Energy relaxation in the Gross-Pitaevskii equation
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] J u l Energy relaxation in the Gross-Pitaevskii equation
Michiel Wouters and Vincenzo Savona Institute of Theoretical Physics, Ecole Polytechnique F´ed´erale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland
We introduce a dissipation term in the Gross-Pitaevskii equation that describes the stimulatedrelaxation of condensed bosons due to scattering with a different type of particles. This situationapplies to Bose-Einstein condensates of quasi-particles in the solid state, such as magnons andexcitons. Our model is compatible with the phenomenology of superfluidity: supercurrents arestable up to a critical speed and decay when they are faster.
PACS numbers: 05.70.Ln, 03.75.Kk, 67.90.+z, 71.36.+c.
The Gross-Pitaevskii equation (GPE) provides an ex-cellent description of an isolated dilute Bose gas at lowtemperature. For example, the GPE accurately describesthe density profiles and frequencies of elementary excita-tions of trapped ultracold atomic bose gases[1].Systems featuring Bose-Einstein Condensation (BEC)of quasi-particles in the solid state, such as (exciton-)polaritons [2] and magnons [3] are however not so wellisolated from their environment as the atomic gases. Insemiconductor microcavities for example, the polaritonsthat form a Bose Einstein condensate undergo scatter-ing with lattice phonons and with high energy excitons.These interactions allow the polaritons to relax towardlower energy.The Boltzmann equation is the easiest way to modelthese scattering dynamics theoretically [4]. The semi-classical approximation involved in this approach be-comes however problematic when the polariton gas entersthe condensed phase. Most importantly, supercurrents,that are a gradient of the phase of the polariton field, aremissed. This precludes e.g. the description of vorticesin polariton condensates [5], that appear naturally in theGPE.In order to allow for a simultaneous description super-currents and of relaxation due to interactions with theenvironment, we will derive in this Letter a dissipationterm that can be added to the GPE. This term modelstransitions between components of the GPE field at dif-ferent frequencies (see Fig. 1). For specificity, we will usein the following the terminology of ‘polaritons’ relaxingthrough ‘phonon scattering’, but our formalism is appli-cable to any dilute Bose gas that dissipates energy intothe environment.An important requirement of the theory is compati-bility with superfluidity. According to the Landau crite-rion [1], the superflow cannot relax due to phonon scat-tering as long as the superfluid velocity is below the criti-cal speed. A Boltzmann description that includes phononscattering violates this requirement: a Bose gas with ini-tially all the particles at finite momentum k = k c will re-lax to a state with the majority of particles in the groundstate k = 0. We will show below that our GPE basedmodel for phonon relaxation is instead fully in agree- distance ene r g y FIG. 1. Schematic illustration of the relaxation of polaritonsdue to scattering with phonons. Several modes in a randompotential are macroscopically occupied. Where they spatiallyoverlap, relaxation takes locally place from the high to thelow frequency modes, as indicated by the arrows. ment with the Landau criterion for stability of superflowslower than the critical speed. In addition, it providesa dynamical model for the dissipation of superflow whenthe speed exceeds the critical one.A kinetic model formulated in terms of transitions be-tween states characterized by their frequencies involvesa coarse graining of the time, analogous to the coarsegraining of space used in the semiclassical Boltzmannequation [6]. We will use the following definition of thetime dependent spectrum of the bose field ψ ( x, t ): ψ ω ( x, t ) = 1 T Z tt − T e iωt ′ ψ ( x, t ′ ) dt ′ , (1)where T is the coarse graining time step. The inversetransform reads ψ ( x, t ) = X ω e − iωt ψ ω ( x, t ) . (2)Spontaneous scattering is a quantum feature that is dif-ficult to describe with a classical field model such as theGPE. We therefore restrict to the stimulated relaxationprocesses: dn ω ( x, t ) = n ω ( x, t ) X ω ′ r ( ω, ω ′ ) n ω ′ ( x, t ) dt, (3)where r ( ω, ω ′ ) is the net scattering rate from the modeat frequency ω ′ to the mode at ω . In order to conserveparticle number, the relaxation rate should change signunder exchange of ω and ω ′ : r ( ω, ω ′ ) = − r ( ω ′ , ω ). Thefirst term in the expansion of the rate as a function ofthe frequency difference is therefore r ( ω, ω ′ ) = κ ( ω ′ − ω ) , (4)where the relaxation constant κ has the dimension of aninverse density. Eq. (4) is for example the form of thescattering rate obtained with the golden rule for the re-laxation of quantum well excitons due to scattering withacoustic phonons [7].Under the change of the density (3), the wave functionvaries as ψ ω ( x, t ) + dψ ω ( x, t ) = s n ω ( x, t ) + dn ω ( x, t ) n ω ( x, t ) ψ ω ( x, t ) , (5)where we used the fact that a stimulated relaxation pro-cess does not change the phase of ψ . Expanding Eq. (5)to first order in dn ω , substituting Eqns. (3) and (4) andusing the inverse transform (2), one obtains for the re-laxation dynamics of the wave function dψ ( x, t ) dt = κ ¯ n ( x, t )2 (cid:20) ¯ µ ( x, t ) − i∂∂t (cid:21) ψ ( x, t ) . (6)In the derivation, boundary terms in the partial integra-tion that scale as 1 /T were neglected, because we con-sider the limit of a large coarse graining time where 1 /T is smaller than any other relevant frequency scale. InEq. (6), ¯ n and ¯ µ are the time averaged density and chem-ical potential respectively:¯ n ( x, t ) = 1 T Z tt − T | ψ ( x, t ′ ) | dt ′ , (7)¯ µ ( x, t ) = 1¯ n ( x, t ) Re (cid:20) T Z tt − T ψ ∗ ( x, t ′ ) i∂∂t ψ ( x, t ′ ) dt ′ (cid:21) . (8)The real part is taken in Eq. (8), because it is readilyshown with the Madelung transformation ψ = √ n e iθ ,that the imaginary part of the integral in Eq. (8) scalesas 1 /T . Eq. (6) gives the term that was sought to de-scribe the stimulated relaxation of a classical Bose fielddue to local scattering with phonon like particles and isthe central result of this Letter.The right hand side in Eq. (6) resembles the frequencydependent amplification term that was introduced inRef. [8] to describe an energy dependent gain mechanism: ∂ψ/∂t = ( P/ Ω K )[Ω K − i∂/∂t ] ψ . Two main differencesshould be emphasized. First, here the relaxation termis proportional to the polariton density ¯ n , whereas inRef. [8], the relaxation is proportional to the gain fromthe reservoir P . The second important difference is thatthe gain cutoff frequency Ω K is replaced by the average frequency ¯ µ . As a consequence, particle loss and gainbalance each other and there is no net gain in Eq. (6).The modes with frequency ω > ¯ µ experience loss, wherethe ones with ω < ¯ µ are amplified. Phonon absorption isthus not included in our model. The relaxation (6) is dueto the interaction with an environment that is effectivelyat zero temperature.As a first application, we add the dissipation term (6)to the GPE for noninteracting particles in two coupledlevels. In the context of polariton condensation, the cou-pled two-level system describes for example the situationof spatially distinct potential wells [9] or the polarizationdegree [10]. The full dynamics is i ∂∂t ψ j ( t ) = − Jψ − j + i κ ¯ n j ( t )2 (cid:20) ¯ µ j ( t ) − i∂∂t (cid:21) ψ j ( t ) , (9)for j = 1 ,
2, where J is the coupling energy. Fig. 2 showsthe time evolution of the densities in the two states, ob-tained from the numerical integration of Eq. (9). Theinitial wave function is taken to have equal amplitudeson the two states and a small phase difference betweenthem. Due to the dissipation, the Rabi oscillations aredamped. Note that the time evolution (9) drives thesystem automatically into the ground state and that theeigenstates of the Hamiltonian were not explicitly neededto compute the phonon scattering rates. The contribu-tion of the excited state to the wave function is shownin the inset of Fig. 2. Its exponential decay is excellentlyreproduced with the rate obtained from Eq. (3).We have numerically checked that the dissipative dy-namics conserves particle number in the limit of a smalltime step in the numerical integration. We correct theresidual deviation by restoring the norm of the wave func-tion after each application of the dissipation operator.We now show that in the case of the interacting ho-mogeneous Bose gas, our description of the dissipationis fully compatible with the phenomenology of persistentsupercurrents. The Gross-Pitaevskii equation for the ho-mogeneous interacting Bose gas supplemented with thephonon scattering term reads i ∂∂t ψ = (cid:20) − ∇ m + g | ψ | + i κ ¯ n (cid:18) ¯ µ − i∂∂t (cid:19)(cid:21) ψ. (10)The usual steady state solution ψ ( x, t ) = √ n c e − iµt + ik c x is still a solution of Eq. (10) with µ = k c / m + gn c ,where k c is the condensate momentum, g the interactionconstant and n c the condensate density.The excitations on top of the condensate can be de-scribed by the wave function ψ ( x, t ) = ψ ( x, t )[1 + ue ikx − iω ( k ) t + v ∗ e − ikx + iω ∗ ( k ) t ]. Linearizing the equationsof motion in u and v is not more difficult than in thestandard case, because to first order in u and v , the av-erage density ¯ n and frequency ¯ µ do not change. In thesmall dissipation limit κn ≪
1, the frequencies of the t[J −1 ]n ∆ θ t[J −1 ] | < ψ | ψ A > | FIG. 2. Dynamics of the relative phase (∆ θ ) and densityon the first well ( n ) for two wells coupled by tunneling anddissipating through the emission of phonons. The initial con-dition was taken to be ψ , = e ± i . Inset: The decay ofthe projection of the wave function on the excited state is ex-cellently reproduced by the exponential decay exp( − Jκn t )(dashed line), where the phonon relaxation constant wastaken κ = 0 .
