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Physical Review A

Analytical expressions for the hydrodynamic excitation spectrum of Bose-Einstein condensates in axially anisotropic traps

Yong-li Ma, S. T. Chui

Published May 2, 2002• 25 citations
Publisher

Abstract

By using decoupling and dimension reduction technologies in equations determing the eigenvalue, we solve analytically the coupled hydrodynamic equation for the eigenvalues of collective excitations and obtain analytical expressions for all modes of a Bose-Einstein condensate in anisotropic axially symmetric harmonic traps in the Thomas-Fermi limit.

DOI
10.1103/physreva.65.053610

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By using decoupling and dimension reduction technologies in equations determing the eigenvalue, we solve analytically the coupled hydrodynamic equation for the eigenvalues of collective excitations and obtain analytical expressions for all modes of a Bose-Einstein condensate in anisotropic axially symmetric harmonic traps in the Thomas-Fermi limit.
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We investigate the collective excitations of a harmonically trapped superfluid Fermi gas at varying coupling strengths across a BCS-BEC crossover. Using a hydrodynamic approach, we solve analytically the eigenvalue problem of collective modes and provide explicit expressions for all eigenvalues and eigenfunctions, which are valid for both BCS and BEC limits and also for the whole crossover regime. Both spherical- and axial-symmetric traps are taken into account, and the features of these collective modes in the BCS-BEC crossover are discussed and compared with available experimental and numerical data.
We propose a method for finding analytical solutions of the Bogoliubov--de Gennes (BdG) equations for the low-lying collective excitations in a harmonically trapped Bose-Einstein condensate beyond the Thomas-Fermi limit. We first use a simple variational wave function for ground state to eliminate the divergence at the boundary layer of the condensate, which appears in the Thomas-Fermi approximation. We then solve the BdG equations analytically and obtain explicit and divergence-free expressions for the eigenvalues and eigenfunctions of the excitations for the traps with spherical and cylindrical symmetries. The solutions of the zero-energy mode of the BdG equations are also presented.
We develop an analytical approach for calculating the Landau damping of collective modes in the Bose-Einstein condensate trapped by an anisotropic harmonic potential. Based on a variational ground-state wave function obtained by solving the time-independent Gross-Pitaevskii equation beyond Thomas-Fermi approximation, we solve the Bogoliubov--de Gennes equations describing thermal excited quasiparticles and provide divergence-free analytical solutions for the Bogoliubov amplitudes and coupling matrix elements that characterize the interaction between the collective modes and the quasiparticles. With these analytical results we evaluate the Landau damping rates of several collective modes for various anisotropic parameters of the trapping potential in terms of the formulas derived from a time-dependent mean-field theory. In addition, we discuss the dependence of the damping rates on temperature, particle number, trapping frequency, and anisotropic parameter of the system, and compare our theoretical results with experimental and numerical data reported in literature.
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.
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