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Journal of Physics Condensed Matter

The frequency response of large-amplitude oscillations of a trapped Bose condensed gas

Yong-li Ma, S. T. Chui

Published Oct 18, 2002• 5 citations
Publisher

Abstract

We study the frequency response of large-amplitude oscillations of a trapped Bose condensed gas. On the basis of the Thomas?Fermi approximation, we deduce the hydrodynamical equation including the excitation source of the velocity drive, and obtain the analytical expression for the frequency as a function of velocity fluctuation amplitude, trap geometry, and symmetry of modes. By solving the expansion equation after switching off the trap, we find a simple relationship between the velocity amplitude and the oscillation amplitude. The theoretical calculations on the response frequencies agree well with the existing experimental observations.

DOI
10.1088/0953-8984/14/43/309

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We study the frequency response of large-amplitude oscillations of a trapped Bose condensed gas. On the basis of the Thomas?Fermi approximation, we deduce the hydrodynamical equation including the excitation source of the velocity drive, and obtain the analytical expression for the frequency as a function of velocity fluctuation amplitude, trap geometry, and symmetry of modes. By solving the expansion equation after switching off the trap, we find a simple relationship between the velocity amplitude and the oscillation amplitude. The theoretical calculations on the response frequencies agree well with the existing experimental observations.
We investigate the behavior of large amplitude oscillations of a trapped Bose-condensed gas of alkali-metal atoms at zero temperature, by solving the equations of hydrodynamics for collective modes. Due to the atom-atom interaction, the equations of motion are nonlinear and give rise to significant frequency shift and mode coupling. We provide analytic expressions for the frequency shift, pointing out the crucial role played by the anisotropy of the confining potential. For special values of the anisotropy parameter the mode coupling is particularly strong and the frequency shift becomes large, revealing a peculiar behavior of the Bose-condensed gas. Consequences on the theory of collapse and revival of collective excitations are also discussed.
We discuss the collective modes of a trapped Bose gas in the hydrodynamic regime where atomic collisions ensure local thermal equilibrium for the distribution function. Starting from the conservation laws, in the linearized limit we derive a closed equation for the velocity fluctuations in a trapped Bose gas above the Bose-Einstein transition temperature. Explicit solutions for a parabolic trap are given. We find that the surface modes above the transition have the same dispersion relation as the one recently obtained by Stringari for the oscillations of the condensate at T=0T\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0T=0 within the Thomas-Fermi approximation. Results are also given for the monopole ``breathing'' mode as well as for the m=0m\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0m=0 excitations which result from the coupling of the monopole and quadrupole modes in an anisotropic parabolic well.
The frequencies of the collective oscillations of a harmonically trapped Fermi gas interacting with large scattering lengths are calculated at zero temperature using hydrodynamic theory. Different regimes are considered, including the molecular Bose-Einstein condensate and the unitarity limit for collisions. We show that the frequency of the radial compressional mode in an elongated trap exhibits a pronounced non-monotonous dependence on the scattering length, reflecting the role of the interactions in the equation of state.
The oscillation frequencies of collective excitations of a trapped Bose-Einstein condensate, when calculated in the mean-field approximation and in the Thomas-Fermi limit, are independent of the scattering length aaa. We calculate the leading corrections to the frequencies from quantum fluctuations around the mean field. The semiclassical correction is proportional to N1/5a6/5{N}^{1/5}{a}^{6/5}N1/5a6/5, where NNN is the number of atoms in the condensate. The correction is positive semidefinite and is zero for surface modes whose eigenfunctions have a vanishing Laplacian. The shift in the frequency of the lowest quadrupole mode for an axially symmetric trap is large enough that it should be measurable in future experiments.
Abstract This chapter provides a general discussion of the dynamic behaviour characterizing a harmonically trapped interacting Bose gas. Explicit results for the frequencies of the collective oscillations were derived by using the hydrodynamic theory of superfluids. Both surface and oscillation of compressional nature are considered and a direct comparison is presented with available experimental results. Sum rules are used to characterize the behaviour of the oscillations. Finite-size as well as beyond-mean-field effects are considered. A special discussion is devoted to the study of large-amplitude oscillations and of the expansion of the gas. The anisotropic expansion predicted by hydrodynamic theory is discussed in a systematic way. An important part of the chapter concerns the study of the dynamic structure factor and its determination in experiments based on two-photon Bragg spectroscopy. Finally, the competition between single-particle and collective modes in the excitation spectrum of a harmonically trapped BEC is discussed.