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arXiv:cond-mat/0203259 · Physical Review A

Thermodynamics of an interacting trapped Bose-Einstein gas in the classical field approximation

Krzysztof Góral, Mariusz Gajda, Kazimierz Rza̧żewski

Published Nov 18, 2002• 67 citations
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Abstract

We present a convenient technique describing the condensate in dynamical equilibrium with the thermal cloud, at temperatures close to the critical one. We show that the whole isolated system may be viewed as a single classical field undergoing nonlinear dynamics leading to a steady state. In our procedure it is the observation process and the finite detection time that allow for splitting the system into the condensate and the thermal cloud.

DOI
10.1103/physreva.66.051602

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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 show that recent treatments of magnetically trapped Bose-Einstein condensates (BECs) at temperature T >0 fit a simple picture: the condensate wavefunction is that of a T = 0 BEC with the appropriate (depleted) number of condensate atoms N 0 , and the surrounding thermal cloud is described by the statistical mechanics of an ideal gas in the combined potentials of the magnetic trap and the cloud-condensate interaction. Even a simple classical description of the cloud reproduces its essential outline, and facilitates easy approximate calculations of BEC properties for T >0.
We optimize the classical field approximation of the version described by M. Brewczyk, M. Gajda, and K. Rz\k{a}\ifmmode \dot{z}\else \.{z}\fi{}ewski [J. Phys. B 40, R1 (2007)] for the oscillations of a Bose gas trapped in a harmonic potential at nonzero temperatures, as experimentally investigated by Jin et al. [Phys. Rev. Lett. 78, 764 (1997)]. Similar to experiment, the system response to external perturbations strongly depends on the initial temperature and the symmetry of perturbation. While for lower temperatures the thermal cloud follows the condensed part, for higher temperatures the thermal atoms oscillate rather with their natural frequency, whereas the condensate exhibits a frequency shift toward the thermal cloud frequency (m=0m=0m=0 mode) or in the opposite direction (m=2m=2m=2 mode). In the latter case, for temperatures approaching critical, we find that the condensate begins to oscillate with the frequency of the thermal atoms, as in the m=0m=0m=0 mode. A broad range of frequencies of the perturbing potential is considered.
The main goal of this paper is to provide a deeper physical understanding of the classical field method—a method that is widely and successfully used for description of the atomic Bose–Einstein condensates at finite temperatures. To this end, we start with a genuine nonlinear classical system—interacting oscillators. We investigate the dynamics and show that the system approaches thermal equilibrium. We show that extra assumptions are necessary to interpret the results in quantum terms. The quantum interpretation is inevitably related to the cut-off of high momenta modes. A proper choice of the cut-off momentum is equivalent to the introduction of the Planck constant. The presented approach sheds more light on the foundations of the classical field method.
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The theory of open quantum systems is central to describing the dynamics and equilibration of dilute-gas Bose-Einstein condensates (BECs). We present an analysis of the linearized stochastic projected Gross-Pitaevskii equation (SPGPE) describing finite-temperature BECs. Our treatment provides an optimal choice for the cutoff that divides the Bose gas into the low-energy coherent region forming a classical wave and the high-energy thermal cloud treated as a reservoir. Moreover, it highlights the relevance of energy damping, the number-conserving scattering between thermal and coherent atoms. We analyze the equilibrium properties and near-equilibrium relaxation of a homogeneous BEC in one, two, and three dimensions at high phase-space density and calculate the autocorrelation function and power spectrum of the density and phase fluctuations. Simulations of the full nonlinear SPGPE are in close agreement and extend our arguments beyond the linear regime. Our work suggests the need for a reexamination of decay processes in BECs studied under the neglect of energy damping.