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arXiv:cond-mat/9709041 · Physical Review A

Excitations of a Bose-condensed gas in anisotropic traps

David A. W. Hutchinson, E. Zaremba

Published Feb 1, 1998• 27 citations
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Abstract

We investigate the zero-temperature collective excitations of a Bose-condensed atomic gas in anisotropic parabolic traps. The condensate density is determined by solving the Gross-Pitaevskii (GP) equation using a spherical harmonic expansion. The GP eigenfunctions are then used to solve the Bogoliubov equations to obtain the collective excitation frequencies and mode densities. The frequencies of the various modes, classified by their parity and the axial angular momentum quantum number, m, are mapped out as a function of the axial anisotropy. Specific emphasis is placed upon the evolution of these modes from the modes in the limit of an isotropic trap.

DOI
10.1103/physreva.57.1280

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We investigate the zero-temperature collective excitations of a Bose-condensed atomic gas in anisotropic parabolic traps. The condensate density is determined by solving the Gross-Pitaevskii (GP) equation using a spherical harmonic expansion. The GP eigenfunctions are then used to solve the Bogoliubov equations to obtain the collective excitation frequencies and mode densities. The frequencies of the various modes, classified by their parity and the axial angular momentum quantum number, m, are mapped out as a function of the axial anisotropy. Specific emphasis is placed upon the evolution of these modes from the modes in the limit of an isotropic trap.
Collective modes of the ground and excited states of Bose?Einstein condensates are calculated by diagonalization of the Bogoliubov?de Gennes matrix in cylindrical coordinates. For an isotropic magnetic trap, the first breathing mode frequencies of the ground and dark soliton excited states are found to be in excellent agreement with a direct solution of the Gross?Pitaevskii equation with a time-dependent driving potential. In addition, the matrix diagonalization yields densities and frequencies for second breathing modes and higher order multipole modes. For an axially symmetric anisotropic trap, a considerable number of collective modes of the ground and dark soliton excited states are calculated. Prolate traps are found to have many more low-frequency multipolar modes than found in oblate traps for both ground and dark soliton excited states.
In the Thomas–Fermi (TF) regime, S. Stringari [Phys. Rev. Lett., 77, 2360 (1996)] established the hydrodynamic normal modes of an interacting dilute Bose-condensed fluid confined in an isotropic harmonic trap. Here, we extend this treatment beyond the TF condensate using the work of B. Hu, G. Huang and Y. Ma [Phys. Rev. A, 69, 063608 (2004)] as a starting point. Both numerical results for low-lying eigenfrequencies, for various angular momentum quantum numbers ℓ and n = 0, and also samples of corresponding eigenfunctions, are presented. Finally, eigenfrequencies are also given for some collective excitations in an axially symmetrical trap.
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.
The mode frequencies of a weakly interacting Bose gas in a magnetic trap are studied as a function of the anisotropy of the trap. As in earlier works, the generalized Hartree-Fock-Bogoliubov equations within the Popov approximation are used for our calculations. A feature of our work is the combined use of a mode expansion in a finite basis and a semiclassical approximation of the highly excited states. The results are applied to check the accuracy of the recently suggested equivalent zero temperature condensate approximation, which involves a much simpler model.
The collective excitations of Bose condensates in anisotropic axially symmetric harmonic traps are investigated in the hydrodynamic and Thomas-Fermi limit. We identify an additional conserved quantity, besides the axial angular momentum and the total energy, and separate the wave equation in elliptic coordinates. The solution is thereby reduced to the algebraic problem of diagonalizing finite-dimensional matrices. The classical quasiparticle dynamics in the local-density approximation for energies of the order of the chemical potential is shown to be chaotic.