05 and the coarse graining time scale T = 20 J − . elementary excitations read : ω ( k ) = ± q ω B ( k ) − ( κn/ + kk c /m − i κn (cid:20) k / m + gnω B ( k ) (cid:21) [ ω B ( k ) ± kk c ] . (11)Note that the correction quadratic in κn to the usual Bo-goliubov dispersion ω B ( k ) = p gnk /m + k / m underthe square root on the first line in Eq. (11) is necessary tosatisfy the Goldstone theorem that requires at least onebranch of elementary excitations to have exactly zero realand imaginary part of the frequency for k → k crit = min[ ω B ( k ) /k ], the elementary excita-tions have a negative imaginary part and the condensateis dynamically stable against decay into a lower energystate. As soon as k c > k crit , the imaginary part of the ex-citations becomes positive and the condensate becomesunstable with respect to decay toward a lower momen-tum. In agreement with the thermodynamical consider-ations made by Landau [1], the phonon scattering is ableto dissipate the supercurrent only when the condensatevelocity is above the critical speed. In the present dy-namical model (10), the energetic instability gives rise toa dynamical one, because the dissipative environment isexplicitly included and describes the kinetics of the su- percurrent’s decay.As a last illustration, we describe the relaxation ofpolaritons in a harmonic trap as observed in Ref. [11].When polaritons were created off center, relaxation to-ward the lower energy state in the middle of the trap wasobserved. This experimental configuration is illustratedin Fig. 3. Another striking evidence of the relaxationmechanism modeled here is the very recent experimentalresult reported by Wertz et al. [12]. A detailed analysisof the latter experiments is under way.In polariton condensates, a steady state can be reachedwhen new particles are continuously injected by opticalexcitation. In order to avoid the pinning of the conden-sate phase by the excitation laser, the additional particlesare created at a frequency different from the condensateone (nonresonant excitation). In the case of a far de-tuned laser, the excitation creates free electron hole pairsthat relax to form an excitonic reservoir. The further re-laxation of the excitons into the lower polariton branchthen provides a gain for the condensate. A steady stateis reached when gain and losses compensate each other.The stimulated part of the scattering into the lower po-lariton branch can be modeled by introducing a gain termin the Gross-Pitaevskii equation [13, 14]. The full dynam-ics of the polariton gas including trapping, nonresonantpumping and relaxation is given by i ∂∂t ψ = − (cid:20) ~ ∇ m + V eff − i γ − R ( n R )] + g | ψ | + i κ ¯ n (cid:18) ¯ µ − i∂∂t (cid:19)(cid:21) ψ. (12)Here γ is the loss rate and the function R ( n R ) describesthe gain of the lower polariton branch thanks to stimu-lated scattering from the nonresonantly excited excitonsin the reservoir with density n R . The exciton density canbe described by the rate equation ddt n R = − γ R n R − R ( n R ) | ψ | + P, (13)where γ R is the reservoir relaxation rate and P the non-resonant pumping rate. The effective potential V eff = V trap + V exc consists of the trapping potential V trap andthe blue shift due to the high energy excitons, propor-tional to the pumping term V exc = G P [15].It is worth pointing out that Eq. (12) actually describestwo relaxation mechanisms: (i) from the excitonic reser-voir into the lower polariton branch and (ii) from high tolow energy polariton states. Only the second relaxationprocess is modeled with the new dissipation mechanismderived here [term on the second line of Eq. (12)].In the absence of the relaxation term, themodel (12),(13) predicts a condensate at a singlefrequency [see Fig. 3(a)]. The frequency of the conden-sate coincides with the potential energy at the center of FIG. 3. Relaxation of a polariton condensate that is nonres-onantly created by a pump profile (dashed line) that is notcentered at the minimum of the effective trapping potential(full line).
