Synchronization crossover of polariton condensates in weakly disordered lattices

H. Ohadi,1, 2, ∗ Y. del Valle-Inclan Redondo,1 A. J. Ramsay,3 Z. Hatzopoulos,4

T. C. H. Liew,5 P. R. Eastham,6 P. G. Savvidis,4, 7, 8 and J. J. Baumberg1, †

1Department of Physics, Cavendish Laboratory, University of Cambridge, Cambridge CB3 0HE, United Kingdom2SUPA, School of Physics and Astronomy, University of St Andrews, St Andrews, KY16 9SS, United Kingdom

3Hitachi Cambridge Laboratory, Hitachi Europe Ltd., Cambridge CB3 0HE, UK4FORTH, Institute of Electronic Structure and Laser, 71110 Heraklion, Crete, Greece

5School of Physical and Mathematical Sciences, Nanyang Technological University 637371, Singapore6School of Physics and CRANN, Trinity College Dublin, Dublin 2, Ireland

7ITMO University, St. Petersburg 197101, Russia8Department of Materials Science and Technology,

University of Crete, 71003 Heraklion, Crete, Greece

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

I. INTRODUCTION

In a lattice of trapped bosons, disorder inhibits coher-ent tunneling so localizing condensation in real space.However on-site repulsive interactions combined with aflow of bosons can tune neighboring condensates into res-onance, thus enhancing tunneling through the trap barri-ers. This subtle interplay is crucial for understanding therich phase diagram composed of Mott-insulator, Bose-glass and superfluid, which acquires further complexitywith pumping and dissipation in nonequilibrium situa-tions. This situation is vital for understanding the ca-pabilities of lattices of condensates to act for instanceas quantum simulators of complex behavior. Exciton-polaritons (polaritons), which are bosons composed ofan admixture of quantum-well exciton and microcavityphoton [1], are an attractive system for studying thenonequilibrium Bose-Hubbard system in two-dimensions[2]. A lattice potential can be achieved either structurallyby depositing thin metallic films [3] and etching [4] themicrocavity or by photo-injecting excitons with a spa-tially patterned laser [5]. In the latter case, the repul-sive on-site interaction can be dynamically tuned throughpolariton-polariton [6] and polariton-exciton nonlineari-ties [7]. For single polaritons the polariton nonlinearityis weaker than the disorder potential, but when polari-tons condense into a macroscopic state [8–10] the collec-tive nonlinearity can be large enough to exhibit effects

∗ ho35@st-andrews.ac.uk† jjb12@cam.ac.uk

such as superfluidity [11, 12] and solitons [13–15]. Po-laritons in one condensate ‘puddle’ can tunnel out anddrive neighboring condensates [16] described through theJosephson mechanism [17]. The mechanism is more com-plicated with disorder-induced energy detunings betweentwo nearest neighbors, in the case of repulsive nonlinear-ity [18] (Fig. 1b). Here there is a critical Josephson tun-neling (Jc), below which an extended condensate sepa-rates into two localized condensates at different energies,forming an unsynchronized (UNSYNC) phase. Howeverabove Jc, the flow of polaritons from the higher-energy tothe lower-energy condensate collapses their energy detun-ing and synchronizes (SYNC) them due to repulsive non-linearities, as the higher energy condensate becomes lesspopulated and the lower energy condensate more popu-lated than before (see lower panel in Fig. 1a). The equiv-alent of a Bose-glass to superfluid crossover is thus ex-pected for a nonequilibrium polariton condensate latticein a disordered potential. This crossover occurs when thephase coherence length, which in a disordered Bose in-sulator grows with increasing density or strength of theJosephson coupling, [19–21] exceeds the overall size ofthe lattice. The Bose-glass to superfluid transition wasfirst observed in Josephson junction arrays [22, 23] andit has recently been observed in thermally equilibratedcold atom lattices [24], but different features emerge forthe nonequilibrium lattice.

Here, we optically trap 25 polariton condensates in-side a 5×5 square lattice within a planar semiconductormicrocavity [25, 26] (Fig. 1a). Optical trapping allowsrapid and facile tuning of the lattice constant as wellas the nearest-neighbor coupling strengths [27]. Due to

arX

iv:1

803.

