Exciton–polariton condensation Contemporary Physics · Bose–Einstein condensation This subsection provides a brief overview of some of the essential features of Bose condensation,

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Exciton–polariton condensationJonathan Keeling a & Natalia G. Berloff ba Scottish Universities Physics Alliance, School of Physics and Astronomy, University of StAndrews, St Andrews, KY16 9SS, UKb Department of Applied Maths and Theoretical Physics, University of Cambridge,Cambridge, CB3 0WA, UK

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Exciton–polariton condensation

Jonathan Keelinga* and Natalia G. Berloffb

aScottish Universities Physics Alliance, School of Physics and Astronomy, University of St Andrews, St Andrews KY16 9SS, UK;bDepartment of Applied Maths and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, UK

(Received 5 October 2010; final version received 17 December 2010)

We review, aiming at an audience of final year undergraduates, the phenomena observed in, and properties of,microcavity exciton–polariton condensates. These are condensates of mixed light and matter, consisting ofsuperpositions of photons in semiconductor microcavities and excitons in quantum wells. Because of the imperfectconfinement of the photon component, exciton–polaritons have a finite lifetime, and have to be continuously re-populated. Therefore, exciton–polariton condensates lie somewhere between equilibrium Bose–Einstein condensatesand lasers. We review in particular the evidence for condensation, the coherence properties studied experimentally,and the wide variety of spatial structures either observed or predicted to exist in exciton–polariton condensates,including quantised vortices and other coherent structures. We also discuss the question of superfluidity in a non-equilibrium system, reviewing both the experimental attempts to investigate superfluidity to date, and the theoreticalsuggestions of how it may be further elucidated.

Keywords: exciton–polariton; BEC; polariton laser; pattern formation; superfluidity

1. Introduction

Microcavity exciton–polaritons are quasi-particles thatresult from the hybridisation of excitons (boundelectron hole pairs) and light confined inside semi-conductor microcavities. At low enough densities, theybehave as bosons according to Bose–Einstein statistics,and so one may investigate Bose–Einstein condensa-tion (BEC) of these particles, and the phenomenaassociated with it, such as increased coherence, super-fluidity, quantised vortices, pattern formation. Oneparticularly notable feature of BEC is that the manyparticle quantum system can be represented by aclassical complex-valued field C, so the dynamics ofthe system can be described by essentially classicalequations of nonlinear physics.

Exciton–polaritons are one of several examples ofquasi-particles inside solids which present opportu-nities to explore condensation and related effects. Oneadvantage of studying condensation of quasi-particlesin solids is that they have effective masses that can becontrolled by the design of the material. These effectivemasses are typically much smaller than atomic masses.Since the critical temperature of the BEC is inverselyproportional to the effective mass, solid state con-densates can often be realised at relatively hightemperatures, and in some cases, even at roomtemperature. This is in contrast with the micro-Kelvintemperatures needed to condense dilute atomic gases.

Because polaritons are quasi-particles that involvelight, they also have a finite lifetime, as light can escapefrom the cavity. This means that any polaritoncondensate requires continuous injection of newpolaritons to balance the loss. As such there isnaturally a close relation to lasing.

This article reviews some of the phenomena ofexciton–polariton condensation, aiming in particularto illustrate the behaviour that has been seen experi-mentally. As the field of microcavity exciton–polaritoncondensation has been a very active field for severalyears, there already exist a large number of reviewarticles [1–7], as well as several books [8–10] andcollections or edited volumes [11–14]. The particularfocus of this article is that it is aimed at an audienceof final year undergraduates, and concentrates ondiscussing the phenomena that have been observed, orare predicted to be observed, in exciton–polaritoncondensates. As such, we do not discuss theoreticaltechniques, but we will try to make connectionsbetween the behaviour of the polariton system andother well-studied fields, such as quantum hydrody-namics [15–17], lasing [18,19], and nonlinear optics andpattern formation [20,21]. Furthermore, by arrangingour discussion according to phenomena, we will notdiscuss the historical development of the subject.

This article is arranged as follows. The remainder ofthis section provides an introduction to Bose–Einstein

*Corresponding author. Email: [email protected]

Contemporary Physics

Vol. 52, No. 2, March–April 2011, 131–151

ISSN 0010-7514 print/ISSN 1366-5812 online

� 2011 Taylor & Francis

DOI: 10.1080/00107514.2010.550120

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condensation, and to microcavity exciton polaritons asa system of quasi-particles in solids where condensa-tion may be studied. Following this, the article is thenarranged around classes of phenomena seen inexciton–polariton systems. Section 2 discusses thosefeatures indicating condensation, including energy andmomentum distribution of polaritons and coherenceproperties. An aspect of condensation which has beenstudied particularly extensively concerns the spatialstructure of condensates, and the effects of finitepolariton lifetime on this structure; these are discussedin Section 3. Section 4 considers features associatedwith exploring which aspects of superfluidity can beseen in polariton systems, an area which is still notfully resolved. Finally, Section 5 concludes, andprovides a brief summary of some of the topics likelyto feature significantly in future work which we havenot had space to discuss in this article.

1.1. Bose–Einstein condensation

This subsection provides a brief overview of some ofthe essential features of Bose condensation, which willplay an important role in the following discussion ofexciton–polariton condensation. More comprehensivediscussions of Bose condensation can be found inmany textbooks [15,16].

Bose–Einstein condensation of a non-interactingBose gas occurs because for a Bose–Einstein distribu-tion in three dimensions, there is a finite density atwhich the chemical potential reaches zero. Thus, as oneincreases density at a fixed temperature, or decreasestemperature at a fixed density, the required chemicalpotential eventually reaches the bottom of the densityof states. Since the Bose–Einstein distribution functionnB(E)¼ [exp[b(E 7 m)] 7 1]71 diverges at E¼ m, thepoint at which m reaches zero then implies that there isa macroscopic population of the E¼ 0 modes. Thetemperature and density at which m first reaches zerothus describes the point where the phase transition tothe condensed state occurs. By finding the density andtemperature at which m¼ 0, one produces the non-interacting transition temperature TBEC¼ (2p�h2/mkB)(r/2.612)2/3, where m is the atomic mass and r is thenumber density.

In more complicated situations, one can rigorouslydefine condensation as the appearance of a singleparticle state with a macroscopic occupation [22].Macroscopic occupation of a single particle state canbe determined by considering the one particle densitymatrix (i.e. the density matrix of the system aftertracing out all but one particle), and determining itseigenvalues. For a regular thermal gas, the eigenvalueswill all be much less than one indicating that mostsingle particle states are occupied by a small number of

particles. As the density increases, or temperaturedecreases, some of the eigenvalues become of orderone, and one has a degenerate quantum gas, where thedifferences between bosonic and fermionic particlesbecome apparent. When condensed, one eigenvaluebecomes macroscopic, indicating that many particlesare in the same state. In a homogeneous system, thisstate can be expected to be the zero momentum state,and this immediately leads to long range coherence.

This macroscopic occupation means that there is asingle complex classical function C that describes theinteracting many particle system. The explicit manyparticle wavefunction has the form cðr1; . . . ; rNÞ ¼QN

i¼1 CðriÞ, i.e. there are many particles in exactly thesame quantum state. The function C(r) can be seen inseveral other ways: it is the eigenvector of the densitymatrix with macroscopic occupation. If C(r) isnormalised so that

RdrjCðrÞj2 ¼ N, it can be seen as

the order parameter for the Bose condensed phase.In the second quantised approach (which we willotherwise avoid in this review), C can be seen asreplacing the quantum field creation and annihilationoperators with a classical field describing the con-densate mode. This replacement is appropriate whenthe occupation of this mode is much larger than oneand so the non-commutativity of the quantum fieldoperators can be neglected.

Such a ‘classical field’ description also occurs whenone goes from quantum electrodynamics to theclassical description of electric and magnetic fields,which obey Maxwell’s equations. Moreover, it hasbeen demonstrated [23,24] using a general analysis ofthe kinetics of a weakly interacting bosonic field, thatall low energy states with macroscopic occupation canbe described as an ensemble of classical fields withcorresponding classical-field equations. The require-ment of weak interaction is essential here, since in astrongly interacting system it is impossible to dividesingle-particle modes into highly occupied and practi-cally empty ones: these modes are always coupled tothe rest of the system.

