Bose-Einstein condensation of photons from the thermodynamic limit to small photon numbers
BBose-Einstein condensation of photons from the thermodynamic limit to small photonnumbers
Robert A. Nyman ∗ and Benjamin T. Walker
1, 2 Quantum Optics and Laser Science group,Blackett Laboratory,Imperial College London, Prince Consort Road, SW7 2BW, United Kingdom Centre for Doctoral Training in Cont olled Quantum Dynamics,Imperial College London, Prince Consort Road, SW7 2BW, United Kingdom
Photons can come to thermal equilibrium at room temperature by scattering multiple times froma fluorescent dye. By confining the light and dye in a microcavity, a minimum energy is set andthe photons can then show Bose-Einstein condensation. We present here the physical principlesunderlying photon thermalization and condensation, and review the literature on the subject. Wethen explore the ‘small’ regime where very few photons are needed for condensation. We comparethermal equilibrium results to a rate-equation model of microlasers, which includes spontaneousemission into the cavity, and we note that small systems result in ambiguity in the definition ofthreshold.
FOREWORD
This article is written in memory of Danny Segal, whowas a colleague of one of us (Rob Nyman) in the Quan-tum Optics and Laser Science group at Imperial Collegefor many years. The topic of this article touches on thesubject of dye lasers, the stuff of nightmares for any AMOphysicist of his generation, but a stronger connection toDanny is that he was very supportive of my applicationfor the fellowship that pushed my career forward, andfunded this research. One of Danny’s quirks was a strongdislike of flying. As a consequence, I had the pleasure ofjoining him on a 24 hour, four-train journey from Londonto Italy to a conference. That’s a lot of time for storytelling and forging memories for life. Danny was one ofthe good guys, and I sorely miss his good humour andadvice.This article presents a gentle introduction to thermal-ization and Bose-Einstein condensation (BEC) of pho-tons in dye-filled microcavities, followed by a review ofthe state of the art. We then note the similarity to mi-crolasers, particularly when there are very few photonsinvolved. We compare a simple non-equilibrium modelfor microlasers with an even simpler thermal equilibriummodel for BEC and show that the models coincide forsimilar values of a ‘smallness’ parameter.
INTRODUCTION TO THERMALIZATION ANDCONDENSATION OF PHOTONS
Bose-Einstein condensation can be defined as macro-scopic occupation of the ground state at thermal equilib-rium [1][2]. It is a natural consequence of the exchangestatistics of identical bosons, and therefore occurs in awide variety of physical systems, such as liquid helium,ultracold atomic gases or electron pairs in superconduc-tors.The Bose-Einstein distribution for non-interacting identical bosons may be familiar to most physicists, butit’s worth looking at again: f ( (cid:15) ) = 1 / [exp ( (cid:15) − µ ) /k B T − , (1)which gives the occupancy f of a state at energy (cid:15) . Ithas two parameters. T is the temperature, which tells usimmediately that we are discussing thermal-equilibriumphenomena. The chemical potential is µ , which is thethermal energy required to add another particle to thesystem from a reservoir, and it dictates the number ofparticles in the system. The chemical potential is alwayslower than the energy of the ground state in the system,but as it approaches from below, the distribution shows adivergence in the ground-state population. That’s BEC.Photons can be brought to thermal equilibrium in ablack box, but their number is not conserved, which im-plies that the chemical potential is either not well definedor strictly zero, depending on your point of view. In ei-ther case, BEC is not possible. It is however possibleto give light a non-zero chemical potential in a mediumwhich has an optical transition between two broad bands,such as a semiconductor (with valence and conductionbands) or a fluorescent dye (with ro-vibrationally broad-ened electronic states), as explained by W¨urfel [3]. Thechemical potential then sets the population of the ex-cited band. There is a kind of chemical equilibrium be-tween photons and excitations in the medium (inducedfor example by optical pumping), with reactions beingabsorption and emission. Thus the chemical potential ofthe photons will equal that of the medium, which is non-zero, provided there are enough absorption and emissionevents, i.e. enough time for the chemical reaction to havetaken place before other loss processes occur or the pho-tons leave the system.Not only will the photons acquire a non-zero chemi-cal potential, they will also reach a thermal equilibriumpopulation, dictated by the ratio of absorption (loss) andemission (gain) of the optically interesting medium. Thatrelation is known as the Kennard-Stepanov [4][5] or Mc- a r X i v : . [ qu a n t - ph ] J un Cumber relation [6]. The ratio is dictated by a principleof detailed balance, through rapid relaxation among thestates within the bands, i.e. vibrational relaxation ofdye molecules, which takes perhaps 100 fs, which is fastcompared to typical spontaneous emission lifetimes of afew ns. It is usually possible to identify a zero-phononline (ZPL) about which the spectra are symmetric witha Boltzmann factor between them: A ( (cid:15) ) F ( (cid:15) ) = e ( (cid:15) − (cid:15) ZPL ) / k B T (2)where F and A are fluorescence emission and absorptionrespectively, normalized to their peak values, (cid:15) is the en-ergy of the light and (cid:15) ZP L the energy of the ZPL [7]. Thetemperature T is the temperature of the the phonons thatcause vibrational relaxation, assumed to be the same asthe temperature of the bulk medium. In Fig. 1 we showthe absorption and emission spectra of Rhodamine 6Gin ethylene glycol, which shows the expected symmetry.Analysis reveals that the Kennard-Stepanov/McCumberrelation Eqn. (2) is very well matched at room tempera-ture over a wide range of wavelengths.
