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Physical Review A

Mean-field approach to excitations of a trapped Bose-Einstein condensate gas at finite temperatures

M. Tohyama

Published Apr 26, 2005• 17 citations
Publisher

Abstract

The monopole and quadrupole oscillations of a Bose-Einstein condensate (BEC) gas trapped in a spherically symmetric potential are calculated using the time-dependent Hartree approximation which enables us to use an orthonormal single-particle basis for the description of both the condensate and noncondensate components. It is found that the monopole mode has a much smaller damping rate than the quadrupole mode. The obtained temperature dependence of the frequency shifts and the damping rates of the collective oscillations are found to be consistent with experimental data obtained for BEC gases in anisotropic trapping potentials. A comparison between the time-dependent Hartree approximation and the time-dependent Hartree-Fock approximation is also made and it is found that there is a rather small difference.

DOI
10.1103/physreva.71.043613

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The monopole and quadrupole oscillations of a Bose-Einstein condensate (BEC) gas trapped in a spherically symmetric potential are calculated using the time-dependent Hartree approximation which enables us to use an orthonormal single-particle basis for the description of both the condensate and noncondensate components. It is found that the monopole mode has a much smaller damping rate than the quadrupole mode. The obtained temperature dependence of the frequency shifts and the damping rates of the collective oscillations are found to be consistent with experimental data obtained for BEC gases in anisotropic trapping potentials. A comparison between the time-dependent Hartree approximation and the time-dependent Hartree-Fock approximation is also made and it is found that there is a rather small difference.
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.
Recent experimental observations of collective excitations of Bose condensed atomic vapours have stimulated interest in the microscopic description of the dynamics of a Bose-Einstein condensate confined in an external potential. We present a finite temperature field theory for collective excitations of trapped Bose-Einstein condensates and use a finite-temperature linear response formalism, which goes beyond the simple mean-field approximation of the Gross-Pitaevskii equation. The effect of the non-condensed thermal atoms we include using perturbation theory in a quasiparticle basis. This presents a simple scheme to understand the interaction between condensate and non-condensed atoms and enables us to include the effect the condensate has on collision dynamics. At first we limit our treatment to the case of a spatially homogeneous Bose gas. We include the effect of pair and triplet anomalous averages and thus obtain a gapless theory for the excitations of a weakly interacting system, which we can link to well known results for Landau and Beliaev damping rates. A gapless theory for trapped systems with a static thermal component follows straightforwardly. We then investigate finite temperature excitations of a condensate in a spherically symmetric harmonic trap. We avoid approximations to the density of states and thus emphasise finite size aspects of the problem. We show that excitations couple strongly to a restricted number of modes, giving rise to resonance structure in their frequency spectra. Where possible we derive energy shifts and lifetimes of excitations. For one particular mode, the breathing mode, the effects of the discreteness of the system are sufficiently pronounced that the simple picture of an energy shift and width fails. Experiments in spherical traps have recently become feasible and should be able to test our detailed quantitative predictions.
We have applied the Popov version of the Hartree-Fock-Bogoliubov (HFB) approximation to calculate the finite-temperature excitation spectrum of a Bose-Einstein condensate (BEC) of 87Rb{}^{87}\mathrm{Rb}87Rb atoms. For lower values of the temperature, we find excellent agreement with recently published experimental data for the JILA time-averaged orbiting potential trap. In contrast to recent comparison of the results of HFB-Popov theory with experimental condensate fractions and specific heats, there is disagreement of the theoretical and recent experimental results near the BEC phase transition temperature.
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.
The mean-field properties of finite-temperature Bose-Einstein gases confined in spherically symmetric harmonic traps are surveyed numerically. The solutions of the Gross-Pitaevskii (GP) and Hartree-Fock-Bogoliubov (HFB) equations for the condensate and low-lying quasiparticle excitations are calculated self-consistently using the discrete variable representation, while the most high-lying states are obtained with a local-density approximation. Consistency of the theory for temperatures through the Bose condensation point Tc{T}_{c}Tc​ requires that the thermodynamic chemical potential differ from the eigenvalue of the GP equation; the appropriate modifications lead to results that are continuous as a function of the particle interactions. The HFB equations are made gapless either by invoking the Popov approximation or by renormalizing the particle interactions. The latter approach effectively reduces the strength of the effective scattering length asc,{a}_{\mathrm{sc}},asc​, increases the number of condensate atoms at each temperature, and raises the value of Tc{T}_{c}Tc​ relative to the Popov approximation. The renormalization effect increases approximately with the log of the atom number, and is most pronounced at temperatures near Tc.{T}_{c}.Tc​. Comparisons with the results of quantum Monte Carlo calculations and various local-density approximations are presented, and experimental consequences are discussed.