Power-law decay of the spatial correlationfunction in exciton-polariton condensatesGeorgios Roumposa,1,2, Michael Lohsea,b, Wolfgang H. Nitschea, Jonathan Keelingc, Marzena Hanna Szymańskad,e,Peter B. Littlewoodf,g, Andreas Löfflerh, Sven Höflingh, Lukas Worschechh, Alfred Forchelh, and Yoshihisa Yamamotoa,i

aE. L. Ginzton Laboratory, Stanford University, Stanford, CA 94305; bFakultät für Physik, Universität Karlsruhe, Karlsruhe 76128, Germany; cScottishUniversities Physics Alliance, School of Physics and Astronomy, University of St. Andrews, St. Andrews, KY16 9SS, United Kingdom; dDepartment ofPhysics, University of Warwick, Coventry, CV4 7AL, United Kingdom; eLondon Centre for Nanotechnology, London, WC1H 0AH, United Kingdom;fArgonne National Laboratory, Argonne, IL 60439; gJames Franck Institute, University of Chicago, Chicago, IL 60637; hTechnische Physik, UniversitätWürzburg, Physikalisches Institut and Wilhelm-Conrad-Röntgen-Research Center for Complex Material Systems, Würzburg 97070, Germany; andiNational Institute of Informatics, Tokyo 153-8505, Japan

Edited by Peter Zoller, University of Innsbruck, Austria, and approved February 27, 2012 (received for review May 20, 2011)

We create a large exciton-polariton condensate and employ aMichelson interferometer setup to characterize the short- and long-distance behavior of the first order spatial correlation function.Our experimental results show distinct features of both the two-dimensional and nonequilibrium characters of the condensate. Wefind that the gaussian short-distance decay is followed by a power-law decay at longer distances, as expected for a two-dimensionalcondensate. The exponent of the power law is measured in therange 0.9–1.2, larger than is possible in equilibrium. We comparethe experimental results to a theoretical model to understand thefeatures required to observe a power law and to clarify the influ-ence of external noise on spatial coherence in nonequilibriumphase transitions. Our results indicate that Berezinskii–Kosterlitz–Thouless-like phase order survives in open-dissipative systems.

quantum well excitons ∣ semiconductor microcavities

The spatial correlation function quantifies the coherence prop-erties of a system (1). In a 3D Bose-condensed gas, long range

order is observed, and the correlation function decays towarda plateau at large distances (2, 3). In the homogeneous 2D Bosegas (4), however, no long range order can be established (5).Instead, Berezinskii–Kosterlitz–Thouless (BKT) theory of theequilibrium interacting gas predicts a transition to a low-tempera-ture superfluid phase, which shows a power-law decay of thecorrelation function (6, 7). Unfortunately, it is frequently hardto directly measure this, and only very recently (8) was indicationof the power-law decay of coherence seen in a 2D atomic gas. Ithas been theoretically predicted (9, 10) that power-law decay ofcoherence survives in the nonequilibrium problem, and it is thisprediction that the current experiment sets out to test.

Exciton polaritons are short-lived quasiparticles formed in asemiconductor quantum well strongly coupled to a planar micro-cavity (11). Each one is a superposition of a quantum well excitonand a microcavity photon, and they behave as 2D bosons belowthe Mott density. Above a threshold particle density, condensa-tion is observed (12). Due to the nonequilibrium nature of polar-iton condensation, understanding its coherence properties isquite revealing regarding the different roles of fluctuations in theequilibrium and nonequilibrium problems.

Previous measurements on polariton condensates have de-monstrated coherence at large distances but were limited by largeexperimental uncertainties (13) or highly disordered samples(14, 15), and the long-distance behavior could not be fully ex-tracted. Recently, the correlation function at large distances wasstudied in 1D condensates confined in a quantum wire (16) and ina valley of the disorder potential (17). In ref. 16, the data wasenergy-resolved so that excited states were filtered out, whilein ref. 17 a rare area on the sample was chosen in which a singlemode condensate is seen. The purpose of both those experimentswas to investigate how long the coherence length of a spectrallyisolated 1D condensate state can be. We, on the other hand, are

interested in the functional form of the correlation function in a2D condensate and how the excitations populated by the pump-ing and decay processes can modify it.

With our setup, we can measure values of gð1ÞðrÞ as low as 0.02,so we can reliably extract the long-distance behavior. We findthat, although true thermal equilibrium is not established, aneffective thermal de Broglie wavelength can still be defined fromthe short-distance gaussian decay of gð1ÞðrÞ. Furthermore, gð1ÞðrÞat long distances r decays according to a power law, in analogy tothe equilibrium BKT superfluid phase. The exponent of thepower-law decay is, however, higher than can be possible withinthe BKT theory. We apply a nonequilibrium theory (9, 10) toidentify the source of the large exponent. We argue that, althoughthe spectrum is modified due to dissipation, the exponent wouldstill have the equilibrium value if the spectrum was thermallypopulated. If, on the other hand, a white noise source acts onthe system and induces a flat occupation of the excited states,the exponent can have a large value, proportional to the noisestrength. We therefore conclude that the pumping and decay pro-cesses, which introduce a nonthermal occupation of the excitedstates, can be responsible for the large value of the exponent.