The existence of long-range phase coherence isalso associated with the idea of breaking of phasesymmetry. This is because in the condensed state alarge number of particles occupy the same quantumstate, and so it is possible to see macroscopicinterference effects, meaning that a well-definedquantum mechanical phase can exist. A pivotalrelation in the theory of condensates is that thevelocity of the condensate flow v(r) at the position ris proportional to the gradient of the phase of thewavefunction C¼ jCjexp if(r)

vðrÞ ¼ �h

mrfðrÞ: ð1Þ

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Several fundamental properties follow from this result.Firstly, as one proceeds around any closed path in asimply-connected condensate, the phase f can onlychange by a multiple of 2p in order that C remainssingle-valued along this path. If the phase f doesindeed change by, say, 2p, then there has to be at leastone point (in 2D) or a line (in 3D) inside this pathwhere f takes any value between 0 and 2p. To preventsuch phase singularity, the amplitude of C has tovanish at this point (line). These points or linesconstitute quantised vortices with the unit of circula-tion

Hv � dl¼ 2p�h/m¼ h/m. Secondly, it follows from

Equation (1) that the condensate cannot supportrotation, since the vorticity o¼r6 v¼ 0 away fromphase singularities. The flow is thus a real example ofpotential flow and, therefore, can be described by theequations of classical irrotational hydrodynamics.Moreover, on distances larger than the size of thevortex core (the length-scale over which the numberdensity of the condensate, jCj2 heals back to theunperturbed value) the compressibility of the flow canbe neglected implying r2f¼ 0. This is again directlyrelated to the motion of point vortices and vortex linesin classical Eulerian fluid. In this representationvortices move as material lines (points) according tothe classical Biot–Savart Law. The same law is used tocompute the magnetic field generated by steady currentin a vacuum.

In two dimensions (which many of the solid statecondensate systems are), the meaning of condensation israther different. For a non-interacting gas in twodimensions, no phase transition occurs. For an inter-acting gas, a transition does occur, but there is nomacroscopically occupied state. Instead, there isa transition between a low temperature superfluid phasewith power law decay of correlations, and ahigh temperature normal phase. This power law decayof correlations occurs due to the effects of longwavelength fluctuations of the condensate phase f(r) –at any non-zero temperature, these fluctuations aresufficiently large that one cannot consider a well--defined phase, and the correlation between the phasesat two distant points has the formhexp[if(r)]exp[7if(r0)]i*jr 7 r0j7Z, with Z¼ kBTm/2pr0�h

2. Although no one single state acquires amacroscopic occupation in an infinite two-dimensionalsystem, there is a notably increased population of arange of low momentum states. If the system is alsoconfined by a trap within the two-dimensional plane,then many features of condensation can be recovered,in particular a single particle state will becomemacroscopically occupied. One may note that thistransition involves many body effects, in that thetransition temperature is much larger than the energylevel spacing for a single particle in the same trap.

Many other physical systems are closely linked withBose condensation. A superconductor is a chargedsuperfluid, where the Bose condensate is formed byCooper pairs – quasi-particles with integer spin(normally zero) consisting of electron pairs. Thewave function of such a charged Bose condensate isonce again a complex classical field. In nonlinear opticsthe order parameter is the electric field of the(polarised) wave. The close connection betweenbroad-aperture lasers, superconductors, BEC, andsuperfluids has been recognised since the late 1960s[25] by establishing that all these systems can bedescribed by the same order parameter equation – thecomplex Ginsburg–Landau equation. This fundamen-tal equation, which can be viewed as the drivendissipative nonlinear Schrodinger equation with re-laxation, will be discussed in Section 3.

1.2 . Introduction to exciton–polaritons

Exciton–polaritons are the quasi-particles that resultwhen one allows strong coupling between excitons andphotons [26,27]. Excitons are the bound states result-ing from the Coulomb interaction between electronsand holes in a semiconductor. Excitons can be createdby shining light on a semiconductor in its ground state,exciting an electron from the valence to the conductionband, leaving a hole in the state the electron wasexcited from. Excitons can therefore also decay, by theelectron and hole recombining to emit a photon.‘Strong coupling’ occurs when the rate at whichphotons are converted to excitons and vice versaexceeds the rate at which photons escape from thesystem. In this case, the normal modes of the systemare not excitons and photons, but rather superposi-tions of these, i.e. polaritons.

The idea of polaritons was originally considered[26,27] in bulk materials. However, for polaritoncondensation, it is necessary to consider microcavitypolaritons. These are formed when the photons areconfined inside a microcavity, i.e. a planar Fabry–Perot resonator formed by two Bragg mirrors. Asillustrated in the cartoon in Figure 1(a), the Braggmirrors which form the cavity consist of quarter-wavelength layers of alternating refractive index, andso they can provide a high quality mirror over a limitedrange of wavelengths of light. By confining photons,one allows polaritons to potentially survive longenough to cool and condense. Furthermore, one canengineer stronger coupling between excitons andphotons by locating the excitons (confined in quantumwells) at anti-nodes of the confined light mode in thecavity.

For photons that are confined inside the micro-cavity, their state can be described by a wave-vector in

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the two-dimensional plane of the cavity, the ‘in-planewavevector’ k, and an index N labelling the excitationnumber in the direction perpendicular to the plane. Interms of these variables, the angular frequency of sucha mode can be written as ok¼ (c/n)[k2 þ (2pN/Lw)

2]1/2,where n is the refractive index, c the speed of light invacuum, and N labels the transverse mode in a cavityof transverse size Lw. For small in-plane wavevector k,this frequency can be expanded to give a quadraticdispersion �hok¼ �ho0 þ �h2k2/2m. The photon massappearing in this expression m¼ �h(n/c)(2pN/Lw) isthus set by the size of the cavity, however this size is inturn fixed by the requirement that the bottom of thephoton band, o0¼ (c/n)(2pN/Lw), should be close toresonance with the energy required to create a boundexciton. For typical materials, this sets the photonmass to be of the order of m¼ 1074 me (where me is thefree electron mass.) Compared to this steep photondispersion, the exciton dispersion can be almostneglected; excitons have a mass set by the electronand hole mass in a semiconductor which is thustypically somewhere between 0.1me and me.

For a sufficiently high quality cavity, the polaritonstates can be found by solving the Schrodingerequations for coupled exciton and photonwavefunctions:

i�h@tCphot

Cex

� �¼

�hok12 g

12 g e

!Cphot

Cex

� �; ð2Þ

where g describes the rate of interconversion betweenexcitons and photons; g thus depends on the dipolematrix element for a single photon to excite a single

exciton. The eigenstates of this equation are the newnormal modes, the lower and upper polaritons,

ELP;UPk ¼ 1

2�ho0 þ eþ �h2k2

2m

� ��

� �ho0 � eþ �h2k2

2m

� �2

þ g2

" #1=235: ð3Þ

The dispersion of these two modes are shown inFigure 1(b). For small in-plane wavevector, the lowerpolariton has an almost quadratic dispersion, with aneffective mass of the order of the photon effective mass(at resonance, �ho0¼ e, one has mpol¼ 2m). At largemomenta, the lower polariton dispersion has a point ofinflection, before eventually approaching the limit ofthe exciton dispersion. Realisation of the strong-coupling regime in semiconductor microcavities[28,29] can be clearly seen experimentally, since thetransmission and reflection of the microcavity as afunction of energy and incident angle can be measured,as discussed below in Section 1.3, allowing directobservation of the modified spectrum.

Polaritons are not, however, ideal, non-interactingbosons. Because of their excitonic component, thereare polariton–polariton interactions. These arise bothdue to the interactions between the charged particlesmaking up the exciton, and because of saturationof the exciton–photon interaction. In a clean sample,the dominant interaction between polaritons at low-momenta is the relatively short ranged electron–electron exchange interaction [30] (i.e. it results fromthe process in which two excitons exchange theirelectrons). For low densities, as a first approximation

Figure 1. (a) Cartoon of a semiconductor microcavity, formed from two distributed Bragg reflectors (upper and lower sections,with alternating dielectric contrast), with a spacer layer in the middle, containing quantum wells at the anti-nodes of the confinedphoton mode. (b) Polariton spectrum (solid lines) resulting from strong coupling between exciton and microcavity photon(dashed lines). Energies are measured relative to the bare exciton energy, and the parameters assumed are g¼ 26 meV,m¼ 36 1075 me, based on typical parameters for CdTe microcavities. The bare exciton energy (and thus the bottom of thecavity photon dispersion) is typically of the order 1.5 eV. The x axis is given in terms of the angle of photons escaping the cavity,as discussed in Section 1.3 ((b) from [5]). Reproduced figure 2 with permission from Keeling et al., Semicond. Sci. Technol. 22(2007) pp. R1–R26. Copyright (2007) by IOP Publishing.


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to the full form of this interaction, one may consider apseudo-potential, U(r) ! Ud(r), describing a contactinteraction between two polaritons. A typical scale forU is of the order 1073 meV mm2. In addition, becausethe strength of coupling between excitons and photonsis comparable to the temperatures at which condensa-tion is studied, the internal structure of the polaritoncannot be entirely neglected. Polaritons also have arelatively short lifetime (of the order of 5 – 10 ps)because the photonic component is not perfectlyconfined by the Bragg mirrors, i.e. their reflectivity isnot perfect. The fact that polaritons escape from themicrocavity is, however, what allows one to experi-mentally image the properties of the condensate, asdiscussed below.

Another complicating feature of polaritons is thatthey have two possible polarisation states, correspond-ing to the left- and right-circularly polarised photonstates. A comprehensive review of polarisation phenom-ena in polaritons can be found in [7], and a review ofpolarisation dynamics in [31]. In many cases, however,coupling between mechanical strain in the sample andthe energy of electron and hole states breaks thispolarisation symmetry and favours a particular linearpolarisation – i.e. a particular superposition of left- andright-circular polarised polaritons. In these cases, thepolarisation degree of freedom can then be ignored. Inthis article, we will only mention polarisation in a fewcases where it leads to interesting new physics.

