42 August 2010 Physics Today © 2010 American Institute of Physics, S-0031-9228-1008-030-0

Most students of physics know about the special prop-erties of Bose–Einstein condensates (BECs) as demonstratedin the two best-known examples: superfluid helium-4, firstreported in 1938,1 and condensates of trapped atomic gases,first observed in 1995.2 (See the article by Wolfgang Ketterlein PHYSICS TODAY, December 1999, page 30.) Many also knowthat superfluid 3He and superconducting metals containBECs of fermion pairs.3 An underlying principle of all thosecondensed-matter systems, known as quantum fluids, is thatan even number of fermions with half-integer spin can becombined to make a composite boson with integer spin. Suchcomposite bosons, like all bosons, have the property thatbelow some critical temperature—roughly the temperatureat which the thermal de Broglie wavelength becomes compa-rable to the distance between the bosons—the total free en-ergy is minimized by having a macroscopic number ofbosons enter a single quantum state and form a macroscopic,coherent matter wave. Remarkably, the effect of interparticlerepulsion is to lead to quantum mechanical exchange inter-actions that make that state robust, since the exchange inter-actions add coherently.3,4

In the past few years, researchers have discovered a newtype of BEC based on the polariton—a quasiparticle that is a

quantum mechanical superposition of a photon and an elec-tronic excitation in a solid. Polariton condensates have nowbeen shown to exhibit the classic hallmarks of a BEC, includinga bimodal momentum distribution,5,6 quantized vortices,7

spontaneous symmetry breaking,8 long-range coherence,9

phase locking,10 and long-range superfluid motion.11 Polari-tons are less familiar than other bosonic quasiparticles, despitehaving been introduced conceptually by John Hopfield manydecades ago and observed experimentally in both bulk solidand quantum-well semiconductor structures. Since the 1990s,however, designs of semiconductor microcavities that confinethe quasiparticles to a two- dimensional sheet have alloweddramatic progress in controlling and observing polariton con-densates. The field has moved rapidly from demonstrations ofcollective, driven many-body states to polariton trapping tothe spontaneous condensates we discuss here. The early workis nicely summarized in reference 12.

A new eigenstateFigure 1a shows a typical structure for polariton experiments.One or more semiconductor layers are used as quantum wells.The polariton effect will occur with just one quantum well,but most recent experiments insert extra, identical quantum

wells to increase the overall coupling.In the thin quantum-well layers, thecharge carriers (electrons and holes)are confined to 2D motion. The quan-tum wells are then placed betweentwo mirrors, typically made of alter-nating dielectric layers in a Bragg re-flector structure; the arrangementmakes an optical cavity that also re-stricts photons to 2D motion.

If there were no cavity, then theabsorption of a photon by a quantumwell at low temperature would yieldan exciton—an electron in the conduc-tion band and a hole in the valenceband, bound together by Coulomb at-traction. An exciton is a compositeboson, just like a Cooper pair of elec-trons in a superconductor, except thatit is metastable, since the electron canemit a photon and fall back into thehole. An exciton is analogous topositronium, in that both are particle–antiparticle pairs with finite lifetimes.

If a quantum well is placed at anantinode of the confined photon modein an optical cavity, and if the energyof the exciton is close to that of a pho-

Polariton condensatesDavid Snoke and Peter Littlewood

A carefully engineered coupling between light and matter could pave theway to a room temperature Bose–Einstein condensate.

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Figure 1. A microcavity polariton experiment. (a) The setup consists of one ormore quantum wells (QW) placed at the antinodes of a confined optical mode in acavity. Typically, the entire structure is made as a single solid-state device, with themirrors of the cavity made of alternating dielectric layers—a structure known as a dis-tributed Bragg reflector (DBR). (b) When coupled in a quantum well, the exciton andphoton modes mix and become two upper and lower polariton (UP and LP) modes,with the energy difference between them defining the Rabi splitting. Mode energies,E, are shown here as a function of in-plane momentum, k||. In general, special caremust be taken to tune the exciton and photon modes to be nearly resonant at k|| = 0.