ResultsIn our study, we use a weak-disorder GaAs-based sample, thesame one as in our recent experiments (18). The condensate isgenerated nonresonantly by the multimode laser, which createsfree electon-hole pairs at an excitation energy approximately100 meV above the lower polariton (LP) energy. Carriers suffermultiple scatterings before reaching the LP energy, so coherenceis established spontaneously in the condensate and cannot beinherited from the laser pump. The laser is continuously on andreplaces LPs that leak out of the microcavity at a ps rate. We areinterested in the limit of the homogeneous 2D polariton gas.For this purpose, we employ a setup based on a refractive beamshaper that forms a large laser excitation spot with uniform in-tensity. There is no confining potential on our sample. Becauseof the short lifetime, however, the condensate density follows thephoton density of the excitation spot, so we can create circularcondensates with almost flat density and diameters ranging from14 μm to 44 μm (see ref. 18 and SI Appendix). LP luminescence inthe steady state is observed through a combination of a long-pass

Author contributions: P.B.L. and Y.Y. designed research; G.R., M.L., W.H.N., J.K., andM.H.S. performed research; A.L., S.H., L.W., and A.F. contributed new reagents/analytictools; G.R. and M.L. analyzed data; and G.R. and J.K. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.1Present address: JILA, University of Colorado, Boulder, CO 803092To whom correspondence should be addressed. E-mail: georgios.roumpos@jila.colorado.edu.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1107970109/-/DCSupplemental.

www.pnas.org/cgi/doi/10.1073/pnas.1107970109 PNAS Early Edition ∣ 1 of 6

PHYS

ICS

Dow

nloa

ded

by g

uest

on

Janu

ary

31, 2

020

and a band-pass interference filter, which reject scattered laserlight without distorting the LP spectrum.

We confirmed that the sample disorder potential is weak in twoways (see SI Appendix). First, the lineshape of the luminescenceat low excitation power is Lorentzian, which is characteristic of ahomogeneously broadened line. Second, we measured a 2D mapof the disorder potential with resolution approximately 1 μm andfound that its spatial fluctuations are indeed weaker than thehomogeneous broadening and also much weaker than the energyshift due to polariton–polariton interactions. Therefore, we canignore the sample disorder in our experiment. The condensate isstill localized in space, though, following the shape of the laserexcitation spot.

The first order spatial correlation function is defined as

gð1Þðr1; t1; r2; t2Þ ¼hψ †

1ψ2iffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffihψ †

1ψ1ihψ †2ψ2i

q [1]

where ψ †i and ψ i are the creation and annihilation field operators

at space-time point ðri; tiÞ. To measure this function, we built aMichelson interferometer setup. A schematic is shown in Fig. 1A.It includes a mirror in one arm, and a right angle prism in theother. We overlap the condensate real-space image with itsreflected version, so that fringes similar to that of Fig. 1B areobserved on the camera. By changing the length of one interfe-rometer arm, as shown in Fig. 1A, the relative phase of the twobeams is shifted. As a result, the intensity measured at one pixelpoint shows a sinusoidal modulation (Fig. 1C). From the data ofFig. 1C, we extract the phase difference of the two images at a

particular pixel point, as well as the fringe visibility. The latter isproportional to the first order correlation function, which is thephysical quantity we are interested in in this experiment.

The prismM2 in Fig. 1A forms the reflection of the condensateimage along the prism axis. Therefore, point ðx; yÞ overlaps witheither ð−x; yÞ, or ðx; −yÞ on the camera, depending on the orien-tation of the prism. This allows us to measure

gð1Þðx; −x; τÞ ≡ hgð1Þðx; y; tþ τ;−x; y; tÞit; [2]

or

gð1Þðy; −y; τÞ ≡ hgð1Þðx; y; tþ τ; x; −y; tÞit; [3]

where hit denotes time average. In this experiment, we are mainlyinterested in interference at τ ¼ 0, so when the time argument isnot mentioned explicitly, we imply τ ¼ 0.

We repeat the procedure explained in Fig. 1 for every pixel, sothat we measure the phase difference between the two interferingimages in addition to the correlation function across the wholespot. Representative data are shown in Fig. 2. Recording boththese quantities allows us to identify useful signal from systematicor random noise. Because the prism displaces the beam that isincident on it, the images from the mirror and the prism arefocused on the camera from different angles, so the two phasefronts are tilted with respect to each other. As a consequence, weexpect to measure a constant phase tilt. This is the case in Fig. 2B,in which the laser power is above threshold and a condensate hasformed. We conclude that our measurement of the correlationfunction in Fig. 2D is reliable over this whole area. On the otherhand, at a pump rate below threshold, only short-range correla-tions exist. Fig. 2A shows that in this case the phase difference ismeasured correctly only over a small area around the center,(jxj ≤ 1 μm). So the measured values of gð1Þðx; −xÞ outside thisarea are not reliable and give an estimate of our measurementuncertainties. As is clear from Fig. 2C, the experimental errorcan be suppressed down to 0.01.

