# Synchronization crossover of polariton condensates in weakly disordered
lattices

H. Ohadi<EMAIL_ADDRESS>Department of Physics, Cavendish Laboratory,
University of Cambridge, Cambridge CB3 0HE, United Kingdom  SUPA, School of
Physics and Astronomy, University of St Andrews, St Andrews, KY16 9SS, United
Kingdom  Y. del Valle-Inclan Redondo  Department of Physics, Cavendish
Laboratory, University of Cambridge, Cambridge CB3 0HE, United Kingdom  A. J.
Ramsay  Hitachi Cambridge Laboratory, Hitachi Europe Ltd., Cambridge CB3 0HE,
UK  Z. Hatzopoulos  FORTH, Institute of Electronic Structure and Laser, 71110
Heraklion, Crete, Greece  T. C. H. Liew  School of Physical and Mathematical
Sciences, Nanyang Technological University 637371, Singapore  P. R. Eastham
School of Physics and CRANN, Trinity College Dublin, Dublin 2, Ireland  P. G.
Savvidis  FORTH, Institute of Electronic Structure and Laser, 71110 Heraklion,
Crete, Greece  ITMO University, St. Petersburg 197101, Russia  Department of
Materials Science and Technology, University of Crete, 71003 Heraklion, Crete,
Greece  J. J. Baumberg<EMAIL_ADDRESS>Department of Physics, Cavendish
Laboratory, University of Cambridge, Cambridge CB3 0HE, United Kingdom

###### Abstract

We demonstrate that the synchronization of a lattice of solid-state
condensates when inter-site tunnelling is switched on, depends strongly on the
weak local disorder. This finding is vital for implementation of condensate
arrays as computation devices. The condensates here are nonlinear bosonic
fluids of exciton-polaritons trapped in a weakly disordered Bose-Hubbard
potential, where the nearest neighboring tunneling rate (Josephson coupling)
can be dynamically tuned. The system can thus be tuned from a localized to a
delocalized fluid as the number density, or the Josephson coupling between
nearest neighbors increases. The localized fluid is observed as a lattice of
unsynchronized condensates emitting at different energies set by the disorder
potential. In the delocalized phase the condensates synchronize, and long-
range order appears, evidenced by narrowing of momentum and energy
distributions, new diffraction peaks in momentum space, and spatial coherence
between condensates. Our work identifies similarities and differences of this
nonequilibrium crossover to the traditional Bose-glass to superfluid
transition in atomic condensates.

##  I Introduction

Figure 1: Below threshold, unsynchronized and synchronized phases. (a)
Schematic of condensation in optical lattice potentials (top panel), and
synchronization of two energy-detuned condensates in a double-well potential
(bottom panel). There is a nonzero energy detuning $\delta E$ at the UNSYNC
phase but this vanishes at the SYNC phase as the Josephson coupling increase.
(b-d) Real-space and momentum space emission of (b) uncondensed polaritons,
(c) unsynchronized (localized), and (d) synchronized (delocalized) condensate
lattice. Pump spots are marked by grey circles in (c,d) and visible by dark
spots in (b). The schematic diagrams above the panels show the trapping
potential $V(x)$, Josephson coupling $J$ and energies of the condensates
$E_{n}$. Momentum space images are in logarithmic scale.