David Snoke is a professor in the department of physics and astronomy at the University of Pittsburgh in Pennsylvania. Peter Littlewoodis a professor in the department of physics at the University of Cambridge in the UK.

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ton in the cavity, then the exciton and photon states coupleto each other. As shown in figure 1b, the standard effect ofquantum mechanical mixing leads to two new eigenstates,each of which is a linear combination of the photon and ex-citon states. Those new eigenstates are the polaritons.12 Tech-nically, they are known as exciton–polaritons. It is also pos-sible to couple photon states to other excitations in solids tomake new eigenstates called phonon–polaritons, plasmon–polaritons, and so forth.

Many people have trouble thinking of polaritons as“real.” But quantum mechanics shows that once the propereigenstates for a system have been defined, they are the cor-rect particles to keep in mind. Quantum field theory goes fur-ther to say that every particle—even familiar ones such as theelectron, proton, and photon—can be viewed as an eigen-state, or excitation resonance, of an underlying field. Newresonances such as polaritons, magnons, phonons, and so on,known as quasiparticles, are not fundamentally different incharacter from elementary particles; they are just highlyrenormalized to have different properties. In particular, po-laritons can be viewed as photons having an effective massand much stronger interactions than photons in a vacuum.Solid-state physics has a long history of generating new typesof particles that can be studied as real objects, including par-ticles with fractional charge (as in the famous fractional quan-tum Hall effect) and quasiparticles that exist only in the pres-ence of other quasiparticles, such as fermion-like excitationsof Bose- condensed Cooper pairs in a superconductor.

A brief but eventful existenceNo mirror is perfect, so in typical experiments a polariton ex-ists for only 5–20 picoseconds inside an optical cavity beforetunneling out to form an external photon. Although that mayseem like a short time, it can be much longer than the inter-particle scattering time—the relevant time scale to reach athermalized quasi equilibrium that can be described by sta-tistical mechanics with approximate number conservation.Polaritons at high density in microcavities have strong repul-sive elastic scattering interactions, which allow them to ther-malize in less than a picosecond.

Polaritons can also move from place to place. They havebeen observed by time- resolved imaging to scatter into a ther-mal momentum distribution, much like a gas of atoms. Andanalogous to the way gases of atoms can be trapped in freespace by magnetic and optical fields, polaritons can betrapped by various methods in confined regions inside asolid. One such method is to apply a stress with the tip of apin to shift the bandgap of the semiconductor, creating a po-tential energy minimum, or “stress trap,” at a point in thesolid (see figure 2). Polaritons in the semiconductor can thenbe made to migrate toward, congregate near, and under cer-tain conditions condense in the trap.

A useful feature of planar microcavities is that the in-plane momentum of the polaritons is preserved when theytunnel into photons that leave the sample. Therefore, the mo-mentum distribution of the polaritons, along with the energyspectrum at each momentum, can be measured by resolvingthe angle of the emission of the external photons. Figure 3ashows the momentum-space narrowing—a canonical Bose–Einstein effect—of polaritons condensing as their density in-creases in a stress trap. (Density can be increased at nearlyconstant temperature by turning up the intensity of the pumplaser.) The spectral energy width also decreases, which indi-cates increased coherence in the polariton state.

The spatial width of the polariton cloud also narrows asthe polaritons increase in density. The polaritons pile into

their ground state, and even taking into account the repulsiveinteractions of the polaritons, their resulting spatial extent issmall compared with the size of the cloud of thermal particlesin the trap (see figure 3b). Thus both spectral and spatial nar-rowing occur for the condensate in a trap. That does not vi-olate the uncertainty principle, because the thermal particlesin the noncondensed state have total uncertainty much largerthan the quantum limit.

Why go to all the trouble to create polaritons in micro-cavities? As BECs, polaritons belong to the same general classas atomic condensates and superfluid He, yet one featuremakes them quite different: They are extremely light—about10−4 electron masses—and lighter mass means that quantumeffects appear at higher temperature.