Phase maps such as those in Fig. 2 have been used to identifylocalized phase defects—namely, quantum vortices (19). Thedata of Fig. 2B show that such localized defects are not presentin our sample. At points with large fringe visibility (nearx ¼ 0 μm), fringes are perfectly parallel, whereas defects that ap-pear for large jxj could be due to a numerical uncertainty in themeasurement of the local phase due to the small fringe visibility.In any case, localized stationary phase defects cannot influencegð1ÞðrÞ, because it is their motion that destroys spatial correlationsand not their mere presence. It has been found that vorticesappear in large disorder samples (19), when a direct externalperturbation is introduced (20), before the condensate reachesits steady state (21), or when the condensate moves against anobstacle (22, 23). None of these conditions is satisfied in our ex-periment. On the other hand, we have found that, under the sameconditions as the current experiment, mobile bound vortex pairsappear spontaneously due to the special form of the pumpingspot and the pumping and decay noise (18). In ref. 18), we foundthat a single mobile bound vortex–antivortex pair is visible in asmall condensate. In the current experiment, we probe largercondensate sizes, so it is likely that several vortex pairs are pre-sent at the same time. Mobile bound vortex pairs are in generalinvisible in time-integrated phase maps, like the one in Fig. 2B,and they are consistent with a power-law decay of gð1ÞðrÞ.

Fig. 3A shows the short-distance dependence of gð1Þðx; −xÞ forthe same pumping power as in Fig. 2A andC. Every dot in Fig. 3Acorresponds to one pixel on the camera, and the x axis is itsdistance from the axis of reflection (slightly tilted with respect tothe columns of the charge-coupled device array). Data at dis-tances jxj > 1 μm is noise, because the measured phase in thisarea is random (Fig. 2A). At shorter distances, we can measuregð1Þðx; −xÞ reliably, and we find that the correlation function has a

x (µm)

y (µ

m)

−10 0 10−15

−10

−5

0

5

10

15B C

A

0

500

1,000

1,500

2,000

2,500

0.5 1 1.5 20

1

2x 10

4

Phase / 2π

Inte

nsi

ty (

a.u

.)

Fig. 1. Michelson interferometer. (A) Schematic of the setup for measure-ment of the correlation function. The laser is linearly polarized, and we re-cord luminescence of the orthogonal linear polarization through a polarizingbeamsplitter (PBS). We then employ a 50-50 nonpolarizing beamsplitter(NPBS), a mirror (M1) and a right-angle prism (M2). The latter creates the re-flection of the original image along one axis, depending on the prism orien-tation. A two-lens microscope setup overlaps the two real space images ofthe polariton condensate on the camera. (B) Typical interference pattern ob-served above the polariton condensation threshold along with a schematicshowing the orientation of the two overlapping images. (C) Blue circles: mea-sured intensity on one pixel of the camera as a function of the prism (M2)position in normalized units. Red line: fitting to a sine function.

2 of 6 ∣ www.pnas.org/cgi/doi/10.1073/pnas.1107970109 Roumpos et al.

Dow

nloa

ded

by g

uest

on

Janu

ary

31, 2

020

gaussian form. This is the same functional dependence as for athermalized Bose gas when the temperature is sufficiently high orthe density sufficiently small (2, 4). In that equilibrium case,the width of the gaussian decay is proportional to the thermalde Broglie wavelength. Although our nonequilibrium system isquite different than the thermalized Bose gas, we will use thisanalogy to define a thermal de Broglie wavelength and thereforealso a temperature. We note that the temperature measured fromthe short-distance behavior of gð1Þðx; −xÞ is a measure of theoccupation of the higher energy part of the spectrum (i.e., theparticle-like part of the spectrum). For an insufficiently therma-lized system, it is quite possible that excitations in different energyranges have different effective temperatures. Therefore, the tem-perature measured this way will not necessarily agree with othermeasures of temperature.

