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arXiv:cond-mat/0007049 · Physical Review A

Collective excitations of degenerate Fermi gases in anisotropic parabolic traps

András Csordás, Robert Graham

Published Dec 12, 2000• 9 citations
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Abstract

We investigate the hydrodynamic low-frequency oscillations of highly degenerate Fermi gases trapped in anisotropic harmonic potentials. Despite the lack of an obvious spatial symmetry the wave equation turns out to be separable in elliptical coordinates, similar to a corresponding result established earlier for Bose condensates. This result is used to give an analytical solution of the anisotropic wave equation for the hydrodynamic modes.

DOI
10.1103/physreva.63.013606

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We investigate the hydrodynamic low-frequency oscillations of highly degenerate Fermi gases trapped in anisotropic harmonic potentials. Despite the lack of an obvious spatial symmetry the wave equation turns out to be separable in elliptical coordinates, similar to a corresponding result established earlier for Bose condensates. This result is used to give an analytical solution of the anisotropic wave equation for the hydrodynamic modes.
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.
A method is developed to calculate all excitations of trapped gases using hydrodynamics at zero temperature for any equation of state $\ensuremath{\mu}=\ensuremath{\mu}(n)$ and for any trapping potential. It is shown that a natural scalar product can be defined for the mode functions, by which the wave operator is Hermitian and the mode functions are orthogonal. It is also shown that the Kohn modes are exact for harmonic trapping in the hydrodynamic theory. Excitations for fermions are calculated in the Bardeen-Cooper-Schrieffer--Bose-Einstein condensation transition region using the equation of state of the mean-field Leggett model for isotropic harmonic trap potential.
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.