Recall that the general condition for superfluid coher-ence is that the thermal de Broglie wavelength of the parti-cles, λ ~ (h2/mkBT)1/2, be comparable to the distance betweenthem. In two dimensions, that interparticle distance is of theorder of 1/n1/2, where n is the particle density, so that the onsetof coherence occurs when kBT ~ nh2/m. Consequently, conden-sation of polaritons has been observed to occur at tens ofkelvin and higher in many semiconductor structures and isexpected to be possible at room temperature. Compare that,for example, with the critical temperature of just 2.17 K forliquid He and the microkelvin temperatures required forBose–Einstein condensation of atomic gases. A team led byJeremy Baumberg in Cambridge, UK has already reportedspontaneous symmetry breaking of the polarization of po-laritons at room temperature in gallium nitride structures.8

For other energy scales relevant to polariton condensates, seee box on page 45.

An unconventional laser Like a laser, a polariton condensate is an open system, con-tinuously pumped in order to support the electron–hole pairdensity, which in turn supports a coherent light field. In bothsystems, an incoherent pump creates free electrons and holes,and in each case coherent light is produced in a spectrallynarrow, beamlike emission. What, then, is the distinction?

The difference, qualitatively, is the degree to which the

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Figure 2. Migrating polaritons. In a stress-trap experiment,polaritons created by a laser at one point in a semiconductor(the bright spot at the top left in this spectral image) flowdown an energy gradient and accumulate in an approximatelyharmonic potential minimum about 35 μm away (the spot atlower right). The potential minimum, indicated here by thedashed white curve, results from a shift in the bandgap of thesemiconductor due to stress applied with the tip of a pin. Theentire migration happens within the polaritons’ lifetime ofaround 15 picoseconds, such that the drift velocity is about108 cm/s. (Adapted from ref. 6.)

44 August 2010 Physics Today www.physicstoday.org

electronic states participate in the coherence. In a laser, the co-herent light adopts the cavity photon mode frequency, whichis fixed by the cavity length and dielectric constant. The roleof the electronic excitations is both to populate the photonmode and to damp it. Consequently, to produce coherent light,the population of electronic excitations must be inverted, sothat stimulated emissions, which yield phase-locked photons,dominate spontaneous emissions, which yield photons of ran-dom phase. In contrast, in a polariton condensate, the phasesof the exciton and photon are locked together as a coherent po-lariton, and the phases of the polaritons are in turn locked to-gether as a coherent condensate. Hence the oscillating lightfield adopts the polariton energy and phase, and there is noneed for inversion; coherent fields can be supported by a ther-mal quasi equilibrium. One practical outcome is that the elec-tron–hole density threshold for coherent emission is muchlower than for a conventional laser in the same semiconductormedium. The term “polariton laser” is often used to refer tocoherent light emission from a polariton condensate that is notquite in full equilibrium with the noncondensed polaritons.

The polariton system allows a continuous transition be-tween condensate and lasing states. Since the continuouspumping introduces random phase perturbations, the con-densed state only forms above a finite polariton densitythreshold, even at the lowest temperatures.13 Theoretical es-timates made at Cambridge, based on experiments by Benoît Deveaud- Plédran’s group at the École PolytechniqueFédérale de Lausanne in Switzerland, and Le Si Dang’s group

at Joseph Fourier University in Grenoble, France, suggest thatthe onset of the condensed state is determined by lossesrather than by temperature. If the pumping further increasesthe density beyond the critical density for condensation, de-phasing eventually decouples the excitonic component fromthe photon component, the polariton energy splitting(known as the Rabi splitting, see figure 1b) collapses, and oneis left with a conventional weak-coupling laser.