450 500 550 600 650Wavelength (nm)0.00.20.40.60.81.0 N o r m a li s e d V a l u e AbsorptionFluorescenceZPL
FIG. 1. Absorption and fluorescence emission spectra forRhodamine 6G in Ethylene Glycol, the solution most used forthermalizing and condensing photons. Both are normalizedto their peak value, and the ZPL is the wavelength at whichthey are equal. The data in this figure are available fromRef. [8].
But BEC requires a unique ground state into whichcondensation can occur. We engineer the density ofstates for the light using a pair of near-planar mirrors,a Fabry-Perot optical cavity. Let us make the mirrorsso close together that the free spectral range, the differ-ence between longitudinal modes of the cavity, is at leastas large as the widths of the absorption and emissionspectra of the dye. Then, only one longitudinal mode isrelevant, but there can be many transverse modes. Thisis known as a microcavity. For photon thermalizationin Rhodamine, the cavity is typically about 8 half wave-lengths long, so we label the longitudinal mode q = 8.The cavity-resonant energy is minimum for light prop-agating parallel to the cavity optical axis. Light prop-agating at an angle must have higher energy to match the boundary conditions for resonance. For small angles,the energy is quadratically proportional to the in-planewavenumber (or momentum) of the light, just like a mas-sive particle with with kinetic energy.We can understand how the transverse modes of thecavity relate to the shape of the mirrors, by consideringa local cavity length, and hence local cavity-resonant en-ergy for the light. Where the mirrors are closer together,the energy is higher, so there is an effective potentialenergy cost, dependent on the mirror shape.Thus the light can be considered as a massive particlemoving in a trapping potential (assuming that the cavityis convex, longer in the middle than the edges), whoseenergy as a function of momentum p and position r is: E ( r , p ) = mc ∗ + p m + V ( r ) (3)Here the effective mass is given by the cavity length L and c ∗ = c/n the speed of light in the intracav-ity medium: m = hn /cλ . The cutoff wavelength λ is the longest wavelength (for light emitted from thecavity) which is resonant with the cavity in the per-tinent longitudinal mode: λ = 2 nL /q . The localpotential energy V ( r ) is given by local deviations ofthe cavity length, δL ( r ) and can be simply written as V ( r ) /mc ∗ = δL ( r ) /L . Experimental apparatus
To trap the light long enough for thermalizationthrough multiple absorption events, the cavity mirrorsmust have reflectivity of at least 99.99% ( <
100 ppmloss). Such mirrors are commercially available, and usedielectric coatings of several pairs of layers. The simplestconfiguration for a cavity uses spherical mirrors, either apair, or one in conjunction with a planar mirror, as shownin Fig. 2. The spherical cavity length variation leads toa harmonic potential, at least close to the longest partof the cavity, where the photons are trapped. Typicallythe mirror radius of curvature is about 0.5 m, leadingto mode spacings (trapping frequencies) around 40 GHz.Because of the curvature and the proximity of the mir-rors, being just a few half-wavelengths apart, at leastone of mirrors is ground down to about 1 mm diameter.Light is pumped at an angle to the optic axis, to takeadvantage of the transmission maximum of the dielectricmirror coating, As a result, the mirrors are often gluedto other components which make alignment of the pumpeasier: see Fig. 2 for one example of how the assemblycan be done.To align the cavity, the mirrors require five degreesof freedom for their relative position and orientation.The separation on the optic axis must be actively con-trolled with nanometre-precision using for example apiezo-electric translation stage. It is very likely that
To detectorsObjective LensSpherical MirrorPlanarMirrorPumpLight Dye andSolvent Photoluminescence
FIG. 2. Top: Schematic of the apparatus required for pho-ton thermalization and condensation. Because the distancebetween the mirrors is short (about 1.5 µ m) and one mirroris curved the planar mirror is ground down to about 1 mmdiameter. To pump at an angle, taking advantage of an angle-dependent transparency of the dielectric mirrors, the planarmirror is built into a simple optical assembly (bottom). the cavity length will need to be actively stabilized, yetscanned so that the resonant wavelength varies by tens ofnanometers. The solution is to shine a collimated beamof narrowband light at the edge of the stop-band of themirrors, at which wavelength they transmit at least anorder of magnitude more than at the wavelengths usedfor thermalized light, and the dye does not strongly ab-sorb this light. HeNe laser light 633 nm wavelength is agood match to mirrors designed for thermalizing light ataround 580 nm using Rhodamine 6G. This narrowbandlight then forms rings, similar to Newton’s rings. Theimages are acquired by a camera, processed to find thering radius, and feedback is applied to actively controlthe cavity length.Light emitted from the cavity is collected by an ob-jective. Our imaging system uses a simple achromaticdoublet in an afocal setup, i.e. producing an image atinfinity. This collimated light is then split, by dichroicmirrors (to extract the stabilization reference light) andby non-polarizing beamsplitters, after which is it sent toa variety of diagnostic