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arXiv:cond-mat/9711250 · International Journal of Modern Physics B

Effective Field Theory Approach to Bose–Einstein Condensation

W. Vincent Liu

Published Aug 20, 1998• 7 citations
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Abstract

We consider the low-energy collective excitations at finite temperature of Bose–Einstein condensed gases (and liquids as well). A most general model-independent effective Lagrangian is written down according to a prescription obtained from the breakdown of the global symmetry U(1). To show how the theory predicts easily, we derive the momentum and temperature dependence of the damping of excitations by means of power counting as an example.

DOI
10.1142/s0217979298001241

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We consider the low-energy collective excitations at finite temperature of Bose–Einstein condensed gases (and liquids as well). A most general model-independent effective Lagrangian is written down according to a prescription obtained from the breakdown of the global symmetry U(1). To show how the theory predicts easily, we derive the momentum and temperature dependence of the damping of excitations by means of power counting as an example.
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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 calculate the frequencies and damping rates of the low-lying collective modes of a Bose-Einstein condensed gas at nonzero temperature. We use a complex nonlinear Schr\"odinger equation to determine the dynamics of the condensate atoms, and couple it to a Boltzmann equation for the noncondensate atoms. In this manner we take into account both collisions between noncondensate-noncondensate and condensate-noncondensate atoms. We solve the linear response of these equations, using a time-dependent Gaussian trial function for the condensate wave function and a truncated power expansion for the deviation function of the thermal cloud. As a result, our calculation turns out to be characterized by two dimensionless parameters proportional to the noncondensate-noncondensate and condensate-noncondensate mean collision times. We find in general quite good agreement with experiment, both for the frequencies and damping of the collective modes.