Recent experiments have clearly identified two distinctthresholds—one for polariton condensation and another forstandard lasing.14,15 Figure 4 shows data from one of those ex-periments, in which stress—applied by the tip of a pin, as inthe experiments of figures 2 and 3—was used to tune thebandgap energy of the quantum-well excitons relative to thecavity photon energy. The resulting shift in the polariton con-densate energy moves in the same direction as the exciton en-ergy shift, which demonstrates that the electronic componentis very much involved in the coherence of the condensate. Bycontrast, in the conventional lasing state, the coherent emis-sion occurs at the cavity photon energy, which does not shiftwith stress.

Other unique propertiesPolaritons in microcavities are 2D, spin-1 bosons with +1 and−1 spin projections, corresponding to left and right circularpolarization in their plane of motion. The nonzero angularmomentum condensate can therefore support so-called halfvortices, which have a π, instead of 2π, rotation in phase

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Figure 3. Momentum, energy, and spatial narrowing. (a) The angular distribution of the light emission from polaritons, shownhere in angle-resolved spectra, maps directly to the polaritons’ in-plane momentum distribution. At low polariton density or, equiva-lently, low pump-laser intensity, the thermalized polariton cloud fills a wide range of momentum states. As the pump intensity in-creases, the density approaches the critical threshold, Bose–Einstein statistics enforce a pileup of the particles near the ground state,and the momentum distribution narrows. The energy width also narrows, which implies increasing coherence. The curvature of thespectra give the effective mass of the polaritons, via the relation ħ2/m = d2E/dk∣∣

2. (b) A similar experiment shows spatial narrowing ofa polariton cloud. Although one might expect an increase in the number of particles to lead to higher pressure and a resulting ex-pansion, the opposite happens. As their density is increased past the critical threshold, polaritons pile up in the ground state, caus-ing the polariton cloud to shrink. (Adapted from ref. 6.)

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along with a π rotation of polarization over the same patharound the vortex core. In a fascinating recent experiment bythe Lausanne group, half vortices were detected as forks ininterference patterns of the light coming from a polaritoncondensate (see figure 5).7 The patterns are observable intime- averaged experiments because the vortices are pinnedat local structural defects.

Another distinctive trait of polaritons is that their dy-namics can be driven externally. In a superconductor, the co-herent phase is visible only to another superconductor, viathe Josephson effect. In a polariton condensate, the conden-sate’s phase can be coupled to that of an external coherentsource of light. It is therefore possible to create coherent po-laritons directly by tuning an external laser to resonate witha polariton mode. Though such a system is not in thermalequilibrium, it shows a type of spontaneous symmetry break-ing, in that the coherent oscillation of the polaritons will havea fixed but undetermined phase. Luis Viña’s group at the Au-tonomous University of Madrid in Spain, along with MauriceSkolnick’s group at the University of Sheffield in the UK, useda resonant external laser to produce superfluid, soliton-liketransport11 and to inject vorticity into a condensate. Well be-fore the demonstration of thermalized condensates, coherentpolariton systems generated by resonant pump lasers hadalso been used to show strong nonlinear effects,12 and a col-laboration between the Viña group and Jacqueline Bloch’sgroup at the CNRS Laboratory of Photonics and Nanostruc-tures in Marcoussis, France, recently observed critical slow-ing of polariton dynamics near the symmetry-breaking phase

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Figure 4. A transition between polariton condensate andconventional lasing states. The upper and lower polariton(UP and LP) energies decrease as the exciton energy istuned—by means of a mechanical stress—through the cav-ity photon resonance. The stress shifts the electronic states ofthe medium while leaving the cavity photon energy largelyunchanged. The energy of the cavity emission at the thresh-old for coherence initially decreases like the exciton state,which indicates that the system is a condensate and that theelectron states of the medium are involved in the coherence.At large detuning—high stress—the excitons and photonsbecome decoupled, and only standard lasing is possible, sothe coherent emission jumps up to track with the cavity pho-ton mode. (Adapted from ref. 15.)