tools. The most important toolsare a camera and a spectrometer.Typical data are shown in Fig. 3. The spectrum shows,at low power, a distribution which is compatible withthe Bose-Einstein distribution, Eqn. (1), at room tem-perature, taking into account the density of states for a FIG. 3. Typical data showing photon BEC. Left: spectra.At low pump powers, the spectrum is compatible with theBoltzmann distribution at room temperature, showing botha cutoff (ground state) and a density of states equivalent to atwo-dimensional harmonic oscillator. At higher powers, extraphotons go into the ground state and the populations of theexcited states saturate. Right: a real-space, real-colour im-age just above threshold, showing a thermal cloud of photonsaround a large population in the centre, where the lowest-energy transverse mode of the cavity is located. two-dimensional (2D) symmetric harmonic trapping po-tential. There is a clear cutoff showing that there isa well-defined lowest-energy mode, in this case around λ = 576 nm. As the pump power is increased, the chem-ical potential approaches zero from below, and the pop-ulation of the lowest mode increases dramatically whileall other modes saturate. BEC is the explanation. Theimage is taken just above threshold. It shows a Gaus-sian fuzz around, which is the non-condensed, thermalcloud, whose size depends mainly on the temperature andthe trapping potential. In the centre is a bright spot,indicating the large occupation of the smallest mode,which is the lowest energy. It is noteworthy that thelight is green (high energy) at the edge, where the po-tential energy is highest, and yellow (low energy) in themiddle. At high pump intensities, the pump must bepulsed with a low duty cycle, so that the population ofa scientifically-uninteresting triplet state [9, 10] is keptlow. Typically, pulses last about 500 ns and are repeatedat about 500 Hz. Things which are like Photon BEC but which arenot Photon BEC
At this point we will digress and discuss three otherkinds of condensates of light, none of which is photonBEC: exciton-polariton condensates; classical wave con-densation of light and BEC of plasmons.There is a large community working with light andsolid-state matter which are strongly-coupled, in the cav-ity QED sense that the coherent coupling is faster thanincoherent mechanisms like spontaneous emission or cav-ity loss, using microcavities. Strongly coupled light-matter systems are known as polaritons. Typically thelight interacts with a quasiparticle made of a boundelectron and hole pair known as an exciton, making anexciton-polariton. In near-planar microcavities, sufficientpump power leads polariton condensation [11]. Conden-sation is considered distinct from lasing in that the exci-tons interact with each other substantially (see Ref. [12],p362), approaching thermal equilibrium, even if imper-fectly. The excitons associated with the condensed po-laritons can be free to move (Wannier excitons, typicalof inorganic semiconductors [13]) or bound to individ-ual sites (Frenkel excitons, typical of organic fluorescentsolids [14, 15]). By contrast, thermalization and BECof photons as described above is performed in the weak-coupling limit, and with liquid-state matter.Classical wave condensation, sometimes known asRayleigh-Jeans condensation, of light is a nonlinear wavephenomenon. Let us consider light with spatial ampli-tude or phase noise, propagating through a nonlinearmedium. Stochastically, the spatial spectrum will redis-tribute to follow a Rayleigh-Jeans distribution in trans-verse wavenumber k : f ( k ) = T / ( (cid:15) k − µ ) where T is theamount of noise which is equivalent to a temperature, (cid:15) k ∝ k the equivalent of kinetic energy, and µ repre-sents the total light power propagating relative to thenonlinearity. It has been achieved using simple imagingoptics [16] and in multimode pulsed lasers [17]. There isno mention of quantization of light or matter here, andindeed the distribution is the high-temperature limit ofBose-Einstein distribution ( (cid:15) k (cid:28) T ), Eqn. (1), providedthat modes are very closely spaced ( (cid:126) → STATE OF THE ART
Having presented the physical principles and experi-mental basics, we now review the history and state ofthe art of Photon BEC.The thermalization of photons in a dye-filled microcav-ity was first shown by Martin Weitz’s group in Bonn [19],far below condensation threshold. Proof of thermaliza-tion relied on showing that the distribution of light in thecavity is largely independent of the details of the pump-ing, e.g. pump light position. They also showed thatthermalization works well only with detuning of the cut-off wavelength close to resonance so that re-absorptionof a cavity photon is likely to happen before loss fromthe cavity. Having proven thermal equilibrium, theycranked up the pump power. Macroscopic occupation of the ground state, much as in Fig. 3 was observed [20].Together with the thermal equilibrium, that is sufficientevidence for most commentators to declare that BEC hasbeen achieved [21]. In addition, they inferred a thermo-optic repulsive interaction between photons, whose value˜ g in dimensionless units is ˜ g ’ × − which is verysmall indeed and indicate that photon BEC is for themost part, an ideal gas of non-interacting bosons.Since these initial observations, there have been a greatnumber of theoretical discussions of how photon