††thanks: These two authors contributed equally to this work††thanks: These
two authors contributed equally to this work††thanks: current address: Hansen
Experimental Physics Laboratory and Department of Neurosurgery, Stanford
University, Stanford, CA 94305††thanks: current address: National Renewable
Energy Laboratory, Golden, CO 80401, USA

# Coherence of a non-equilibrium polariton condensate across the interaction-
mediated phase transition

P. Comaron<EMAIL_ADDRESS>Department of Physics and Astronomy,
University College London, Gower Street, London, WC1E 6BT, United Kingdom  E.
Estrecho<EMAIL_ADDRESS>ARC Centre of Excellence in Future Low-
Energy Electronics Technologies and Department of Quantum Science and
Technology, Research School of Physics, The Australian National University,
Canberra, ACT 2601, Australia  M. Wurdack  ARC Centre of Excellence in Future
Low-Energy Electronics Technologies and Department of Quantum Science and
Technology, Research School of Physics, The Australian National University,
Canberra, ACT 2601, Australia  M. Pieczarka  ARC Centre of Excellence in
Future Low-Energy Electronics Technologies and Department of Quantum Science
and Technology, Research School of Physics, The Australian National
University, Canberra, ACT 2601, Australia  Department of Experimental Physics,
Faculty of Fundamental Problems of Technology, Wroclaw University of Science
and Technology, Wyb. Wyspiańskiego 27, 50-370 Wrocław Poland  M. Steger
Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, PA
15260, USA  D. W. Snoke  Department of Physics and Astronomy, University of
Pittsburgh, Pittsburgh, PA 15260, USA  K. West  Department of Electrical
Engineering, Princeton University, Princeton, NJ 08544, USA  L. N. Pfeiffer
Department of Electrical Engineering, Princeton University, Princeton, NJ
08544, USA  A. G. Truscott  Department of Quantum Science and Technology,
Research School of Physics, The Australian National University, Canberra, ACT
2601, Australia  M. Matuszewski  Institute of Physics, Polish Academy of
Sciences, Al. Lotnikow 32/46, 02-668 Warsaw, Poland  Center for Theoretical
Physics, Polish Academy of Sciences, Al. Lotnikow 32/46, 02-668 Warsaw, Poland
M. Szymanska<EMAIL_ADDRESS>Department of Physics and Astronomy,
University College London, Gower Street, London, WC1E 6BT, United Kingdom  E.
A. Ostrovskaya<EMAIL_ADDRESS>ARC Centre of Excellence in
Future Low-Energy Electronics Technologies and Department of Quantum Science
and Technology, Research School of Physics, The Australian National
University, Canberra, ACT 2601, Australia

###### Abstract

The emergence of spatial coherence in a confined two-dimensional Bose gas of
exciton polaritons with tuneable interactions offers a unique opportunity to
explore the role of interactions in a phase transition in a driven-dissipative
quantum system, where both the phase transition and thermalisation are
mediated by interactions. We investigate, experimentally and numerically, the
phase correlations and steady-state properties of the gas over a wide range of
interaction strengths by varying the photonic/excitonic fraction of the
polaritons and their density. We find that the first order spatial coherence
function exhibits algebraic decay consistent with the
Berezinskii–Kosterlitz–Thouless (BKT) phase transition. Surprisingly, the
exponent of the algebraic decay is inversely proportional to the coherent
density of polaritons, in analogy to equilibrium superfluids above the BKT
transition, but with a different proportionality constant. Our work paves the
way for future investigations of the phenomenon of phase transitions and
superfluidity in a driven-dissipative setting.

Introduction

The physics of collective phenomena in quantum many-body systems is strongly
connected to dimensionality, interactions and to the intrinsic
equilibrium/nonequilibrium nature of the system. A gas of identical bosons is
a perfect example of the crucial role that these factors play in the onset of
order Hohenberg (1967). The three-dimensional (3D) gas of bosons can undergo a
Bose–Einstein condensation (BEC), which results in a state of matter
exhibiting long-range order. While thermal fluctuation preclude the formation
of the long-range order in a two-dimensional (2D) gas, it can instead undergo
the Berezinskii–Kosterlitz–Thouless (BKT) transition and exhibit superfluidity
below a finite critical temperature. Within the superfluid phase, the long-
range decay exponent of the spatial correlations of the system is predicted to
be a function of temperature and the superfluid density Berezinskii (1971);
Kosterlitz and Thouless (1973). In order to achieve a BEC phase transition in
2D, the bosons have to be non-interacting and trapped Dalfovo  _et al._
(1999). Both BEC and BKT transitions have been explored in the system of cold
atomic gases by tuning the system’s dimensionality or the interaction strength
using Feshbach resonances Hadzibabic  _et al._ (2008); Fletcher  _et al._
(2015); Chomaz  _et al._ (2015); T. P. Simula and Hutchinson (2005); Bisset
_et al._ (2009); Hadzibabic and Dalibard (2011); Nazarenko  _et al._ (2014);
Comaron  _et al._ (2019).

2D bosonic systems, such as cavity photons Klaers  _et al._ (2010) or exciton-
polaritons Byrnes  _et al._ (2014); Carusotto and Ciuti (2013), represent a
class of systems with driving and dissipation, where the full thermalisation
(energy relaxation) may not be achieved Deng  _et al._ (2010) and the physics
is inherently nonequilibrium. While cavity photons closely resemble an “ideal”
gas, exciton polaritons (polaritons herein) arise from the strong coupling of
cavity photons and excitons in a semiconductor and interact strongly due to
their excitonic component. Theoretical Comaron  _et al._ (2021); Mei  _et al._
(2021); Gladilin and Wouters (2019, 2023), and experimental Nitsche  _et al._
(2014); Caputo  _et al._ (2017) investigations demonstrated the relevance of
the interaction-dominated BKT physics to polariton systems under nonresonant
pumping, when the coherence in the system develops spontaneously. However, so
far it remains unclear how the coherence in the intrinsically driven-
dissipative polariton system depends on the interactions, how it is affected
by confinement, and whether there exists, in analogy with conservative
systems, any relation between the decay of coherence and the superfluid
fraction. These questions are also interesting because both the thermalisation
(energy relaxation) and the phase transition in a polariton system are driven
by interactions Deng  _et al._ (2010).

