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arXiv:cond-mat/0111413

Density Matrix of a Bose--Einstein Condensate: Steady--State versus Mean--Field Approach

Wang, S. J., Maria Carolina Nemes, A. N. Salgueiro, Hans A. Weidenmüller

Published Nov 21, 2001• 0 citations
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Abstract

We compare the equilibrium solution for the condensate obtained in the mean--field approximation to the master equation for sympathetic cooling with the one obtained by Scully for a system in contact with a heat bath with the help of an analogy with the laser. While the mean--field approach yields analytical formulas for the approach towards equilibrium and for the equilibrium solution, it neglects the correlations between occupation numbers of different single--particle states which are approximately kept in Scully's approach. Such neglect is admissible as long as the fraction of Bosons in the condensate does not exceed a few percent or so.

DOI
10.48550/arxiv.cond-mat/0111413

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We compare the equilibrium solution for the condensate obtained in the mean--field approximation to the master equation for sympathetic cooling with the one obtained by Scully for a system in contact with a heat bath with the help of an analogy with the laser. While the mean--field approach yields analytical formulas for the approach towards equilibrium and for the equilibrium solution, it neglects the correlations between occupation numbers of different single--particle states which are approximately kept in Scully's approach. Such neglect is admissible as long as the fraction of Bosons in the condensate does not exceed a few percent or so.
We use the mean-field approximation to simplify the master equation for sympathetic cooling of bosons. For the mean single-particle occupation numbers, this approach yields the same equations as the factorization assumption introduced in an early paper. The stationary or equilibrium solution of the resulting master equation for the one-body density matrix shows that the mean-field approximation breaks down whenever the fraction of condensate bosons exceeds 10% or so of the total. Using group-theoretical methods, we also solve the time-dependent master equation for the one-body density matrix. Given the time dependence of the mean single-particle occupation numbers, this solution is obtained by quadratures. It tends asymptotically towards the equilibrium solution.
We present a microscopic derivation of the master equation for the condensate density matrix for an interacting Bogoliubov-Bose gas of N atoms. We choose the interaction Hamiltonian in a special way that substantially simplifies the master equation, yielding no coupling between diagonal and off-diagonal terms. The present formulation allows us to solve the problem analytically in a steady state and obtain the expression for the distribution function and equilibrium condensate fluctuations. For the first two central moments, our results are equivalent to those obtained in the canonical-ensemble quasiparticle formalism [V. V. Kocharovsky, Vl. V. Kocharovsky, and M. O. Scully, Phys. Rev. Lett. 84, 2306 (2000); Phys. Rev. A 61, 053606 (2000)], in the low-temperature range where these papers are valid, but also give an accurate description at high temperatures. The present analysis for an interacting Bose gas is as accurate as the master equation approach of Kocharovsky et al. [Phys. Rev. A 61, 023609 (2000)] is for an ideal gas.
A nonequilibrium approach to the dynamics and statistics of the condensate of an ideal N-atom Bose gas cooling via interaction with a thermal reservoir using the canonical ensemble is developed. We derive simple analytical expressions for the canonical partition function and equilibrium distribution of the number of atoms in the ground state of a trap under different approximations, and compare them with exact numerical results. The N-particle constraint associated with the canonical ensemble is usually a burden. In the words of Kittel, ``in the investigation of the Bose-Einstein\dots{}laws it is very inconvenient to impose the restriction that the number of particles in the subsystem shall be held constant.'' But in the present approach, based on the analogy between a second-order phase transition and laser threshold behavior, the N-particle constraint makes the problem easier. We emphasize that the present work provides another example of a case in which equilibrium (detailed balance) solutions to nonequilibrium equations of motion provide a useful supplementary approach to conventional statistical mechanics. We also discuss some dynamical and mesoscopic aspects of Bose-Einstein condensation. The conclusion is that the present analytical (but approximate) results, based on a nonequilibrium approach, are in excellent agreement with exact (but numerical) results. The present analysis has much in common with the quantum theory of the laser.
We study the mean-field theory, and the properties of fluctuations, in an out of equilibrium Bose-Fermi system, across the transition to a quantum condensed phase. The system is driven out of equilibrium by coupling to multiple baths, which are not in equilibrium with each other, and thus drive a flux of particles through the system. We derive the self-consistency condition for a uniform condensed steady state. This condition can be compared both to the laser rate equation and to the Gross-Pitaevskii equation of an equilibrium condensate. We study fluctuations about the steady state and discuss how the multiple baths interact to set the system's distribution function. In the condensed system, there is a soft phase (Bogoliubov, Goldstone) mode, diffusive at small momenta due to the presence of pump and decay, and we discuss how one may determine the field-field correlation functions properly including such soft phase modes. In the infinite system, the correlation functions differ both from the laser and from an equilibrium condensate; we discuss how in a finite system, the laser limit may be recovered.