In Fig. 3B we plot the effective wavelength λeff as a function ofpumping power. If σ is the standard deviation of the gaussian fitfor gð1Þðx; −xÞ, λeff ¼ 2

ffiffiffiffiffiffi2π

pσ in analogy to the thermal de Broglie

wavelength. λeff shows a smooth increase for increasing pumping

power with no obvious threshold, analogous to the theory ofequilibrium noninteracting 2D Bose gas as the particle density isincreased (4). We performed the same experiment for two ortho-gonal prism orientations as shown in the legend of Fig. 3B. Inone case we measured gð1Þðx; −xÞ, whereas in the other case wemeasured gð1Þðy; −yÞ. We found that λeff is shorter along the y axisand attribute this difference to a small asymmetry of the laserpumping spot. The occupation of excited states (which deter-mines λeff) depends on their spatial overlap with the laser pump-ing spot, so states of equal energy are not always equallypopulated. This asymmetry shows that λeff is not simply relatedto the cryostat temperature and depends on the spatial andenergy profiles of the high-energy states involved in producingthis correlation length. We also note that the resolution limit ofour imaging setup is approximately 1 μm, hence the measure-ment of λeff at small pumping power is resolution-limited.

It is known that an ideal autocorrelation measurement with aMichelson interferometer provides the same information as anideal measurement of the spectrum. In particular, gð1Þðx; −x; tÞis the Fourier transform of the power spectrum in momentumspace Sðk; ωÞ (24). However, systematic noise in measurement ofSðk; ωÞ currently makes the direct measurement of gð1Þðx; −x; tÞthe only way to reliably extract λeff of Fig. 3B as well as the power-law decay at long distances to be explained later. The Fourier-transform relationship between gð1Þðx; −x; tÞ and Sðk; ωÞ is illu-strated in Fig. 4. The measured gð1Þðx; −x; tÞ at very low pumpingpower is shown in Fig. 4A. At time delay t ¼ 0, it has a gaussianform as a function of x, but for increasing t it broadens andacquires a multipeak structure. This unusual space-time depen-dence is reproduced by the numerical Fourier transform (Fig. 4C)of measured Sðk; ωÞ (Fig. 4B). As explained in SI Appendix,measurement of the time dependence of gð1Þðx; −x; tÞ is limited byinhomogeneous broadening due to time-integrated data, so itcannot provide an estimate of the homogeneous dephasing time.

At long distances, the behavior of the correlation function atzero time delay t ¼ 0 is no longer gaussian. We found that it is

0.7×Pth

x (µm)

y (µ

m)

−10 0 10

−10

0

10

2.4×Pth

x (µm)

y (µ

m)

−20 −10 0 10 20

−20

−10

0

10

20

−15 −10 −5 0 5 10 1510

−3

10−2

10−1

100

x (µm)

g(1

) (x,−

x)

−15 −10 −5 0 5 10 1510

−3

10−2

10−1

100

x (µm)

g(1

) (x,−

x)

A B

C D

Fig. 2. Phase map measured for laser power (A) below and (B) above the threshold power Pth. The prism in the Michelson interferometer is orientedhorizontally. The schematics on the top right of A and B show the orientation of the two interfering images. (C and D) Measured gð1Þðx; −xÞ correspondingto A and B, respectively, averaged over the y axis inside the excitation spot area of 19-μm radius. Blue circles are experimental data. The continuous red anddashed yellow fitting lines are explained in Figs. 3 and 6, respectively.

−3 −2 −1 0 1 2 310

−3

10−2

10−1

100

37.3 mW

x (µm)

g(1

) (x,−

x)

0 100 200 300 4001

2

3

4

5

6

Pumping power (mW)

λ eff(µ

m)

A B

Fig. 3. (A) Decay of gð1Þðx; −xÞ at short distances. Blue dots are experimentaldata, the red line is a gaussian fit. Data at jxj > 1 μm is noise. (B) Effective deBroglie wavelength λeff as a function of laser pumping power. λeff is extractedfrom the width of the gaussian fit as shown in A. Blue circles and red squarescorrespond to orthogonal orientations of the prism in the Michelson inter-ferometer (see text). The condensation threshold is at approximately 55 mW.

Roumpos et al. PNAS Early Edition ∣ 3 of 6

PHYS

ICS

Dow

nloa

ded

by g

uest

on

Janu

ary

31, 2

020

influenced by the edge of the condensate. In Fig. 5, we plot themeasured gð1ÞðΔxÞ ¼ gð1Þðj2xjÞ ≡ gð1Þðx; −xÞ at pumping powerP ∼ 3 × Pth for increasing pumping spot radius. The measuredgð1ÞðΔxÞ at long distances decreases as the spot size is increasedand eventually converges to a power-law decay for large con-densates.

We note that the condensate size is slightly smaller than thepump laser spot radius (3–4 μm smaller from each side for a largespot). Because of the repulsive interaction between polaritons,and between polaritons and reservoir excitons, the large densityof the condensate and reservoir creates an antitrapping potentialthat pushes LPs away from the center. This effect is stronger for agaussian or a very small pumping spot and in long-lifetime sam-ples (16, 25), whereas in the present experiment it only influencesLPs that are close to the edge.