In this work, we investigate the phase coherence, i.e. the first-order spatial
correlation function, of a confined polariton condensate in an optical
microcavity. The confinement allows us to explore the coherence features at
high densities, when the interaction energy dominates over the kinetic energy.
This is in contrast to the previous work, where the signatures of the BKT
transition were detected by studying the coherence of a low-density polariton
flow Caputo  _et al._ (2017). In particular, we investigate the coherence
behaviour while varying the polariton interaction energy, which is
proportional to the polariton density and the strength of interparticle
interactions. The variation of the interaction strength is achieved by
changing the energy detuning between the cavity photon and the exciton, which
changes the excitonic fraction of the polariton. The larger excitonic fraction
corresponds to stronger interparticle interaction. In turn, the condensate
density is controlled by the power of the optical pump.

Our study reveals that the trapped, condensed polaritons exhibit different
regimes of spatial coherence depending on the interaction strength. Full
coherence across the spatial extent of the system is reached for stronger
interacting (more excitonic) polaritons. The study of coherence dependence on
the density for a fixed interaction strength (fixed excitonic fraction) shows
similar behaviour, as the interaction energy grows with increasing density.
However, at very high excitonic fractions and pump powers the coherence
decreases due an increase in various decoherence mechanisms. Importantly, in
the limit of weakly interacting (more photonic) polaritons, full
thermalisation does not occur, and multi-mode condensation takes place instead
of a ground state BEC predicted for a non-interacting trapped 2D bosonic
system Dalfovo  _et al._ (1999). This confirms that the thermalisation in this
system is driven by interactions and is inhibited in the photonic regime,
making it difficult to reach the weakly or non-interacting BEC state. Since
both condensation and thermalisation processes are strongly interaction-
driven, observation in polariton condensates of a clean crossover (pioneered
in cold atoms using Feshbach resonances Chin  _et al._ (2010) from the
interacting BKT to the non-interacting 2D BEC regimes by tuning the
interaction strength is challenging.

To further characterise the transition to the BKT phase in this driven-
disspative system, we introduce the concept of “coherent fraction” of the
polariton condensate, and study the relation to the power-law exponent,
describing the decay of the spatial first-order correlation function. Our
analysis demonstrates growth and decay of the coherent fraction with the
increasing interaction energy, consistent with the non-monotonic behaviour of
the first-order spatial correlation function. Surprisingly, our findings also
suggest a clear relationship between the power-law exponent, the coherent
fraction and the effective temperature of a polariton system, in analogy with
the relationship between the power-law exponent and superfluid fraction for
quantum gases in thermal equilibrium Prokof’ev and Svistunov (2002).

Figure 1: Experimental configuration. a, Sketch of the excitation scheme where
a ring-shaped laser excites the sample and creates a ring-shaped reservoir of
excitons with the density $n_{r}$ that traps the top-hat condensate with the
density $n_{c}$. b, Comparison of the cross-sections of the measured intensity
of the condensate PL $I$ (proportional to the polariton density) and the
first-order correlation function $|g^{(1)}|$ showing that the spatial extent
of $|g^{(1)}|$ is limited by the size of the condensate. Scale bar: 10 $\mu$m.
c, Example experimental image of the real-space distribution of the
condensate. Scale bar: 20 $\mu$m. d, Measured 2D $g^{(1)}(\mathbf{r})$ of the
condensate in c, acquired using a modified Michelson interferometer (see
‘Methods’). e, Example density dependence of the blueshift and healing length.
The color represents the normalized spectra for each density datapoint. The
excitonic fraction here is $|X|^{2}=0.34$. Figure 2: Density dependence of
$g^{(1)}(\Delta x)$. a, Logarithmic plots of $g^{(1)}$ at $|X|^{2}=0.37$ for
different densities. Dots are experimental data and solid lines are the power-
law fits (see text). b $g^{(1)}(\Delta x)$ calculated using the stochastic
model (2) with the experimental parameters. Extracted c power-law exponents
$\alpha$ and d coherence lengths $\ell$, respectively, as a function of
density for different excitonic fractions $|X|^{2}$. In c and d we report as a
dashed line the equilibrium limit for the algebraic exponent $\alpha=0.25$
Kosterlitz and Thouless (1973) and the box size, respectively.

Results

We study polariton condensates confined in an optically-induced circular trap
Estrecho  _et al._ (2019); Askitopoulos  _et al._ (2013); Pieczarka  _et al._
(2020); Sun  _et al._ (2017a, b); Alnatah  _et al._ (2024) in the steady-state
regime. An off-resonant quasi-continuous wave (cw) excitation laser, shaped
into a ring profile, pumps the microcavity sample with embedded GaAs quantum
wells (see Methods), creating a similarly shaped excitonic reservoir that
feeds and repels polaritons. As schematically shown in Fig. 1(a), above a
critical pump power (reservoir density), this configuration produces a trapped
condensate inside the ring. The constant replenishment of the reservoir by the
excitation laser ensures that the condensate density is relatively constant
despite the losses due to the short polariton lifetime ($\sim$100 ps). The
polaritons decay by emitting photons, which we detect in the experiment, and
the intensity of the photoluminescence (PL) is directly proportional to the
polariton density [Fig. 1(b)]. The real space image [Fig. 1(c)], momentum
space distribution, spectra, and first-order correlation functions of the PL
are used to infer the information about the polariton condensate.

At high densities of the condensate, when its interaction energy dominates
over the kinetic energy, the density profile shows a relatively flat-top
distribution [Fig. 1(b)]. By virtue of the Thomas-Fermi approximation, this
reflects the flat, box-like profile of the underlying trapping potential
Estrecho  _et al._ (2019); Pieczarka  _et al._ (2020). The uniform profile of
both the condensate and the reservoir-induced potential ensures that the
densities of the condensate and the reservoir are spatially homogeneous near
the centre of the trap. Note that this regime only works in our experiments
for sufficiently large traps with a diameter around $30\leavevmode\nobreak\
\mu$m. Smaller traps lead to quasi-harmonic potentials while very large traps
need high excitation powers, beyond our experimental capability, to reach the
high-density regime. In this work, we probe interaction energies up to 700
$\mu\mathrm{eV}$ (see SI).

