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arXiv:cond-mat/9807051 · Physical Review A

Collisionless modes of a trapped Bose gas

M. Bijlsma, Henk T. C. Stoof

Published Nov 1, 1999• 48 citations
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Abstract

We calculate the excitation frequencies of low-lying modes of a trapped Bose-condensed gas at nonzero temperatures. In our calculation we include the dynamics of the noncondensed cloud, and find agreement with experimental results if we assume that in the experiment both the in-phase and out-of-phase monopole modes are excited simultaneously. In order to explore whether this is indeed the correct explanation, we also calculate how strongly the modes couple to a perturbation of the external trapping potential.

DOI
10.1103/physreva.60.3973

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We calculate the excitation frequencies of low-lying modes of a trapped Bose-condensed gas at nonzero temperatures. In our calculation we include the dynamics of the noncondensed cloud, and find agreement with experimental results if we assume that in the experiment both the in-phase and out-of-phase monopole modes are excited simultaneously. In order to explore whether this is indeed the correct explanation, we also calculate how strongly the modes couple to a perturbation of the external trapping potential.
We calculate the frequencies and damping rates of the low-lying collective modes of a Bose-Einstein condensed gas at nonzero temperature. We use a complex nonlinear Schr\"odinger equation to determine the dynamics of the condensate atoms, and couple it to a Boltzmann equation for the noncondensate atoms. In this manner we take into account both collisions between noncondensate-noncondensate and condensate-noncondensate atoms. We solve the linear response of these equations, using a time-dependent Gaussian trial function for the condensate wave function and a truncated power expansion for the deviation function of the thermal cloud. As a result, our calculation turns out to be characterized by two dimensionless parameters proportional to the noncondensate-noncondensate and condensate-noncondensate mean collision times. We find in general quite good agreement with experiment, both for the frequencies and damping of the collective modes.
We present a detailed study of the low-lying collective excitations of a spherically trapped Bose-Fermi mixture at finite temperature in the collisionless regime. The excitation frequencies of the condensate are calculated self-consistently using the static Hartree-Fock-Bogoliubov theory within the Popov approximation. The frequency shifts and damping rates due to the coupled dynamics of the condensate, noncondensate, and degenerate Fermi gas are also taken into account by means of the random-phase approximation and linear-response theory. In our treatment, the dipole excitation remains close to the bare trapping frequency for all temperatures considered, and thus is consistent with the generalized Kohn theorem. We discuss in some detail the behavior of monopole and quadrupole excitations as a function of the Bose-Fermi coupling. At nonzero temperatures we find that, as the mixture moves towards spatial separation with increasing Bose-Fermi coupling, the damping rate of the monopole (quadrupole) excitation increases (decreases). This provides us a useful signature to identify the phase transition of spatial separation.
We study the low-energy collective oscillations of a dilute Bose gas at finite temperature in the collisionless regime. By using a time-dependent mean-field scheme we derive for the dynamics of the condensate and noncondensate components a set of coupled equations, which we solve perturbatively to second order in the interaction coupling constant. This approach is equivalent to the finite-temperature extension of the Beliaev approximation and includes corrections to the Gross-Pitaevskii theory due to both quantum and thermal fluctuations. For a homogeneous system we explicitly calculate the temperature dependence of the velocity of propagation and damping rate of zero sound. In the case of harmonically trapped systems in the thermodynamic limit, we calculate, as a function of temperature, the frequency shift of the low-energy compressional and surface modes.
Recently it was shown that the Landau-Khalatnikov two-fluid hydrodynamics describes the collision-dominated region of a trapped Bose condensate interacting with a thermal cloud. We use these equations to discuss the low frequency hydrodynamic collective modes in a trapped Bose gas at finite temperatures. We derive variational expressions based on these equations for both the frequency and damping of collective modes. A new feature is our use of frequency-dependent transport coefficients, which produce a natural cutoff by eliminating the collisionless low-density tail of the thermal cloud. Above the superfluid transition, our expression for the damping in trapped inhomogeneous gases is analogous to the result first obtained by Landau and Lifshitz for uniform classical fluids. We also use the moment method to discuss the crossover from the collisionless to the hydrodynamic region. Recent data for the monopole-quadrupole mode in the hydrodynamic region of a trapped gas of metastable 4He{}^{4}\mathrm{He}4He is discussed. We also present calculations for the damping of the analogous m=0m=0m=0 monopole-quadrupole condensate mode in the superfluid phase.
We present a detailed study of the temperature dependence of the condensate and noncondensate density profiles of a Bose-condensed gas in a parabolic trap. These quantities are calculated self-consistently using the Hartree-Fock-Bogoliubov equations within the Popov approximation. Below the Bose-Einstein transition the excitation frequencies have a relatively weak temperature dependence even though the condensate is strongly depleted. As the condensate density goes to zero through the transition, the excitation frequencies are strongly affected and approach the frequencies of a noninteracting trapped gas in the high temperature limit.