In the case of a large condensate, we should recover the limitof (infinitely large) homogeneous polariton gas. Therefore, weconsider a pumping spot radius R0 ¼ 19 μm. In Fig. 6A, we plotthe correlation function gð1ÞðΔxÞ versus Δx as the pumping poweris increased. Only short-range correlations exist for small pump-

ing power, whereas above the condensation threshold of approxi-mately 55 mW (4.8 kW∕cm2), substantial phase coherenceappears across the whole spot. The functional form of the long-distance decay is measured to be a power law over about onedecade, as can be seen in Fig. 6B, in which we plot the data atone specific laser power. We fit the data to a function gð1ÞðΔxÞ ¼ðλp∕ΔxÞap and plot the exponent ap as a function of pumpingpower in Fig. 6C. It is found to be in the range 0.9–1.2. λp isa parameter with units of length and is not related to λeff , whichis plotted in Fig. 3B*.

It has been claimed that a criterion for polariton condensationis the appearance of a second threshold as the pumping power isincreased (26, 27). The state after the first threshold was called a“polariton BEC,” “polariton laser,” or “polariton condensate,”whereas the state after the second threshold has not been fullyunderstood yet. It might be a Bardeen–Cooper–Schrieffer (BCS)crossover (28, 29), photon BEC (30), or photon laser (26). Thisdouble threshold behavior has been observed in micropillar struc-tures (26), and using a stress trap (27). In the supplementaryinformation of ref. 18, we also reported the observation of doublethreshold using the same sample and excitation conditions as inthe present experiment. We found that in our sample the windowof intensities between the two thresholds is not very wide, andcan only be witnessed using a flat laser excitation spot. This spotcreates a uniform polariton density over a large area, as opposedto the more common gaussian spot, where the density changes alot across the pumping spot.

Finally, we repeated the same measurement of gð1Þðx; −xÞusing an identical sample at a temperature of 200 K. Becauseof the small binding energy, the GaAs excitonic effect is weakat this temperature. Also, the lasing energy was well above thebandgap. Therefore, only standard photon lasing was possible.In this case, we only found exponential decay of the correlationfunction and no power law. The details of this measurement arereported in SI Appendix. This suggests that the interactions ofthe strongly coupled exciton-polaritons are essential in the obser-vation of the reported phenomena

P ~0.002 Pth

t (ps)

x (µ

m)

−2 0 2 4

−4

−3

−2

−1

0

1

2

3

4 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

P ~0.004 Pth

kx (µm−1)

Wav

elen

gth

(µm

)

−4 −2 0 2 4

0.7685

0.769

0.7695

0.77

0.7705

0.771

0.7715

t (ps)

x (µ

m)

−2 0 2 4

−4

−2

0

2

4

A

B C

Fig. 4. (A) Measured gð1Þðx; −x; tÞ for very low pumping power. (B) Mea-sured momentum-space spectrum Sðkx; ℏωÞ for very low pumping power.As explained in the text, gð1Þðx; −x; tÞ is the Fourier transform of Sðkx; ℏωÞ.(C) Fourier transform of the experimental data shown in B. The result indeedreproduces accurately A. In B and C, the data is plotted in linear color scale inarbitrary units.

10−1

100

101

10−2

10−1

100

∆x (µm)

g(1

) (∆x)

R0=7µm

R0=12µm

R0=19µm

R0=22µm

Fig. 5. Measured gð1ÞðΔxÞ vs.Δx for various pumping spot radii R0. All data istaken above threshold and is chosen such that λeff ∼ 4.1 μm. As the conden-sate size increases, gð1ÞðΔxÞ converges to a power-law decay.

10−1

100

101

10−3

10−2

10−1

100

∆x (µm)

g(1

) (∆x)

37mW 48mW 60mW 76mW 95mW118mW134mW149mW170mW190mW214mW237mW260mW283mW343mW

10−1

100

101

10−2

10−1

100

∆x (µm)

g(1

) (∆x)

134 mW

x>0x<0fit

0 100 200 300 4000.9

0.95

1

1.05

1.1

1.15

1.2

1.25

Pumping power (mW)

A

B C

Fig. 6. (A) gð1ÞðΔxÞ vs. Δx for increasing laser power. The laser pumping spotradius is R0 ¼ 19 μm and the threshold power Pth ¼ 55 mW. (B) gð1ÞðΔxÞ vs.Δx for one particular laser power and for x both positive (blue circles) andnegative (red squares). Dashed line is a power-law fit. (C) Exponent ap of thepower-law decay as a function of laser power.

*See SI Appendix for a discussion of λp , data at different detunings, the orthogonal prismorientation, as well as for time-resolved data.

4 of 6 ∣ www.pnas.org/cgi/doi/10.1073/pnas.1107970109 Roumpos et al.