A wedge in the microcavity sample, arising during the fabrication process,
leads to a spatial variation of the exciton-photon detuning Sun  _et al._
(2017b); Estrecho  _et al._ (2019), which translates to the variation of the
excitonic fraction quantified by the Hopfield coefficient $0\leq|X|^{2}\leq 1$
(see ’Methods’). The range of the excitonic Hopfield coefficient accessible in
our sample is $0.21\leq|X|^{2}\leq 0.6$. The excitonic fraction in the
polariton quasiparticle is varied within this range by moving the laser
excitation spot across the sample.

We use a modified Michelson interferometer Caputo  _et al._ (2018) to measure
the two-dimensional first-order correlation function |$g^{(1)}(\Delta x,\Delta
y)$| (see ’Methods’), where $(\Delta x,\Delta y)$ is the displacement from the
autocorrelation point. An example of $|g^{(1)}|$ is shown in Fig. 1(b) for an
excitonic fraction of $|X|^{2}=0.37$ with the peak density of $n=4.13\times
10^{3}\mu\mathrm{m}^{-2}$. It features a peak at the autocorrelation point
$\Delta\mathbf{r}=0$, which characterises short-range correlations. The slowly
decaying shoulder at larger distances characterises the long-range order in
this system which is the main interest of this work. By comparing the
$|g^{(1)}|$ to the condensate profile, one can clearly see in Figs. 1(c),(d)
that the long-range order is limited only by the size of the condensate. Note
that despite the small inhomogeneities of the measured condensate profile,
which can arise from both imperfections of the imaging setup and sample
disorder, the measured $|g^{(1)}|$ remains smooth, which strongly suggests
that our measurement is not affected by the small perturbations of the steady-
state density.

Figure 3: Excitonic fraction dependence of $g^{(1)}(\Delta x)$. a, Logarithmic
plots of $g^{(1)}$ at constant density $n\approx 1\times
10^{3}\mu\mathrm{m}^{-2}$ for different excitonic fractions $|X|^{2}$. Dots
are experimental data and solid lines are the power-law fits. Theoretical
results are reported in panel b. Extracted c power-law exponents $\alpha$ and
d coherence lengths $\ell$ as a function of the excitonic fraction.
Experimental data (solid blue line and blue filled circles) compared to the
simulation data (solid blue line and blue bare squares). The pump strength in
simulations is shown by a black solid line and squares.

Density dependence of coherence at a fixed excitonic fraction. – To understand
how the condensate interaction energy affects the first-order correlation
function, we performed a series of measurements at different excitation powers
above the condensation threshold, resulting in different condensate densities
$n$. Here, $n$ is the average density in a small area around the centre of the
condensate.

It is important to highlight that the size of the condensate (diameter of
$\approx 30\leavevmode\nobreak\ \mu m$) is an order of magnitude larger than
the healing length $\xi=\hbar/\sqrt{2gnm}$, where $g$ is the polariton-
polariton interaction strength Estrecho  _et al._ (2019), $m$ is the polariton
mass, and $\hbar$ is the Planck’s constant. The healing length ranges from 6
$\mu$m down to 1 $\mu$m at the probed densities, while the measured condensate
blueshift can reach $800\mu\mathrm{eV}$ as shown in Fig. 1(e). The condensate
interaction energy $gn$ is less than this due to nonvanishing reservoir
density inside the trap Pieczarka  _et al._ (2020); Estrecho  _et al._ (2021);
Pieczarka  _et al._ (2022) (see SI). The small healing lengths relative to the
system size guarantee that we are indeed probing the effect of interactions on
the long-range correlations of the condensate. If $\xi\sim D$, the long-range
universal behaviour in $g^{(1)}$ is masked by other short-range effects.

Figure 2a presents the measured $|g^{(1)}|$ at a fixed value of the excitonic
fraction $|X|^{2}=0.37$, which clearly shows that the long-range coherence
increases with density. At low density, $|g^{(1)}(\Delta x)|$ quickly decays
with distance $\Delta x$. This spatial decay decreases with increasing density
until $|g^{(1)}|$ becomes almost flat.

To quantify the effect of interactions on the coherence length, we fit the
experimental data using the functions Dalfovo  _et al._ (1999)

$g^{(1)}_{\text{exp}}(r)\propto e^{-r/\ell}\quad\text{and}\quad
g^{(1)}_{\text{alg}}(r)\propto r^{-\alpha}\,.$ (1)

According to the BKT theory of a spatially isotropic condensate, the fast,
exponential (slow, power-law) decay of the first-order spatial coherence
function should apply in the disordered (quasi-ordered) phase, below (above)
the critical point $P/P_{th}$, where $P_{th}$ is the pump power corresponding
to the condensation threshold in our system. Kosterlitz and Thouless (1973).
For each data sets of Fig. 2(a), we show the fitting to Eqs. (1) as solid blue
and dashed red lines for the exponential and power-law fitting, respectively.
The power-law exponents $\alpha$ and the coherence lengths $\ell$ (extracted
as fitting parameters) are depicted in Fig. 2(c),(d). At larger densities,
$\ell$ becomes much larger than the trap size (dashed line) while $\alpha$
decreases asymptotically to zero, signaling full coherence across the system.

