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Physical Review Letters

Bose-Einstein Condensation Quantum Kinetics for a Gas of Interacting Excitons

O. M. Schmitt, D. B. Tran Thoai, L. Bányai, P. Gartner, H. Haug

Published Apr 23, 2001• 25 citations
Publisher

Abstract

A quantum kinetics of the Bose-Einstein condensation in the self-consistent (s.c.) Hartree-Fock-Bogoliubov (HFB) model of the interacting Bose gas is formulated and numerically solved for the example of excitons scattering with a thermal bath of acoustic phonons. The theory describes the condensation in real time starting from a nonequilibrium initial state towards the equilibrium HFB solution. The s.c. changes of the spectrum are automatically incorporated in the scattering terms.

DOI
10.1103/physrevlett.86.3839

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arXiv:cond-mat/9812281•Physical Review A•Jul 1, 1999•49 citations
A quantum kinetics of the Bose-Einstein condensation in the self-consistent (s.c.) Hartree-Fock-Bogoliubov (HFB) model of the interacting Bose gas is formulated and numerically solved for the example of excitons scattering with a thermal bath of acoustic phonons. The theory describes the condensation in real time starting from a nonequilibrium initial state towards the equilibrium HFB solution. The s.c. changes of the spectrum are automatically incorporated in the scattering terms.
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.
The time-dependent correlation function of the order parameter of Bose-Einstein condensation is calculated for a dilute interacting exciton gas cooled by a cold phonon bath. This calculation is achieved by combining two theoretical concepts: (i) a non-Markovian quantum kinetics, which is necessary in order to include the changing single-particle spectrum during the condensation self-consistently, and (ii) the stochastic non-Markovian Langevin fluctuations which are connected with the dissipative scattering integrals. A linearization around the mean quantum kinetics is used in order to determine the order parameter's fluctuations around its mean value. Similarly as in the laser theory the linewidth or the coherence time of the coherent order parameter amplitude is determined. The results show a clear reduction of the linewidth of the coherent exciton amplitude as the system cools down in time through the threshold of the Bose-Einstein condensation. The observation of this line narrowing by nonlinear optical spectroscopy would give a direct indication of the temporal build-up of an exciton condensate in a semiconductor.
We investigate the thermodynamic properties of an interacting Bose gas with a condensate within the energy-functional formulation of the Hartree-Fock-Bogoliubov (HFB) approach. For a contact interaction, we derive a self-consistent solution to the HFB equations that intrinsically eliminates divergence. This solution characterizes the equilibrium state featuring a condensate of correlated pairs of particles. We analyze the temperature dependence of key thermodynamic quantities such as condensate density, chemical potential, entropy, pressure, specific heat capacity at constant volume, and isothermal compressibility and compare them with predictions from the Popov approximation (PA). We predict that the transition temperature shifts to higher values due interactions, with the HFB approach yielding a larger shift than the PA. Analysis of the compressibility indicates that a pure pair condensate is unstable, and the stable equilibrium corresponds to only a mixture of single-particle and pair condensates.
The dephasing and condensation kinetics of bosonic excitons after coherent pulse excitation is studied in the vicinity of a Bose-Einstein condensation (BEC). For a system of excitons interacting with acoustic phonons various excitation conditions are investigated. Significant qualitative changes of the polarization dephasing in the vicinity of the BEC are discussed. Furthermore the kinetics of excitons, interacting via a two particle interaction, is studied. Both the bosonic mean-field terms and the scattering terms are calculated on a microscopic level. As in the phonon-model the dephasing rate of the exciton–exciton scattering diminishes considerably in the vicinity of the BEC. All results are suited as experimentally observable signatures of an excitonic BEC.
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.
Using the Kadanoff-Baym nonequilibrium Green's-function formalism, we derive the self-consistent Hartree-Fock-Bogoliubov (HFB) collisionless kinetic equations and the associated equation of motion for the condensate wave function for a trapped Bose-condensed gas. Our work generalizes earlier work by Kane and Kadanoff for a uniform Bose gas. We include the off-diagonal (anomalous) pair correlations, and thus we have to introduce an off-diagonal distribution function in addition to the normal (diagonal) distribution function. This results in two coupled kinetic equations. If the off-diagonal distribution function can be neglected as a higher-order contribution, we obtain the semiclassical kinetic equation recently used by Zaremba, Griffin, and Nikuni (based on the simpler Popov approximation). We discuss the static local equilibrium solution of our coupled HFB kinetic equations within the semiclassical approximation. We also verify that a solution is the rigid in-phase oscillation of the equilibrium condensate and noncondensate density profiles, oscillating with the trap frequency.