In a lattice of trapped bosons, disorder inhibits coherent tunneling so
localizing condensation in real space. However on-site repulsive interactions
combined with a flow of bosons can tune neighboring condensates into
resonance, thus enhancing tunneling through the trap barriers. This subtle
interplay is crucial for understanding the rich phase diagram composed of
Mott-insulator, Bose-glass and superfluid, which acquires further complexity
with pumping and dissipation in nonequilibrium situations. This situation is
vital for understanding the capabilities of lattices of condensates to act for
instance as quantum simulators of complex behavior. Exciton-polaritons
(polaritons), which are bosons composed of an admixture of quantum-well
exciton and microcavity photon Kavokin  _et al._ (2007), are an attractive
system for studying the nonequilibrium Bose-Hubbard system in two-dimensions
Carusotto and Ciuti (2013). A lattice potential can be achieved either
structurally by depositing thin metallic films Lai  _et al._ (2007) and
etching Galbiati  _et al._ (2012) the microcavity or by photo-injecting
excitons with a spatially patterned laser Wertz  _et al._ (2010). In the
latter case, the repulsive on-site interaction can be dynamically tuned
through polariton-polariton Takemura  _et al._ (2014) and polariton-exciton
nonlinearities Gao  _et al._ (2012). For single polaritons the polariton
nonlinearity is weaker than the disorder potential, but when polaritons
condense into a macroscopic state Deng  _et al._ (2002); Kasprzak  _et al._
(2006); Baumberg  _et al._ (2008) the collective nonlinearity can be large
enough to exhibit effects such as superfluidity Amo  _et al._ (2009); Lerario
_et al._ (2017) and solitons Amo  _et al._ (2011); Sich  _et al._ (2012);
Walker  _et al._ (2017). Polaritons in one condensate ‘puddle’ can tunnel out
and drive neighboring condensates Baas  _et al._ (2008) described through the
Josephson mechanism Lagoudakis  _et al._ (2010). The mechanism is more
complicated with disorder-induced energy detunings between two nearest
neighbors, in the case of repulsive nonlinearity Wouters (2008) (Fig. 1b).
Here there is a critical Josephson tunneling ($J_{c}$), below which an
extended condensate separates into two localized condensates at different
energies, forming an unsynchronized (UNSYNC) phase. However above $J_{c}$, the
flow of polaritons from the higher-energy to the lower-energy condensate
collapses their energy detuning and synchronizes (SYNC) them due to repulsive
nonlinearities, as the higher energy condensate becomes less populated and the
lower energy condensate more populated than before (see lower panel in Fig.
1a). The equivalent of a Bose-glass to superfluid crossover is thus expected
for a nonequilibrium polariton condensate lattice in a disordered potential.
This crossover occurs when the phase coherence length, which in a disordered
Bose insulator grows with increasing density or strength of the Josephson
coupling, Malpuech  _et al._ (2007); Janot  _et al._ (2013); Nelsen  _et al._
(2013) exceeds the overall size of the lattice. The Bose-glass to superfluid
transition was first observed in Josephson junction arrays Geerligs  _et al._
(1989); Fisher  _et al._ (1989) and it has recently been observed in thermally
equilibrated cold atom lattices Meldgin  _et al._ (2016), but different
features emerge for the nonequilibrium lattice.

Here, we optically trap 25 polariton condensates inside a 5$\times$5 square
lattice within a planar semiconductor microcavity Cristofolini  _et al._
(2013); Askitopoulos  _et al._ (2013) (Fig. 1a). Optical trapping allows rapid
and facile tuning of the lattice constant as well as the nearest-neighbor
coupling strengths Ohadi  _et al._ (2016). Due to residual weak disorder in
the microcavity, polaritons at different sites condense at slightly different
energies Ohadi  _et al._ (2017). When the lattice constant is large, the
nearest neighbor coupling is small and we observe an unsynchronized (UNSYNC)
phase evidenced by a broad peak in the momentum space corresponding to that of
a single site condensate as well as a distribution of condensate energies, and
no long-range order. However, as the nearest-neighbor coupling increases by
reducing the lattice constant or increasing the condensate number density (for
higher excitation powers), neighboring condensates collapse their energy
detunings due to polariton-polariton and polariton-exciton nonlinearities,
until they all synchronize (SYNC) resulting in an extended state with long-
range order. This appears as progressive narrowing of the momentum space peak
and appearance of sets of diffraction peaks signifying a phase-locked fluid
with a single delocalized macroscopic wavefunction. Simulations show that
disorder plays a key role in the crossover width and the decay of long-range
order.