1.3. Experiments on exciton–polaritons

The basic outline of almost all experiments on exciton–polariton condensates are similar: light is used tocreate excitations inside the semiconductor microcav-ity, these excitations relax and scatter to form apolariton condensate, and because of the finitepolariton lifetime, light then escapes the cavity and isdetected. Since the escaping light conserves the energyand in-plane momentum of the polaritons, it allowsone to image both the real-space and momentum-spaceshape of the condensate, the polariton dispersion,the line-shape, and the coherence properties of thepolaritons. The conserved in-plane momentum of thephoton can be written in terms of its angle of emissionas ck¼Ek

LP sin(y), hence one may refer to the polaritonmomentum, wavevector or emission angle somewhatinterchangeably. Figure 2 illustrates schematically thevarious setups for imaging and coherence measure-ments on the condensate.

Experiments on polaritons are performed in anumber of materials, and with a variety of differentways of injecting polaritons. The choice of semicon-ductor material that the cavity and mirrors are madefrom affects the various parameters describing

polaritons, such as the exciton binding energy, theexciton–photon coupling, the effective exciton–excitoninteraction, and the photon lifetime. For most of theexperiments discussed in this review, the system iseither made from CdTe, doped with Mg or Mn tomake the Bragg mirrors and the cavity, or GaAs dopedwith Al or In.

The way in which new polaritons are injected has amore pronounced effect on the behaviour that can beobserved. In the same way that the light escaping fromthe cavity conserves the energy and in-plane momen-tum of polaritons, one can create polaritons with apump laser incident on the cavity, and thereby controlwhat states are populated by varying the energy andin-plane momentum (i.e. incident angle) of the pump.Schemes to inject polaritons include: directly creatingzero momentum polaritons with a coherent pumplaser; coherently creating polaritons at a ‘magic angle’from where they can parametrically scatter directlyinto the ground state; coherently creating polaritonsat large angles so that many scattering events arerequired to reach the ground state; using a pump laserdetuned from the polariton resonance to createincoherent populations of electrons and holes whichsubsequently relax; and in some recent experiments,injecting electrons and holes by electrical currents in alight-emitting diode configuration. This list is orderedstarting from cases where the coherence, polarisation,and details of the pump have the most profound effecton the properties of the resulting polariton state, and

Figure 2. Schematic of an experiment studying polaritons,pumped either coherently or incoherently, with a choice ofpumping angle. The content of the box labelled ‘Detector’depends on what is to be measured, some possible options forthe measurement scheme are shown in panels (b)–(d). Thesedifferent options image either (b) the real and momentumspace images of the polaritons (depending on whether theCCD is placed in the image plane or Fourier plane), (c)energy resolved images of the polaritons using aspectrometer, or (d) first-order coherence of the polaritonsusing an interferometer. For a more detailed illustration of aparticular experimental set up, see e.g. [M. Richard, Quasi-condensation of polarisation in II-VI CdTe-basedmicrocavities under non-coherent excitation, Ph.D. Thesis,Universite Joseph Fourier, Grenoble, 2004. Available athttp://tel.archives-ouvertes.fr/tel-00009088/fr/].


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ending with cases where such details of the pump havealmost no effect. However, with the exception ofcoherently populating the zero-momentum state, allthe other methods can be said to show some type ofcondensation, and so we will discuss experiments withall these cases of polariton injection below.

1.4. Other solid state quantum condensates

Exciton–polaritons are not the only quasi-particles insolids in which condensation has been sought orrealised, and so this subsection provides a very briefsummary of some of the other quasi-particles in solidswhich have been studied.

Magnons, which are elementary excitations –quantised spin waves – of a magnetic system havealso been observed to condense. Magnon condensationhas been seen in two quite different systems; ferro-magnetic insulators [32–35] and within superfluidphases of 3He [36,37]. Similar to the state of non-condensed Bose gas, the ‘normal’ states of thesemagnetic materials, without magnon condensation,are disordered paramagnetic states. In the magnoncondensate, spins develop a common global frequencyand phase of precession.

Magnon condensation in room temperature ferro-magnetic insulators, in particular yttrium-iron garnet(YIG) films, requires a combination of an in-planemagnetic field, and driving with microwave radiation. Amicrowave photon excites two primary magnons thatthen relax, forming a magnon gas. As for the incoher-ently pumped polariton system, the chemical potentialincreases with pump power.When themicrowave powerexceeds a threshold value, the magnon populationcondenses in the minimum energy state. However, formagnons, this minimum energy occurs at non-zeromomentum. This is due to the combined effects of themagnetic exchange interaction and the magnetic dipoleinteractions. The formation and structure of condensatecan be analysed by scattering of light. A quasi-equilibrium state with a non-zero chemical potentialcan be realised because the magnon lifetimes, set by thespin–lattice relaxation times (*1 ms), are long comparedto the magnon–magnon thermalisation time (*100 ns).Magnon condensation in helium is studied using nuclearmagnetic resonance to create and then probe themagnoncondensate.

Excitons on their own have also long beenconsidered as candidates for condensation, (see, e.g.the articles in [38]). In such experiments, there is noFabry–Perot cavity, and it is preferable to reduce theexciton–photon coupling strength, so that the excitonhas a long lifetime before it decays by recombination.Various different ways to achieve long lifetimeshave been considered. One way involves choosing

excitonic states for which recombination is dipoleforbidden, meaning that selection rules forbid the decayof excitons; this is the case in cuprous oxide [39]. Analternate way to enhance exciton lifetimes is to spatiallyseparate electrons and holes, by having two parallelquantum wells, and apply an electric field to trapelectrons and holes in different wells [40–42]. Theseexciton systems have shown a significant change in theirbehaviour at a temperature where condensation wouldbe expected, including evidence of enhanced spatialcoherence [43], however, a full understanding of theobserved behaviour remains to be found. A review ofmuch of the physics of such excitonic systems is given in[44]. There are also close connections between excitonsin coupled quantum wells, formed of bound electron–hole pairs, and the properties of quantum-Hall bilayers[45,46] – these systems consist of electron–electron pairsin strong perpendicular magnetic fields.

2. Condensation, Bose stimulation and coherence

Condensation – in the broad sense described above –of exciton polaritons manifests itself by macroscopicaccumulation in low momentum states, increasedtemporal and spatial coherence, and evidence of Bosestimulation in producing the macroscopic occupationof low momentum states. This section will review theseunderlying features of condensation; features whichserve as prerequisites for the phenomena discussed insubsequent sections.

2.1. Stimulated scattering

When microcavity systems are pumped sufficientlystrongly, a threshold is seen above which the emissionat low momenta increases sharply. The mechanism ofsuch accumulation differs according to how the systemis pumped. In particular, it matters whether pumpingcreates an incoherent population of polaritons whichthen scatter to the ground state, or whether polaritonsare created coherently at wave-vectors which allow forparametric scattering. We will thus next discuss thesetwo cases separately.

2.1.1. Incoherent pumping

In the first case, with an initially incoherent polaritonpopulation, the threshold behaviour is relativelysimple. The relaxation of polaritons towards thelow momentum state becomes stimulated due to theincreasing population of the low momentum states[47]; i.e. the bosonic nature of polaritons leads tofinal state stimulation of the relaxation. One mayalso directly probe the stimulated scattering towardthe ground state by seeing how the system responds


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to an additional coherent pump at zero momentum[48,49]. By measuring the response of either thepopulation at higher angles from which polaritonsare scattered, or the state to which they are scattered,one can see that scattering is stimulated by the finalstate population.

2.1.2. Parametric scattering

If polaritons are injected coherently at angles near thepoint of inflection in the lower polariton spectrum,a rather different behaviour is seen. As shown inFigure 3(a), there is a momentum state from which apair of polaritons can scatter into one zero momentumand one high momentum state in a process whichconserves energy and momentum. This means that fora pump at this angle (sometimes called the ‘magicangle’), and a weak probe at zero momentum, therewill be strong amplification of the probe [51,52] due tostimulated scattering. This process is referred to asoptical parametric amplification (OPA), as it isanalogous to parametric amplification in nonlinearoptics [18,19], where a pump beam can scatter in anenergy and momentum conserving way into a signaland idler mode, and a probe in the signal mode wouldbe amplified.

