potential for the excitons,
proportional to $\mathrm{Re}[g]|\psi^{X}_{j}|^{2}$. We note that for small
$\Omega_{2}$ exciton sites which are indexed by $j=2n$ are almost isolated
from the rest of the lattice. Since excitons at these sites do not experience
photonic decay $\gamma_{C}$, their decay is controlled by the imaginary part
of the nonlinear term $g|\psi^{X}_{j}|^{2}$. This term is much smaller than
the decay of photonic states $\gamma_{C}$ in the case of the weakly nonlinear
regime that we consider here. Hence, the $j=2n$ excitonic sites experience the
highest gain, which results in a steady state where density is mostly
localized at these sites at small $\Omega_{2}$. We show an example of steady
state formation starting from a random initial condition in Fig. 4(b).

As $\Omega_{2}$ increases, these disconnected excitonic states
($\psi^{X}_{2n}$) couple strongly with the other states. The populations in
the other sites in the steady state increase while the population in
$\psi^{X}_{2,n}$ decreases. At some critical $\Omega_{2}$ the split band gap
is closed and the real part of the energy eigenvalue becomes zero, as visible
in Fig. 4(a). At this point, the eigenstate of the system with the highest
imaginary part of the energy belongs to $E=0$ flat band. This has a profound
effect on the dynamics of the system. As shown in Fig. 4(c), states that are
spontaneously formed from a random initial condition result in a different
random density distribution after some certain time. This random density
distribution is long lived and locally resembles the density of a flat band
state. This can be seen as a reminiscence of flat band CLS of the linear Lieb
model, see Fig. 2(d). Further increasing $\Omega_{2}$ with other parameters
fixed leads to a situation where no steady state exists since all the
eigenstates have negative imaginary part. In this case any initial
distribution decays to zero. A qualitatively different situation occurs where
the photon decay is decreased, as shown in Fig. 4(d). In this case the system
converges to a steady state with a periodic, regular density distribution with
all sites occupied. The initial dynamics shows a clear signature of collective
Rabi oscillations in the lattice, which decays after a sufficiently long time.

##  VI conclusion

In summary, we propose that a lattice system with coupled light and matter
modes can be used to realize a synthetic comb lattice by employing the
internal degree of freedom due to light-matter coupling. This idea can be
readily realized in exciton-polariton systems, where several methods of tuning
light-matter coupling strength have been shown. We demonstrated that local
engineering of light-matter coupling provides a way to explore a dissipative
phase transition between the regimes of dispersive and flat band phases. The
transition is accompanied by the appearance of exceptional points in the
spectrum. Importantly, in the dissipative case the flat band regime is not
restricted to a limiting case in the parameter space, but exists in a range of
values of light-matter coupling, which makes the phenomenon much more robust
than in the Hermitian case. We also showed that the existence of a flat band
has a profound effect on the states of the system after long evolution in the
regime of lasing, enabling a straightforward experimental observation. The
proposed method can be generalized to higher dimensional lattices, and the
number of sites in the synthetic lattice can be increased by considering
additional light and matter states. This opens the way to explore non-
Hermitian systems in higher dimensions.

## Acknowledgement

AR, MK, and MM acknowledge support from National Science Center, Poland, Grant
No. 2016/22/E/ST3/00045, A.O. acknowledges 2019/35/N/ST3/01379.

## Appendix

The dynamics of exciton-polaritons are known to be described by mean-field
Gross-Pitaevskii equations:

$\displaystyle i\hbar\partial_{t}\psi^{C}=$
$\displaystyle-i\hbar\gamma_{C}\psi^{C}+\hbar\Omega\psi^{X}\,,$ (4a)
$\displaystyle i\hbar\partial_{t}\psi^{X}=$
$\displaystyle(-\delta-i\hbar\Gamma_{X})\psi^{X}+\hbar\Omega\psi^{C}+\alpha_{1}|\psi^{X}|^{2}\psi^{X}$
$\displaystyle+g_{R}n_{R}\psi^{X}+i\hbar Rn_{R}\psi^{X}\,,$ (4b)
$\displaystyle\partial_{t}n_{R}=$ $\displaystyle
P-(\gamma_{R}+R|\psi^{X}|^{2})n_{R}\,,$ (4c)

where $n_{R}$ is the density of the exciton reservoir, $P(\gamma_{R})$ is the
rate of pumping to (decay from) a reservoir. There are three parameters that
take into account interactions: $\alpha_{1}$, $g_{R}$ and $R$. The energy
detuning between the exciton and the photon field is given by $\delta$. By
using the adiabatic approximation [57] one can assume that
$n_{R}\approx\frac{P}{\gamma_{R}}(1-\frac{R|\psi^{X}|^{2}}{\gamma_{R}})$, the
effective equation for the exciton field becomes

$\displaystyle i\hbar\partial_{t}\psi^{X}=$
$\displaystyle-i\hbar\gamma_{X}\psi^{X}+\hbar\Omega\psi^{C}+g|\psi^{X}|^{2}\psi^{X}\,,$
(5)

where we introduce
$-i\hbar\gamma_{X}=-\delta-i\hbar\Gamma_{X}+P/\gamma_{R}(g_{R}+i\hbar R)$,
$g=\alpha_{1}-\frac{g_{R}PR}{\gamma_{R}^{2}}-i\frac{\hbar
R^{2}P}{\gamma_{R}^{2}}$. Equation (5) has been used to simulate the exciton
field in the main text.

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