0319

5v1

[co

nd-m

at.o

ther

] 8

Mar

201

8

2

En

Rea

l spa

ceM

omen

tum

spa

ce

Unsynchronized SynchronizedBelow threshold

Unsynchronized Synchronized

(a) (b) (c) (d)

pump

microcavity

8

FIG. 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 anonzero energy detuning δE 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) syn-chronized (delocalized) condensate lattice. Pump spots are marked by grey circles in (c,d) and visible by dark spots in (b). Theschematic diagrams above the panels show the trapping potential V (x), Josephson coupling J and energies of the condensatesEn. Momentum space images are in logarithmic scale.

residual weak disorder in the microcavity, polaritons atdifferent sites condense at slightly different energies [28].When the lattice constant is large, the nearest neigh-bor coupling is small and we observe an unsynchronized(UNSYNC) phase evidenced by a broad peak in themomentum space corresponding to that of a single sitecondensate as well as a distribution of condensate ener-gies, and no long-range order. However, as the nearest-neighbor coupling increases by reducing the lattice con-stant or increasing the condensate number density (forhigher excitation powers), neighboring condensates col-lapse their energy detunings due to polariton-polaritonand polariton-exciton nonlinearities, until they all syn-chronize (SYNC) resulting in an extended state withlong-range order. This appears as progressive narrow-ing of the momentum space peak and appearance of setsof diffraction peaks signifying a phase-locked fluid witha single delocalized macroscopic wavefunction. Simula-tions show that disorder plays a key role in the crossoverwidth and the decay of long-range order.

II. FORMATION OF TRAPPEDCONDENSATES IN A DISORDERED

POTENTIAL

We create polariton condensates by the nonresonantoptical excitation of a microcavity composed of GaAsquantum-wells sandwiched between two distributedBragg reflectors (DBR). The cavity top (bottom) DBRis made of 32 (35) pairs of Al0.15Ga0.85As/AlAs layers of

57.2 nm/65.4 nm. Four sets of three 10 nm GaAs quan-tum wells separated by 10 nm thick layers of Al0.3Ga0.7Asare placed at the maxima of the cavity light field. The5λ/2 (583 nm) cavity is made of Al0.3Ga0.7As. Photonsin the cavity are strongly coupled to quantum-well exci-tons to form mixed light-matter bosonic polaritons.

The quasi continuous wave pump is a single-modeTi:Sapphire laser tuned to the first Bragg mode∼100 meV above the polariton energy. A spatial lightmodulator is used to spatially pattern the pump beaminto a square lattice. A 0.4 NA objective is used forimaging the pattern onto the sample. A cooled CCDand a 0.55 m spectrometer is used for imaging and en-ergy resolving the emission.

The nonresonant excitation patterned in a square ge-ometry [29] initially creates a plasma of hot electronswhich then relax in energy and form bound excitons. Theexcitons in this ‘reservoir’ eventually relax to form polari-tons. Polaritons are repelled from reservoirs generated ateach pump spot due to repulsive exciton-polariton andpolariton-polariton interactions. The resulting repulsivepotential causes polaritons to roll off and gather at theminima of the square potential. Polaritons condense inthe ground state by stimulated scattering once the den-sity at each center surpasses a critical threshold and forma macroscopic state in each optical trap [25, 26, 30].

As the trapped condensate is moved across the semi-conductor sample, the energy of the condensate variesby a standard deviation of ∼30 µeV (see Appendix A).This ∼30 µeV disorder potential is approximately 10% ofthe confining potential from the optical trap [26], which

3

(a)

UNSYNC

UNSYNC

SYNC

Lattice constant (a)

real-space momentum space real-space momentum space

5 7 9 11

1

2

3

4

5

6

SYNC

(b)

UNSYNC SYNC

Power (Pth)

0.5 1.0 1.5 2.01

10

single-sitelattice

FIG. 2. Synchronization phase diagram. Dependence of δk ·aversus (a) lattice constant, and (b) power. Top panel in (a)shows real-space and momentum-space intensity (log scale)at a =7.8 µm (left) and a =11.8 µm. (b) Power dependencecomparison of δk · a for full lattice (dark blue) and a spa-tially filtered single condensate (light blue). Real-space scaleis 10 µm and momentum space scale is 1 µm−1.

remains relatively constant after condensation.

III. NARROWING OF MOMENTUM

Condensate lattices are created by patterning the opti-cal excitation into a square lattice of 6×6 Gaussian spots(marked by grey circles in Fig. 1c,d) using a spatial lightmodulator. The lattice constant (a) of the trapping po-tential can be continuously varied while monitoring thetotal emission of the condensate lattice in momentumspace (see also Supplemental Movie 1). At low excita-tion powers incoherent polaritons confined by the highestpoints of the lattice potential (dark spots in Fig. 1b) arecreated with a very broad momentum distribution (fullwidth half max = δk ' 2.1 µm−1, k is the in-plane po-lariton momentum). At a threshold power (P = Pth) po-laritons condense in the lattice sites. When the conden-sates are far apart (a > 12 µm) and the coupling betweenthem is small we observe a broad peak (δk ' 0.5 µm−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 constantis reduced to a ' 8 µm when a new diffraction pattern isobserved. It has a narrow center peak (δk ' 0.14 µm−1),with 4 primary interference maxima at opposite cornersof the reciprocal lattice with k ·a = 2π×0.96, suggestingthe lattice is phase locked (Fig. 1d, and also Appendix B).

