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arXiv:cond-mat/9712002 · Physical Review A

Quantum kinetic theory. III. Quantum kinetic master equation for strongly condensed trapped systems

Crispin W. Gardiner, P. Zoller

Published Jul 1, 1998• 108 citations
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Abstract

We extend quantum kinetic theory to deal with a strongly Bose-condensed atomic vapor in a trap. The method assumes that the majority of the vapor is not condensed, and acts as a bath of heat and atoms for the condensate. The condensate is described by the particle-number-conserving Bogoliubov method developed by one of the authors. We derive equations which describe the fluctuations of particle number and phase, and the growth of the Bose-Einstein condensate. The equilibrium state of the condensate is a mixture of states with different numbers of particles and quasiparticles. It is not a quantum superposition of states with different numbers of particles---nevertheless, the stationary state exhibits the property of off-diagonal long-range order, to the extent that this concept makes sense in a tightly trapped condensate.

DOI
10.1103/physreva.58.536

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•Dec 31, 2001•0 citations
We extend quantum kinetic theory to deal with a strongly Bose-condensed atomic vapor in a trap. The method assumes that the majority of the vapor is not condensed, and acts as a bath of heat and atoms for the condensate. The condensate is described by the particle-number-conserving Bogoliubov method developed by one of the authors. We derive equations which describe the fluctuations of particle number and phase, and the growth of the Bose-Einstein condensate. The equilibrium state of the condensate is a mixture of states with different numbers of particles and quasiparticles. It is not a quantum superposition of states with different numbers of particles---nevertheless, the stationary state exhibits the property of off-diagonal long-range order, to the extent that this concept makes sense in a tightly trapped condensate.
Summary form only given.We use quantum kinetic theory to calculate the steady-state and the fluctuations of a trapped Bose-Einstein condensate at finite temperature. We divide the system into two parts. The first part consists of the condensate particles. They are described quantum mechanically using the number-conserving Bogoliubov method that allows us to characterize the condensate by the number of condensed particles. The second part are the noncondensed particles. They are treated as a classical gas in thermal equilibrium. Using quantum kinetic theory we find a master equation for the reduced density operator of the Bose-Einstein condensate. We calculate the steady-state of the system and investigate the effect of one-, two- and three-particle losses on the condensate. Using linearized Ito equations we find expressions for the intensity fluctuations and the amplitude fluctuations in the condensate. The effect of the trap losses on the particle statistics is also studied. We then assume that the condensate particles are coherently pumped into an untrapped internal energy level and fall down in gravity. For this situation we derive the relation between the statistics of the trapped condensate particles and the statistics of the particles arriving at an atom detector positioned below the condensate.
We use the quantum kinetic theory to calculate the steady state and fluctuations of a trapped Bose-Einstein condensate at a finite temperature. The system is divided in a condensate and a noncondensate part. A quantum-mechanical description based on the number-conserving Bogoliubov method is used for describing the condensate part. The noncondensed particles are treated as a classical gas in thermal equilibrium with temperature TTT and chemical potential $\ensuremath{\mu}$. We find a master equation for the reduced density operator of the Bose-Einstein condensate, calculate the steady state of the system, and investigate the effect of one-, two-, and three-particle losses on the condensate. Using linearized Ito equations, we find expressions for the intensity fluctuations and the amplitude fluctuations in the condensate. A Lorentzian line shape is found for the intensity correlation function that is characterized by a time constant ${\ensuremath{\gamma}}_{I}^{\ensuremath{-}1}$ derived in the paper. For the amplitude correlation function, we find ballistic behavior for time differences smaller than ${\ensuremath{\gamma}}_{I}^{\ensuremath{-}1}$, and diffusive behavior for larger time differences.
A detailed analysis of the growth of a Bose-Einstein condensate is given, based on quantum kinetic theory, in which we take account of the evolution of the occupations of lower trap levels, and of the full Bose-Einstein formula for the occupations of higher trap levels, as well as the Bose-stimulated direct transfer of atoms to the condensate level introduced by Gardiner et al. [Phys. Rev. Lett. 79, 1793 (1997); 81, 5266 (1998)]. We find good agreement with experiment at higher temperatures, but at lower temperatures the experimentally observed growth rate is somewhat more rapid. We also confirm the picture of the ``kinetic'' region of evolution, introduced by Kagan, Svistunov, and Shlyapnikov (Zh. Eksp. Teor. Fiz. 101, 528 (1992) [Sov. Phys. JETP 75, 387 (1992)]), for the time up to the initiation of the condensate. The behavior after initiation essentially follows our original growth equation, but with a substantially increased rate coefficient. Our modeling of growth implicitly gives a model of the spatial shape of the density profile of the condensate-vapor system as the condensate grows, and thus provides an alternative to the present phenomenological fitting procedure, based on the sum of a zero-chemical potential vapor and a Thomas-Fermi-shaped condensate. Our method gives substantially different results for condensate numbers and temperatures obtained from phenomenological fits, but fits the published column density data very well.
We present results of simulations of a quantum Boltzmann master equation (QBME) describing the kinetics of a dilute Bose gas confined in a trapping potential in the regime of Bose condensation. The QBME is the simplest version of a quantum kinetic master equation derived in previous work. We consider two cases of trapping potentials: a three-dimensional square-well potential with periodic boundary conditions and an isotropic harmonic oscillator. We discuss the stationary solutions and relaxation to equilibrium. In particular, we calculate particle distribution functions, fluctuations in the occupation numbers, the time between collisions, and the mean occupation numbers of the one-particle states in the regime of onset of Bose condensation.
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.
This paper extends an earlier quantum kinetics treatment for dilute, weakly interacting, partially Bose-Einstein condensed gases, presented by the author elsewhere (Proukakis N P and Burnett K 1996 J. Res. Natl Inst. Stand. Technol. 101 457), by consistently treating the dynamics of the uncondensed atoms to the same level of approximation as the condensed atoms. Our method is based on a hierarchy of coupled equations of motion for the condensate mean field and fluctuations around this mean field, truncated to second order in the (effective two-body) interatomic potential, and with suitable decoupling approximations for higher-order correlations. By applying perturbation theory in the Hartree-Fock-Bogoliubov basis, we re-derive the quantum kinetic theory of Walser et al (Walser R, Williams J, Cooper J and Holland M 1999 Phys. Rev. A 59 3878), which further indicates the consistency of our treatment with the Kadanoff-Baym non-equilibrium Green function formalism for trapped gases.