##  II Formation of trapped condensates in a disordered potential

We create polariton condensates by the nonresonant optical excitation of a
microcavity composed of GaAs quantum-wells sandwiched between two distributed
Bragg reflectors (DBR). The cavity top (bottom) DBR is made of 32 (35) pairs
of Al0.15Ga0.85As/AlAs layers of
$57.2\text{\,}\mathrm{nm}$/$65.4\text{\,}\mathrm{nm}$. Four sets of three
$10\text{\,}\mathrm{n}\mathrm{m}$ GaAs quantum wells separated by
$10\text{\,}\mathrm{n}\mathrm{m}$ thick layers of Al0.3Ga0.7As are placed at
the maxima of the cavity light field. The 5$\lambda$/2
($583\text{\,}\mathrm{nm}$) cavity is made of Al0.3Ga0.7As. Photons in the
cavity are strongly coupled to quantum-well excitons to form mixed light-
matter bosonic polaritons.

The quasi continuous wave pump is a single-mode Ti:Sapphire laser tuned to the
first Bragg mode $\sim 100\text{\,}\mathrm{meV}$ above the polariton energy. A
spatial light modulator is used to spatially pattern the pump beam into a
square lattice. A 0.4 NA objective is used for imaging the pattern onto the
sample. A cooled CCD and a $0.55\text{\,}\mathrm{m}$ spectrometer is used for
imaging and energy resolving the emission.

The nonresonant excitation patterned in a square geometry Ohadi  _et al._
(2015) initially creates a plasma of hot electrons which then relax in energy
and form bound excitons. The excitons in this ‘reservoir’ eventually relax to
form polaritons. Polaritons are repelled from reservoirs generated at each
pump spot due to repulsive exciton-polariton and polariton-polariton
interactions. The resulting repulsive potential causes polaritons to roll off
and gather at the minima of the square potential. Polaritons condense in the
ground state by stimulated scattering once the density at each center
surpasses a critical threshold and form a macroscopic state in each optical
trap Tsotsis  _et al._ (2012); Cristofolini  _et al._ (2013); Askitopoulos
_et al._ (2013).

As the trapped condensate is moved across the semiconductor sample, the energy
of the condensate varies by a standard deviation of $\sim
30\text{\,}\mathrm{\SIUnitSymbolMicro eV}$ (see Appendix A). This $\sim
30\text{\,}\mathrm{\SIUnitSymbolMicro eV}$ disorder potential is approximately
10% of the confining potential from the optical trap Askitopoulos  _et al._
(2013), which remains relatively constant after condensation.

##  III Narrowing of momentum

Figure 2: Synchronization phase diagram. Dependence of $\delta k\cdot a$
versus (a) lattice constant, and (b) power. Top panel in (a) shows real-space
and momentum-space intensity (log scale) at
$a=$$7.8\text{\,}\mathrm{\SIUnitSymbolMicro m}$ (left) and
$a=$$11.8\text{\,}\mathrm{\SIUnitSymbolMicro m}$. (b) Power dependence
comparison of $\delta k\cdot a$ for full lattice (dark blue) and a spatially
filtered single condensate (light blue). Real-space scale is
$10\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and momentum space scale is
$1\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$.

Condensate lattices are created by patterning the optical excitation into a
square lattice of 6$\times$6 Gaussian spots (marked by grey circles in Fig.
1c,d) using a spatial light modulator. The lattice constant ($a$) of the
trapping potential can be continuously varied while monitoring the total
emission of the condensate lattice in momentum space (see also Supplemental
Movie 1). At low excitation powers incoherent polaritons confined by the
highest points of the lattice potential (dark spots in Fig. 1b) are created
with a very broad momentum distribution (full width half max = $\delta
k\simeq$2.1\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}^{-1}$$, $k$ is the
in-plane polariton momentum). At a threshold power ($P=P_{\text{th}}$)
polaritons condense in the lattice sites. When the condensates are far apart
($a>$12\text{\,}\mathrm{\SIUnitSymbolMicro m}$$) and the coupling between them
is small we observe a broad peak ($\delta
k\simeq$0.5\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}^{-1}$$) centered at
zero momentum with no structure, and of a similar width to the emission from a
single condensate (Fig 1c). However, this changes as the lattice constant is
reduced to $a\simeq$8\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ when a new
diffraction pattern is observed. It has a narrow center peak ($\delta
k\simeq$0.14\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}^{-1}$$), with 4
primary interference maxima at opposite corners of the reciprocal lattice with
$\mathbf{k}\cdot\mathbf{a}=2\pi\times 0.96$, suggesting the lattice is phase
locked (Fig. 1d, and also Appendix B).