Tracking the width of the zero-momentum peak (δk)with lattice constant (a) reveals the synchronizationcrossover (Fig. 2). For a > 12 µm, the condensates areweakly coupled and δk is fixed to that of a single latticesite. For 8 < a < 12 µm, δk reduces as the lattice con-stant a decreases until reaching a value of δk · a ∼ 1 ata = 8 µm. This is the value expected for a 5-slit diffrac-tion grating, indicating phase coherence across the entire5×5 lattice. For smaller separations, δk · a remains con-stant until a ' 4 µm where the trap size becomes com-parable to the reservoir width. In this regime, the trapsmerge into a single large inhomogeneous pump spot.

The power dependence of δk for a fixed lattice con-stant (a = 8 µm) shows the evolution of the long-rangecoherence with increasing density (Fig. 2b, also Supple-mental Movie 2). We compare this to emission from asingle condensate using spatial filtering for the same con-ditions. As the lattice power approaches the single-sitecondensation threshold (P = Pth) polaritons condense ineach lattice site and δk drops sharply. At threshold thereis a condensate at each site and the zero-momentum peakhas a width similar to that of the spatially filtered singlecondensate, δk1. This is because there is no long-rangephase correlation and the lattice is in the unsynchronizedphase. However, as power increases further, δk graduallydecreases until it plateaus to δk ' δk1/4, at P ' 2Pth.This value again matches with a 5-slit grating (see SI.B),indicating long-range phase correlation of a delocalizedfluid across the entire lattice.

IV. BUILDUP OF LONG-RANGE COHERENCEAND COLLAPSE OF ENERGY DETUNING

The gradual decrease in momentum width δk withpower implies that phase correlations in the lattice con-tinuously increase with power as the coherence length ofthe order parameter ψ gradually increases. It has beenshown that in a driven-dissipative condensate with dis-order, the phase correlation length Lφ, over which thefirst order correlation function drops by 1/e, increaseswith Josephson coupling and condensates density [20].To confirm this we measure the spatial first-order co-herence function g(1) at zero time delay. The latticeemission is sent to a modified Mach-Zehnder interfer-ometer with a retroreflector in one arm. This inter-feres emission from opposite points relative to the centralcondensate (placing the origin at the white dashed cir-cle 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 Lφ being the phase cor-

4

10 μm

th

(a) (b)

A

C

BD

E

F

th

0.7 P th1.1 P th1.7 P th2.1 P th2.3 P th

A B C D E F

1 /μm

Real space Momentum space

UN

SY

NC

SY

NC

ExperimentSimulation

UNSYNC

SYNC

2 Pth

Pth

(c) (d)

0 1 2 30.0

0.2

0.4

0.6

0.8

2 Pth

1.4 Pth

1.0 1.5 2.0Power ( ) P th

0

2

4

6

8

()

2.2 Pth

1.6 Pth

1.2 Pth

1.1 Pth

0.9 Pth

simulation with disord.simulation without disord.

experiment

Coh

eren

ce le

ngth

,

FIG. 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) experimentand (c) simulations. In the experiment, an average is taken over six site separations marked A-F and their corresponding mirrorsin (a), with separations of (0.5, 2,

√2, 4, 2

√5, 4√

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 = Pth)and SYNC (P = 2Pth) phases. The simulated disorder ratio is 10%. (d) Coherence length (Lφ) vs power in simulations (withand without 10% disorder) and in experiment. Simulation curves are averages of 25 randomly generated disorder potentials.

0.0

0.5

1.0

2.00 2.25 2.50 2.75Blueshift (meV)

0.0

0.5

1.0

Nor

m. i

nten

sity

(a.u

.)