We recall that in the canonical BKT transition Kosterlitz and Thouless (1973),
the power-law decay of the coherence function is expected to exhibit a
crossover in $\alpha$ from $\alpha_{c}=0.25$ at the critical point towards $0$
for large densities. In the driven-dissipative case, it has been previously
shown that the actual $\alpha_{c}$ may assume larger values due to the non-
equilibrium nature of the polariton system introduced by the pump fluctuations
Roumpos  _et al._ (2012); Dagvadorj  _et al._ (2015). Aiming to identify the
transition point, we proceed by comparing the root mean square (RMS) of the
fitting curves, a method employed in a previous work to ascertain the phase of
the system Comaron  _et al._ (2021). The analysis of the RMS of the fits
(reported in the SI) reveals that the confined geometry introduces boundary
effects resulting in a reduction of phase coherence. Moreover, even at very
large densities, where power-law decay is expected, the exponential and power-
law fits are still comparable. As a consequence, the ratio between the RMS of
the fits cannot be used to identify the critical point Comaron  _et al._
(2021).

Figure 4: Dependence on excitonic fraction and polariton density. Extracted
power-law exponents $\alpha$ (top) and coherence lengths $\ell$ (bottom), as a
function of (a) density and (b) excitonic fraction, for different experimental
series. Simulation results are presented as green squares. c Algebraic
exponent $\alpha$ and d coherence lenght $\ell$ as carpet plots in the
density-excitonic-fraction plane. The highlighted region in c is the region of
interest in Fig. 5(f-i).

We support our experimental finding with the numerical modeling of the
polariton system using the stochastic driven-dissipative Gross-Pitaevskii
equation Wouters and Carusotto (2007). As described in “Methods”, we first
evolve the model to reach a steady-state. Then, the first-order correlation
function (see Methods) is numerically computed and plotted in Fig. 3(b) for
different densities. The resulting outcomes are then fitted to exponential and
power-law functions. Parameters from the fits are extracted and compared to
the experimental values in Fig. 2(c,d). Overall, a good qualitative agreement
between the theory and the experiment is found, especially for larger pump
powers (condensate densities).

It is important to note that the slowest decay of correlations is attained,
both in experiment and in simulations, only up to $n\sim 2\times
10^{3}\leavevmode\nobreak\ \mathrm{\mu m^{-2}}$. For larger densities, we
observe a loss of coherence, followed by an increase of $\alpha$ and a
reduction of $\ell$. The plot of the numerically computed pump power required
to achieve a particular value of the condensate density, shown by a black
solid line in Fig. 2(d), confirms that the system needs a dramatically
stronger pump power to reach larger densities. Strong external drive leads to
a strong decoherence of the condensate. Pump sources, in fact, are expected to
introduce noise contributions leading to dephasing at all powers Zhang  _et
al._ (2022); Vallée  _et al._ (1992); Love  _et al._ (2008). At stronger
powers we expect these to be large. In addition, the dephasing may also arise
from Takemura  _et al._ (2015) thermal fluctuations, the incoherent excitonic
reservoir Kavokin  _et al._ (2017); Haug  _et al._ (2010), and polariton-
polariton interactions Kim  _et al._ (2016). In our numerical simulations, the
pump noise is explicitly included in the noise term $dW$ in Eq. (2) (see
Methods), which accounts for the quantum fluctuations of the polariton field
Carusotto and Ciuti (2013) .

Figure 5: Thermalisation, coherent fraction, and dephasing. a Momentum space
distribution at different densities for three representative excitonic
fractions $|X|^{2}=0.21$, $0.34$, $0.60$. The averaged-particle-number
momentum distribution at different densities: $n\sim$ b
$10^{3}\mathrm{\mu}m^{-2}$, c $2\times 10^{3}\mathrm{\mu}m^{-2}$ and d
$3\times 10^{3}\mathrm{\mu}m^{-2}$. e The coherent fraction extracted from the
momentum distributions of polaritons. f, g, h At low densities the energy-
distributions of polariton number (dots and solid lines) are fitted (dashed
lines) with the Bose-Einstein distribution. RMS (root mean squares) of the
fits are reported in i.

Dependence of coherence on the excitonic fraction at a fixed polariton
density. – To further understand the dependence of the spatial coherence on
interactions, we change the excitonic fraction $|X|^{2}$ (i.e. the interaction
strength, see Methods) and measure coherence for different polariton densities
$n$. We then investigate the behavior of spatial coherence as a function of
$|X|^{2}$ for a fixed polariton density $n\sim 10^{3}\mathrm{\mu m^{-2}}$. We
analyse the data for the coherence, similarly to the analysis presented in
Fig. 2, and display the outcomes in Fig. 3. Panel 3(a) shows that the system
undergoes a crossover from low to high coherence as the excitonic fraction of
the polariton increases, quantified by the fitting curves and parameters shown
in 3(a) and (c),(d), respectively. This result clearly illustrates the
interaction-driven nature of the coherence crossover for our system.
Interestingly, at large excitonic fractions, the system shows a lack of
coherence that we attributed to the dephasing mechanisms observed at large
pump powers and discussed in the previous section. In fact, to reach similar
densities, larger excitonic fractions require stronger pumps. This trend is
clearly revealed by our simulations as shown in Fig. 3(d), which depicts the
growth of the required pump power as a function of the excitonic fraction when
the system density is kept constant. Stronger dephasing may be induced at
larger excitonic fractions by stronger polariton-polariton and polariton-
reservoir interactions, which increase with the excitonic Hopfield
coefficient. Furthermore, the trapping potential becomes tighter at larger
excitonic fraction, resulting in a stronger overlap of the condensate with the
reservoir, as previously shown in similar experiments Estrecho  _et al._
(2019). The largest $|X|^{2}$ probed here already shows this effect. The
experimental results shown in Fig. 3(a,c) and (d), are corroborated by the
numerical simulations: the numerically extracted spatial coherence function,
calculated at a fixed density ($n\sim 10^{3}\mu m^{-2}$) for different values
of $|X|^{2}$, are shown in Fig. 3(b). The extracted coherence lengths and
power-law exponents are then compared to the experimental data in Fig. 3
(c,d). Remarkably, we find very good agreement between the theory and
experiment in the limits of both small and large excitonic fractions.

