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physica status solidi (b)

Phonon-Assisted Relaxation Kinetics of a Degenerate Bose Gas

A. L. Ivanov, Claudia Ell, H. Haug

Published Mar 1, 1998• 9 citations
Publisher

Abstract

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.

DOI
10.1002/(sici)1521-3951(199803)206:1<235::aid-pssb235>3.0.co;2-0

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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.
The relaxation kinetics of an ideal Bose gas coupled to a phonon bath at temperatures below or equal to the critical temperature Tc{\mathrm{T}}_{\mathrm{c}}Tc​ is given within a unique scenario. During the first transient stage, which lasts a few characteristic scattering times, the initial distribution disappears. The following nonexponential relaxation towards a quantum degenerate equilibrium state with a Bose-Einstein condensate is a slow adiabatic process. For this second kinetic stage we find an analytic generic solution of the Boltzmann equation. The generic solution is independent of the initial distribution and completely defined by the ratio Tc{\mathrm{T}}_{\mathrm{c}}Tc​/Tb{\mathrm{T}}_{\mathrm{b}}Tb​ of the critical and bath temperatures.
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.
Quasiequilibrium relaxational thermodynamics is developed to understand LA-phonon-assisted thermalization of Bose-Einstein distributed excitons in quantum wells. We study quantum-statistical effects in the relaxational dynamics of the effective temperature of excitons T=T(t).T=T(t).T=T(t). When T is less than the degeneracy temperature T0,{T}_{0},T0​, well-developed Bose-Einstein statistics of quantum-well excitons leads to nonexponential and density-dependent thermalization. At low bath temperatures ${T}_{b}\ensuremath{\rightarrow}0,$ the thermalization of quantum statistically degenerate excitons effectively slows down and $T(t)\ensuremath{\propto}1/\mathrm{ln}t.$ We also analyze the optical decay of Bose-Einstein distributed excitons in perfect quantum wells, and show how nonclassical statistics influences the effective lifetime ${\ensuremath{\tau}}_{\mathrm{opt}}.$ In particular, ${\ensuremath{\tau}}_{\mathrm{opt}}$ of a strongly degenerate gas of excitons is given by $2{\ensuremath{\tau}}_{R},$ where ${\ensuremath{\tau}}_{R}$ is the intrinsic radiative lifetime of quasi-two-dimensional excitons. Kinetics of resonant photoluminescence of quantum-well excitons during their thermalization is studied within the thermodynamic approach and taking into account Bose-Einstein statistics. We find density-dependent photoluminescence dynamics of statistically degenerate excitons. Numerical modeling of the thermalization and photoluminescence kinetics of quasi-two-dimensional excitons are given for ${\mathrm{G}\mathrm{a}\mathrm{A}\mathrm{s}/\mathrm{A}\mathrm{l}}_{x}{\mathrm{Ga}}_{1\ensuremath{-}x}\mathrm{As}$ quantum wells.
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.