Tracking the width of the zero-momentum peak ($\delta k$) with lattice
constant ($a$) reveals the synchronization crossover (Fig. 2). For
$a>$12\text{\,}\mathrm{\SIUnitSymbolMicro m}$$, the condensates are weakly
coupled and $\delta k$ is fixed to that of a single lattice site. For
$8<a<$12\text{\,}\mathrm{\SIUnitSymbolMicro m}$$, $\delta k$ reduces as the
lattice constant $a$ decreases until reaching a value of $\delta k\cdot a\sim
1$ at $a=$8\text{\,}\mathrm{\SIUnitSymbolMicro m}$$. This is the value
expected for a 5-slit diffraction grating, indicating phase coherence across
the entire 5$\times$5 lattice. For smaller separations, $\delta k\cdot a$
remains constant until $a\simeq$4\text{\,}\mathrm{\SIUnitSymbolMicro m}$$
where the trap size becomes comparable to the reservoir width. In this regime,
the traps merge into a single large inhomogeneous pump spot.

The power dependence of $\delta k$ for a fixed lattice constant
($a=$8\text{\,}\mathrm{\SIUnitSymbolMicro m}$$) shows the evolution of the
long-range coherence with increasing density (Fig. 2b, also Supplemental Movie
2). We compare this to emission from a single condensate using spatial
filtering for the same conditions. As the lattice power approaches the single-
site condensation threshold ($P=P_{\text{th}}$) polaritons condense in each
lattice site and $\delta k$ drops sharply. At threshold there is a condensate
at each site and the zero-momentum peak has a width similar to that of the
spatially filtered single condensate, $\delta k_{1}$. This is because there is
no long-range phase correlation and the lattice is in the unsynchronized
phase. However, as power increases further, $\delta k$ gradually decreases
until it plateaus to $\delta k\simeq\delta k_{1}/4$, at $P\simeq
2P_{\text{th}}$. This value again matches with a 5-slit grating (see SI.B),
indicating long-range phase correlation of a delocalized fluid across the
entire lattice.

Figure 3: First order correlation function. (a) Real-space interferograms for
the unsynchronized (top) and synchronized (bottom) phases. (b,c) Power
dependence of $|g^{(1)}(r,-r)|$ at various lattice position $r$, as a function
of total lattice power in (b) experiment and (c) simulations. In the
experiment, an average is taken over six site separations marked A-F and their
corresponding mirrors in (a), with separations of
$(0.5,2,\sqrt{2},4,2\sqrt{5},4\sqrt{2})a$. In simulations, the average is
extracted for all site separations in the lattice. Lines are exponential fits.
Panels to the right of (c) are the time-averaged momentum space emission for
UNSYNC ($P=P_{\text{th}}$) and SYNC ($P=2P_{\text{th}}$) phases. The simulated
disorder ratio is 10%. (d) Coherence length ($\mathcal{L}_{\phi}$) vs power in
simulations (with and without 10% disorder) and in experiment. Simulation
curves are averages of 25 randomly generated disorder potentials.

##  IV Buildup of long-range coherence and collapse of energy detuning

Figure 4: Condensate energies as a function of power. (a) Energy spectra of a
column of 5 condensates crossing the center of the lattice for
$P=P_{\text{th}}$ (top panel) and $P=2P_{\text{th}}$ (bottom panel). (b)
Standard deviation of condensate energies ($\sigma(E)$) as a function of power
(light blue) and the corresponding $\delta k\cdot a$ (dark blue).