Both increasing the excitonic fraction of the polaritons at a fixed density
and increasing the density at a fixed excitonic fraction, leads to the growth
of the interaction energy. To better visualize the coherence behaviour in the
space of these two parameters, in Fig. 4 we report the experimentally
extracted $\alpha$ and $\ell$ as a function of density, Fig. 4(a), and as a
function of excitonic fraction, Fig. 4(b). Panels (c) and (d) illustrate these
parameters as a surface in the $(n,|X|^{2})$ space, showing the onset of
coherence at lengths above the system size (shown as a dashed red line). In
summary, in the confined system, coherence is enhanced when increasing the
interaction energy by means of increasing either density or the excitonic
fraction. The crossover to the full coherence takes place approximately at the
system size, as predicted by the theory of trapped conservative quantum gases
Dalfovo  _et al._ (1999). In the limits of both large densities and excitonic
fractions, the polariton condensate experiences dephasing mechanisms (due to
the pump noise, polariton-polariton and polariton-exciton intercations),
resulting in a reduction of coherence.

Coherent fraction and thermalisation. – We further investigate possible
relationship between the coherence and the thermalisation properties of our
system. We start by obtaining energy-resolved momentum-space imaging using a
monochromator coupled with a CCD camera. The CCD intensity,
$I_{\mathrm{CCD}}$, of the central section of each image is collected as a
function of momenta, and recorded for different densities (pump powers) and
excitonic fractions (exciton-photon detunings). In Fig. 5(a) we report the
normalised $I_{\mathrm{CCD}}$ for three cases: $|X|^{2}=0.21$, $0.34$, $0.60$.
At low $|X|^{2}$ and in the low density regime, the system is in a multimode
regime. At large $|X|^{2}$, the system displays a single-mode occupation at
all densities. This strongly suggests that a larger excitonic fraction leads
to a more efficient energy relaxation towards the ground state Estrecho  _et
al._ (2019). In the context of canonical conservative condensates, quasi-long-
range coherence leads to the formation of different coherent components
Pitaevskii and Stringari (2003): i) the condensate fraction, defined as the
normalized density of particles which populate the zero-momentum $(k=0)$ mode,
ii) the superfluid density defined à la Nelson-and-Kosterlitz Nelson and
Kosterlitz (1977) and iii) the quasicondensate, defined as a condensate with a
fluctuating phase Kagan  _et al._ (1987). The superfluid density, in
particular, can be directly linked to the power-law exponent $\alpha$
charactersising the BKT transition Kosterlitz and Thouless (1973); Prokof’ev
and Svistunov (2002). For a non-equilibrium, open-dissipative condensate
studied here, extraction of $n_{s}$ is diffcult. Instead, we introduce the
notion of “coherent density" $n_{\mathrm{coh}}$, defined as the fraction of
particles below a certain momentum cut-off $k_{\mathrm{cut}}(|X|^{2})$, the
latter obtained by Fourier-transforming the effective trapping potential that
the system experiences at each value of the excitonic fraction.

In order to extract the coherent fraction of the condensate, we first obtain
the particle distribution per momentum state, $\mathcal{N}(k)$, which is
proportional to the CCD intensity, i.e., $\mathcal{N}(k)=\mathcal{A}\times
I_{\mathrm{CCD}}$ taking into account the correct geometry of the system
Pieczarka  _et al._ (2019) (see SI). Figures 5(b,c,d) show $I_{\mathrm{CCD}}$
for different $|X|^{2}$ at fixed densities $n=1$, $2$ and $3\times
10^{3}\mathrm{\mu m^{-2}}$. From these distributions, we extract the coherent
fraction $n_{\mathrm{coh}}/n_{\mathrm{tot}}$ in the (high-density) single-mode
regime, and plot its behavior as a function of $|X|^{2}$ at different values
of the total density $n_{\mathrm{tot}}$ in Figure 5(e). It is evident that the
coherent fraction is increasing as a function of density and excitonic
fraction (i.e., interaction strength). We note that a similar dependence on
the interaction strength is expected for the superfluid fraction in a
conservative system Prokof’ev and Svistunov (2002). For $|X|^{2}$ larger than
$\sim 0.35$ the coherent fraction saturates at
$n_{\mathrm{coh}}/n_{\mathrm{tot}}\sim 0.9$. At much larger excitonic
fractions, the coherent fraction decreases, in agreement with the trend shown
by the first-order correlation functions discussed in the previous sections.

In the second part of this analysis, we discuss thermalisation properties of
the system. First, we move from momentum to energy space, employing the
energy-momentum mapping provided by the dispersion relation of the lower-
polariton branch
$E_{\mathrm{LP}}(k)=1/2[E_{\mathrm{X}}(k)+E_{\mathrm{C}}(k)-\sqrt{\hbar^{2}\Omega^{2}+\delta^{2}}]$,
where $E_{\mathrm{X}}(k)$ and $E_{\mathrm{C}}(k)$ are the energies of the
exciton and cavity photon, respectively,
$\delta(k)=E_{\mathrm{C}}-E_{\mathrm{X}}$ is the energy detuning, and
$\hbar\Omega$ is the Rabi splitting energy. We tentatively fit the data
$\mathcal{N}(E)$ to the Bose-Einstein (BE) distribution of the average
particle number $\mathcal{N}_{\mathrm{BE}}(E)=1/(\exp((E-\mu)/k_{B}T)-1)$,
where $T$ and $\mu$ are the temperature and chemical potential of the
polaritons, respectively, and $k_{B}$ is the Boltzmann constant. Here we treat
the temperature $T$ and chemical potential $\mu$ as free parameters. We are
careful to apply this fitting only at low densities and high excitonic
fraction, over a sufficiently populated tail of the modes which otherwise is
obscured by a very bright (and peaked) main mode. This area is highlighted as
a faint red shadowed region in Fig. 4(c)

In Fig. 5(f,g,h), we report the particle distributions in energy, with fitting
curves as dashed lines. In panel (i) we show the RMS of the BE distributions;
these give a quantitative tool for determining the goodness of the fit to a
thermal distribution. Our results suggest that the fitting to a thermal tail
improves as the densities and excitonic fractions increase. Note that at
larger densities ($n\sim 10^{3}\mathrm{\mu m^{2}}$), the fitting procedure
deteriorates for larger $|X|^{2}$; this is in connection with the increase of
the $k=0$ peak which, at large excitonic fractions, dominates over the other
modes. Finally, we study the relation between the power-law exponent $\alpha$
and the coherent density $n_{\mathrm{coh}}$, with the aim to draw a parallel
with the relationship between $\alpha$ and the superfuid density $n_{s}$ in
conservative systems. In the canonical BKT theory, the two quantities are
related by $\alpha=1/n_{s}\lambda^{2}$ where $\lambda=h/\sqrt{(2\pi mk_{B}T)}$
is the thermal de-Broglie length Dalfovo  _et al._ (1999).

