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arXiv:cond-mat/9911018 · Physical Review A

Shifts and widths of collective excitations in trapped Bose gases determined by the dielectric formalism

Jürgen Reidl, András Csordás, Robert Graham, P. Szépfalusy

Published Mar 10, 2000• 52 citations
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Abstract

We present predictions for temperature-dependent shifts and damping rates. They are obtained by applying the dielectric formalism to set up a self-consistent model of a trapped Bose gas which can be shown to satisfy generalized Ward identities. Within the framework of the model we use lowest-order perturbation theory to determine the first-order correction to the results of Hartree-Fock-Bogoliubov-Popov theory for the complex collective excitation frequencies, and present numerical results for the temperature dependence of the damping rates and the frequency shifts. Good agreement with the experimental values measured by Jin et al. [Phys. Rev. Lett. 77, 420 (1996)] are found for the $m=2$ mode, while we find disagreements in the shifts for $m=0.$ The latter point to the necessity of a nonperturbative treatment for an explanation of the temperature dependence of the $m=0$ shifts.

DOI
10.1103/physreva.61.043606

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We present predictions for temperature-dependent shifts and damping rates. They are obtained by applying the dielectric formalism to set up a self-consistent model of a trapped Bose gas which can be shown to satisfy generalized Ward identities. Within the framework of the model we use lowest-order perturbation theory to determine the first-order correction to the results of Hartree-Fock-Bogoliubov-Popov theory for the complex collective excitation frequencies, and present numerical results for the temperature dependence of the damping rates and the frequency shifts. Good agreement with the experimental values measured by Jin et al. [Phys. Rev. Lett. 77, 420 (1996)] are found for the m=2m=2m=2 mode, while we find disagreements in the shifts for m=0.m=0.m=0. The latter point to the necessity of a nonperturbative treatment for an explanation of the temperature dependence of the m=0m=0m=0 shifts.
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 present predictions for the frequencies of collective modes of trapped Bose-condensed 87Rb{}^{87}\mathrm{Rb}87Rb atoms at finite temperature. Our treatment includes a self-consistent treatment of the effects upon the mean field from finite- TTT 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=2m\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2m=2 excitation frequencies observed in recent experiments as the critical temperature is approached.
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.
We develop a finite-temperature perturbation theory (beyond the mean field) for a Bose-condensed gas and calculate temperature-dependent damping rates and energy shifts for Bogolyubov excitations of any energy. The theory is generalized for the case of excitations in a spatially inhomogeneous (trapped) Bose-condensed gas, where we emphasize the principal importance of inhomogeneity of the condensate density profile and develop the method of calculating the self-energy functions. The use of the theory is demonstrated by calculating the damping rates and energy shifts of low-energy quasiclassical excitations, i.e., the quasiclassical excitations with energies much smaller than the mean-field interaction between particles. In this case the boundary region of the condensate plays a crucial role and the result for the damping rates and energy shifts is completely different from that in spatially homogeneous gases. We also analyze the frequency shifts and damping of sound waves in cylindrical Bose condensates and discuss the role of damping in the recent MIT experiment on the sound propagation.
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.
Following the idea of the density-functional approach, we develop a generalized Bogoliubov theory of an interacting Bose gas confined in a one-dimensional harmonic trap by using a local chemical potential, calculated with the Lieb-Liniger exact solution, as the exchange energy. At zero temperature, we use the theory to describe collective modes of a finite-particle system in all interaction regimes from the ideal gas limit to the mean-field Thomas-Fermi regime and to the strongly interacting Tonks-Girardeau regime. At finite temperature, we investigate the temperature dependence of collective modes in the weak-coupling regime by means of a Hartree-Fock-Bogoliubov theory with Popov approximation. By emphasizing the effects of finite particle number and nonzero temperature on collective-mode frequencies, we compare our results with a recent experimental measurement [E. Haller et al., Science 325, 1224 (2009)] and some previous theoretical predictions. We show that the experimental data are still not fully explained within current theoretical framework.