Dow

nloa

ded

by g

uest

on

Janu

ary

31, 2

020

DiscussionIn ref. 31, it was found that excitation with a low-noise singlemode laser revealed the formation of multimode condensation,and the different condensate modes could be spectrally sepa-rated. The authors of ref. 31 argued that, if one wants to measurethe intrinsic linewidth of polariton condensates, single mode laserexcitation and energy-resolved data are required. Indeed, it hasbeen shown (32) that the temporal coherence properties can beunderstood based on the idea that laser intensity noise introducespopulation fluctuations, which modulate the interaction energyaccordingly, leading to decoherence. However, it is not clearhow pump and decay noise influences spatial coherence. Undersingle mode laser excitations, and when the lowest-energy stateis spectrally isolated, long coherence lengths can be observed(15–17, 33). In the current experiment, we are interested inhow robust spatial correlations are when excitations are included.We study the “worst case” scenario of multimode laser excitation,which gives broader spectra compared to single mode laser. How-ever, as shown in SI Appendix, single mode laser excitation givessimilar results in energy-integrated data. We note that laser phasenoise cannot be an issue in our experiment, because the laser en-ergy is approximately 100 meV above the LP energy, so the gen-erated quasiparticles suffer multiple scatterings before formingthe condensate.

Focusing on the multimode laser excitation case, let us nowexplore the interpretation of the power law that we observeand consider what it means for the properties of the nonequili-brium polariton condensate. In particular, we discuss under whatconditions a power-law decay should be seen and what may con-trol the value and pump power dependence of the observed ex-ponent. As has been discussed previously (9, 10, 34), power-lawdecay of spatial correlations are not an artifact of equilibriumcondensates but survive more generally in a nonequilibrium con-densate. Because the power-law decay at long distances arisesfrom the long wavelength collective modes, this statement isnot trivial, because dissipation can modify the spectrum at longwavelengths (9, 10, 34).

Let us first recall the results that would apply if one were toconsider an equilibrium interacting 2D Bose gas. In this case,the exponent is given by ap ¼ 1∕nsλ2 ≤ 1∕4, where ns is thesuperfluid density and λ is the thermal de Broglie wavelengthλ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2πℏ2∕mkBT

p. The restriction ap < 1∕4 occurs because in-

creasing temperature has two effects: It excites long wavelengthphase fluctuations, which are responsible for the power-law de-cay, and it can also excite vortex pairs. The maximum value ofap occurs at the transition when vortex–antivortex pairs unbind,so that vortices would proliferate, and cause the BKT transition toa phase with short-range correlations. The observation here of apower law ap > 1∕4 implies that effects beyond thermal equili-brium are required to explain the data; i.e. there is noise that ex-cites phase fluctuations without leading to vortex proliferation. Inaddition, because the equilibrium exponent ap ∝ 1∕ns, one wouldexpect the exponent to decrease with pump power, as the conden-sate density increases; the absence of such a decrease again im-plies effects beyond thermal equilibrium are relevant and suggeststhat pumping noise is indeed affecting the observed exponent.

Whereas the existence of power-law decay in a nonequilibriumcondensate was discussed previously, the value of the exponentand its pump power dependence were not given in those previousworks. Using the formalism described in refs. 9 and 10, the ex-ponent can be found by calculating

g1ðΔ ~xÞ ∝ exp�Z

d2kð2πÞ2 ð1 − ei ~k·Δ ~xÞf ðkÞ

�[4]

where f ðkÞ ¼ ∫ ðdω∕2πÞiðDKϕϕ −DR

ϕϕ þDAϕϕÞ, and DK;R;A

ϕϕ ðk; ωÞare the Keldysh, retarded, and advanced Green’s functions for

phase fluctuations. The advantage of writing the correlation func-tion in this formal way is that it allows one to disentangle theeffects of changes to the spectrum of long wavelength excitationsfrom the effects of how this spectrum is populated.

The retarded and advanced Green’s functions are independentof how the spectrum is occupied, and following quite general ar-guments (9, 10, 34) we can prove that they have poles describingthe low energy spectrum ωk ≃ −iγ�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiϵkðϵk þ 2μÞ − γ2

pwhere

ϵk ¼ ℏ2k2∕2m is the long wavelength polariton dispersion, μthe chemical potential (or blueshift) and γ is the linewidth.† De-spite the modification of the equilibrium spectrum introduced byγ, it nevertheless remains the case that if this spectrum is occupiedthermally (i.e., if the Keldysh Green’s function is chosen to obeythe equilibrium fluctuation dissipation theorem), one finds †

f thermalðkÞ ≃1

ns

Zdω2π

4γμkBTjω2 þ 2iγω − 2μϵkj2

≃mkBTnsℏ2k2

; [5]

which is independent of γ and matches the equilibrium form off ðkÞ. Thus, despite the modifications to the long wavelength spec-trum, a sufficiently thermalized polariton condensate has theequilibrium exponent.