In addition to parametric amplification, it is alsopossible to realise optical parametric oscillation(OPO), where there is no probe beam present. In thiscase, above a threshold pump power, the state withpolaritons only at the pump wave-vector becomesunstable, and the population of the signal and idlermodes grows [51,53]. To understand the conditionsunder which OPO can be realised, one must takeaccount of the energy shifts that occur with increasing

density. These energy shifts mean that both thedetuning between the pump laser frequency and thepolariton state at the pump wave-vector is affected bythe polariton density. The energy shifts also affect thedetuning between the pump state and the signal andidler states. If one detunes the pump laser below thepolariton mode, then as pump intensity increases,resonance becomes increasingly worse, and this limitsthe ability of parametric scattering to occur. If thepump is detuned above the polariton mode, the energyshift drives the system towards resonance. This leads toanother kind of instability of the pump-wavevector-only state, Kerr bistability. This bistability occursbetween two possible states, one with the pump modehaving a low occupation, and thus a small blueshift,and so remaining far from resonance; the other witha higher occupation, and thus a larger blueshift,pushing this state into resonance with the pump (seeFigure 3(b)). With an appropriate choice of detuning,these two states may also compete with the parametricinstability, hence there is a range of pumping strengthsin Figure 3(c) where there are both multiple states inwhich only the pump mode is occupied (grey, solidline) as well as states with all three modes (signal, idlerand pump) occupied [50,54] (red dashed line).

While the parametric oscillator state looks verydifferent from an equilibrium Bose condensate, it doeshowever share with it the important idea of breaking ofphase symmetry. Although the external pump sets thephase of the polariton field at the pump wavevector,the parametric scattering to signal and idler fieldsallows for an arbitrary phase fsignal 7 fidler, i.e. theonly requirement is that fsignal þ fidler¼ 2fpump. Thisfree phase allows in principle for some of thephenomena normally associated with condensation

Figure 3. (a) Lower polariton spectrum, illustrating the idea of parametric scattering; pairs of polaritons scatter toward a signalstate (near zero momentum) and a higher momentum idler state. Grey dashed lines represent the phonon relaxation mechanismfor incoherently injected polaritons (see Section 2.2). (b) Illustration of the conditions for Kerr bistability, where the interactioninduced blue-shift of the polariton can drive it closer to resonance with a coherent pump. (c) Input–output relation for the pumpfield, showing the pump only state (grey, solid line) and the parametric oscillation (OPO) state (red, dashed line). The pump-onlystate (grey) displays Kerr bistability for a narrow range of pump strengths, while the OPO state exists for a wider range ofpumping strength. Also shown is the signal field intensity (blue dashed line) for the OPO state. One should note that the linesshown represent steady states of the equations of motion, however, these steady states may either be stable or unstable. After [50].


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to be seen in this very non-equilibrium system. Inparticular, the spectrum for fluctuations on top of theOPO state shares features with the spectrum of acondensate [4,54,55] – these will be discussed furtherwhen discussing superfluidity in Section 4.

2.2. Momentum distribution and thermalisation

For incoherent pumping, observation of stimulatedrelaxation and accumulation of polaritons in lowmomentum modes leaves open several questionsregarding the nature of this state. Because polaritonshave a finite lifetime, the resulting steady state is abalance between particle loss, replenishment by thepump, and processes whereby polaritons thermaliseand cool by collisions among themselves and interac-tions with other particles such as phonons. Thesteepness of the polariton dispersion at low energies,which is responsible for the elevated transitiontemperatures, also means that the process of coolingvia phonon emission is slow. Phonons have a shallowdispersion, so momentum and energy conservationrequire emission of phonons with small energies (seedashed lines in Figure 3(a)). This means many phononsare required for such relaxation, leading to a ‘bottle-neck effect’ [56], in which polaritons accumulate at thepoint where the dispersion switches from exciton-liketo photon-like.

This bottleneck effect prevented the investigation ofcoherence properties for a long time, by hamperinglarge accumulation of polaritons at low energies, andproducing very non-thermal distributions of polari-tons. A combination of experimental improvement and

theoretical simulations using the Boltzmann equation[57–59] led to the observation that the bottleneck effectcould be considerably reduced by a combination ofdetuning the photon slightly above the exciton modeand increasing the polariton density, at the expense ofincreasing the polariton temperature. The detuningleads to more exciton-like lower polaritons, increasingthe strength of the interaction between polaritons.Combined with higher densities, this allows polariton–polariton scattering to thermalise the polariton dis-tribution. One should, however, note that while suchcollisions establish a thermal distribution, they do notremove energy from the system – thermalisation andcooling are separate processes.

By reducing the bottleneck effect, it becamepossible to see a reasonably thermalised distributionof polaritons, along with an accumulation of polar-itons in low energy modes [60] (see Figure 4)1. Theprocesses of thermalisation and cooling have also beenstudied experimentally. By observing the dynamics ofthe polariton distribution [61] following a pulsedexcitation, one may determine the time required toreach a quasi-thermal momentum distribution, and seehow this depends on the exciton–photon detuning. Onemay also consider the system with a continuous pump[62], and study how the threshold pump powerdepends on detuning and on temperature of thesemiconductor. Comparison to Boltzmann equationmodelling then allows one to interpret this dependencein terms of the two effects of detuning the photonabove the exciton: increased rate of thermalisation andreduced critical temperature for condensation (due tothe increased polariton mass).

Figure 4. (a) Upper row: Energy and momentum distribution, Lower row: momentum distribution of polaritons, for threedifferent pump powers, from below to above threshold. (b) Energy distribution with increasing pump power. At threshold, anexponential (i.e. Maxwell–Boltzmann) distribution describes the population, and above threshold, polaritons accumulated in lowenergy states, without any appreciable change to the temperature describing the decay of the tail. Adapted from [60], results arefor a CdTe microcavity, held at a cryostat temperature of 5 K.


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2.2.1. Non-equilibrium condensation versus lasing

Since polariton condensates are non-equilibrium stea-dy states emitting coherent light, the question ofwhether they are more properly described as con-densates or as lasers is one that is frequently asked.There are several criteria by which one mightdistinguish equilibrium condensation from simplelasers, and for most of these criteria, polaritoncondensates are somewhere between the two extremes.

Considering the criterion of whether there is athermal distribution, the polariton distribution is setby a balance of pumping, decay and relaxation. Thesmaller both the decay and pumping rates become,the closer the system approaches the equilibrium state.In this sense, there is a smooth crossover betweenequilibrium Bose–Einstein condensation and thebehaviour of a polariton condensate. With a non-vanishing pump rate, the kinetics of thermalisationinvolves stimulated scattering into the ground state.One may thus try to describe the system by writing the‘quantum’ Boltzmann equation, i.e. a kinetic equation,describing the time evolution of the populations of thestates, taking account of stimulated scattering due tothe population of the states particles are scattered into.For a vanishing pump rate, the steady state of thisequation is just the equilibrium Bose–Einstein distri-bution. Lasers also involve stimulated scattering, buttypically have a much larger pump rate, so are furtheraway from equilibrium. This illustrates how a smoothcrossover between lasing a condensation might occur,however, there are also some characteristic differencesbetween polariton condensates and typical photonlasers. Firstly, for polaritons, the stimulated scatteringis within the set of polariton modes, rather than thestimulated emission of photons which occurs forregular lasers. Secondly, polariton condensation canoccur without any inversion of the gain medium, whichis required for normal laser operation, i.e. polaritoncondensation occurs with a quasi-thermal distributionof polaritons in the system, while regular lasing wouldrequire an inverted (negative temperature) distributionof some part of the gain medium in order for gain toexceed absorption. This idea provided one of theoriginal motivations for polariton condensation [63].

One may also note that the same microcavitysystems that show exciton–polariton condensation canalso show regular lasing, when they cross over fromstrong to weak coupling at higher temperatures andpumping strengths. In order to check whether such atransition to weak coupling has occurred, one shouldinvestigate whether the emission follows the lowerpolariton dispersion, or the bare photon dispersion.Because lower polaritons have repulsive interactions,the energy of the low momentum states does increase

with increasing density, but near threshold, this shift istypically small compared to the polariton splitting.

2.3. Coherence and correlation measurements

As a polariton condensate produces coherent light, itis of interest to characterise this coherence, i.e. tomeasure how the coherence decays in time and inspace. Such measurements do several things: theyprovide further evidence that there is a sharp thresholdfor the appearance of coherent light; they providemeasurable quantities which allow one to position thepolariton system between simple lasers and equilibriumcondensates; and they can provide information aboutthe kinetics of the polariton system. Such measure-ments aim to determine the first- and second-ordercorrelation functions of the electromagnetic field:

g1ðr; r0; t; t0Þ ¼hE�ðr0; t0ÞEðr; tÞi

hEðr0; t0Þ2ihEðr; tÞ2ih i1=2 ; ð4Þ

g2ðr; r0; t; t0Þ ¼hE�ðr0; t0ÞE�ðr; tÞEðr; tÞEðr0; t0Þi

hEðr0; t0Þ2ihEðr; tÞ2i: ð5Þ

In an infinite steady state system, such correlationsclearly depend only on jr 7 r0 j, t 7 t0, but since mostexperiments to date involve relatively small clouds,coherence depends also on the position within thecloud. In the following, except where explicitly noted,we consider coherence measurements of the incoher-ently pumped system, as this case has been studiedmore extensively.

