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arXiv:cond-mat/9805050 · Physical Review Letters

Gapless Finite- T Theory of Collective Modes of a Trapped Gas

David A. W. Hutchinson, R. J. Dodd, Keith R. Burnett

Published Sep 14, 1998• 97 citations
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Abstract

We present predictions for the frequencies of collective modes of trapped Bose-condensed ${}^{87}\mathrm{Rb}$ atoms at finite temperature. Our treatment includes a self-consistent treatment of the effects upon the mean field from finite- $T$ excitations and the anomalous average. This is the first gapless [A. Griffin, Phys. Rev. B 53, 9341 (1996)] calculation of this type for a trapped Bose-Einstein condensed gas. The corrections quantitatively account for the downward shift in the $m\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2$ excitation frequencies observed in recent experiments as the critical temperature is approached.

DOI
10.1103/physrevlett.81.2198

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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 develop a formalism based on the projected Gross-Pitaevskii equation to simulate the finite-temperature collective mode experiments of Jin et al. [Phys. Rev. Lett. 78, 764 (1997)]. We examine the m=0m=0m=0 and m=2m=2m=2 quadrupolar modes on the temperature range 0.51Tc−−0.83Tc0.51{T}_{c}--0.83{T}_{c}0.51Tc​−−0.83Tc​ and calculate the frequencies of, and phase between, the condensate and noncondensate modes, and the condensate mode damping rate. This study is the first quantitative comparison of the projected Gross-Pitaevskii equation to experimental results in a dynamical regime.
Using the hydrodynamic theory of superfluids and the Lee-Huang-Yang equation of state for interacting Bose gases, we derive the first correction to the collective frequencies of a trapped gas, due to effects beyond the mean field approximation. The corresponding frequency shift, which is calculated at zero temperature and for large NNN, is compared with other corrections due to finite size, nonlinearity, and temperature. We show that for reasonable choices of the relevant parameters of the system, the non-mean-field correction is the leading contribution and amounts to about 11%1. The role of the deformation of the trap is also discussed.
We describe a method for studying the dynamics of a condensate in a weakly interacting, dilute Bose gas which is based on a finite-temperature extension of the Gross-Pitaevskii equation. We present a general finite-temperature linear response formalism for nonuniform systems and give explicit results for homogeneous ones, where we can link our treatment to well-known results. This treatment leads to a consistent V2{V}^{2}V2 calculation of the excitations for a weakly interacting Bose gas, including Landau and Beliaev damping as well as collisional lifetimes. We do not make use of the Popov approximation, but include pair and triplet anomalous averages to second order in the interaction potential and thus obtain a gapless theory for the excitations of a weakly interacting homogeneous gas. We also show how the formalism can be used to obtain self-consistent theories. We explicitly derive gapless equations for a weakly coupled finite-temperature condensate with a static thermal component and give a clear criterion of their validity in trapped systems.
We present a finite temperature field theory for collective excitations of trapped Bose condensates which includes the dynamics of the thermal cloud. In spherical traps we show that excitations couple strongly to a small number of modes, giving resonance structure in their frequency spectra. Where possible, we derive energy shifts and lifetimes of excitations. For the l = 0 mode we show that the simple picture of a decay rate fails, which should be observable in suitable experiments. It also suggests a possible explanation for the anomalous behavior of the m = 0 mode observed in anisotropic traps.
We study the collective modes of a one-dimensional harmonically trapped Bose-Einstein condensate in the presence of the anomalous density using the time-dependent Hartree-Fock-Bogoliubov theory. Within the hydrodynamic equations, we derive analytical expressions for the mode frequencies and the density fluctuations of the anomalous density which constitutes the minority component at very low temperature and feels an effective external potential exerted by the majority component, i.e., the condensate. On the other hand, we numerically examine the temperature dependence of the breathing mode oscillations of the condensate at finite temperature in the weak-coupling regime. At zero temperature, we compare our predictions with available experimental data, theoretical treatments, and Monte carlo simulations in all interaction regimes and the remaining hindrances are emphasized. We show that the anomalous correlations have a non-negligible role on the collective modes at both zero and finite temperatures.