In order to explain the larger exponent observed, and the flatdependence on pump power, we consider a crude model of a sys-tem with excess pumping noise as an opposite extreme to thethermalized case. We thus consider a case where the occupationof excitations is set by a Markovian noise source of strength ζ.Namely, we take the inverse Keldysh Green’s function to be en-ergy independent.† This differs significantly from thermal noisecorrelations, which are frequency dependent, and diverge at thechemical potential. The measured spectra shown in SI Appendixare broad and their linewidth increases as the pumping power isincreased. This occupation of excited states could be induced byan energy-independent noise source whose strength increaseswith the pumping power. In this case, the function f ðkÞ is givenby

f noiseðkÞ ≃1

ns

Zdω2π

2ζðμ2 þ γ2Þjω2 þ 2iγω − 2μϵkj2

[6]

which, despite the changed occupation spectrum, still yields apower-law decay. The exponent becomes ap ¼ ðmζ∕2πℏ2nsÞ½ðμ2 þ γ2Þ∕2μγ�. Because this has the form ap ∝ ζ∕ns then if thenoise strength and polariton density both increase with the pumppower, then this would explain the absence of a 1∕ns decrease ofthe exponent, as seen on Fig. 6C.

One point not addressed so far regards the process of vortexproliferation in a noisy nonequilibrium condensate. As pumpnoise increases, it is likely eventually to lead to proliferationof vortices, and a transition to a state with only short-range cor-relations, just as occurs at high temperatures in equilibrium.

In conclusion, the measured power-law decay of the correla-tion function suggests that some form of the BKT superfluidphase survives in nonequilibrium condensates; namely, phaseflucutations are excited but no vortices. The large value of theexponent implies that, in the current experiment, this orderedphase is more robust against external noise than would beexpected in equilibrium, in which equipartition holds. We conjec-ture that the main noise source is pump and decay noise, whichcreate a nonthermal occupation of excited states, and apply anonequilibrium theory to show that a power-law decay with alarge exponent is possible in an open system with excess noise.One may anticipate that sufficient noise could induce vortexproliferation and a transition to short-range coherence. This

†See SI Appendix for further details.

Roumpos et al. PNAS Early Edition ∣ 5 of 6

PHYS

ICS

Dow

nloa

ded

by g

uest

on

Janu

ary

31, 2

020

fascinating possibility remains an open question for futurestudies.

Materials and MethodsOur GaAs-based sample shows a Rabi splitting of 2ℏΩRabi ¼ 14 meV and LPlifetime of τLP ∼ 2–4 ps near photon-exciton detuning δ ¼ 0, where the datapresented in this paper is taken. From the curvature of the measured energyversus momentum dispersion at low pumping power, the LP effective masswas found to be m� ¼ 9.5 × 10−5me at this detuning, where me is the elec-tron rest mass. The sample is the same as in our recent experiments (18, 33),and the experimental setup is very similar to ref. 18. We pump the systemwith a multimode Ti-Sapphire laser operated in the continuous wave mode,combinedwith a chopper that creates 0.5-ms pulses at 100-Hz repetition rate.All powers quoted in the text and SI Appendix refer to the unchopped laser

beam. We employ a commercial refractive beam shaper to generate a flat-top pumping profile of varying size. TheMichelson interferometer consists ofa 50-50 nonpolarizing cube beamsplitter, a dielectric mirror in the first arm,and an uncoated glass right angle prism in the second one. The position ofthe prism is controlled by a combination of a translation stage and a piezo-electric actuator.

ACKNOWLEDGMENTS. We acknowledge support from National Science Foun-dation Grant ECCS-09 25549, Navy/SPAWAR Grant N66001-09-1-2024, FIRST,MEXT, Special Coordination Funds for Promoting Science and Technology,EPSRC, Department of Energy support under FWP 70069, and the State ofBavaria. G.R. thanks M. D. Fraser, F. M. Marchetti, I. Carusotto, and F. Ametfor helpful discussions.

1. Chaikin PM, Lubensky TC (2000) Principles of Condensed Matter Physics (CambridgeUniv Press).

2. Pitaevskii L, Stringari S (2003) Bose-Einstein Condensation (Oxford Univ Press).3. Bloch I, Hänsch TW, Esslinger T (2000) Measurement of the spatial coherence of a

trapped Bose gas at the phase transition. Nature 403:166–170.4. Hadzibabic Z, Dalibard J (2011) Two-dimensional bose fluids: An atomic physics

perspective. Rivista del Nuovo Cimento 34:389.5. Hohenberg PC (1967) Existence of long-range order in one and two dimensions. Phys

Rev 158:383–386.6. Berezinskii VL (1972) Destruction of long-range order in one-dimensional and two-

dimensional systems possessing a continuous symmetry group. ii. quantum systems.Sov Phys JETP-USSR 34:1144–1156.

7. Kosterlitz JM, Thouless DJ (1973) Ordering, metastability and phase transitions intwo-dimensional systems. J Phys C Solid State 6:1181–1203.