2.3.1. Temporal coherence

Temporal coherence – i.e. g1(t) : g1(r, r, t þ t, t) caneither be measured directly, using e.g. an interferom-eter (see Figure 2(d)), or can be inferred from theline-width of the polariton emission, by using theWiener–Khinchin theorem to relate the power spec-trum to the temporal correlation function. As such,increased temporal coherence should be seen by a linenarrowing above threshold, which is indeed seen inexperiments [60,64]. However, at yet higher powers,the line-width was then seen to increase. Such an effect,due to the phase noise induced by interactions betweenparticles, had been anticipated [65,66], but the broad-ening observed in experiments [60,64] was later foundto be due to an extraneous source of noise. In [60,64],a multi-mode pump laser was used, meaning that theintensity of injected polaritons fluctuated in time.Because of interactions between the condensate polar-itons, and the higher energy excitonic states created by


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the laser, noise in the pumping intensity translates tofluctuations of the energy of the condensate, broad-ening the line-width.

Experiments using a single mode pumping laserrevealed a coherence time that increased above thresh-old and then remained constant [67] (see Figure 5(b)).These experiments showed a Gaussian time depen-dence of the correlation function which can beexplained [68] by considering the dynamics of con-densate density fluctuations, which in turn produceenergy shifts, reducing the phase coherence. If oneallows for a non-zero relaxation time for the densityfluctuations tr, then one has a contribution to thedecay g1(t)/ exp[7Ur0tr

2(exp(7t/tr) þ t/tr7 1)],where U is the polariton–polariton interaction and r0is the condensate number density. The dependence ont/tr accounts for the spectrum of intensity fluctuationsdue to the non-zero relaxation time. This approach,along with [65,66] focused on the dynamics of thesingle condensate mode.

For an equilibrium two-dimensional condensate, thefirst-order coherence is expected to show power lawdecay of correlations as a function of either distance ortime, with a form g1(t)* t7Z, Z¼ kBTm/2pr0�h

2, thesame power law as discussed for spatial correlations

in Section 1.1. Attempts to calculate the coherence decayin a non-equilibrium two-dimensional system have beenmade in [55,69]. In a finite system, these approacheseventually reproduce the Gaussian form of the singlemode problem discussed above. Other approaches tocalculating the coherence function have made use ofsimulations of the quantum kinetic equation [70].

Further information on the processes that lead todephasing, and which thus control the coherence time,can be found by measuring the intensity–intensitycorrelation function, i.e. g2(t): g2(r, r, t þ t, t). For athermal state, g2 (t)¼ 2 (by Wick’s theorem), while fora coherent state, g2(t¼ 0)¼ 1, and g2(t) then tendstoward 2 with increasing delay time. Measuring g2(0)therefore can be used as a confirmation of appearanceor disappearance of a coherent state. Early measure-ments [71] were limited by their finite time resolution,meaning that the experiment actually recordsð1=tmÞ

R tm0 dt0g2ðt0Þ, where tm is the measurement

time. This averaging makes interpretation of suchresults challenging. A combination of improved timeresolution [72], and use of single mode rather thanmulti-mode pumping [67] as above gives a muchclearer signature of g2(t), and allows extraction ofthe time-scale for relaxation of intensity fluctuation.Again, such results have been reproduced by singlemode models [68] and by quantum kinetic approaches[70]. There have also been investigations of the growthand decay of the line-width, and degree of coherence,i.e. studying the time dependence of g1(r, r, t, t) follow-ing pulsed excitation [73].

2.3.2. Spatial coherence, g1(|rj) : g1(r0 þ r, r0, t, t)

Spatial coherence is investigated by interfering lightfrom different points in real space. One way this can bestudied is by interfering an image of the condensatewith a rotated copy of the same image. As such, eachpoint on the detector corresponds to light emergingfrom two different points on the sample, one from therotated image, and one from the non-rotated image.By inserting a delay for one copy of the image, onethen has either constructive or destructive interference,depending on the delay (as in Figure 2). The visibilityof these interference fringes depends on the coherencebetween the light coming from the two different pointson the sample. The fringe visibility thus provides amap of the coherence of the condensate. At the pointon the detector corresponding to the centre of rotation,the light from both images comes from the same pointon the sample and so the coherence is maximum.As one moves away from the centre, the distancebetween the two points on the sample increases, andthe coherence decreases. Such a map of coherence,as measured in [60] is shown in the lower panels of

Figure 5. Top: temporal coherence properties of CdTewhen pumped with a single mode laser, from [67]. Reprintedfigure 3 with permission from Love et al., Phys. Rev. Lett.101 (2008), 067404. Copyright (2008) by the AmericanPhysical Society. Bottom: spatial coherence of CdTe, from[60], showing maps of fringe visibility as a function ofdisplacement between source points on the sample. Leftpanel is below threshold, right figure is above threshold.


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Figure 5. One should note that this technique givesa map of the unnormalised coherence, G1(2jrj)¼hE*(r, t)E(7r, t)i. An alternative way to study spatialcoherence is with a Young’s double slit experiment[74,75], which examines how the fringe visibility isaffected by the spacing between the two slits.

In most of the experiments to date, the small size ofthe polariton condensate has made it hard to extractuseful information about the decay of spatial coher-ence (e.g. the coherence map shown in Figure 5 mainlyreflects the density profile of the condensate (seeSection 3 for a discussion). However, recent experi-ments with higher quality samples, having bothless disorder, and longer polariton lifetimes, haveallowed for notably larger condensates [76] in whichcoherence can be measured on sufficient length-scalesto investigate the functional form of decay of spatialcoherence.

3. Spatial pattern formation

Spontaneous emergence of spatio-temporal order –pattern formation – in non-equilibrium systemsrepresents one of the key mechanisms of self-organisa-tion in nature [77]. Many different physical systems(chemical, biological, hydrodynamical, optical, etc.)are described by similar order parameter equationsand, therefore, may share similar pattern formingproperties. Often, two types of patterns are distin-guished: localised (coherent) structures and extended(possibly periodic) solutions.

There are several aspects of polariton condensatesthat suggest that they are capable of pattern forming.Polariton condensates are non-equilibrium steadystates, meaning that spatial inhomogeneity generallyimplies the existence of steady state currents, which inturn modify the spatial pattern. Polaritons also have anon-trivial dispersion, with a point of inflection (seeSection 4.1), meaning that the coherently pumpedsystem can display interesting solitonic structures,that result from a complex interplay between disper-sion, dissipation, forcing and nonlinear interactions.Finally, polaritons can live in a non-trivial potentiallandscape, either due to intrinsic disorder in thematerial, or due to deliberately designed potentials.These potentials may induce stabilising(or destabilising) currents further facilitating symme-try breaking in the system. Both coherently andincoherently pumped systems can show (differentkinds of) interesting pattern formation, and so bothwill be discussed below.

Much of this spatial pattern formation can beunderstood in terms of the complex Ginsburg–Landauequation (cGLE) [15,78] – the universal equation thatdescribes the behaviour of systems in the vicinity of an

instability and symmetry-breaking. The cGLE for theorder parameter c(r, t) takes a generic form:

i@tc ¼ c1r2cþ c2jcj2cþ c3c; ð6Þ

where c1, c2 and c3 are complex parameters.For an equilibrium condensate system c1, c2 and c3

are real, leading to an ubiquitous equation of nonlinearphysics called the nonlinear Schrodinger equation(NLSE) (also known as the Gross–Pitaevskii equa-tion). For a weakly interacting Bose gas, this equationcan be derived microscopically from the Heisenbergrepresentation of the many-body Hamiltonian. Toemphasise the generic nature of this fundamentalequation we introduce it here by using a fully classicalargument. From general considerations of a symmetrybreaking transition in an equilibrium system, one canconsider an energy functional that reaches a minimumin a stationary state. The energy of this state can bewritten as E ¼

RLdr, where the Lagrangian L can be

expressed through the order parameter field c and itsspatial derivatives. The Lagrangian containing onlyalgebraic functions of c and of its first derivative iscalled the Ginsburg–Landau (GL) Lagrangian. Thelowest order (i.e. simplest) GL Lagrangian thatcorresponds to a state of broken symmetry, andpreserves the symmetry of the system under globalphase rotations of the order parameter c, isL ¼ 1

2 jrcj2 þ 1

4 ð1� jcj2Þ2: The quartic potential re-

presents the simplest form for which the disorderedstate jcj ¼ 0 can become unstable leading to sponta-neous symmetry breaking and a new ordered state(corresponding to jcj ¼ 1) emerges. The dynamicequation of motion is then obtained directly as theEuler–Lagrange equation coming from this Lagran-gian yielding the NLSE

�i@tc ¼ r2cþ ð1� jcj2Þc: ð7Þ

It is also instructive to see that the NLSE can beobtained as a non-relativistic limit of the Klein–Gordon equation – the simplest equation consistentwith special relativity [17].