UNSYNC

SYNC

(a) (b)

1.25 1.50 1.75 2.00Power ( )

10

15

20

25

30

1

2

3

4

UNSYNC SYNC

FIG. 4. Condensate energies as a function of power. (a) En-ergy spectra of a column of 5 condensates crossing the centerof the lattice for P = Pth (top panel) and P = 2Pth (bottompanel). (b) Standard deviation of condensate energies (σ(E))as a function of power (light blue) and the corresponding δk ·a(dark blue).

relation length where g(1) drops to 1/e. The interfero-grams (Fig. 3a) clearly differ between the localized (toppanel, P = 1.4Pth) and the delocalized phase (bottompanel, P = 2Pth). Single-site coherence builds up im-mediately after condensation. However, the coherencebetween different lattice sites turns on at a power thatincreases with separation (see also Supplemental Movie3). This first-order spatial coherence function g(1)(−r, r)clearly changes at different powers (Fig. 3b). Before con-densation (P < Pth), Lφ ' 0. At single-site condensa-tion (UNSYNC, P = 1.1Pth), order is confined to onesite. As power increases (1.1Pth < P < 2.1Pth), Lφ in-creases gradually until it reaches ∼8 lattice constants (atP = 2.1Pth), but decays at higher powers due to the ap-pearance of higher order modes (Fig. 3d). We note thatthe increase of g(1) (and the reduction of δk) in steps is

likely due to a percolation effect where domains of con-densates phase-lock in discrete steps.

Measuring the site energies of condensates in the syn-chronization crossover reveals the reason behind this be-havior. At threshold, condensates form at slightly dif-ferent energies (Fig. 4a). The energy distribution ob-served in the UNSYNC phase is a result of spatial in-homogeneities in the sample. However, as the power in-creases the condensate energies converge as long-rangeorder propagates across the lattice, and at the SYNCphase they condense to one energy (Fig. 4b).

V. THEORETICAL MODEL AND NUMERICALSIMULATIONS

Our system can be modelled using the stochasticdriven-dissipative Gross-Pitaevskii (GP) equation whichis a nonlinear Schrodinger equation including pumping,dissipation, energy relaxation and Langevin noise to de-scribe fluctuations:

i~dψ

dt=

(−~2∇2

2m+ V0 + gRn+ gpP + α|ψ|2

)ψ,

+i

2(~Rn− ~γ)ψ − i~Λψ + ~

√γ +Rn

dW

dt(1)

dn

dt=P − (γR +R|ψ|2)n. (2)

Here, ψ is the wavefunction, n is the reservoir density,m is the polariton mass, R is the scattering rate fromthe reservoir to the condensate, γ is the polariton de-cay rate, gR and gp describe energy repulsion due to thepresence of the reservoir and pump, α is the strengthof polariton-polariton interactions, P is the non- reso-nant pumping rate, γR is the reservoir decay rate and

5

(a)

(d)

(b) (c) (g) (i)

(j)(h)(f)(e)

lattice, δksingle, δk

lattice, Isingle, I

10 1

1.0 1.5 2.0

1.0 1.5 2.0

0.0

0.5

1.0

Power ( ) P th

10 −1

10 0

10 1

10 2

Den

sity

40

80

120

10 1

10 2

40

80

120

10 1

10 2

−1

0

1

−1

0

1

0.00

0.15

0.30

a=9 μm

a=15 μm

0.0 2.5 5.00.0

0.5

1.0

FIG. 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.

Λ is a spatial dependent function matching the pumpprofile, which accounts for a phenomenological energyrelaxation. Langevin noise is given by dW which is acomplex Gaussian random variable characterized by cor-relation function 〈dW ∗dW 〉 = dt. Here, we also adda static disorder potential V0 accounting for the sam-ple inhomogeneity. Disorder ratio is defined as DOR =V0/(gRn+gPP +α|ψ|2). The first order correlation func-tion is given by:

g(1)(r,−r) =〈ψ(r)∗ψ(−r)〉√〈|ψ(r)|2〉〈|ψ(−r)|2〉

. (3)

Simulating the condensate lattice in a randomly gen-erated disorder potential demonstrates the crucial roleof disorder in the synchronization crossover, as shown inFig. 5. [31] Here, the disorder ratio (DOR) defined asthe mean ratio of a Gaussian-distributed disorder poten-tial (Fig. 5g) to the confinement potential by the pumppattern is ∼10% (as in the experiment). The condensatelattice crosses from unsynchronized to synchronized asthe lattice constant is shrunk, with similar signatures tothe experiment. In the UNSYNC phase when a = 15 µm[Fig. 5(a-c)], momentum-space peak is broad, conden-sates phases are unlocked and Lφ ' a (blue line inFig. 5h). This changes in the SYNC phase as the lattice isshrunk to a = 9 µm: the momentum-space peak narrowsand new diffraction peaks appear, condensates phase-lock[Fig. 5(d-f)], and Lφ goes beyond the lattice dimensions(red line in Fig. 5h). The power dependence of the SYNClattice shows gradual narrowing of the momentum peakand buildup of long-range coherence [Fig. 5(i,j)].

