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arXiv:quant-ph/9707037 · Physical Review Letters

Kinetics of Bose-Einstein Condensation in a Trap

Crispin W. Gardiner, P. Zoller, R. J. Ballagh, Matthew J. Davis

Published Sep 8, 1997• 137 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevlett.79.1793

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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 n\ifmmode˙\else˙\fi=2W+(n)[(1\ensuremath−e\ensuremathμn\ensuremath−\ensuremathμ/kT)n+1]\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]n\ifmmode˙\else˙\fi=2W+(n)[(1\ensuremath−e\ensuremathμn​\ensuremath−\ensuremathμ/kT)n+1] is derived, in which the difference between the condensate chemical potential \ensuremathμn{\ensuremath{\mu}}_{n}\ensuremathμn​ and the bath chemical potential \ensuremathμ\ensuremath{\mu}\ensuremathμ 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.
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 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.
We derive a nonequilibrium finite-temperature kinetic theory for a binary mixture of two interacting atomic Bose-Einstein condensates and use it to explore the degree of hydrodynamicity attainable in realistic experimental geometries. Based on the standard separation-of-time-scales argument of kinetic theory, the dynamics of the condensates of the multicomponent system are shown to be described by dissipative Gross-Pitaevskii equations self-consistently coupled to corresponding quantum Boltzmann equations for the noncondensate atoms: On top of the usual mean-field contributions, our scheme identifies a total of eight distinct collisional processes, whose dynamical interplay is expected to be responsible for the system's equilibration. In order to provide their first characterization, we perform a detailed numerical analysis of the role of trap frequency and geometry on collisional rates for experimentally accessible mixtures of $^{87}\mathrm{Rb}\ensuremath{-}^{41}\mathrm{K}$ and $^{87}\mathrm{Rb}\ensuremath{-}^{85}\mathrm{Rb}$, discussing the extent to which the system may approach the hydrodynamic regime with regard to some of those processes as a guide for future experimental investigations of ultracold Bose gas mixtures.
We describe the transition of NNN weakly interacting atoms into a Bose-Einstein condensate within a number-conserving quantum master equation theory. Based on the separation of time scales for condensate formation and noncondensate thermalization, we derive a master equation for the condensate subsystem in the presence of the noncondensate environment under the inclusion of all two-body interaction processes. We numerically monitor the condensate particle number distribution during condensate formation, and derive a condition under which the unique equilibrium steady state of a dilute, weakly interacting Bose-Einstein condensate is given by a Gibbs-Boltzmann thermal state of NNN noninteracting atoms.
Evaporative cooling of trapped atoms is studied based on the quantum kinetic theory. We extend the classical analysis of Luiten et al. [Phys. Rev. A 53, 381 (1996)] to treat Bose statistics of alkali-metal atoms. Dynamics in evaporative cooling process is described by a kinetic equation for the truncated Bose-Einstein distribution function under the assumption that the system is close to the thermal equilibrium. This approach is applicable to the ``slow'' evaporative cooling normally adopted in most experiments for the efficient production of Bose-Einstein condensates. Time-evolution calculations explain the current experiment well with sodium atoms using a slow evaporative cooling and demonstrate a rapid growth of condensate in the cooling process.