8. Hadzibabic Z, Krüger , CheneauM, Battelier B, Dalibard J (2006) Berezinskii-Kosterlitz-Thouless crossover in a trapped atomic gas. Nature 441:1118–1121.

9. Szymańska MH, Keeling J, Littlewood PB (2006) Nonequilibrium quantum condensa-tion in an incoherently pumped dissipative system. Phys Rev Lett 96:230602.

10. Szymańska MH, Keeling J, Littlewood PB (2007) Mean-field theory and fluctuationspectrum of a pumped decaying Bose-Fermi system across the quantum condensationtransition. Phys Rev B Condens Matter Mater Phys 75:195331.

11. Weisbuch C, Nishioka M, Ishikawa A, Arakawa Y (1992) Observation of the coupledexciton-photon mode splitting in a semiconductor quantummicrocavity. Phys Rev Lett69:3314–3317.

12. Deng H, Haug H, Yamamoto Y (2010) Exciton-polariton Bose-Einstein condensation.Rev Mod Phys 82:1489–1537.

13. Deng H, Solomon GS, Hey R, Ploog KH, Yamamoto Y (2007) Spatial coherence of apolariton condensate. Phys Rev Lett 99:126403.

14. Kasprzak J, et al. (2006) Bose-Einstein condensation of exciton-polaritons. Nature443:409–414.

15. Krizhanovskii DN, et al. (2009) Coexisting nonequilibrium condensates with long-range spatial coherence in semiconductor microcavities. Phys Rev B Condens MatterMater Phys 80:045317.

16. Wertz E, et al. (2010) Spontaneous formation and optical manipulation of extendedpolariton condensates. Nat Phys 6:860–864.

17. Manni F, et al. (2011) Polariton condensation in a onedimensional disordered poten-tial. Phys Rev Lett 106:176401.

18. Roumpos G, et al. (2011) Single vortex-antivortex pair in an exciton polariton conden-sate. Nat Phys 7:129–133.

19. Lagoudakis KG, et al. (2008) Quantized vortices in an exciton-polariton condensate.Nat Phys 4:706–710.

20. Tosi G, et al. (2011) Onset and dynamics of vortex-antivortex pairs in polariton opticalparametric oscillator superfluids. Phys Rev Lett 107:036401.

21. Lagoudakis KG, et al. (2011) Probing the dynamics of spontaneous quantum vortices inpolariton superfluids. Phys Rev Lett 106:115301.

22. Amo A, et al. (2011) Polariton superfluids reveal quantum hydrodynamic solitons.Science 332:1167–1170.

23. Nardin G, et al. (2011) Hydrodynamic nucleation of quantized vortex pairs in a polar-iton quantum fluid. Nat Phys 7:635–641.

24. Weiner AM (2009) Ultrafast Optics (John Wiley & Sons, Hoboken, NJ).25. Wouters M, Carusotto I, Ciuti C (2008) Spatial and spectral shape of inhomogeneous

nonequilibrium exciton-polariton condensates. Phys Rev B CondensMatterMater Phys77:115340.

26. Bajoni D, et al. (2008) Polariton laser using single micropillar GaAs-GaAlAs semicon-ductor cavities. Phys Rev Lett 100:047401.

27. Balili R, Nelsen B, Snoke DW, Snoke L, West K (2009) Role of the stress trap in thepolariton quasiequilibrium condensation in gaas microcavities. Phys Rev B CondensMatter Mater Phys 79:075319.

28. Keeling J, Eastham PR, Szymanska MH, Littlewood PB (2005) BCS-BEC crossover in asystem of microcavity polaritons. Phys Rev B Condens Matter Mater Phys 72:115320.

29. Byrnes T, Horikiri T, Ishida N, Yamamoto Y (2010) BCS wave-function approach to theBEC-BCS crossover of exciton-polariton condensates. Phys Rev Lett 105:186402.

30. Klaers J, Schmitt J, Vewinger F, Weitz M (2010) Bose-Einstein condensation of photonsin an optical microcavity. Nature 468:545–548.

31. Love APD, et al. (2008) Intrinsic decoherence mechanisms in the microcavity polaritoncondensate. Phys Rev Lett 101:067404.

32. Whittaker DM, Eastham PR (2009) Coherence properties of the microcavity polaritoncondensate. EPL 87:27002.

33. Roumpos G, Nitsche WH, Höfling S, Forchel A, Yamamoto Y (2010) Gaininduced trap-ping of microcavity exciton polariton condensates. Phys Rev Lett 104:126403.

34. Wouters M, Carusotto I (2006) Absence of long-range coherence in the parametricemission of photonic wires. Phys Rev B Condens Matter Mater Phys 74:245316.

6 of 6 ∣ www.pnas.org/cgi/doi/10.1073/pnas.1107970109 Roumpos et al.

Dow

nloa

ded

by g

uest

on

Janu

ary

31, 2

020