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arXiv:cond-mat/0311411 · Physical Review A

Collective oscillations of a confined Bose gas at finite temperature in the random-phase approximation

Xia-Ji Liu, Hui Hu, Anna Minguzzi, M. P. Tosi

Published Apr 9, 2004• 30 citations
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Abstract

We present a theory for the linear dynamics of a weakly interacting Bose gas confined inside a harmonic trap at finite temperature. The theory treats the motions of the condensate and of the noncondensate on an equal footing within a generalized random-phase approximation, which (i) extends the second-order Beliaev-Popov approach by allowing for the dynamical coupling between fluctuations in the thermal cloud, and (ii) reduces to an earlier random-phase scheme when the anomalous density fluctuations are omitted. Numerical calculations of the low-lying spectra in the case of isotropic confinement show that the present theory obeys with high accuracy the generalized Kohn theorem for the dipolar excitations and demonstrate that combined normal and anomalous density fluctuations play an important role in the monopolar excitations of the condensate. Mean-field theory is instead found to yield accurate results for the quadrupolar modes of the condensate. Although the restriction to spherical confinement prevents quantitative comparisons with measured spectra, it appears that the non-mean-field effects that we examine may be relevant to explain the features exhibited by the breathing mode as a function of temperature in the experiments carried out at JILA on a gas of $^{87}\mathrm{Rb}$ atoms.

DOI
10.1103/physreva.69.043605

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We present a theory for the linear dynamics of a weakly interacting Bose gas confined inside a harmonic trap at finite temperature. The theory treats the motions of the condensate and of the noncondensate on an equal footing within a generalized random-phase approximation, which (i) extends the second-order Beliaev-Popov approach by allowing for the dynamical coupling between fluctuations in the thermal cloud, and (ii) reduces to an earlier random-phase scheme when the anomalous density fluctuations are omitted. Numerical calculations of the low-lying spectra in the case of isotropic confinement show that the present theory obeys with high accuracy the generalized Kohn theorem for the dipolar excitations and demonstrate that combined normal and anomalous density fluctuations play an important role in the monopolar excitations of the condensate. Mean-field theory is instead found to yield accurate results for the quadrupolar modes of the condensate. Although the restriction to spherical confinement prevents quantitative comparisons with measured spectra, it appears that the non-mean-field effects that we examine may be relevant to explain the features exhibited by the breathing mode as a function of temperature in the experiments carried out at JILA on a gas of 87Rb^{87}\mathrm{Rb}87Rb atoms.
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.
A linear response framework is set up for the evaluation of collective excitations in a confined vapour of interacting Bose atoms at finite temperature. Focusing on the currently relevant case of contact interactions between the atoms, the theory is developed within a random-phase approximation with exchange. This approach is naturally introduced in a two-fluid description by expressing the density response of both the condensate and the non-condensate in terms of the response of a Hartree - Fock reference gas to the self-consistent Hartree - Fock potentials. Such an approximate account of correlations (i) preserves an interplay between the condensate and the non-condensate through off-diagonal components of the response, which instead vanish in the Hartree - Fock - Bogoliubov approximation; and (ii) yields a common resonant structure for the four partial response functions. The theory reduces to the temperature-dependent Hartree - Fock - Bogoliubov - Popov approximation for the fluctuations of the condensate when its coupling with the density fluctuations of the non-condensate is neglected. Analytic results are presented which are amenable to numerical calculations and to inclusion of damping rates.
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 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.
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 a theory of the linear response of a Bose-Einstein-condensed gas to external perturbations at finite temperature. The theory developed here is the basis of a recent quantitative explanation of the measurements of condensate excitations and decay rates made at JILA [D. S. Jin et al., Phys. Rev. Lett. 78, 764 (1997)]. The formalism is based on a dynamic, number-conserving, mean-field scheme and is valid in the collisionless limit of well-defined quasiparticles. The theory is gapless, consistent with the generalized Kohn theorem for the dipole modes, and includes the time-dependent normal and anomalous averages, Beliaev and Landau processes, and all relevant finite-size effects. The important physical process where the thermal cloud is driven directly by the external perturbation is explicitly included. This is required for consistency with the dipole modes and is also needed to explain the JILA observations.