The gradual decrease in momentum width $\delta k$ with power implies that
phase correlations in the lattice continuously increase with power as the
coherence length of the order parameter $\psi$ gradually increases. It has
been shown that in a driven-dissipative condensate with disorder, the phase
correlation length $\mathcal{L}_{\phi}$, over which the first order
correlation function drops by $1/e$, increases with Josephson coupling and
condensates density Janot  _et al._ (2013). To confirm this we measure the
spatial first-order coherence function $g^{(1)}$ at zero time delay. The
lattice emission is sent to a modified Mach-Zehnder interferometer with a
retroreflector in one arm. This interferes emission from opposite points
relative to the central condensate (placing the origin at the white dashed
circle in Fig. 3a) so that emission from $r$ interferes with $-r$. The
contrast of these interferograms then gives $|g^{(1)}(r,-r)|=|g^{(1)}(-r,r)|$
with $\mathcal{L}_{\phi}$ being the phase correlation length where $g^{(1)}$
drops to $1/e$. The interferograms (Fig. 3a) clearly differ between the
localized (top panel, $P=1.4P_{\text{th}}$) and the delocalized phase (bottom
panel, $P=2P_{\text{th}}$). Single-site coherence builds up immediately after
condensation. However, the coherence between different lattice sites turns on
at a power that increases with separation (see also Supplemental Movie 3).
This first-order spatial coherence function $g^{(1)}(-r,r)$ clearly changes at
different powers (Fig. 3b). Before condensation ($P<P_{\text{th}}$),
$\mathcal{L}_{\phi}\simeq 0$. At single-site condensation (UNSYNC,
$P=1.1P_{\text{th}}$), order is confined to one site. As power increases
($1.1P_{\text{th}}<P<2.1P_{\text{th}}$), $\mathcal{L}_{\phi}$ increases
gradually until it reaches $\sim$8 lattice constants (at
$P=2.1P_{\text{th}}$), but decays at higher powers due to the appearance of
higher order modes (Fig. 3d). We note that the increase of $g^{(1)}$ (and the
reduction of $\delta k$) in steps is likely due to a percolation effect where
domains of condensates phase-lock in discrete steps.

Measuring the site energies of condensates in the synchronization crossover
reveals the reason behind this behavior. At threshold, condensates form at
slightly different energies (Fig. 4a). The energy distribution observed in the
UNSYNC phase is a result of spatial inhomogeneities in the sample. However, as
the power increases the condensate energies converge as long-range order
propagates across the lattice, and at the SYNC phase they condense to one
energy (Fig. 4b).

##  V Theoretical model and numerical simulations

Figure 5: Time-averaged momentum space, real space and phase of an
unsynchronized lattice (a-c) and synchronized lattice (d-f). (g) Static
potential accounting for sample inhomogeneity. (h) $g^{(1)}$ versus length for
UNSYNC (blue) and SYNC phase (red). (i) Power dependence of density (light
blue) and momentum width (dark blue) for a synchronized lattice (solid line)
and a single site (dotted line). (j) Power dependence of $g^{(1)}$ for
different lattice lengths.

Our system can be modelled using the stochastic driven-dissipative Gross-
Pitaevskii (GP) equation which is a nonlinear Schrödinger equation including
pumping, dissipation, energy relaxation and Langevin noise to describe
fluctuations:

$\displaystyle i\hbar\frac{d\psi}{dt}=$
$\displaystyle\left(-\frac{\hbar^{2}\nabla^{2}}{2m}+V_{0}+g_{R}n+g_{p}P+\alpha|\psi|^{2}\right)\psi,$
$\displaystyle+\frac{i}{2}(\hbar
Rn-\hbar\gamma)\psi-i\hbar\Lambda\psi+\hbar\sqrt{\gamma+Rn}\frac{dW}{dt}$ (1)
$\displaystyle\frac{dn}{dt}=$ $\displaystyle P-(\gamma_{R}+R|\psi|^{2})n.$ (2)