The power-law exponent and the coherent fraction of the condensate in our
system are plotted in Fig. 6(a,b), in the small-density and high excitonic
component limit accessible from the previous analysis. Figure 6(c) clearly
depicts the linear dependence $\alpha=1/{n_{\mathrm{coh}}}\lambda^{2}$ for the
data sets obtained in our experiment. Note that this linear behaviour persists
across almost two orders of magnitude for both $\alpha$ and
$1/{n_{\mathrm{coh}}}\lambda^{2}$. It also shows a proportionality constant
diverging from the expected value of 1 by two orders of magnitude, a deviation
that may be due to the driven-dissipative and/or confined nature of the
system.

Figure 6: Power-law exponent and coherent fraction. a Algebraic exponents
$\alpha$ and b coherent fraction $n_{\mathrm{coh}}$ in the low-density and
high excitonic fraction regime (highlighted in Fig. 4 c,d as a dashed area),
for different densities (see legend in c). c Algebraic exponent shows an
inverse proportionality to the coherent fraction, in analogy with the theory
for conservative superfluids, where $\alpha=1/n_{\mathrm{s}}\lambda^{2}$,
where $\lambda=h/\sqrt{2\pi mk_{B}T}$ is the thermal de-Broglie wavelength,
and $n_{\mathrm{s}}$ is the superfluid density. The dashed line corresponds to
$\alpha=10^{2}/(n_{\mathrm{s}}\lambda^{2})$. Linear scales plot is shown in
the inset.

Discussion

In this work, we investigate the role of interactions and energy relaxation on
the development of spatial coherence in the confined polariton system. Our
experimental and numerical findings confirm that, in the driven-dissipative
scenario of polariton condensation, the system follows a BKT-type phase
transition, which is driven by interactions. As the interaction energy is
tuned across a wide range (up to 700 $\mu$eV) by varying the density and
interparticle interactions, we observe different regimes of coherence. In the
high-density regime of stronger interacting, highly excitonic polaritons full
coherence across the spatial size of the system can be reached. However, the
coherence is degraded above certain densities and excitonic fractions. Our
results suggest that this behavior is driven by the dephasing due to the
fluctuations of the laser pump, which are stronger at higher pump powers, as
well as due to polariton-polariton interactions and interactions of polaritons
with the reservoir excitons. This is exacerbated in the experiments due to the
tightening of the reservoir-induced trap at higher excitonic fractions
Estrecho  _et al._ (2019), resulting in stronger reservoir-condensate overlap.

In the low-density regime, we clearly see the short-range thermal component of
the first-order correlation function Krüger  _et al._ (2007). In the limit of
highly photonic polaritons, we observe multi-mode condensation due to
inefficient energy relaxation towards the ground state. We find that the
fitting to BE statistics improves for stronger interactions and at larger
densities, proving that thermalisation processes in non-equilibrium condensate
take place via interactions. We note that this inhibited energy relaxation
towards the ground state precludes the observation of a BEC phase in confined,
2D gas of weakly interacting, highly photonic polaritons.

Finally, we tentatively investigate the relationship between the power-law
exponent of $g^{(1)}(r)$, the condensate coherent fraction, and the effective
temperature of the system. Remarkably, we find that the power-law exponent is
directly proportional to the temperature of the system, which defines the
thermal de Broglie wavelength of a polariton, and is inversely proportional to
the coherent density, in analogy to the dependence of the power-law exponent
on the superfluid density in the conservative case. However, we note that the
system temperature in conservative systems is extracted from the fitting with
a thermal distribution, while in a driven-dissipative system, full
equilibration is not guaranteed and the BE fitting is useful only in a limited
subset of the system’s parameters. To our knowledge, prior to this study, no
investigation had explored this particular relation in the context of
nonequilibrium, driven-dissipative quantum systems. Further experimental
effort could address the issue of the extraction of the superfluid density and
its relation to the coherent density defined here.

Summarising, our work represents an important step towards the
characterisation of interaction-mediated BKT phase transition and coherence in
the driven-dissipative quantum system, across a range of interaction energy.
Our findings are particularly relevant to the discussion about superfluid
properties Juggins  _et al._ (2018), universal behaviour Fontaine  _et al._
(2022), and turbulent features Panico  _et al._ (2023) in driven-dissipative
quantum systems.

Methods

Sample and experimental setup. We use a high Q-factor GaAs/AlGaAs planar
microcavity with 7-nm GaAs quantum wells embedded in a 3$\lambda$/2-cavity,
similar to the samples used in Refs. Steger  _et al._ (2015); Caputo  _et al._
(2018). The thickness gradient of the sample enables control of the exciton-
photon detuning by probing different positions on the sample. The 720-nm off-
resonant continuous-wave (CW) excitation laser is spatially structured into a
ring profile using an axicon, a pair of lenses, and a 50X 0.5-NA objective.
The same objective collects the polariton emission which goes through a relay
of lenses for real-space imaging with a SCMOS camera (Andor Zyla 4.2). The
camera images the interference fringes used to measure the first-order
correlation function. A monochromator (Princeton Instruments IsoPlane SCT 320)
coupled with a CCD camera (Andor iXon Ultra 888) enables energy-resolved real
and momentum-space imaging.

