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arXiv:cond-mat/9612060 · Physical Review A

Collapses and revivals of collective excitations in trapped Bose condensates

R. Graham, Daniel F. Walls, Matthew J. Collett, Martin Fliesser, Ewan M. Wright

Published Jan 1, 1998• 17 citations
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Abstract

We present a quantum theory of low-lying excitations in a trapped Bose condensate with finite particle numbers. We find that even at zero temperature condensate number fluctuations and/or fluctuations of the excitation frequency due to quantum uncertainties of the mode occupation lead to a collapse of the collective modes due to dephasing. Coherent revivals of the collective excitations are predicted on a much longer time scale. Depletion of collective modes due to second-harmonic generation is discussed.

DOI
10.1103/physreva.57.503

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We present a quantum theory of low-lying excitations in a trapped Bose condensate with finite particle numbers. We find that even at zero temperature condensate number fluctuations and/or fluctuations of the excitation frequency due to quantum uncertainties of the mode occupation lead to a collapse of the collective modes due to dephasing. Coherent revivals of the collective excitations are predicted on a much longer time scale. Depletion of collective modes due to second-harmonic generation is discussed.
We study the energies and decay of elementary excitations in weakly interacting Bose-Einstein condensates within a finite-temperature gapless second-order theory. The energy shifts for the high-lying collective modes turn out to be systematically negative compared with the Hartree-Fock-Bogoliubov-Popov approximation and the decay of the low-lying modes is found to exhibit collapse and revival effects. In addition, perturbation theory is used to qualitatively explain the experimentally observed Beliaev decay process of the scissors mode.
Quantum evolution of a collective mode of a Bose-Einstein condensate containing a finite number N of particles can demonstrate the phenomena of collapses and revivals. The characteristic collapse time depends on the scattering length, the initial amplitude of the mode, and N. The corresponding time values have been derived analytically under certain approximations and numerically for the parabolic atomic trap. The revival of the mode amplitude at a time of several seconds, as direct evidence of the effect, can occur if the normal component is significantly suppressed.
We discuss some recent work on the dephasing of Bose-Einstein condensates of interacting atoms due to fluctuations in the chemical potential μ. At vanishing temperature such fluctuations are due to the uncertainty of the number of particles in the condensate caused by the depletion effect. The dephasing is in this case a non-diffusive (and in principle reversible) effect called ‘collapse’ (and ‘revival’). Above a certain cross-over temperature the dephasing proceeds mainly via a diffusive process which follows in time a short and ineffective phase collapse. The phase-diffusion constant is dominated by the temperature dependent condensate-number fluctuations caused by the scattering of low-energy excitations off the condensate. We also discuss the (in general subdominant) contribution of the occupation-number fluctuations in quasi-particle states in the thermal cloud. For simplicity we discuss here results for a box-like trapping potential.
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.
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.