Here, $\psi$ is the wavefunction, $n$ is the reservoir density, $m$ is the
polariton mass, $R$ is the scattering rate from the reservoir to the
condensate, $\gamma$ is the polariton decay rate, $g_{R}$ and $g_{p}$ describe
energy repulsion due to the presence of the reservoir and pump, $\alpha$ is
the strength of polariton-polariton interactions, $P$ is the non- resonant
pumping rate, $\gamma_{R}$ is the reservoir decay rate and $\Lambda$ is a
spatial dependent function matching the pump profile, which accounts for a
phenomenological energy relaxation. Langevin noise is given by $dW$ which is a
complex Gaussian random variable characterized by correlation function
$\langle dW^{*}dW\rangle=dt$. Here, we also add a static disorder potential
$V_{0}$ accounting for the sample inhomogeneity. Disorder ratio is defined as
$DOR=\overline{V_{0}}/(g_{R}n+g_{P}P+\alpha|\psi|^{2}).$ The first order
correlation function is given by:

$g^{(1)}(\mathbf{r},\mathbf{-r})=\frac{\langle\psi(\mathbf{r})^{*}\psi(\mathbf{-r})\rangle}{\sqrt{\langle|\psi(\mathbf{r})|^{2}\rangle\langle|\psi(\mathbf{-r})|^{2}\rangle}}.$
(3)

Simulating the condensate lattice in a randomly generated disorder potential
demonstrates the crucial role of disorder in the synchronization crossover, as
shown in Fig. 5.
[Parameters]$\alpha=$3\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{eV}\mathrm{\SIUnitSymbolMicro}\mathrm{m}^{2}$$;
$g_{r}=2\alpha$; $g_{P}=0.6\alpha$; $\Lambda=0.2$; $m^{*}=5.1\times
10^{-5}m_{e}$;
$R=$0.01\text{\,}\mathrm{p}\mathrm{s}^{-1}\mathrm{\SIUnitSymbolMicro}\mathrm{m}^{2}$$;
$\gamma=$0.1\text{\,}\mathrm{p}\mathrm{s}^{-1}\mathrm{\SIUnitSymbolMicro}\mathrm{m}^{2}$$;
$\gamma_{R}=62.5\gamma$; Here, the disorder ratio ($DOR$) defined as the mean
ratio of a Gaussian-distributed disorder potential (Fig. 5g) to the
confinement potential by the pump pattern is ${\sim}10\%$ (as in the
experiment). The condensate lattice crosses from unsynchronized to
synchronized as the lattice constant is shrunk, with similar signatures to the
experiment. In the UNSYNC phase when $a=$15\text{\,}\mathrm{\SIUnitSymbolMicro
m}$$ [Fig. 5(a-c)], momentum-space peak is broad, condensates phases are
unlocked and $\mathcal{L}_{\phi}\simeq a$ (blue line in Fig. 5h). This changes
in the SYNC phase as the lattice is shrunk to
$a=$9\text{\,}\mathrm{\SIUnitSymbolMicro m}$$: the momentum-space peak narrows
and new diffraction peaks appear, condensates phase-lock [Fig. 5(d-f)], and
$\mathcal{L}_{\phi}$ goes beyond the lattice dimensions (red line in Fig. 5h).
The power dependence of the SYNC lattice shows gradual narrowing of the
momentum peak and buildup of long-range coherence [Fig. 5(i,j)].