Tuning the excitonic fraction. The excitonic fraction $|X|^{2}$ is tuned by
probing areas of the samples with different photon-exciton detuning, $\Delta$.
The sample has a gradient in detuning due to a wedge in the effective cavity
length Steger  _et al._ (2015) resulting in a gradient in cavity photon
energy. The detuning is calculated from the polariton dispersion, measured at
very low excitation power (below condensation threshold) for each position on
the sample, and using previously known parameters Estrecho  _et al._ (2019);
Pieczarka  _et al._ (2020). We then extract the excitonic fraction from the
detuning as $|X|^{2}=1/2\left(1+\Delta/\sqrt{\Delta^{2}+\Omega^{2}}\right)$,
where the Rabi splitting is about $\Omega=15.9$ meV Pieczarka  _et al._
(2020).

Coherence measurement setup. We employ a modified interferometer Caputo  _et
al._ (2018) consisting of a 50:50 beam splitter, a flat mirror, and a hollow
retroreflector to measure the spatial first-order correlation function
$|g^{(1)}|$. The setup combines the polariton real-space emission
$I(\mathbf{r})=|\psi(\mathbf{r})|^{2}$ with its inverted image
$I(-\mathbf{r})=|\psi(-\mathbf{r})|^{2}$ onto a sCMOS camera where the
interferogram image $F(\mathbf{r})$ forms. The mirrors are aligned such that
the autocorrelation point, $(\Delta x,\Delta y))=0$, is at the center of the
real-space distribution. The first-order correlation function is then
extracted using the equation

$|g^{(1)}(\mathbf{r},-\mathbf{r};\Delta
t=0)|=V(\mathbf{r})\frac{I(\mathbf{r})+I(-\mathbf{r})}{2\sqrt{I(\mathbf{r})I(-\mathbf{r})}}$

where $V={(F_{max}-F_{min})}/{(F_{max}+F_{min})}$ is the visibility of the
interference fringes extracted by introducing sub-micron delays using a piezo
actuator Caputo  _et al._ (2018). The mirrors are mounted on motorized stages
to ensure near-zero temporal delay $\Delta t\approx 0$. A piezo actuator
provides a series of sub-wavelength delays near $\Delta t\approx 0$ to
accurately determine the spatial variation of the fringe visibility,
$V(\mathbf{r})$. To filter out low-frequency noise coming from the mechanical
vibrations of the whole experimental setup, the CW laser is chopped down to
$\sim$10-100-$\mu s$ pulse lengths depending on the signal-to-noise ratio.
This time scale is orders of magnitude longer than the polariton lifetime
($\sim$100-ps) and the pulse rise time ($\sim$50 ns), and hence is sufficient
to capture the steady-state regime.

Modelling of the polariton field. We describe the effective dynamics of the
polariton fluid through the equation of motions for the two-dimensional
polariton field $\psi=\psi(\bm{r},t)$ as a function of the position
$\bm{r}=(x,y)$ and time $t$ within the truncated Wigner approximationCarusotto
and Ciuti (2013), which reads ($\hbar=1$) Wouters and Carusotto (2007);
Chiocchetta and Carusotto (2013); Comaron  _et al._ (2018):

$id\psi=dt\bigg{[}\left(i\beta-1\right)\frac{\nabla^{2}}{2m_{pol}}+g_{c}|{\psi}|^{2}_{-}+\\\
+\frac{i}{2}\frac{R}{\gamma_{R}}\bigg{(}\frac{P(\textbf{r})}{1+\frac{R}{\gamma_{R}}|{\psi}|^{2}_{-}}-\gamma\bigg{)}\bigg{]}\psi+dW$
(2)

where $m$ is the polariton mass, $\gamma_{c}$ and $\gamma_{R}$ are the
polariton and excitonic reservoir decay rates, $P(\textbf{r})$ the circularly-
shaped external drive, $g_{c}$ is the polariton-polariton interaction
strength. $R$ is the scattering rate of reservoir particles into the
condensate. The renormalised density
$|{\psi}|^{2}_{-}\equiv\left(\left|{\psi}\right|^{2}-{1}/{2dV}\right)$
includes the subtraction of the Wigner commutator contribution (where
$dV=a^{2}$ is the element of volume of our 2D grid with the lattice spacing
$a$). The zero-mean white Wigner noise $dW$ fulfils the condition $\langle
dW^{*}(\vec{r},t)dW(\vec{r}^{\prime},t)\rangle=[((R/\gamma_{R})P+\gamma_{c})/2]\delta_{\vec{r},\vec{r}^{\prime}}dt$.
The equation above includes the phenomenological term proportional to the
constant $\beta$, which quantifies the rate of energy relaxation in the system
Chiocchetta and Carusotto (2013); Wouters  _et al._ (2010); Wouters and
Carusotto (2010); Comaron  _et al._ (2018); Zamora  _et al._ (2020). The model
Eq. (2) corresponds to the adiabatic approximation limit of the generalised
polariton equations of motion coupled to an external reservoir Wouters and
Carusotto (2007), which is justified once the reservoir is able to
adiabatically follow the evolution of the condensate and
$\gamma_{c}\ll\gamma_{R}$ Bobrovska and Matuszewski (2015).

Using the parameters adopted in Ref. Estrecho  _et al._ (2018) to model an
experimental system similar to that described here, in this work we vary the
excitonic fraction $|X|^{2}$ in order to model the steady-state coherence
across the range of interaction strengths. It is therefore crucial to note
that most of the parameters in Eq. (2) depend on the excitonic Hopfield
coefficient $X$. The latter also defines the value of the excitonic fraction
$|X|^{2}$, and depends on the exciton–photon detuning $\Delta$ and the Rabi
splitting $\hbar\Omega$ as $|X|^{2}\leavevmode\nobreak\ =\leavevmode\nobreak\
{1}/{2}\left(1+{\Delta}/{\sqrt{4\hbar^{2}\Omega^{2}+\Delta^{2}}},\right)$. The
polariton decay rate depends on the excitonic fraction as
$\gamma_{c}=(1-|X|^{2})\gamma_{ph}$, where $\gamma_{ph}=1/\tau_{ph}$
corresponds to the inverse of the photon lifetime. The constants $g$ and
$g_{R}$ characterise the strengths of polariton-polariton and polariton-
reservoir interactions, which become stronger for polaritons with a larger
excitonic fraction. They can be estimated $g_{c}=g_{ex}|X|^{4}$,
$g_{R}=g_{ex}|X|^{2}$ Bleu  _et al._ (2020), with $g_{ex}$ the exciton-exciton
interaction. Following the discussion in Ref. Estrecho  _et al._ (2018), we
assume that the stimulated scattering rate $R$ from the reservoir into the
polariton states is more efficient for more excitonic polaritons: from our
results we find that $R=R_{0}(g_{c}/g_{R})^{2}$, namely $R$ scales
quadratically with $|X|^{2}$.