We find that the power dependence of correlation func-tion strongly depends on the spatial profile of the disorderpotential. For this reason, |g(1)| simulations are averaged

for 25 randomly-generated disorder potentials, as shownin Fig. 3c. The general trend stays the same: in theUNSYNC phase g(1) does not expand beyond a singlesite, whereas in the SYNC phase g(1) builds up through-out the lattice with an exponential drop off (Fig. 3c,d)(see Appendix C for larger lattices). In the synchronizedphase, the Josephson coupling is strong enough to over-come the disorder potential and phase-lock the conden-sates to form a single macroscopic state. Without staticdisorder this synchronization crossover is sharp and thecoherence length is 3 orders of magnitude longer than inthe experiment (Fig. 3d). The minimization of the widthof this synchronization crossover with power can thus beused to search for an optical potential that minimizesthe static disorder potential, which has applications inenabling using such arrays as simulators. We note thatdisorder is inherent to any supported array of conden-sates, suggesting these observations will be universal toall implementations.

VI. CONCLUDING REMARKS

We demonstrated a controllable crossover from unsyn-chronized to synchronized in a driven-dissipative polari-ton condensate lattice with a background disorder poten-tial. The crossover occurs either by increasing the densityof all condensates or the nearest-neighbor Josephson cou-pling. The synchronized (delocalized) phase is accompa-nied by the appearance of long-range order and narrow-ing of the central peak in momentum space, whereas inthe unsynchronized phase order remains local, and themomentum space resembles the emission from a singlecondensate. Using simulations of driven-dissipative GP

6

equations, we find that in the absence of disorder a sharpsynchronization transition with long-range spatial coher-ence is observed. However, the introduction of disorderresults in a softer crossover from unsynchronized to syn-chronized regimes with finite range for the spatial coher-ence.

In thermalized cold atoms in a spatially disorderedBose-Hubbard potential, there is the compressible Bose-glass (insulator) crossover phase between the Mott in-sulator and superfluid phases [23, 24, 32, 33]. There, al-though the condensates share a global chemical potentialthere is no long-range spatial coherence. Instead, the sys-tem forms puddles or domains of coherent condensates,but there is no coherence between neighboring puddles.As the tunneling rate exceeds the disorder potential, thesuperfluid puddles coalesce until there exists a global su-perfluid with a spatial coherence length that exceeds thesize of the system.

By contrast, in the nonequilibrium case studied here,in the crossover regime the puddles or domains have dif-ferent chemical potentials, or emission energies. Further-more, in the superfluid phase the spatial coherence hasa finite range that increases with pump power and tun-neling rate. Hence driven-dissipative generalization ofsuperfluidity can only be observed in a system of finitesize, as discussed theoretically in ref. [20]. Unless ac-tively compensated [28], this may have consequences forthe scale-up of simulators based on lattices of exciton-polariton condensates.

With our capability to optically compensate for disor-der [28], it would be interesting to study the interplaybetween on-demand disorder strengths and the synchro-nization crossover, and their relation to the characteristiclength-scales of the first-order spatial correlation. In ad-dition to fundamental interest, synchronization of arraysof polariton lasers for example, would be advantageousfor creating high-power density coherent sources at lowpump density. Conventional laser diodes locked by injec-tion coupling can synchronize only in a narrow range ofparameters [34], due to their large carrier induced red-shift. By contrast, we demonstrate that with a carrierinduced blue-shift, synchronization readily occurs, sug-gesting that polariton laser approaches may be highlyprofitable.

ACKNOWLEDGMENTS

We thank Jonathan Keeling, Peter Kirton and Ul-rich Schneider for fruitful discussions. We acknowl-edge Grants No. EPSRC EP/L027151/1, ERC LINASS320503, and Leverhulme Trust Grant No. VP1-2013-011.PS acknowledges support from ITMO Fellowship Pro-gram and megaGrant No. 14.Y26.31.0015 of the Min-istry of Education and Science of Russian Federation andGreece-Russia bilateral ‘POLISIMULATOR’ project onQuantum Technologies funded by Greek GSRT. PE ac-knowledges support from SFI Grant No. 15/IACA/3402.

FIG. 6. Background disorder potential

TL was supported by the Singaporean Ministry of Edu-cation (MOE2017-T2-1-001).

Appendix A: Disorder potential

Background disorder potential V0 is measured exper-imentally by moving a single trapped condensate overthe sample and measuring its energy, as shown in Fig. 6.Standard deviation of disorder potential is ∼29 µeV.