BEChappens, and what one expects its properties to be. Mea-surements have been rarer, with only the Weitz groupand ours publishing experimental articles on the topic,with Dries van Oosten’s group (Utrecht) having more re-cently achieved photon BEC.There have been a small number of review articles onphoton BEC, some of which explain the concepts of pho-ton BEC for a non-specialist audience [22]. Jan Klaers’stutorials [23, 24] provide an excellent introduction to thefield. Schmitt et al. [25] is more up to date. There are afew chapters of the book on Universal Themes of Bose-Einstein Condensation [26] which are relevant to photonBEC and available open-access, most notably Ref. [27].In this section, we attempt a more comprehensive review,covering the majority of the published literature directlyon the topic of photon BEC, even if some of the theorywork may have little hope of experimental implementa-tion.
Observed phenomena
After their first observation of thermalization and con-densation of photons, the Weitz group attempted to movefrom a liquid to a solid-state sample of dye dissolvedin a UV-setting polymer, cured while inside a micro-cavity [28]. The thermalization functions equally well,although the concentration of dye is limited by fluores-cence quenching, as explored in Ref. [29]. While they didobserve BEC, it was not reproducible as the dye photo-bleached at high pump intensity in a matter of seconds.They have also made progress by dissolving materialswith large thermo-optic effects to the dye solution, andthen locally heating. The resultant position-dependentrefractive index translates to a controllable potential en-ergy landscape for the microcavity photons [30].They then measured both second-order correlationsand number fluctuations of the condensate mode [31].With thermalizing photons it is possible to interpo-late between canonical and grand-canonical ensemblesby changing the cavity detuning from resonance, effec-tively changing the ratio of photons to molecular exci-tations. The molecular excitations form a reservoir. Inthe canonical ensemble, far detuned from resonance, thephoton number is large relative to the square-root of theexcitation number, and fluctuations are largely Poisso-nian. By contrast, close to resonance, the reservoir of ex-citations is large, and the fluctuations super-Poissonian.The result is that the condensate number can fluctuatewildly, even leading to the Grand Canonical Catastrophewhere frequently there are no photons at all in the con-densate. While this work was guided by earlier statisticalmodelling [32], the measured correlations have also beenexplained through photon-photon interactions [33]. Theconclusion is that the photon-photon interaction is cer-tainly weak (˜ g < − ), depends on the detuning fromresonance, and that perhaps counter-intuitively the fewermolecules involved, the stronger the interactions.Two studies indicated how thermalization happens,and how it breaks down. At Imperial College, we pro-duced the first photon condensates outside the Weitzgroup, and showed how simple parameters such as theshape of the pump spot affect the distribution of pho-tons [34]. Light is imperfectly redistributed from pumpspot towards the thermal equilibrium distribution. Onecan achieve condensation with very low pump powersusing a small spot, but only for larger spots does thespatial distribution of photons match thermal equilib-rium. Using a streak camera and 15-ps pump, Schmitt et al. [35] observed the dynamics of thermalization ofphotons, showing how thermalization happens on thetimescale of photon absorption by dye molecules.BEC is a thermodynamic phase transition. Damm etal. [36] measured the internal energy of the photons (fromthe spectrum) and defined an equivalent for heat capac-ity, as a function of not absolute temperature but tem-perature relative to threshold for condensation. Fromthis, they inferred a heat capacity, which shows a lambdatransition characteristic of BEC.Condensates are typically characterized by their long-range coherence, first hinted at in photon BEC by a sin-gle image in Ref. [23]. Marelic et al. [37] systematicallystudied stationary first-order coherence using imaging in-terferometers with slow cameras. They showed that non-dissipative thermal Bose gas theory describes the datawell below and just above threshold, with the conden-sate showing long-range spatial and temporal coherence.Below threshold, the thermal cloud has a position depen-dent potential energy, which makes for interesting imagesbut complicated analysis: see Fig. 4. Far above thresh-old the coherence decreases, which can be explained bymultimode condensation, in which several modes becomemacroscopically occupied.Non-stationary measurements of the phase of the con-densate mode show phase slips [38], associated with theoccurrences of fluctuations in population number downto zero photons. These phase slips are a clear exampleof spontaneous phase symmetry breaking in a driven-dissipative system: when the population is large, thephase diffuses only slowly, but with small populations thephase is ill-defined and can jump discontinuously. Thecondensate re-forms with a spontaneously chosen phase. FIG. 4. Image of the interference pattern of a thermal cloudof photons away from the interferometer white-light fringe.Rings appear because the potential energy landscape is rota-tionally symmetric, with increasing energy near the edges.