The circularly symmetric pump profile $P(\textbf{r})$ is implemented as the
sum of two contributions: i) the profile
$P(\textbf{r})=P_{0}{\gamma_{c}}/{R}[e^{-\left({\mathcal{E}}/{2\sigma^{2}}\right)^{2}}]$,
where $P_{0}$ is the pump strength, and
$\mathcal{E}=x^{2}/a^{2}+y^{2}/a^{2}-1$ is the circle equation with radius
$R_{P}$, and ii) a weak flat pump $\bar{P}=\alpha_{p}P_{0}\gamma_{c}/R$, with
$\alpha_{p}<1$ positive and small, at the center of the ellipse for
$\textbf{r}<R_{P}$: with the latter, we account for the presence of a non-
vanishing reservoir density at the center of the trap. We note that the
inclusion of such a term is important for enhancing the onset of coherence at
the centre of the trap. We also find that the experimental data are adequately
modeled by relaxation mechanisms Wouters and Carusotto (2010); Wouters (2012);
Wouters  _et al._ (2010) (controlled by the paramater $\beta)$ being more
efficient for larger excitonic fractions as
$\beta=\beta_{0}+\beta_{1}|X|^{2}$. In order to match the experimental
measurements, we use the following parameters: $m=3.6\times 10^{-5}m_{e}$ with
$m_{e}$ the electron mass, $\tau_{ph}=135$\mathrm{p}\mathrm{s}$$,
$\gamma_{R}=10^{-3}\mathrm{ps}^{-1}$, $g_{ex}\leavevmode\nobreak\
=1.12\leavevmode\nobreak\ $\mu\mathrm{e}\mathrm{V}\mu\mathrm{m}^{2}$$,
$R_{0}=1.4\times 10^{-3}$\mu\mathrm{m}^{2}\mathrm{p}\mathrm{s}^{-1}$$. The
parameters of the pump ring read: the radius
$R_{P}=R_{in}+(R_{out}-R_{in})/2$, with $R_{out}=15$\mu\mathrm{m}$$ and
$R_{in}=13$\mu\mathrm{m}$$, and $\sigma_{ring}=0.63$\mu\mathrm{m}$$,
corresponding to a FWHM of the laser of $1.5$\mu\mathrm{m}$$. For the
relaxation coefficients we use $\beta_{0}\approx 0.1$ and $\beta_{1}\approx
0.5$. The weak flat pump at the center of the trap has an amplitude parameter
$\alpha_{p}=0.1$.

We simulate the dynamics of the polariton system by numerically integrating
the stochastic differential equations Eq. (2) for the polariton field. The
numerical integration is performed on a two-dimensional lattice with Dirichlet
boundary conditions. The lattice is composed of $128^{2}$ grid points, with
the lattice spacing $\Delta x=0.63\mu m$. The lattice spacing is chosen to lie
within the validity regime of the truncated Wigner methods used for the
description of the stochastic field equations Carusotto and Ciuti (2013);
Comaron  _et al._ (2018), but large enough to capture the macroscopic physics
of the system. We notice that it also introduces a cut-off
$k_{cut}\propto\Delta x^{-1}$ in the momentum representation of the field. The
integration of Eq. (2) in time is performed by using the XMDS2 software
framework Dennis  _et al._ (2013). Specifically, we use a fixed time step
which ensures stochastic noise consistency, and a fourth-order Runge-Kutta
algorithm.

Once the steady-state is achieved, to obtain a smooth first-order correlation
function, $g^{(1)}(r)$, we average over both the correlators at the $x=0$ and
$y=0$ positions, using the expression
$g^{(1)}(r)\equiv[g_{x}^{(1)}(r)+g_{y}^{(1)}(r)]/2$ where
$g_{x}^{(1)}(r)={\langle\psi^{*}_{i,0}\psi_{i+r,0}\rangle_{\mathcal{N}}}/{\sqrt{\langle|\psi_{i,0}|^{2}\rangle_{\mathcal{N}}\langle|\psi_{i+r,0}|^{2}\rangle_{\mathcal{N}}}}$
and
$g_{y}^{(1)}(r)={\langle\psi^{*}_{0,j}\psi_{0,j+r}\rangle_{\mathcal{N}}}/{\sqrt{\langle|\psi_{0,j}|^{2}\rangle_{\mathcal{N}}\langle|\psi_{0,j+r}|^{2}\rangle_{\mathcal{N}}}}$.
Here, $\psi_{i,j}=\psi(x_{i},y_{j})$ and the average
$\langle\dots\rangle_{\mathcal{N}}$ is performed over the number $\mathcal{N}$
of stochastic realisations. All the results presented in our study are
converged with respect to the number of stochastic realisations
$\mathcal{N}=100$.

Acknowledgements
We thank H. Weinberger for fruitful discussions and for proofreading the text.
This research made use of the HPC Computing service at University College of
London. We thank F. Ducluzeau and F. Sidoli for computational assistance. PC
and MSW acknowledge financial support from EPSRC (Grants No. EP/S019669/1, and
No. EP/V026496/1). MM acknowledges support from the National Science Center,
Poland grant No. 2021/43/B/ST3/00752. EE, MW, and EAO acknowledge support from
the Australian Research Council (ARC) through the Centre of Excellence Grant
CE170100039. EE acknowledges support from the ARC Discovery Early Career
Researcher Award (DE220100712). MW acknowledges support by Schmidt Science
Fellows, in partnership with Rhodes Trust.

Additional information
Supplementary Information accompanies this paper at (link)

Competing interests: The authors declare no competing interests.

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