In non-equilibrium systems, such as exciton–polariton condensates, c1, c2 and c3 can becomecomplex with the imaginary parts representing theprocesses of pumping and dissipation. The equationfor the macroscopically occupied polariton stateC(r, t)becomes:

i�h@tC ¼ EðirÞ þUjCj2 þ VðrÞh i

C

þ i Pcohðr; tÞ þ PincðrÞ � k� sjCj2� �

Ch i

; ð8Þ


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where we restored the dimensions omitted in writingEquations (6)–(7). Schematically this includes: thepolariton dispersion, E(k) (which may be approxi-mated by a quadratic dispersion E(k) ’ �h2k2/2mpol ifonly low momentum modes are significantly occupied,leading to the Laplacian of C in real space), thestrength of the d-function interaction (pseudo)poten-tial U (as discussed in Section 1.2), an external potentialV(r), a coherent pump field Pcoh(r) exp(iopt) whichtries to fix the phase and time dependence of themacroscopically occupied mode C(r, t), as well as anincoherent pump field Pinc(r), which populates polar-itons without fixing their phase; k and s describe linearand nonlinear losses, respectively. In almost allexperiments, only one of Pcoh or Pinc are present, butnot both.

For the case of incoherent pumping it is necessarythat there should be some mechanism whereby theeffective pump strength would decrease as the densityjC(r)j2 increases, either by considering separate dy-namics of a population of reservoir excitons, or bydirectly including nonlinear dissipation, such an effectcan be approximately modelled by the nonlinear term7sjCj2C in Equation (8). If no such nonlinear processexisted then the system would be unstable to unlimitedcondensate growth as soon as Pinc4 k. The micro-scopic mechanism leading to the nonlinear term is amixture of depletion of the population of reservoirexcitons, and may also include an effect due to anincrease of condensate chemical potential reducingscattering from the reservoir [79].

From the form of Equation (8), one may immedi-ately see one reason why spatial pattern formation is ofparticular interest in a non-equilibrium condensate.The combination of spatial inhomogeneity and pump-ing tend to produce states with steady state currents.Since the net gain depends on local density (asreservoir diffusion can be neglected2), inhomogeneousdensity distributions imply different rates of gain orloss at different positions, so requiring currents toconnect these regions. One immediate possible con-sequence of this is finite momentum condensation,when the profile of pumping demands steady statecurrents, which we will discuss below. While alternateways exist to describe spatial structure, the cGLEequation provides a simple way to understand how theinterplay of pumping and trapping potentials can beresponsible for pattern formation, leading to kinds ofpatterns and coherent structures that would not occurin thermal equilibrium. Some examples of thesepatterns and structures include multimode condensa-tion, spontaneous creation of vortices and localisedsolitary waves. We will discuss the properties andobservations of these structures in the remainder ofthis section.

3.1. Condensation at finite momentum

A particularly simple way to create a non-trivial spatialpattern in a condensate, and one which in fact wasunintentionally realised in several early experiments isto pump with a sufficiently small pumping spot, andwith the exciton–photon detuning close to, or justgreater than zero [80]. This system has been studiedboth with quantum kinetic equations [81] and with theabove Ginsburg–Landau approach [82]. In the latterapproach, one can understand transparently thereason for the finite momentum condensation: due torepulsion the high density at the centre of the pumpspot shifts the polariton energy. If one wants to find asteady state profile C(r, t)¼C0(r)exp(7imt), then oneis forced to have m¼UjC0(r¼ 0)j2 at the centre of thespot. Away from the centre, the density decreases (asloss exceeds pumping), and so to maintain the sameenergy, one requires C0(r) ’ exp(ik � r)jC0(r)j with�h2k2/2mpol¼ m7UjC(r)j2. Alternatively, this can bere-interpreted as strong repulsion repelling polaritons,giving them a finite outward velocity, thus producinga condensate at a ring of finite momentum values. Thetwo descriptions are equivalent. Note that the finitemomentum ring still shows features such as coherence[80], in that light emitted at different azimuthal angles(from polaritons with different directions of in-planemomenta) shows interference fringes. Similar conden-sation at finite momenta has also been seen recentlyin a clean, long polariton lifetime, 1D microcavitystructure [76]. In this case, for some conditions ofpumping, more than one condensate was seen at once;we will discuss this below in Section 3.2.

3.2. Multimode coexistence

In the thermal equilibrium condensed state of weaklyrepulsive bosons, there is only one macroscopicallyoccupied mode, i.e. there is no ‘fragmentation’ [15].Many experiments on incoherently pumped polaritonsystems have however shown simultaneous coexistenceof several macroscopically occupied modes [67,76,83,84]; i.e. strong emission is visible at a number ofdistinct frequencies, indicating a state Cðr; tÞ ¼P

n CnðrÞexp ð�imntÞ. Spectrally resolving the emissionand then determining the real and momentum spaceprofiles of the resultant signal allows one to determinethe functions Cn(r). These mode profiles show thatthe occupied modes are related to the series of singleparticle modes in a disorder-induced effective trap(for experiments on CdTe, as in [67,83,84]), or fromdifferent plane-wave-like states in a 1D waveguidegeometry (for the cleaner GaAs system of [76]). Thereis, however, evidence in momentum space images ofthese condensates that the occupied modes are strongly


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influenced by pumping and decay. Without pumpingand decay, the system would be time-reversal sym-metric, and so the real-space wavefunctions can bemade real, implying that C(k)¼C*(7k), and so theintensity pattern in momentum space, jC(k)j2 shouldshow mirror symmetry. Measurements of the momen-tum space profile of the coexisting macroscopicallyoccupied states shows an asymmetry in momentumspace [84,85]. Thus, the modes occupied in such amultimode condensate seem to be related to the singleparticle modes, but modified by the particle flux due topumping and decay.

To understand what conditions are required forcoexistence, one may first consider competition be-tween the different single particle states which are fedby the same reservoir of incoherently created excitons[86]. In this approach, multimode coexistence ispossible if the density profiles of the modes havesufficiently small overlap that they can be simulta-neously fed by different spatial regions of the reservoir.There are, however, other features that can affectwhether multimode coexistence will occur: synchroni-sation of modes, and relaxation effects.

Synchronisation refers to the fact that if particlescan coherently move between the different spatialmode profiles, then this leads to a term that favourstwo different spatial modes having their phases locktogether, and oscillating at a common frequency [87].To describe this, suppose one starts with a systemC1(r)exp(7iy1) þ C2(r)exp(7iy2), where a simplecoexistence of independent modes would have y1¼ m1t,y2¼ m2t, then the question of whether two potentiallycoexisting modes synchronise or not depends onwhether y12¼ y17 y2 takes a fixed value, or a timedependent value. The equation satisfied by y12 can bereduced to the form [87,88]:

d2y12dt2þ g

dy12dt� D12

� �¼ �Juð�n1�n2Þ1=2sin ðy12Þ; ð9Þ

where g is a damping term arising from the pumpingand decay rates, D12 is the energy difference betweenthe single particle states, J is the rate of tunnellingbetween the two states, �n1;2 are the average populationsof the two modes and u is an effective averageinteraction strength between particles in one stateand particles in the other. If D12 > Juð�n1�n2Þ1=2 theny12 ’ D12t, and one has multimode coexistence, butfor small enough D12, the modes lock together. In thiscase, there can, however, be interesting dynamics ofcollective oscillations of polaritons between the twostates after a pulsed excitation; this has been investi-gated using disorder localised states in CdTe [89].

Recent work has also begun to investigate the effectof energy relaxation on the possibility of multimode

condensation. In its simplest form, the cGLE does notaccount for relaxation, in that the extent to whichmodes are populated does not necessarily favour lowerenergy modes. Experiments on extended 1D wave-guides appear to suggest a process of relaxation isimportant, in allowing multimode condensation ofmodes that have no overlap with the pump spot, butwhich might couple via intermediate macroscopicallyoccupied states [76,90]. Such relaxation can beincorporated in the cGLE by adding a term propor-tional to 7i@tC inside the second bracket on the right-hand side of Equation (8) [91]. This approach is verysimilar to how relaxation has been incorporated toaccount for the interactions of a superfluid with athermal cloud [92] in superfluid helium.

3.3. Vortices

In equilibrium condensates, the observation of quan-tised vortices has been viewed as evidence for amacroscopically occupied quantum state, as such astate can only be made to rotate by inserting phasesingularities, i.e. vortices, in the wavefunction.A superfluid condensate that has been set into rotationalso demonstrates the ability of a superfluid to showpersistent metastable flow; i.e. to remain in a metastablestate for astronomically long times if the transition tothe ground state would involve introducing phase twistsinto the macroscopically occupied wavefunction. Inpolariton condensates, several other aspects of vorticesand rotation are interesting: in particular the interplaybetween steady state currents and vortex formation, andthe ability to induce vortices by appropriate pumping,for both coherently and incoherently pumped systems.