Further experiments have explored the momentum dis-tribution of thermalized light [39] showing that the pho-tons interact only weakly with themselves (˜ g < − ) andwith the molecules. More recently, a third group, thatof Dries van Oosten in Utrecht, have achieved photonBEC. Their preliminary results show that the conden-sate is polarized, whereas the thermal cloud around it isunpolarized [40]. Theoretical models
The theory works on photon condensates can looselybe divided into those that assume approximate thermalequilibrium and those that don’t. There are works whichtake a fully quantized approach to fluctuations and thereare semiclassical mean-field models, and there are thosethat apply statistical mechanics.
A nonequilibrium model of photon condensation
Foremost among the nonequilibrium models is the Kir-ton and Keeling model, as first presented in Ref. [41].The model starts from conservative dynamics based ona standard cavity QED model, the Jaynes-Cummingsmodel, with multiple emitters and multiple light modes,and with the addition of a phonon degree of freedom as-sociated with each molecule. Molecular electronic stateis coupled to vibrational state via a Huang-Rhys parame-ter (a continuum generalization of a Franck-Condon fac-tor) since the molecule shape is slightly different betweenground and excited electronic states. Drive and dissipa-tion are then included via standard Markovian assump-tions.The resulting master equation contains terms whichinclude cavity loss, pumping of molecules by an externalsource, decay of molecules via spontaneous emission outof the cavity, and most crucially both emission of lightinto the cavity and absorption from the cavity. The lattertwo terms come with amplitudes which depend on the ab-sorption and emission strength of the dye, and it is theseprocesses which lead to thermalization of light. The mas-ter equation, realistically involving millions of moleculesand thousands of light modes, is far too unwieldy to solvedirectly, but the averages of populations can be solved forquite efficiently. The solutions of the rate equation forphoton populations show thermalization and condensa-tion matching the Bose-Einstein distribution when therate at which cavity photons scatter from dye moleculesis larger than the cavity loss rate. For larger loss rates,a mode can show threshold behaviour, but it is not nec-essarily the ground state, indicating lasing rather thanBEC.Kirton and Keeling elaborated further results of theirmodel [42], looking at the dynamics of photon popula-tions after a pulsed pumping event, and evaluating bothfirst- and second-order correlations for individual photonmodes. In response to observations of the breakdownof thermalization due to inhomogeneous pumping, inboth stationary [34] and time-resolved experiments [35],they modified their model to include spatial distributionsof pumping and molecular excitation [43]. The resultsmatch the salient points of the experimental data. Theywere able to show that the multimode condensation seenin Ref. [37] was due to imperfect clamping of the molec-ular excited-state population in regions adjacent to thecentral condensate light mode which leaves the possibilityof positive gain for other modes.Hesten et al. used the Kirton and Keeling modelto explore a large parameter space, describing a non-equilibrium phase diagram for dye-microcavity pho-tons [44]. The phase diagram proved to be particularlyrich, with many possible multimode condensate phasesin the crossover between well-thermalized BEC and un-thermalized laser states. In particular, they predict de-condensation, where population in a mode decreases withincreasing pumping rate, due to mode competition.A full master equation approach using just a smallnumber of light modes can be tractable. Kopylov etal. [45] have worked with two modes, which is enoughmodes to draw conclusions about condensation but notabout thermalization.