Polariton condensates—and Bose–Einstein condensates in gen-eral—are characterized by several relevant energy scales in addi-tion to the critical temperature, Tc, determined by the thermal deBroglie wavelength condition, kBTc ~ nh2/m, as described in thetext. One energy scale is the average repulsion between particles,of order nU, where n is the particle density, and U is the character-istic interaction energy per unit density between two particles. Ifthat energy is small compared with the thermal energy, kBT, thesystem is weakly interacting; if it is comparable to or larger thankBT, it is strongly interacting. Cold atoms in optical traps andpolaritons in microcavities are weakly interacting at typical con-densation densities, whereas the atoms in liquid helium arestrongly interacting.

Another natural energy scale is the binding energy of thefermions that make up the composite bosons, h2/μa2. Here μ is thereduced mass of the fermion pair and a the pair correlationlength—the characteristic distance between the fermions thatmake up the composite boson. For polaritons, μ is on the order ofthe electron mass—about 104 times larger than the polaritonmass—so when the condition for polariton condensation sets in,the excitons are rarely in contact; that is, na2 ≪ 1. Equivalently, thebinding energy is large compared with kBT, so the compositebosons act to all intents and purposes as pure bosons. The sameis true for all bosonic atom condensates, including liquid He. If na2

becomes larger than 1, then in the high-temperature limit a Motttransition to an uncorrelated fermionic plasma will occur, whereasat low temperature a Bardeen-Cooper-Schrieffer transition to acoherent state of weakly correlated fermions can occur (see thearticle by Carlos Sá de Melo in PHYSICS TODAY, October 2008, page45). Such behavior describes Cooper pairs in superconductorsand can also happen with pairs of fermionic atoms in optical traps.

The preceding energy scales are important for all Bose–Ein-stein condensates; for polaritons, there are two others. One is theenergy of the splitting between the lower and upper polariton

states, known as the Rabi splitting (see figure 1b). If that energy isless than kBT, thermal excitations to the upper state will disruptthe coherent superposition of the excitons and photons in thepolaritons, since the exciton–photon combination of the upperpolariton state is exactly out of phase with that of the lower,ground state. If one keeps the temperature below that point andincreases the polariton density, the interparticle separationbecomes much smaller than the de Broglie wavelength, whichgives the length scale over which the polaritons can be spatiallylocated. In a sense, such polaritons will be much closer to thezero-temperature limit—in which mean-field theory applies—than atomic gases at much lower absolute temperatures. Theresult is a system with weak but long-range interactions. This canoccur as density increases, even while the polariton density is wellbelow the critical density for a Mott transition, that is, while theinterparticle distance is much larger than the exciton radius a.

The final relevant energy scale for polaritons is the energybroadening due to their finite lifetimes. It can be understoodsimply as the Heisenberg energy uncertainty, ΔE ~ h/Δt, due tothe finite time, Δt, spent in one state. The Q factor—the averagenumber of round trips a photon makes before it escapes throughthe cavity mirrors—for the polariton microcavity structures istypically 2000 or greater, which means that the lifetime broaden-ing of the cavity mode is much less than the Rabi splittingbetween the two polariton eigenstates. The polaritons are there-fore well-defined particles. That so-called strong coupling is oneimportant distinction between a polariton mode and a conven-tional laser mode. Nonetheless, polaritons, given their finite life-times, are continually lost, and losses generate noise due to thefluctuation–dissipation theorem. Therefore, condensation willnot occur at arbitrarily low densities. In the condensed state,losses will also subtly alter the long-wavelength and low-fre-quency collective behavior of the condensate from that of a truesuperfluid.

