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arXiv:cond-mat/9804108 · Physical Review A

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

Hualin Shi, Wei‐Mou Zheng

Published Feb 1, 1999• 8 citations
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Abstract

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.

DOI
10.1103/physreva.59.1562

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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.
We derive a simple formula for the finite-temperature shifts of the collective excitations of a Bose-Einstein condensate. To test the validity of our treatment we apply it to explain the anomalous behavior of the ``m=2'' excitation frequency in the recent JILA experiments (D. S. Jin et al, PRL, 78 (1997) 764), where this frequency does not approach its ``noninteracting'' value at ω=2ωρω= 2 ω_ρω=2ωρ​ but, instead, decreases with temperature. It turns out that the effect is mainly governed by a resonance between the condensate excitation frequency and one of the quantum eigenfrequencies of the thermal cloud. Comparison of our predictions with the JILA experimental data shows a good agreement.
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.
We extend to finite temperature the study of collective excitations of a Bose-Einstein condensate in a dilute gas of 87Rb{}^{87}\mathrm{Rb}87Rb. Measurements of two modes with different angular momenta show unexpected temperature-dependent frequency shifts, with very different behavior for the two symmetries; in addition there is a sharp feature in the temperature dependence of one mode. The damping of these excitations exhibits dramatic temperature dependence, with condensate modes at temperatures near the transition damping even faster than analogous noncondensate oscillations.
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