Panel (a) : In the absence of a relaxation mech-anism ( κ = 0), the polaritons accelerate ballistically whenmoving toward the bottom of the potential: The energy re-solved real space distribution shows a single frequency for thecondensate. Panel (b) : When the dissipation mechanism ispresent ( κ = 0 . µ m ), the polaritons relax to the bottomof the trap. The condensate frequency in the center of thetrap is different from the condensate frequency in the pump-ing area. The red dashed line shows the excitation intensity P and the full black line represents the effective potential V eff .Other parameters: ~ γ = 0 . m LP / ~ = meV − µ m − , ~ P = 1meV µ m − , ~ γ R = 0 . ~ R ( n R ) = (1meV µ m )n R . the pumping spot. Away from the pumping spot, thepotential energy is converted into kinetic energy.On the other hand, when the relaxation is included[Fig 3(b)], a condensate also appears at an energy closeto the bottom of the trap. It is important to highlightthat the steady state solution consists of two condensatesat well defined frequencies. As our mean field modelonly describes the coherent part of the bosonic field, thecondensate line width vanishes.Fig. 2 (b) is globally in agreement with the experimen-tal observations in Ref. [11]. Differently from our sim-ulations though, in experiments polariton luminescencewas observed at all frequencies between the pump regionand ground state [11]. Possibly microcavity disorder andspontaneous relaxation are responsible for these devia- tions.Physically, it is likely that a large contribution to therelaxation within the lower polariton branch comes fromthe scattering of polaritons with reservoir excitons. In-deed, estimates based on the golden rule for exciton-phonon scattering [7] yield a relaxation constant of theorder of κ ≈ − µ m . Our numerical simulationsshow relaxation toward the bottom of the lower polari-ton branch for κn & .
1. If exciton-phonon scatteringwas the only relaxation mechanism, it would require apolariton density n = 10 µ m − , much larger than theexperimental estimates [11].Further applications of the relaxation in the descrip-tion of polariton condensates include the relaxation ofnonresonantly excited polariton condensates in periodicpotentials [16, 17], in one dimensional polariton wires [12]and the interplay of phonon relaxation with parametricscattering [18]. Our model may also be useful to describethe interaction of the thermal cloud with the condensedpart of an equilibrium Bose gas at finite temperature [19].We are indebted to B. Pietka, I. Carusotto, K.Lagoudakis, T. Liew and F. Manni for stimulating dis-cussions. [1] L.P. Pitaevskii and S. Stringari, Bose-Einstein Conden-sation , Clarendon Press Oxford (2003).[2] J. Keeling et al. , Semicond. Sci. Technol. R1 (2007).[3] O. Demokritov, et al. , Nature , 430 (2006).[4] H. Deng, H. Haug, and Y. Yamamoto, Rev. Mod. Phys. , 1489 (2010).[5] K. G. Lagoudakis et al. , Nature Physics , 706 (2008).[6] H. Haug and A. Jauho, Quantum kinetics in trans-port and optics of semiconductors , Springer-Verlag Berlin(2008).[7] H.T. Cao et al. , Phys. Rev. B , 245325 (2004).[8] M. Wouters and I. Carusotto, Phys. Rev. Lett. ,020602 (2010).[9] K.G. Lagoudakis, et al. , arXiv:1004.2216.[10] I. A. Shelyky et al. , Semicond. Sci. Technol. , 013001(2010).[11] R. Balili et al. , Science , 1007 (2007).[12] E. Wertz et al. arXiv:1004.4084.[13] M. Wouters and I. Carusotto, Phys. Rev. Lett. ,140402 (2007).[14] J. Keeling and N. G. Berloff, Phys. Rev. Lett. ,250401 (2008).[15] M. Wouters, I. Carusotto, and C. Ciuti, Phys. Rev. B , 115340 (2008)[16] C. W. Lai et al. , Nature , 529 (2007).[17] E. Cerda et al. , submitted.[18] J. J. Baumberg, et al. , Phys. Rev. B , R16247 (2000);R.M. Stevenson et al. Phys. Rev. Lett. 85, 3680 (2000);R. Houdr´e et al. , Phys. Rev. Lett. 85, 2793 (2000).[19] H. T. C. Stoof and M. J. Bijlsma, J. Low Temp. Phys. , 431 (2001); C. W. Gardiner and P. Zoller, Phys.Rev. A , 2902 (1997); E. Zaremba, T. Nikuni, and A.Griffin, J. Low Temp. Phys.116