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Physical review. B, Condensed matter

Population dynamics of a Bose gas near saturation

David W. Snoke, James P. Wolfe

Published Mar 1, 1989• 142 citations
Publisher

Abstract

At low density a gas of bosons achieves a classical equilibrium distribution of energies within just a few characteristic scattering times. At sufficiently high densities, stimulated emission comes into play, increasing the rate of scattering into low-energy states. This results in the well-known Bose-Einstein equilibrium distribution. In this paper we show---by numerical calculation---that for a weakly interacting Bose gas the number of characteristic scattering times required to achieve an equilibrium Bose-Einstein distribution increases with increasing degeneracy. However, we find that the characteristic scattering time for the degenerate gas is significantly shortened by the stimulated emission, with the net result that equilibration times may be comparable to the classical case. These results have a bearing on recent experiments that produced degenerate Bose distributions of excitons. A basic question from these experiments is whether Bose-Einstein condensation can occur within the particle lifetime. Our calculations indicate that condensation may take place. We also show in this paper that exciton-phonon interactions will not significantly increase the rate of approach to condensation.

DOI
10.1103/physrevb.39.4030

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At low density a gas of bosons achieves a classical equilibrium distribution of energies within just a few characteristic scattering times. At sufficiently high densities, stimulated emission comes into play, increasing the rate of scattering into low-energy states. This results in the well-known Bose-Einstein equilibrium distribution. In this paper we show---by numerical calculation---that for a weakly interacting Bose gas the number of characteristic scattering times required to achieve an equilibrium Bose-Einstein distribution increases with increasing degeneracy. However, we find that the characteristic scattering time for the degenerate gas is significantly shortened by the stimulated emission, with the net result that equilibration times may be comparable to the classical case. These results have a bearing on recent experiments that produced degenerate Bose distributions of excitons. A basic question from these experiments is whether Bose-Einstein condensation can occur within the particle lifetime. Our calculations indicate that condensation may take place. We also show in this paper that exciton-phonon interactions will not significantly increase the rate of approach to condensation.
Condensate Formation in the Bose Gas An important question in connection with the experiments aimed at achieving BEC of a weakly interacting Bose gas with a finite lifetime is the time needed for the formation of the condensate. It is not at all evident that the condensation can always take place within the lifetime of the system. For the case of liquid He, the assumption of equilibrium seems satisfied because 4 He atoms, which do not decay, undergo roughly 10 11 interparticle-scattering events per second, assuming classical hard-sphere scattering. In other boson systems, however, nonequilibrium effects may have a significant effect. For excitons in a semiconductor, the lifetime of the particle may be only a few hundred scattering times. Spin-polarized hydrogen also has a finite lifetime to nonpolarized states [1], and alkali atoms can recombine into molecules [2]. Naively judging from classical behavior, one might assume that hundreds of scattering times may seem more than adequate time to establish equilibrium. The case of Bose condensation is unique, however, since a macroscopic number of particles must enter a single quantum state that by random-scattering processes alone is highly improbable. Therefore we need not assume that the time for Bose condensation is short. Early experiments with orthoexcitons in CU 2 O seemed to indicate that the excitons might not have enough time to condense within their lifetime [3]. As discussed in Subsection 1.4.3, the orthoexciton density increased to approach the BEC boundary given in Eq. (1.18) and highly degenerate Bose-Einstein momentum distributions were observed, but a significant condensate fraction did not appear. Instead the gas increased its temperature to accommodate extra particles, reaching temperatures well above the lattice temperature.
The relaxation kinetics of an ideal Bose gas coupled to a phonon thermostat at temperature Tb below or equal to the critical temperature Tc consists of an initial transient and the following adiabatic stage. The transient depends on the initial distribution of bosons at t = 0 and lasts a few characteristic scattering times. The following nonexponential relaxation towards a quantum degenerate equilibrium state with a Bose-Einstein (BE) condensate is a slow adiabatic process. We find a generic solution of the phonon-assisted Boltzmann kinetics at the adiabatic stage. This solution is completely defined by the ratio Tc/Tb. Here, one has a unique example of the fundamental solution of Boltzmann kinetics that is insensitive to the initial distribution. For Tb → Tc from above, we develop a theory of the thermodynamic critical slowing-down which also leads to the nonexponential thermalization of bosons. We apply our results to the relaxation kinetics of para-excitons in Cu2O and reinterpret the recent reports on the observation of BE condensation of para-excitons.
Photo-excited excitons, i.e. closely bound electron-hole pairs, are in the low-density regime approximately bosons. For sufficiently low temperatures and sufficiently high densities of the exciton gas, a nonequilibrium phase transition in the form of a Bose-Einstein condensation (BEC) can be expected. Various levels of the description of the kinetics of relaxation and condensation due to the coupling of the excitons to a cold reservoir of acoustic phonons are discussed. For noninteracting excitons a rate equation approach with the exciton Boltzmann equation can be used. For interacting excitons the nonmarkovian quantum kinetics is required for the description of the BEC in order to incorporate the changes of the energy spectrum during the condensation self-consistently. Finally colored Langevin fluctuations can be used to calculate the phase diffusion of the condensate.
We report on the experimental observation of stimulated cooling in the non-equilibrium Bose-Einstein condensate (BEC) of weakly interacting exciton-polaritons from approximately room temperature down to 20K. By resolving the condensate in energy-momentum space and performing interferometric measurements, we distinguish the condensate from thermalized particles yet occupying excited states macroscopically. In contrast to the analytical quantum theories of non-equilibrium BEC [Shishkov et al., Phys. Rev. Lett. 128, 065301 (2022)], we observe segmentation of the particle density along the excited states into two fractions both following Bose-Einstein distribution, albeit with different effective temperatures and chemical potentials. Our results indicate that the temperature of the weakly interacting Bose gas is universally set by the density-dependent chemical potential, revealing a defining property of non-equilibrium BECs. Finally, we demonstrate that the stimulated nature of the cooling process directly governs the emergence of quantum coherence of the condensate and shapes the dissipative properties of the excited states.
We study the kinetics of condensation in a trapped two-dimensional exciton gas using the quantum Boltzmann equation. We assume that the exciton gas interacts with a lattice that is kept at a constant temperature. Our simulations allow us to estimate and compare the condensation time for Bose gases with and without interactions. We find that anti-Stokes Raman-type two-phonon processes dominate the relaxation rate of low-energy states for a noninteracting gas. The presence of dissipation from the low-energy states results in a heating effect for the exciton gas which in turn increases the mean number of Bose particles required for the onset of condensation. Our results indicate that the condensation time scale in trapped two-dimensional excitons is determined by the phonon emission rate into the ground state. This time scale is orders of magnitude shorter than what is predicted for three-dimensional excitons. The steady state in a dissipative gas is a nonequilibrium condensate, or equivalently an exciton matter laser, where the exciton temperature remains above the lattice temperature.
We discuss Bose-Einstein condensation in a trapped atomic gas and analyze how the sign of the scattering length a and the ratio \ensuremath{\eta} of the interaction between particles to the level spacing in the trap influence the behavior of the condensate wave function ${\ensuremath{\psi}}_{0}$. We find that for a<0a<0a<0 and $\ensuremath{\eta}\ensuremath{\ll}1$ it is possible to form a metastable Bose condensate, with a long characteristic lifetime with respect to contraction and transitions of particles to excited trap states. For $\ensuremath{\eta}\ensuremath{\gg}1$ a negative scattering length prevents the formation of the condensate. If a>0a>0a>0, then an increase of density is accompanied by the evolution of ${\ensuremath{\psi}}_{0}$ to a comparatively wide quasihomogeneous distribution.