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Phase Transitions

Quantum Kinetics of the Exciton Bose-Einstein Condensation

H. Haug

Published Jan 1, 2002• 1 citation
Publisher

Abstract

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.

DOI
10.1080/0141159021000008954

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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 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 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 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.
The kinetics of the formation of coherent correlation properties associated with Bose condensation is studied in detail. The evolution of a nonequilibrium state with no “condensate-seed” is related to a hierarchy of relaxation times. At the first stage, a particle flux in energy space toward low energies sets in. The evolution in this case is described by a nonlinear Boltzmann equation, with a characteristic time given by interparticle collisions. When the particles which will later form the condensate have a kinetic energy which is less than the potential energy, a quasicondensate starts to form. In this stage, fluctuations of the density (but not of the phase) are suppressed and short-range coherent correlation properties are governed by the equation of motion for a quasiclassical complex field. The next stage is connected with the formation of the long-range order. The time for forming topological order and therefore genuine superfluidity proves to be dependent on the system size. The off-diagonal long-range order, arising after the attenuation of long-wave phase fluctuations, has a size-dependent relaxation time as well. Introduction The problem of Bose–Einstein Condensation (BEC) kinetics, being interesting in itself, has acquired a special significance in connection with experimental efforts to observe this condensation in a number of systems with particles with a finite lifetime. Such systems include spin-polarized atomic hydrogen [1], excitons [2] and biexcitons [3] in semiconductors and, more recently, laser-cooled atomic systems [4].
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.