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arXiv:cond-mat/9912420 · Physical Review A

Quantum kinetic theory. VI. The growth of a Bose-Einstein condensate

M. D. Lee, Crispin W. Gardiner

Published Aug 11, 2000• 52 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.62.033606

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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.
In a major extension of our previous model [Phys. Rev. Lett. 79, 1793 (1997)] of condensate growth, 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. We find good agreement with experiment, especially at higher temperatures. We also confirm the picture of the ``kinetic'' region of evolution, introduced by Kagan et al., 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 W+{W}^{+}W+.
We extend earlier models of the growth of a Bose-Einstein condensate c.w. [Gardiner et al., Phys. Rev. Lett. 79, 1793 (1997); e-print cond-mat/9801027; Phys. Rev. Lett. 81, 5266 (1998)] to include the full dynamical effects of the thermal cloud by numerically solving a modified quantum Boltzmann equation. We determine the regime in which the assumptions of the simple model of Gardiner et al. [Phys. Rev. Lett. 81, 5266 (1998)] are a reasonable approximation, and compare our results with those that were earlier compared with experimental data. We find good agreement with our earlier modeling, except at higher condensate fractions, for which a significant speedup is found. We also investigate the effect of the final temperature on condensate growth, and find that this has a surprisingly small effect. The particular discrepancy between theory and experiment found in our earlier model remains, since the speedup found in these computations does not occur in the parameter regime specified in the experiment.
A major extension of the model of condensate growth introduced by us is made to take account of the evolution of the occupations of lower trap levels (quasiparticles) by scattering processes, and of the full Bose-Einstein formula for the occupations of higher trap levels, which are assumed to have a time independent occupation. The principal effect is a speedup of the growth rate by somewhat less than an order of magnitude, the precise value depending on the the assumptions made on scattering and transition rates for the quasiparticle levels.
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.
In trapped Bose-Einstein condensates (BECs), \emph{condensate growth} refers to the process in which an increasing number of quasi-particles are immediately transferred from the non-condensate state (the thermal cloud) into the condensate state following the initial formation of the BEC. Despite its physical significance, this phenomenon has not yet been studied rigorously from a mathematical standpoint. In this work, we investigate a kinetic equation whose collision operator includes three types of wave interactions: one corresponding to a 3-wave process, and two classified as 4-wave processes. This wave kinetic equation models the evolution of the density function of the thermal cloud. We establish the immediate formation of condensation in solutions to this equation, thus providing a rigorous demonstration of the condensate growth phenomenon.
We extend the earlier model of condensate growth of Davis et al (Davis M J, Gardiner C W and Ballagh R J 2000 Phys. Rev. A 62 063608) to include the effect of gravity in a magnetic trap. We carry out calculations to model the experiment reported by Köhl et al (Köhl M, Davis M J, Gardiner C W, Hänsch T and Esslinger T 2001 Preprint cond-mat/0106642) who study the formation of a rubidium Bose-Einstein condensate for a range of evaporative cooling parameters. We find that, in the regime where our model is valid, the theoretical curves agree with all the experimental data with no fitting parameters. However, for the slowest cooling of the gas the theoretical curve deviates significantly from the experimental curves. It is possible that this discrepancy may be related to the formation of a quasicondensate.