3.3.1. Vortices with incoherent pumping

Numerical simulations of an incoherently pumpedsystem, with toroidal pumping and no trappingpotential have suggested it should be possible to inducea rotating polariton condensate if it is seeded with asufficiently strong coherent pump to inject angularmomentum. The rotation then persists for some timeuntil vortex motion out of the polariton cloud stops therotation [93]. Other numerical simulations have sug-gested that in the presence of a harmonic trappingpotential, one can in fact engineer situations where thereverse behaviour occurs – the non-rotating solution ismade unstable by the current flow in the inhomoge-neous density profile, and the solution with vortices isthe stable solution [94]. Figure 6(a) shows the simulationstarting from an originally non-rotating cloud, withvortices moving in from the edge of the polariton cloud.Experimentally, vortices have been seen in incoherentlypumped systems, almost certainly arising due to current


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flows caused by the intrinsic disorder potential in CdTe[95]. Such quantised vortices can be clearly observed ininterference images (see schematic experiment in Figure2), where phase winding at a point leads to a fork in theinterference pattern, see Figure 6(b). In contrast withequilibrium trapped condensates, in which vorticesspiral out of the condensate due to the Magnus force(force in the direction orthogonal to the densitygradients) and interactions with the thermal cloud[96], the exciton–polariton condensates’ vortices arestabilised (pinned) by inward fluxes at the local minimaof the disordered potential.

3.3.2. Vortices and the polarisation degree of freedom

If one accounts for the polarisation degree of freedomof polaritons then one may consider more interestingkinds of vortex. The simplest description of a polarisedpolariton condensate neglects weaker effects such asthe frequency difference between photon modes withtransverse electric and transverse magnetic polarisa-tions (TE–TM splitting), and the anisotropy wherebythe strain field splits the energies of different linearlypolarised exciton states. This leaves, however, a strongeffect due to interactions between polaritons: therepulsion between opposite polarisation polaritons isweaker than that between equal polarisation polar-itons. Therefore, the ground state has equal popula-tions of left- and right-polarised polaritons; i.e. it is alinearly polarised state [97,98]. If one then considersvortices on top of this polarised state, one finds thatthe minimum energy configuration for a single vortex,balancing the interaction energy with kinetic energy, isa ‘half vortex’ [99], i.e. either a vortex of either left- orright-circular polarisation.

The left- and right-circular vortices are not howeverentirely independent; the TE–TM splitting in a cavityadds a term HTE–TM/Cl*(@x7 i@y)

2Crþ Cr*(@xþi@y)

2Cl to the Hamiltonian of the equilibrium system.

This produces a long-range interaction between left-and right-vortices with opposite circulation (i.e. betweena left vortex and right anti-vortex, or vice versa) [100].Experiments have seen such individual half vortices ofleft- and right-circular polarisation as well as alignedhalf vortices of opposite circulation at various fixedpositions in a disorder potential [101]. These experi-ments studied the interference patterns of light after ithad passed through a polarising filter, thereby observingthat there are points where the left-circular light has avortex but the right-circular light has no vortex, or viceversa. Any energetic splitting between linear polarisa-tion states can destabilise half vortices; as mentionedearlier, such a splitting can arise due to strain fields, andmay also occur intrinsically in quantum wells in non-centrosymmetric semiconductors. The existence of adisorder potential and associated supercurrents mayhelp to stabilise half vortices. It also was shownnumerically [88] that it is possible to stabilise the co-existing right- and left-circular vortices of oppositecirculation, by applying a magnetic field perpendicularto the cavity. A full understanding of the conditionsunder which half vortices are stable, and the featurespresent in the experiment which allows their observa-tion, remains an open question [102,103].

3.3.3. Vortices with parametric pumping

With coherent pumping in the OPO configuration,since there is a free phase between the phase of thesignal and that of the idler, it is possible to seedvortices in this relative phase. This means that onecan have a situation where, e.g. there is a vortex inthe signal beam, and an associated anti-vortex in theidler [104]. Experiments have shown that a vortex inthe signal can be observed in the presence of a probebeam containing a vortex at either the signal or idlermomentum [85]. Further experiments have alsostudied the behaviour following a pulsed probe

Figure 6. (a) Time evolution of vortex lattice formation in a numerical simulation in a harmonic trap (after [94]). Reprintedfigure 2 permission from Keeling and Berloff, Phys. Rev. Lett. 100 (2008), 250401. Copyright (2008) by the American PhysicalSociety. (b) Fringe pattern from two displaced images of a condensate containing a vortex. The appearance of a fork in the fringepattern indicates a point where the phase winds by 2p about this point (from [95], experiments on the same CdTe sample as inFigure 4 and Figure 5(b), at a cryostat temperature of 4.2 K).


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beam that seeds a vortex [105]. If such a pulsed probeis injected in a coherently pumped system belowthreshold (i.e. with a pump power too low to showOPO), then a large transient amplified signal is seen,and the vortex survives as long as the transient signal.If the system was pumped above threshold, then thevortex may either survive for long times, or decay (byescaping out of the pump spot). To create a long livedvortex in this way requires a sufficiently strong probe.

3.4. Confining polaritons

One way to explicitly manipulate the pattern creationis to control the external potential V (r) by producing aspatial variation of either photon or exciton energy. Tocreate a spatial variation of the photon energy one canmodify the mirrors. Varying the width of the cavity,one can create shallow in-plane traps for photons[106]. One can also shift the photon energy by addingmetallic layers on top of the Bragg mirrors [75]. Verystrong confinement of photons can also be producedby etching to produce a narrow pillar [107], and relyingon dielectric contrast for reflection from the edge of thepillar. A spatial variation of exciton energy can beproduced by applying localised stress to the sample,which modifies the electronic band structure, and thuschanges the energy of the excitonic states [108].

A different approach to engineering an effectivetrapping potential is to use the repulsion betweenpolaritons to dynamically create an effective potential[76,109,110]. For incoherent injection, there is typicallya large population of reservoir excitons at highenergies, and the low energy polariton states will berepelled by these states, as well as by other low energypolaritons. Thus, a spatial profile of the pump field hastwo effects: it both creates a spatially dependent sourcefor the polariton condensate and also creates apotential that affects the polariton condensate.

Even when no potential is deliberately created,there may be an effective potential due to disorder.This can either come from spatial variation of theexciton energy, i.e. roughness of the quantum wellinterfaces [111], or from spatial variation of the photon[112] energy, roughness of the Bragg mirrors. Whileboth excitonic and photonic disorders are present, thephotonic disorder seems to play the dominant role intrapping polaritons in most current experiments;the relative role of both types of disorder is reviewedin [113].

4. Superfluidity

One of the more remarkable features of quantumcondensates is superfluidity; the ability of an interactingcondensate to flow without mechanical resistance, as

long as the flow is below a critical velocity. This occursbecause the condensate is unable to respond to thetransverse drag force exerted by the walls of a contain-er. This in turn arises from a combination of twoeffects. Firstly, as mentioned in the introduction, thecondensate can only be made to rotate by creatingvortices, and so it is not able to respond to thetransverse force arising from the wall. Secondly, as willbe discussed further below, the spectrum of singleparticle excitations in the presence of a condensate islinear, defining a critical velocity. For speeds lower thanthis critical velocity, it is not energetically favourable tocreate excitations. At non-zero temperatures, a popula-tion of quasi-particle excitations will however alreadybe present, and these quasi-particles may respond todrag forces. This leads to a two-fluid description, wherethe behaviour of the system can be described by acombination of a normal and a superfluid part. The factthat the condensate can only be made to rotate byexciting vortices also means that if one has a non-simply-connected geometry (e.g. a ring of superfluid)that is made to rotate, and then cooled below thetransition, the condensate persists in a rotating state.

In an equilibrium interacting condensate, all thephenomena named in the previous paragraph are seentogether as aspects of superfluidity (for a morecomplete review of aspects of equilibrium superfluiditysee e.g. [16,114]). However, in non-equilibrium polar-iton condensates it is not necessarily the case that theseaspects need all appear together. For example, theexcitation spectrum of single particle excitations can besignificantly modified by finite particle lifetime, requir-ing a re-defining of a critical velocity, yet, as alreadydiscussed above, quantised vortices can still be clearlyobserved in polariton condensates. Table 1 shows apossible checklist of different aspects of superfluidityshown by equilibrium condensates, and what has beenobserved in both coherently and incoherently pumpedpolariton systems. Some of these aspects of super-fluidity, such as the observation of quantised vortices,and the possibilities for solitary wave propagationhave already been discussed in Sections 3.3 and 3.1,respectively. The remainder of this section will discussvarious other aspects of superfluidity that have beenstudied in polariton condensates.

4.1. Spectrum

In an equilibrium condensate, the Bogoliubov spec-trum comes from considering fluctuations of the formCðr; tÞ¼ exp ð�imt=�hÞ½C0 þ

Pk uk expð�ixktþ ik � rÞþ

vk exp ðix�kt� ik � rÞ�, and finding a self-consistent set ofequations for uk, vk and the frequency xk. Substitutingthis ansatz for fluctuations into Equation (8) withoutany pumping or decay, and with a quadratic


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dispersion, the result for small k is the real mode-energy spectrum �hxk ’ �hk(m/mpol)

1/2 with m¼UjC0j2.This linear dispersion defines the critical velocity suchthat the energy in a moving frame xk! xk7 v � kremains positive.