Quantum field-theory models
There are several papers treating near-equilibrium as agiven, and using quantum field theory techniques such as Schwinger-Keldysh [46] or quantum Langevin [47] tech-niques to access not only the average behaviour butalso fluctuations. These techniques are needed to dealwith the fact that photon BEC is driven-dissipative sys-tem with both pumping and loss processes (like exciton-polariton condensates), rather than a conservative sys-tem (like atomic BEC).The theory group of Henk Stoof have applied theSchwinger-Keldysh to calculate the effects of drive anddissipation on both temporal [48] and spatial [49] coher-ence. They have also shown how interacting photons ina lattice potential behave differently from the superfluid-Mott insulator transition known from conservative sys-tems [50].The fluctuations of a system at thermal equilibriumare understood to be related to compressibilities and sus-ceptibilities via the temperature in what are known asfluctuation-dissipation relations. Chiochetta et al. [51]propose testing the fluctuation-relations as a means ofquantifying how close driven-dissipative systems like pho-ton BEC come to true thermal equilibrium.Snoke and Girvin [52] point out that it is rather un-usual that coherence in a photon BEC can build up de-spite the absence of direct photon-photon interactions.They show that, remarkably, the coherence is generatedthrough incoherent interactions with the thermal bath ofmolecular vibrations.
Mean-field models
The equation of motion for the condensate order pa-rameter is typically derived in the same way as for non-linear optical systems, and is sometimes known as theLugiato-Lefever equation [53], or a dissipative Gross-Pitaevskii equation or a complex Ginzburg-Landau equa-tion. There are various nearly-equivalent forms, whichinclude: − i (cid:126) ∂ψ∂t = (cid:20) V ( r ) − (cid:126) m ∇ ⊥ + g | ψ | + i (cid:0) γ net − Γ | ψ | (cid:1)(cid:21) ψ (4)where ψ is the order parameter, which is the electric fieldof the condensate mode; g the strength of interactions; γ net is the difference between the pump rate and cav-ity decay rate; Γ describes the saturation of molecularexcited states; and m and V are effective mass and po-tential as described earlier. The effective kinetic energy ∇ ⊥ comes from diffraction of light. The dissipative termsmodify beyond-mean-field properties such as correlationsand depend on the fact that the pump light is incoherentwith the condensate mode [11]. Excluding the dissipativeterms, the order parameter equation reduces to Eqn. (3)for plane waves, ignoring the rest-mass energy term.Similar equations were first derived for multimodelasers, then applied to light in microcavities [54]. Its solu-tions are Bogoliubov modes of sound [55, 56] or collectivebreathing [57] or scissors [58] modes. It can be derivedfrom Maxwell’s equations [59], and coincides with themean of the equations coming from quantum field treat-ments [46, 47]. Interactions in photon BEC are expectedto include retarded thermo-optic effects, and nonlocal ef-fects have also been considered [60]. Statistical models
It is possible to treat photon BEC with non-quantumformalisms from statistical mechanics or laser rate equa-tions, where quantum effects only come in throughbosonic stimulation or exchange statistics. Average pop-ulations for effectively two-dimensional [61] and one-dimensional [62, 63] landscapes are readily calculatedfrom the Bose-Einstein distribution.Fluctuations in numbers of photons are correctly cal-culated only if the finite size of the reservoir of molecularexcitations is taken into account [32, 64]. When takinginto account polarization modes of the light, there is atleast one prediction that the second-order correlationsof condensed light could show sub-Poissonian statistics(anti-bunched) with not unreasonable parameters [65].Although the approximation that photons do not interactamong themselves is both simplifying and usually appli-cable to photon BEC, it has been shown that interactingphotons should show non-Gaussian statistics, and per-haps suppress the Grand Canonical Catastrophe [66, 67].
Suggestions for alternative systems for photoncondensation
So far, experiments on photon BEC have been re-stricted to near-planar microcavities filled with one ofa small number of fluorescent dyes (mostly Rhodamine6G and Perylene Red) in liquid water or ethylene glycol,with the exception of Ref [28] in a UV-set polymer. Therequirements of the thermalizing medium are rather gen-eral: satisfying the Kennard-Stepanov/McCumber re-lation, having a good fluorescence quantum yield andstrong absorption at high concentrations. Other dyesand perhaps colloidal quantum dots are obvious candi-date replacement materials. Suitable media may also in-clude optomechanical devices [68]. Preliminary measure-ments suggest that both molecular gases at high pressurewith ultraviolet light [69] and erbium-doped fibres in theinfrared [70] would be suitable media. BEC of photonsthermalizing by scattering from plasmas is probably theoldest of all the proposals [71] but still relevant [72].The optical environment need only provide a mini-mum energy mode and a gap as well as retaining pho-tons longer than the re-scatter time from the thermaliz-ing medium. For example, planar photonic crystals filled with semiconductors have been proposed for photon ther-malization and condensation [58], as have arrays of su-perconducting qubits [73].There are a few outlandish theoretical proposals tocombine photon BEC with quantum optomechanics [74,75] or atomic BEC [76, 77] but, while not technically im-possible, it is unlikely that anyone will go to the effort toimplement the ideas experimentally.One unusual proposal [78] interpolates between theclassical Rayleigh-Jeans condensation of waves and quan-tum BEC. Light with spatial noise propagates in anon-linear medium with a gradient of linear refractiveindex perpendicular to the propagation direction, andlight is made to selectively leak out for large transversewavenumbers. This system is then formally equivalentto evaporative cooling of trapped, interacting bosons intwo dimensions, where the propagation direction playsthe role of time.