We find that the power dependence of correlation function strongly depends on
the spatial profile of the disorder potential. For this reason, $|g^{(1)}|$
simulations are averaged for 25 randomly-generated disorder potentials, as
shown in Fig. 3c. The general trend stays the same: in the UNSYNC phase
$g^{(1)}$ does not expand beyond a single site, whereas in the SYNC phase
$g^{(1)}$ builds up throughout the lattice with an exponential drop off (Fig.
3c,d) (see Appendix C for larger lattices). In the synchronized phase, the
Josephson coupling is strong enough to overcome the disorder potential and
phase-lock the condensates to form a single macroscopic state. Without static
disorder this synchronization crossover is sharp and the coherence length is 3
orders of magnitude longer than in the experiment (Fig. 3d). The minimization
of the width of this synchronization crossover with power can thus be used to
search for an optical potential that minimizes the static disorder potential,
which has applications in enabling using such arrays as simulators. We note
that disorder is inherent to any supported array of condensates, suggesting
these observations will be universal to all implementations.

##  VI Concluding remarks

We demonstrated a controllable crossover from unsynchronized to synchronized
in a driven-dissipative polariton condensate lattice with a background
disorder potential. The crossover occurs either by increasing the density of
all condensates or the nearest-neighbor Josephson coupling. The synchronized
(delocalized) phase is accompanied by the appearance of long-range order and
narrowing of the central peak in momentum space, whereas in the unsynchronized
phase order remains local, and the momentum space resembles the emission from
a single condensate. Using simulations of driven-dissipative GP equations, we
find that in the absence of disorder a sharp synchronization transition with
long-range spatial coherence is observed. However, the introduction of
disorder results in a softer crossover from unsynchronized to synchronized
regimes with finite range for the spatial coherence.

In thermalized cold atoms in a spatially disordered Bose-Hubbard potential,
there is the compressible Bose-glass (insulator) crossover phase between the
Mott insulator and superfluid phases Fisher  _et al._ (1989); Krauth  _et al._
(1991); Fallani  _et al._ (2007); Meldgin  _et al._ (2016). There, although
the condensates share a global chemical potential there is no long-range
spatial coherence. Instead, the system forms puddles or domains of coherent
condensates, but there is no coherence between neighboring puddles. As the
tunneling rate exceeds the disorder potential, the superfluid puddles coalesce
until there exists a global superfluid with a spatial coherence length that
exceeds the size of the system.

By contrast, in the nonequilibrium case studied here, in the crossover regime
the puddles or domains have different chemical potentials, or emission
energies. Furthermore, in the superfluid phase the spatial coherence has a
finite range that increases with pump power and tunneling rate. Hence driven-
dissipative generalization of superfluidity can only be observed in a system
of finite size, as discussed theoretically in ref. Janot  _et al._ (2013).
Unless actively compensated Ohadi  _et al._ (2017), this may have consequences
for the scale-up of simulators based on lattices of exciton-polariton
condensates.

With our capability to optically compensate for disorder Ohadi  _et al._
(2017), it would be interesting to study the interplay between on-demand
disorder strengths and the synchronization crossover, and their relation to
the characteristic length-scales of the first-order spatial correlation. In
addition to fundamental interest, synchronization of arrays of polariton
lasers for example, would be advantageous for creating high-power density
coherent sources at low pump density. Conventional laser diodes locked by
injection coupling can synchronize only in a narrow range of parameters Winful
and Wang (1988), due to their large carrier induced red-shift. By contrast, we
demonstrate that with a carrier induced blue-shift, synchronization readily
occurs, suggesting that polariton laser approaches may be highly profitable.

## Acknowledgments

We thank Jonathan Keeling, Peter Kirton and Ulrich Schneider for fruitful
discussions. We acknowledge Grants No. EPSRC EP/L027151/1, ERC LINASS 320503,
and Leverhulme Trust Grant No. VP1-2013-011. PS acknowledges support from ITMO
Fellowship Program and megaGrant No. 14.Y26.31.0015 of the Ministry of
Education and Science of Russian Federation and Greece-Russia bilateral
‘POLISIMULATOR’ project on Quantum Technologies funded by Greek GSRT. PE
acknowledges support from SFI Grant No. 15/IACA/3402. TL was supported by the
Singaporean Ministry of Education (MOE2017-T2-1-001).