Polariton energy scales

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transition in a polaritonic optical parametric oscillator.16

Many polariton experiments do not have a single exter-nally controlled trap, but instead have several traps due tomultiple potential minima in a disordered landscape. In a re-cent experiment, the Lausanne and Grenoble groups observedseveral coexisting condensates in adjacent local minima.10 Atlow density, each condensate in a separate minimum had itsown phase. At higher density, when the repulsive interactionsof the polaritons caused them to fill up their localized regionsand begin to interact with particles in neighboring regions, theseparate condensates locked phases. In another experiment,the Sheffield and Grenoble groups observed overlapping con-densates with slightly different frequencies and wavefunc-tions. 9 Such systems contain polariton currents, driven by dif-fusion and gradients in the potential energy landscape. Thenonlinear dynamics of coupled condensates is becoming an in-teresting area for further research. Since the microcavity po-laritons form a truly 2D gas, they may also present opportuni-ties to study the crossover from a trapped condensate withstable phase to a translationally invariant superfluid withstrongly fluctuating phase.

The polaritons’ lifetimes are long enough to lead to sub-stantial effects in the excitation spectrum of the polariton con-densate. The classic feature of a superfluid is the develop-ment of a linear collective sound mode—the so-calledBogoliubov mode—which is predicted to be visible as a shiftin the polariton emission spectrum above threshold; Yoshi-hisa Yamamoto’s group at Stanford University has reportedexperimental evidence of such a shift.17 However, theory pre-dicts that polariton condensates will decohere on the longestlength scales; that is, the Bogoliubov mode should becomediffusive. That length scale is estimated to be long enoughthat it is not easily visible in current experiments. Still, it isan important fundamental difference from other BECs.

The mathematical pictureTheoretical approaches to the study of polariton condensates

fall into three broad classes.13 At the lowest densities, micro-scopic models treat polaritons as weakly interacting bosons.As the density increases, the relative proportion of photon toexciton in the ground state grows beyond the low-density 50–50 superposition case. Beyond the crossover to a dense sys-tem, it is more convenient to work with models that have anexplicit long-range photon- mediated interaction between lo-calized exciton states. Such models are amenable to beingtreated as open systems coupled to baths and can also readilyincorporate the effects of excitonic disorder. That approachallows quantitative connection to experiments and the even-tual crossover to standard lasing when, at yet higher densi-ties, the exciton and the photon decouple.

The details of the microscopic models, however, are toocumbersome to describe large-scale spatial structure, and acoarse-grained description of order-parameter structure anddynamics is needed. As is the case with atomic condensates(see the article by Keith Burnett and colleagues in PHYSICSTODAY, December 1999, page 37), a Gross– Pitaevskii or Ginzburg– Landau approach is preferred,4 but modified to in-clude dissipation, dynamics, and flow. A Gross–Pitaevskiiequation is simply a Schrödinger equation for a matter wave,having a potential energy term that is linear with local parti-cle density or, in other words, proportional to the square ofthe wavefunction, ∣Ψ∣2; a Ginzburg–Landau equation is thetime- independent version.

In a Gross–Pitaevskii or Ginzburg–Landau equation, thepolariton condensate is treated as a classical wave; the polari-ton–polariton interactions correspond to a highly nonlinearterm in the wave equation. Much of the behavior seen in mi-crocavity polariton experiments with resonant excitation cantherefore be interpreted as extreme nonlinear optical effects.For example, optical parametric oscillator effects, includingentangled photon pair generation, have been demonstratedin polariton condensates with continuous-wave excitation atlow average power; in contrast, standard optical parametricoscillator devices require intense, ultrafast laser pulses. Not

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Figure 5. “Half vortices” in polariton condensates. (a) In a path around a so-called half vortex—starting, say, at the leftmostpoint of the outer dashed ring—the phase of the polariton wavefunction rotates by 180°, not 360°. The half-phase rotation is ac-companied by a 180° rotation in polarization. In the schematic, red lines and ellipses indicate the polarization; blue vectors—repre-senting an instantaneous field, which over time traces out the wavefunction—indicate the phase. Far from the core the polariza-tion is linear, toward the center it becomes elliptical, and at the core it is circular. (b) Such a vortex can be detected as a fork in theinterference pattern that is produced when the polariton condensate light emission is sent through a circular polarizing filter. Thefork (circled in red) is seen if the filter passes one handedness, but it is absent if it passes the other, a signature of half vortices.(Adapted from ref. 7.)