The spectrum of single particle excitations in a non-equilibrium polariton condensate has been theoreti-cally studied extensively, both for incoherent [69,117]and parametric pumping [55,118]. The basic behaviouris quite similar for both cases because both involve thesame ingredients, of having a free phase (such as thatbetween the signal and idler modes) and a finiteparticle lifetime. For incoherent pumping, with a netpumping rate Z, the above linear dispersion becomesinstead �hxk ’ 7i�hZ þ (m�h2k2/mpol7 �h2Z2)1/2. Thisform means that the real part of the spectrum is zerofor k5 Z(mpol/m)

1/2. The diffusive behaviour at small kmeans that a simplistic identification of superfluidity asa property of the real part of the spectrum xk wouldimply that the non-equilibrium system is not super-fluid. However, since the separate aspects of super-fluidity are not necessarily related in the same way as inequilibrium, such a statement is premature.

There is an important difference in the role of thespectrum between equilibrium and non-equilibriumcondensates. In equilibrium, one is interested in theenergetic criterion of whether creating quasi-particlescosts or gains energy; in a general non-equilibrium case,the question is rather whether superfluid flow isdynamically stable [119]. If one allows a degree ofthermalisation by adding a relaxation mechanism, theenergetic and dynamic criteria for stability then becomelinked [91], in that the combination of relaxationaldynamics and excitations which would reduce energycan combine to give a dynamical instability above acritical velocity.

Experiments have only begun to explore thespectrum, and as yet do not have sufficient resolutionto clarify differences between equilibrium and non-equilibrium spectra. The experiments so far areconsistent with there being a change between aquadratic spectrum in the non-condensed state and aBogoliubov-like form when condensed [120].

4.2. Scattering from disorder

A more direct probe of superfluidity is to consider thebehaviour of polaritons flowing in a disorderedpotential, and ask whether the polaritons sufferfrictional drag. For flow past a weak potential, therewill be a simple connection between the observedbehaviour, and the spectrum discussed above, in thatthe linear response of the polariton system to the weakpotential will be involved. However, it is experimentallysimpler to investigate scattering off a strong disorderpotential, which can complicate the interpretation ofexperimental results. A further source of complication isthat even if the system were partially superfluid (i.e.superfluid, but not at zero temperature, so a two-fluiddescription was required), then the observed behaviourwould be a mixture of superfluid and normal behaviour.There would thus still be drag on a defect, and similarlythere would still be some scattering off disorder.

Experiments on such scattering have only beenperformed with resonant pumping, and in two quitedistinct experimental configurations, which we willdescribe.

4.2.1. Polariton ‘bullets’ in parametrically pumpedsystems

The experiments in [121] considered a parametricallypumped system (i.e. an injected signal), below thethreshold for OPO. On top of a steady pump beam, aweak idler beam was injected, which caused stimulatedparametric scattering to the signal and idler states. Thepacket of polaritons in the signal state can be createdwith a non-zero group velocity, and thus be made topropagate througha larger regionwhere thepumpexists.One can then study how such packets of polaritons inter-act with disorder. No scattering is seen, and the packetmaintains a well-defined single group wavevector.

4.2.2. Rayleigh scattering of pump beam

An alternate approach to studying how the excitationspectrum affects the propagation of polariton

Table 1. Superfluidity ‘checklist’, adapted and updated from [115], showing the phenomena expected to be seen, or that havebeen seen in different classes of potentially superfluid systems (see also [116]).

Quantisedvortices

Landaucriticalvelocity

Metastablepersistent

flowTwo-fluid

hydrodynamics

Localthermaleqbm.

Solitarywaves

Superfluid 4He/cold atom BEC � � � � � �Non-interacting BEC � 7 7 7 � 7

Classical irrotational fluid 7 � 7 � � �Incoherently pumped polariton condensates � 7 ? ? 7 ?Parametrically pumped polariton condensates � 7 ? ? 7 �


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wavepackets can be found by considering a differentexperiment. The experiment in [122], following aproposal of Carusotto and Ciuti [123], uses a singlepump beam, in a configuration where there is neither aKerr nor parametric instability, and so the polaritonpopulation rises smoothly with pumppower.When sucha polariton beam interacts with disorder, it canonly scatter to other momentum states if such statesexist. Because of the nonlinear interactions amongpolaritons, at large enough densities, the spectrummay become linear around the pump wavector andthereby remove other states that the systemmight scatterinto. In experiment it was observed that for the range ofpump wavevectors and pump intensities where thislinearisation should occur, scattering off disorderdisappeared.

5. Conclusions and outlook

In the last few years, experiments have gone beyondsimply attempting to prove that exciton–polaritoncondensates exist, and instead have begun to explorethe properties they display. Future experiments seemlikely to further explore these properties. There arealso likely to be significant developments in severalareas we have not been able to discuss here. Onedirection concerns polariton condensation in organicmaterials supporting polaritons, which have recentlyshown polariton lasing [124,125]. These materials areinteresting both because the exciton states involved arerather different from those in inorganic semiconduc-tors, being far more localised, and because they offerthe promise of condensation at far higher tempera-tures. There are also inorganic materials, wide band-gap semiconductors such as ZnO and GaN, whichhave much stronger exciton–photon coupling, and sooffer the possibility of stable polaritons at roomtemperatures [126–128]. These materials would there-fore offer the possibility of making collective quantumcoherence far easier to realise. It is therefore imperativeto understand what kind of coherence these systems doproduce, and how this may be used. (Note added inproof: very recently experiments on organic dyes inoptical cavities with weak light-matter coupling havereported a thermalised ‘‘photon condensate’’ at roomtemperature [J. Klaers, J. Schmitt, F. Vewinger, andM. Weitz, Bose–Einstein condensation of photons inan optical microcavity, Nature 468 (2010) p. 545].)Another, related, area concerns replacing opticalpumping with electrical injection [129,130], with theaim of being able to use polariton lasing in integratedoptoelectronic devices.

Another subject which we have only briefly men-tioned is that of studying effects associated with thepolarisation degree of freedom of the polaritons. In

Section 3.3.2,wementioned one effect of the polarisationdegree of freedom, that of allowing the possibility ofhaving separate vortices of left and right polarisation. Inaddition, there canbe interesting effectswhere an appliedmagnetic field favours circular polarisation, and thuscompetes both with the interactions that favour linearpolarisation, and with any anisotropy from strain fields[88,131–133]. There are also a rich variety of phenomenaassociated with spin dynamics in the process ofparametric scattering. These arise from competitionbetween a number of effects: polarisation rotation due tothe anisotropy and TE–TM splitting, polarisationrotation due to the effective magnetic field produced bypolariton–polariton interactions, and spin-dependentrates of parametric scattering which arise inevitablyfrom the the spin-dependent polariton–polariton inter-action. The variety of behaviour that can be seen due tothese effects are discussed extensively in [7].

In this review we have discussed many experimentalobservations and theoretical predictions of variousphenomena that take place in exciton–polaritoncondensates. Due to the coupled light–matter natureof exciton–polaritons, they are capable of showingbehaviour both related to that of equilibrium con-densates and lasers. They can also show kinds ofbehaviour that differ from both of these systems, asmanifested, for instance, by the dynamics of vortices,the temporal coherence properties, or the behaviourseen in the parametrically pumped system. The finitepolariton lifetimes puts these condensates in a regimedistinct from cold atomic gases or superfluid helium.At the same time, the behaviour is not that of a simplelaser, as can be seen from the lack of populationinversion, and the clear effects of polariton–polaritoninteractions. By differing from both the simple laserand the equilibrium condensate, exciton polaritoncondensates offer the opportunity to study novelaspects of condensation, pattern formation and super-fluidity. They, therefore, prompt profound questionsabout what superfluidity can mean in a non-equili-brium system, and offer a venue to explore the range ofconditions under which collective quantum coherencecan be achieved.

Notes

1. Earlier experiments had also seen aspects of thisbehaviour, e.g. [134].

2. Given that the reservoir states of higher energy excitonshave a much higher mass than the polaritons, and giventhe existence of disorder in typical quantum wells, it isreasonable to assume that the reservoir excitons do notdiffuse, thus the dynamics of the reservoir, balancingcreation by the pumping laser with scattering into lowerenergy polariton states, can be assumed to be spatiallylocal.


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Notes on contributors

Jonathan Keeling is a Lecturer in Theo-retical Condensed Matter Physics in theScottish Universities Physics Alliance atthe University of St Andrews, and anEPSRC Career Acceleration Fellow. Pre-viously he was a research fellow atPembroke College Cambridge and aLindemann Fellow at the MassachusettsInstitute of Technology. He did his

undergraduate and graduate degrees at the University ofCambridge. He has worked mainly on exciton–polaritoncondensation, and more recently on problems of many-bodyquantum optics in coupled light–matter systems.

Natalia Berloff is a Reader in Mathema-tical Physics in the Department ofApplied Mathematics and TheoreticalPhysics at the University of Cambridge.Previously she was an assistant professorat University of California in LosAngeles. She received her undergraduatedegree from Moscow State Universityand a Ph.D. in Mathematics fromFlorida State University. She is a fluid

dynamicist working on quantum fluids.

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Exciton–polariton condensation Contemporary Physics · Bose–Einstein condensation This subsection provides a brief overview of some of the essential features of Bose condensation,