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arXiv:cond-mat/0109462

Temperature dependent shifts in collective excitations of a gaseous Bose-Einstein condensate

Juan José García‐Ripoll

Published Sep 25, 2001• 0 citations
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Abstract

The moment method is applied to the quantum theory for a trapped dilute gas, obtaining equations for the evolution of the cloud. These equations proof the existence of undamped oscillations in a two-dimensional harmonic trap with radial symmetry. In all other cases we can, with physically reasonable approximations, solve the equations and find the frequencies of the collective modes for all temperatures, from the pure Bose-Einstein condensate up to the normal gas.

DOI
10.48550/arxiv.cond-mat/0109462

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We observe phononlike excitations of a Bose-Einstein condensate (BEC) in a dilute atomic gas. 87{}^{87}87Rb atoms are optically trapped and precooled, loaded into a magnetic trap, and then evaporatively cooled through the BEC phase transition to form a condensate. We excite the condensate by applying an inhomogeneous oscillatory force with adjustable frequency and symmetry. We have observed modes with different angular momenta and different energies and have studied how their characteristics depend on interaction energy. We find that the condensate excitations persist longer than their counterparts in uncondensed clouds.
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.
By including the contribution of the thermal cloud in the Lagrangian of the condensate of a Bose gas, we extend the time-dependent variational method at zero temperature to study temperature-dependent low collective excitation modes. A Gaussian trial wave function of the condensate and a static distribution density of the thermal cloud are used and analytical expressions for temperature-dependent excitation frequencies obtained. Theoretical results are compared with measurements in the JILA and MIT experiments.