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all of the polariton experiments can be described by a simplenonlinear wave equation, however, since fluctuations, dissi-pation, and flow can sometimes play a major role.

The polariton express In the past decade, researchers have attempted experimentson Bose–Einstein condensation of various other types of qua-siparticles in solids, including magnons, spin excitations(“tripletons”), long- lifetime excitons in coupled quantumwells, and correlated electron–hole pairs at high magneticfield in coupled quantum wells.18 All of those, like polaritons,have lifetimes long enough to be described by near- equilibrium statistics.

Researchers in the field of polariton condensates havehad to overcome the difficulty of preparing thermalized sys-tems in short time scales, a task complicated by the fact thatpolariton coupling to the atomic lattice via phonon emissionand absorption is weak. Weak coupling makes cooling diffi-cult, since the only avenue available for polaritons to get ridof energy is through phonon emission—vibrational motionof the lattice atoms, which diffuses away. The relatively weakinteractions between polaritons and the atomic lattice of thesemiconductor may be an advantage, however, in next- generation experiments aimed at studying quantum dynam-ics in a many-body system.

Polariton condensate work is now expanding and in-cludes a European network of researchers headed by AlexeyKavokin at the University of Southampton, UK. The workpromises to lead to unique optical applications and newphysics. Coherent polaritons in microcavities have alreadybeen used for such nonlinear optical effects as optically gatedamplification, optical spin Hall transport, and optical paramet-ric generation of entangled photon pairs.12 Current work onelectrical pumping instead of the widely used optical pumpingmay open a further array of potential applications, such aslow- threshold coherent light generation, optical transistors(modulation of one light beam by another), and entangled-photon-pair emitters. The polariton express is gathering speed.This work has been supported by the NSF and the EPSRC.

References1. A. Griffin, Excitations in a Bose-Condensed Liquid, Cambridge U.

Press, Cambridge, UK (1993), p.1.2. L. Pitaevskii, S. Stringari, Bose–Einstein Condensation, Oxford U.

Press, New York (2003), p.1.3. A. J. Leggett, Quantum Liquids: Bose Condensation and Cooper Pair-

ing in Condensed-Matter Systems, Oxford U. Press, New York(2006).

4. See, for example, D. W. Snoke, Solid State Physics: Essential Con-cepts, Addison-Wesley, San Francisco (2009), sec. 11.1.3; see alsoM. Combescot, D. W. Snoke, Phys. Rev. B 78, 144303 (2008).

5. J. Kasprzak et al., Nature 443, 409 (2006).6. R. Balili et al., Science 316, 1007 (2007).7. K. G. Lagoudakis et al., Science 326, 974 (2009); Y. G. Rubo, Phys.

Rev. Lett. 99, 106401 (2007). For a review, see M. Richard et al.,Int. J. Nanotech. 7, 668 (2010).

8. J. J. Baumberg et al., Phys. Rev. Lett. 101, 136409 (2008).9. A. P. D. Love et al., Phys. Rev. Lett. 101, 067404 (2008).

10. A. Baas et al., Phys. Rev. Lett. 100, 170401 (2008).11. A. Amo et al., Nature 457, 291 (2009).12. A. Kavokin et al., Microcavities, Oxford U. Press, New York,

(2007). 13. For reviews of basic polariton condensate theory, see J. Keeling

et al., Semicond. Sci. Technol. 22, R1 (2007); M. Wouters, I. Caru-sotto, Phys. Rev. Lett. 99, 140402 (2007).

14. D. Bajoni et al., Phys. Rev. Lett. 100, 047401 (2008).15. R. Balili et al., Phys. Rev. B 79, 075319 (2009).16. D. Ballarini et al., Phys. Rev. Lett. 102, 056402 (2009).17. S. Utsunomiya et al., Nat. Phys. 4, 700 (2008).18. D. Snoke, Nature 443, 403 (2006). ■