##  Appendix A Disorder potential

Figure 6: Background disorder potential

Background disorder potential $V_{0}$ is measured experimentally by moving a
single trapped condensate over the sample and measuring its energy, as shown
in Fig. 6. Standard deviation of disorder potential is $\sim
29\text{\,}\mathrm{\SIUnitSymbolMicro eV}$.

##  Appendix B Phenomenological model

Figure 7: Intensity of the synchronized phase in real (a) and reciprocal (b)
space. (c) Intensity of the UNSYNC phase in momentum space. The only parameter
is $L/a=0.22$.

Let us assume a coherent superposition of Gaussian wavefunctions localized at
each lattice point:

$A\sum_{n,m}e^{-\left((x-na)^{2}+(y-ma)^{2}\right)/(2L^{2})},$ (4)

where $A$ is an overall amplitude, $a$ is the lattice constant, and $L$
defines the width of each Gaussian. In reciprocal space the wavefunction is
given by the Fourier transform:

$\displaystyle\tilde{\psi}(k_{x},k_{y})$
$\displaystyle=\frac{1}{2\pi}\int^{\infty}_{-\infty}\int^{\infty}_{-\infty}\psi(x,y)e^{i(k_{x}x+k_{y}y)}dxdy$
$\displaystyle=AL^{2}e^{-(k_{x}^{2}+k_{y}^{2})L^{2}/2}\sum_{n,m}e^{ia(k_{x}n+k_{y}m)}.$
(5)

The corresponding intensities of the wavefunctions in real and reciprocal
space are shown in Fig. 7. In the UNSYNC phase there is no coherence between
different Gaussian spots. In this case the intensity in real space is given
by:

$I(x,y)=|A|^{2}\sum_{n,m}e^{-\left((x-na)^{2}+(y-ma)^{2}\right)/L^{2}}.$ (6)

The intensity in reciprocal space is obtained taking the Fourier transform of
each Gaussian spot separately, and summing the intensities rather than the
amplitudes:

$\tilde{I}(k_{x},k_{y})=NM|A|^{2}L^{4}e^{-(k_{x}^{2}+k_{y}^{2})/L^{2}}.$ (7)

Since there is no coherence between Gaussian spots, the interference term
appearing in Eq. 5 is no longer present in Eq. 7. The sum over $n$ and $m$ has
been reduced to the total number of spots, $NM$, in the square lattice. The
corresponding intensity of the wavefunctions in the momentum space is shown in
Fig. 7c. This model obviously cannot describe why the crossover is not sharp
and why the coherence length decays over length. For that we model the system
using a stochastic driven-dissipative GP equation.

Figure 8: Fraunhofer diffraction of a single slit (light blue) and 5 slits
(dark blue) with $b/a=0.35$. Figure 9: Average first order correlation
function $g^{(1)}(r/a)$ vs. length $r$ for various lattice constants of a
15$\times$15 condensate lattice.

We note here the similarities between the synchronized lattice intensity
profile, and Fraunhofer diffraction from many slits given by Born  _et al._
(1999):

$I=I_{0}\left(\frac{\sin{\beta}}{\beta}\right)^{2}\left(\frac{\sin{N\alpha}}{\sin{\alpha}}\right)^{2},$
(8)

where $\beta=\frac{1}{2}k_{\parallel}b$, $\alpha=\frac{1}{2}k_{\parallel}a$.
Here, $a$ is the slit separation, $b$ is the slit width, $N$ is the number of
slits and $I_{0}$ includes all the constants to give maximum intensity from a
single slit (Fig. 8). From Eq. 8, we can see that the first intensity minimum
occurs at $ka=2\pi/N$. For 5 slits, full width at half maximum of center peak
is $\delta k\cdot a\simeq 1.13$.

##  Appendix C Large lattice simulations with disorder

The dependence of first order correlation function with the lattice constant
for a $15\times 15$ lattice in a random disorder potential ($DOR=10\%$) shows
an exponential decay with a coherence length $\mathcal{L}_{\phi}$ which grows
as the lattice constant $a$ reduces, as shown in Fig. 9.

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