Appendix B: Phenomenological model

−2 0 2x a /

−2

0

2

ya /

ψ x

0.0

0.5

1.0

−1 0 1ak x

−1

0

1

ak y

log( )ψ k

5

7

9

−1 0 1ak x

−1

0

1ak

ylog( )ψ k

5

7

9

(a) (b) (c)SYNC UNSYNC

FIG. 7. Intensity of the synchronized phase in real (a) andreciprocal (b) space. (c) Intensity of the UNSYNC phase inmomentum space. The only parameter is L/a = 0.22.

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

A∑n,m

e−((x−na)2+(y−ma)2)/(2L2), (B1)

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

ψ(kx, ky) =1

2π

∫ ∞−∞

∫ ∞−∞

ψ(x, y)ei(kxx+kyy)dxdy

= AL2e−(k2x+k

2y)L

2/2∑n,m

eia(kxn+kym). (B2)

The corresponding intensities of the wavefunctions in realand reciprocal space are shown in Fig. 7. In the UNSYNC

7

x 25

FIG. 8. Fraunhofer diffraction of a single slit (light blue) and5 slits (dark blue) with b/a = 0.35.

0 2 4 6 8 10 12 14 1610

−2

10 −1

10 0

a=8a=9a=9.5a=10a=10.5a=11a=11.5

FIG. 9. Average first order correlation function g(1)(r/a) vs.length r for various lattice constants of a 15×15 condensatelattice.

phase there is no coherence between different Gaussianspots. In this case the intensity in real space is given by:

I(x, y) = |A|2∑n,m

e−((x−na)2+(y−ma)2)/L2

. (B3)

The intensity in reciprocal space is obtained taking the

Fourier transform of each Gaussian spot separately, andsumming the intensities rather than the amplitudes:

I(kx, ky) = NM |A|2L4e−(k2x+k

2y)/L

2

. (B4)

Since there is no coherence between Gaussian spots, theinterference term appearing in Eq. B2 is no longer presentin Eq. B4. The sum over n and m has been reduced tothe total number of spots, NM , in the square lattice.The corresponding intensity of the wavefunctions in themomentum space is shown in Fig. 7c. This model obvi-ously cannot describe why the crossover is not sharp andwhy the coherence length decays over length. For thatwe model the system using a stochastic driven-dissipativeGP equation.

We note here the similarities between the synchronizedlattice intensity profile, and Fraunhofer diffraction frommany slits given by [35]:

I = I0

(sinβ

β

)2(sinNα

sinα

)2

, (B5)

where β = 12k‖b, α = 1

2k‖a. Here, a is the slit separation,b is the slit width, N is the number of slits and I0 includesall the constants to give maximum intensity from a singleslit (Fig. 8). From Eq. B5, we can see that the firstintensity minimum occurs at ka = 2π/N . For 5 slits, fullwidth at half maximum of center peak is δk · a ' 1.13.

Appendix C: Large lattice simulations with disorder

The dependence of first order correlation function withthe lattice constant for a 15 × 15 lattice in a randomdisorder potential (DOR = 10%) shows an exponentialdecay with a coherence length Lφ which grows as thelattice constant a reduces, as shown in Fig. 9.

[1] Alexey V. Kavokin, Jeremy Baumberg, GuillaumeMalpuech, and Fabrice P. Laussy, Microcavities (OxfordUniv. Press, Oxford, 2007).

[2] Iacopo Carusotto and Cristiano Ciuti, “Quantum fluidsof light,” Rev. Mod. Phys. 85, 299–366 (2013).

[3] C. W. Lai, N. Y. Kim, S. Utsunomiya, G. Roumpos,H. Deng, M. D. Fraser, T. Byrnes, P. Recher, N. Kumada,T. Fujisawa, and Y. Yamamoto, “Coherent zero-stateand [pgr]-state in an exciton-polariton condensate array,”Nature 450, 529–532 (2007).

[4] Marta Galbiati, Lydie Ferrier, Dmitry D. Solnyshkov,Dimitrii Tanese, Esther Wertz, Alberto Amo, MarcoAbbarchi, Pascale Senellart, Isabelle Sagnes, AristideLemaıtre, Elisabeth Galopin, Guillaume Malpuech, andJacqueline Bloch, “Polariton Condensation in PhotonicMolecules,” Phys. Rev. Lett. 108, 126403 (2012).