WHEN PHOTON BEC GETS SMALL
A question that is often asked about photon BEC is‘how is it not a laser’? There are many answers, but itis not unreasonable to argue that photon BEC systemsare a very special case of a laser, where the re-scatter ofphotons is rapid enough to redistribute the light amongmany cavity modes. But if we look at very tight con-finement, i.e. small mirror radii of curvature, the modespacing can become as large as the thermal energy, andonly one cavity mode has significant occupation. In thisregime, photons in dye-filled microcavities can exhibitBEC with very small numbers of photons far away fromthe thermodynamic limit, and they can also act as mi-crolasers, where the spontaneous emission is more likelyto go into a cavity mode than in free space.In this section we will first see how the concept ofthreshold in an equilibrium, Bose-Einstein distributionbecomes ambiguous for large mode spacings. We willthen take a look at a simple rate-equation model whichshows microlasers exhibit remarkably similar effects.
Tiny Bose-Einstein condensates
The thermal-equilibrium Bose-Einstein distributionEqn. (1) is the very simplest statistical model relevantfor photon condensation. In Fig. 5 we show the result fora two-dimensional harmonic oscillator potential of angu-lar frequency ω , where the states involved are discretewith the i th state having and energy i × (cid:126) ω and degen-eracy i + 1. We choose a chemical potential, and fromthat calculate the total population and the ground-statenumber, as displayed. A well-known result is that thetotal number of particles in the system at threshold is N C = (cid:0) π / (cid:1) ( (cid:126) ω/k B T ) − . -2 -1 Total population n tot -2 -1 G r o un d - s t a t e p o p u l a t i o n n ω/k B T FIG. 5. The ground-state population in the Bose-Einsteindistribution in a two-dimensional harmonic oscillator poten-tial, as a function of total particle number. When the modespacing is small compared to the temperature, the thresholdtends a thermodynamic (sharp) transition. Conversely, forvery large mode spacings, only one mode is occupied and nothreshold is apparent in the population of the ground state.
With a small mode spacing (or equivalently a hightemperature), the threshold is deep and narrow in thesense that there is a large jump in population for a smallchange in total population. In the thermodynamic limit,of infinitesimal mode spacing, the threshold is infinitelysharp, and is a true phase transition. On the other hand,for small mode spacing (low temperature), the differencebetween below- and above-threshold populations is in-distinct, and there is a wide range of population whereit is not clear if the system is above or below thresh-old: the threshold is broad and shallow. For extremelysmall systems, there is just one mode with non-negligiblepopulation, so the population of that mode is equal tothe total. In that case, there is no threshold in termsof average population, although there may be distinctivecorrelation or fluctuation behaviour.
