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Physical review. B, Condensed matter

Trapping kinetics of excitons in a potential well

Andreas Schmitt, L. Bányai, H. Haug

Published May 2, 2001• 7 citations
Publisher

Abstract

We describe the kinetics of bosonic excitons in a potential trap by rate equations in the basis of the single-particle eigenfunctions. We consider the interaction of the excitons with a thermal bath of acoustic phonons. Starting with an initial spatially homogeneous exciton distribution in the continuum, we show both a subcritical and a supercritical scenario for a spherically symmetric model potential. In particular, we observe for the case of ${\mathrm{Cu}}_{2}\mathrm{O}$ the buildup of a Bose-Einstein condensate within a microsecond time scale. However, the condensation time depends strongly on the depth and width of the trapping potential and the initial exciton distribution.

DOI
10.1103/physrevb.63.205113

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We describe the kinetics of bosonic excitons in a potential trap by rate equations in the basis of the single-particle eigenfunctions. We consider the interaction of the excitons with a thermal bath of acoustic phonons. Starting with an initial spatially homogeneous exciton distribution in the continuum, we show both a subcritical and a supercritical scenario for a spherically symmetric model potential. In particular, we observe for the case of Cu2O{\mathrm{Cu}}_{2}\mathrm{O}Cu2​O the buildup of a Bose-Einstein condensate within a microsecond time scale. However, the condensation time depends strongly on the depth and width of the trapping potential and the initial exciton distribution.
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.
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 formation process of a Bose-Einstein condensate in a trap is described using a master equation based on quantum kinetic theory, which can be well approximated by a description using only the condensate mode in interaction with a thermalized bath of noncondensate atoms. A rate equation of the form $\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{n}{\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2W}^{+}(n)[\left({1\ensuremath{-}e}^{{{\ensuremath{\mu}}_{n}\ensuremath{-}\ensuremath{\mu}}/\mathrm{kT}}\right)n+1]$ is derived, in which the difference between the condensate chemical potential ${\ensuremath{\mu}}_{n}$ and the bath chemical potential $\ensuremath{\mu}$ gives the essential behavior. Solutions of this equation give a characteristic latency period for condensate formation and appear to be consistent with the observed behavior of both rubidium and sodium condensate formation.
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.