[5] E. Wertz, L. Ferrier, D. D. Solnyshkov, R. Johne, D. San-vitto, A. Lemaıtre, I. Sagnes, R. Grousson, A. V. Ka-vokin, P. Senellart, G. Malpuech, and J. Bloch, “Spon-taneous formation and optical manipulation of extendedpolariton condensates,” Nature Phys. 6, 860–864 (2010).

[6] N. Takemura, S. Trebaol, M. Wouters, M. T. Portella-Oberli, and B. Deveaud, “Polaritonic feshbach reso-nance,” Nature Phys. 10, 500–504 (2014).

[7] T. Gao, P. S. Eldridge, T. C. H. Liew, S. I. Tsintzos,G. Stavrinidis, G. Deligeorgis, Z. Hatzopoulos, and P. G.Savvidis, “Polariton condensate transistor switch,” Phys.Rev. B 85, 235102 (2012).

[8] Hui Deng, Gregor Weihs, Charles Santori, JacquelineBloch, and Yoshihisa Yamamoto, “Condensation ofsemiconductor microcavity exciton polaritons,” Science298, 199–202 (2002).

8

[9] J. Kasprzak, M. Richard, S. Kundermann, A. Baas,P. Jeambrun, J M J Keeling, F M Marchetti, M HSzymanska, R. Andre, J L Staehli, et al., “Bose-einsteincondensation of exciton polaritons.” Nature 443, 409–414 (2006).

[10] J. Baumberg, A. Kavokin, S. Christopoulos, A. Grundy,R. Butte, G. Christmann, D. Solnyshkov, G. Malpuech,G. Baldassarri Hoger von Hogersthal, E. Feltin,et al., “Spontaneous polarization buildup in a room-temperature polariton laser,” Phys. Rev. Lett. 101,136409 (2008).

[11] Alberto Amo, Jerome Lefrere, Simon Pigeon, ClaireAdrados, Cristiano Ciuti, Iacopo Carusotto, RomualdHoudre, Elisabeth Giacobino, and Alberto Bramati,“Superfluidity of polaritons in semiconductor microcavi-ties,” Nature Phys. 5, 805–810 (2009).

[12] Giovanni Lerario, Antonio Fieramosca, Fabio Barachati,Dario Ballarini, Konstantinos S. Daskalakis, Lorenzo Do-minici, Milena De Giorgi, Stefan A. Maier, GiuseppeGigli, Stephane Kena-Cohen, and Daniele Sanvitto,“Room-temperature superfluidity in a polariton conden-sate,” Nature Phys. 13, 837 (2017).

[13] A. Amo, S. Pigeon, D. Sanvitto, V. G Sala, R. Hivet,I. Carusotto, F. Pisanello, G. Lemenager, R. Houdre,E. Giacobino, C. Ciuti, and A. Bramati, “Polariton Su-perfluids Reveal Quantum Hydrodynamic Solitons,” Sci-ence 332, 1167–1170 (2011).

[14] M. Sich, D. N. Krizhanovskii, M. S. Skolnick, A. V. Gor-bach, R. Hartley, D. V. Skryabin, E. A. Cerda-Mendez,K. Biermann, R. Hey, and P. V. Santos, “Observationof bright polariton solitons in a semiconductor microcav-ity,” Nature Photonics 6, 50–55 (2012).

[15] P. M. Walker, L. Tinkler, B. Royall, D. V. Skryabin,I. Farrer, D. A. Ritchie, M. S. Skolnick, and D. N.Krizhanovskii, “Dark Solitons in High Velocity Waveg-uide Polariton Fluids,” Phys. Rev. Lett. 119, 097403(2017).

[16] A. Baas, K. G. Lagoudakis, M. Richard, R. Andre,Le Si Dang, and B. Deveaud-Pledran, “Synchronizedand Desynchronized Phases of Exciton-Polariton Con-densates in the Presence of Disorder,” Phys. Rev. Lett.100, 170401 (2008).

[17] K. G. Lagoudakis, B. Pietka, M. Wouters, R. Andre,and B. Deveaud-Pledran, “Coherent oscillations in anexciton-polariton josephson junction,” Phys. Rev. Lett.105, 120403 (2010).

[18] Michiel Wouters, “Synchronized and desynchronizedphases of coupled nonequilibrium exciton-polariton con-densates,” Phys. Rev. B 77, 121302 (2008).

[19] G. Malpuech, D. D. Solnyshkov, H. Ouerdane, M. M.Glazov, and I. Shelykh, “Bose Glass and SuperfluidPhases of Cavity Polaritons,” Phys. Rev. Lett. 98,206402 (2007).

[20] Alexander Janot, Timo Hyart, Paul R. Eastham, andBernd Rosenow, “Superfluid Stiffness of a Driven Dissi-pative Condensate with Disorder,” Phys. Rev. Lett. 111,230403 (2013).