Microlasers
Photon BEC takes place inside microcavities, wherethe spontaneous emission from the dye molecules is mod-ified by the resonator, an effect known as the Purcelleffect, which can lead to enhancement (on resonance)or reduction (off resonance) of the spontaneous emissionrate [12]. The factor by which the emission is sped up de-pends on the cavity parameters (the Purcell Factor, F P )and on the exact position of the molecule in the cavitymode, i.e. the emission rate for a molecule at a node ofa cavity mode is very different from that of a moleculeat an antinode. The latter factor can vary greatly, so itis difficult to make better than order-of-magnitude esti- mates for the overall emission enhancement. F P notablydepends inversely on the cavity mode volume: smallercavities result in large modifications to the emission rate,provided they have large quality factors.Lasers using microcavities are parameterized princi-pally by the fraction of spontaneous emission directedinto the one cavity mode of interest, given the symbol β .With Purcell enhancement, β = F P / (1 + F P ). For largelaser systems, where the cavity does not markedly affectthe spontaneous emission rate, F P (cid:28) β (cid:28) P inter-acting with a number of molecular excitations N is:˙ P = [ γβN − κ ] P + γβN (5)˙ N = R p − γN − γβN P (6)where γ is the total spontaneous emission rate includ-ing the effects of the cavity, and R p the pumpingrate. Recasting Eqn. (5) we find ˙ P = γβN ( P + 1) − κP where the terms in parentheses make clear the roles ofstimulated ( P ) and spontaneous (+1) emission. Thismodel neglects re-absorption by the fluorescent medium,non-radiative loss, saturation of excited state popula-tion and fluctuations but still captures the essential be-haviours [79–85].The equations are readily solved for the steady statepopulation: P = ( βρ −
1) + p (1 − βρ ) + 4 β ρ β (7)where ρ = R p /κ , the rate at which molecules are excitedin units of the cavity loss rate. The positive root of thequadratic equation is taken since P > β = 10 − – 10 − . For small β , the lasershows a clear threshold, with a large jump in popula-tion for a small change in pump rate. When more of thespontaneous emission is directed into the cavity mode, β → -2 -1 Pump rate per cavity lifetime ρ -2 -1 P h o t o n nu m b e r P β FIG. 6. Mode population in the microlaser model as afunction of pump rate in units of cavity lifetime, for variousfractions β of spontaneous emission directed into the cavitymode (as opposed to into free space). When β is small , thethreshold tends a sharp transition. Conversely, for β → Which system is smaller?
Figs. 5 and 6 show that tiny Bose-Einstein conden-sates and microlasers exhibit very similar behaviours interms of reduction and broadening of threshold when theparameter indicates that the system is size, respectively k B T / (cid:126) ω and 1 /β , becomes small. In Fig. 7 we show re-sults of the models side-by-side. A value of β is set, andthen (cid:126) ω/k B T adjusted to match mode population in thelimit of low pump rate or total population. For smallsystems, the two models very nearly coincide, althoughthere are deviations for larger parameters. It is there-fore difficult to distinguish BEC from lasing, althoughsaturation of excited state populations, as in Fig. 3, maybe a hint. The number of modes thermally available inthe BEC model is approximately ( k B T / (cid:126) ω ) , and per-haps it is this parameter which should be more directlycompared to 1 /β . In this respect, BEC is an exclusivelymulti-mode phenomenon, but if there is only one occu-pied cavity mode, then there really is not much differencebetween BEC and lasing.Microcavities suitable for photon BEC can be con-structed using focussed ion beam milling to pattern theconfining potential through the surface shape [86]. No-tably, Ref. [87] shows a solid-state dye microlaser op-erating in a regime of strong re-absorption, showingfeatures reminiscent of thermalization and BEC. Withsmall radii of curvature, near-single-mode operation, i.e. k B T / (cid:126) ω ∼
1, is certainly possible. Very small modevolumes and high quality factors can be simultaneouslyachieved [10]. While the bare Purcell factor can be large F p (cid:29)
1, in a fluorescent dye, rapid dephasing due to -2 -1 n tot or ρ -2 -1 n o r P β = 0 . ω/k B T = 0 . β = 0 . ω/k B T = 0 . FIG. 7. Comparison of the two models, BEC and microlaser. n tot and n are the total population and ground-state popu-lation (BEC model); ρ and P are pump rate and mode pop-ulation (microlaser model). For a chosen β , we set (cid:126) ω/k B T to match the low-number population, and there are no otheradjustable parameters. vibrational relaxation makes it difficult to predict ex-actly what proportion of the spontaneous emission willbe emitted into the cavity mode. Nonetheless, lasers us-ing these microcavities should show β parameters whichapproach unity. It seems that is possible to make a devicewhich can be tuned between tiny BEC and tiny laser, bytuning for example the re-scattering rate via the detuningfrom the molecular resonance. At threshold, the photonnumbers will be rather similar, with barely more thanone photon in the lowest-energy cavity mode, despite thedifferent physical origins of the threshold behaviour. CONCLUSIONS
We have seen how photons can be made to thermalizeand condense at room temperature. There is a grow-ing body of literature on this subject, which is connectedto wider fields of driven-dissipative condensates of light.When we push the concept of photon BEC to fewer pho-tons, we run into ideas from microlasers, and the distinc-tion between the two concepts becomes blurred, despitethe fact that BEC is an equilibrium phenomenon and las-ing is dynamic. In this regime of few photons, we expectto find interesting quantum correlations among the pho-tons which may lead to applications of photon BEC inquantum information processing.
ACKNOWLEDGEMENTS
We thank the UK Engineering and Physical SciencesResearch Council for supporting this work through fel-0lowship EP/J017027/1 and the Controlled Quantum Dy-namics CDT EP/L016524/1 which was co-directed byDanny for many years. ∗ Correspondence to [email protected][1] R. Pathria,
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