[21] Bryan Nelsen, Gangqiang Liu, Mark Steger, David W.Snoke, Ryan Balili, Ken West, and Loren Pfeiffer, “Dissi-pationless flow and sharp threshold of a polariton conden-sate with long lifetime,” Phys. Rev. X 3, 041015 (2013).

[22] L. J. Geerligs, M. Peters, L. E. M. de Groot, A. Ver-bruggen, and J. E. Mooij, “Charging effects and quan-tum coherence in regular Josephson junction arrays,”Physical Review Letters 63, 326–329 (1989).

[23] Matthew P. A. Fisher, Peter B. Weichman, G. Grin-stein, and Daniel S. Fisher, “Boson localization andthe superfluid-insulator transition,” Phys. Rev. B 40, 546(1989).

[24] Carolyn Meldgin, Ushnish Ray, Philip Russ, David Chen,David M. Ceperley, and Brian DeMarco, “Probing thebose glass-superfluid transition using quantum quenchesof disorder,” Nature Phys. 12, 646–649 (2016).

[25] P. Cristofolini, A. Dreismann, G. Christmann,G. Franchetti, N. G. Berloff, P. Tsotsis, Z. Hat-zopoulos, P. G. Savvidis, and J. J. Baumberg, “Opticalsuperfluid phase transitions and trapping of polaritoncondensates,” Phys. Rev. Lett. 110, 186403 (2013).

[26] A. Askitopoulos, H. Ohadi, A. V. Kavokin, Z. Hatzopou-los, P. G. Savvidis, and P. G. Lagoudakis, “Polaritoncondensation in an optically induced two-dimensional po-tential,” Phys. Rev. B 88, 041308 (2013).

[27] H. Ohadi, Y. del Valle-Inclan Redondo, A. Dreismann,Y. G. Rubo, F. Pinsker, S. I. Tsintzos, Z. Hatzopou-los, P. G. Savvidis, and J. J. Baumberg, “Tunable mag-netic alignment between trapped exciton-polariton con-densates,” Phys. Rev. Lett. 116, 106403 (2016).

[28] H. Ohadi, A. J. Ramsay, H. Sigurdsson, Y. del Valle-Inclan Redondo, S. I. Tsintzos, Z. Hatzopoulos, T. C. H.Liew, I. A. Shelykh, Y. G. Rubo, P. G. Savvidis, andJ. J. Baumberg, “Spin Order and Phase Transitions inChains of Polariton Condensates,” Phys. Rev. Lett. 119,067401 (2017).

[29] H. Ohadi, A. Dreismann, Y. G. Rubo, F. Pinsker, Y. delValle-Inclan Redondo, S. I. Tsintzos, Z. Hatzopoulos,P. G. Savvidis, and J. J. Baumberg, “Spontaneous SpinBifurcations and Ferromagnetic Phase Transitions in aSpinor Exciton-Polariton Condensate,” Phys. Rev. X 5,031002 (2015).

[30] P Tsotsis, P S Eldridge, T Gao, S I Tsintzos, Z Hat-zopoulos, and P G Savvidis, “Lasing threshold doublingat the crossover from strong to weak coupling regime inGaAs microcavity,” N. J. Phys. 14, 023060 (2012).

[31] Parameters, α = 3 µeVµm2; gr = 2α; gP = 0.6α;Λ = 0.2; m∗ = 5.1 × 10−5me; R = 0.01 ps−1µm2;γ = 0.1 ps−1µm2; γR = 62.5γ;.

[32] Werner Krauth, Nandini Trivedi, and David Ceperley,“Superfluid-insulator transition in disordered boson sys-tems,” Phys. Rev. Lett. 67, 2307–2310 (1991).

[33] L. Fallani, J. E. Lye, V. Guarrera, C. Fort, and M. Ingus-cio, “Ultracold atoms in a disordered crystal of light: To-wards a bose glass,” Phys. Rev. Lett. 98, 130404 (2007).

[34] H. G. Winful and S. S. Wang, “Stability of phase lockingin coupled semiconductor laser arrays,” Applied PhysicsLetters 53, 1894–1896 (1988).

[35] Max Born, Emil Wolf, A. B. Bhatia, P. C. Clemmow,D. Gabor, A. R. Stokes, A. M. Taylor, P. A. Wayman,and W. L. Wilcock, Principles of Optics: Electromag-netic Theory of Propagation, Interference and Diffractionof Light (Cambridge University Press, Cambridge ; NewYork, 1999).