Collective excitations in Bose-Einstein condensates in triaxially anisotropic parabolic traps

Abstract

The wave equation of low-frequency density waves in Bose-Einstein condensates at vanishing temperature in arbitrarily anisotropic harmonic traps is separable in elliptic coordinates, provided the condensate can be treated in the Thomas-Fermi approximation. We present a complete solution of the mode functions, which are polynomials of finite order, and their eigenfrequencies, which are characterized by three-integer quantum numbers. ͓S1050-2947͑99͒01602-9͔

Collective excitations of degenerate Fermi gases in anisotropic parabolic traps

“…Amoruso et al [13] also investigated the collective excitations of trapped degenerate Fermi-gases both in the hydrodynamic and the collisionless regime for isotropic and, for some dipolar and quadrupolar modes, also for axially symmetric parabolic traps. New in the present work is the treatment of the completely anisotropic case, where we follow the lines of our earlier treatment of the bosonic case [10]. The paper is organized as follows.…”

“…eq. (4.24) of [10]). To find the minima of the potential (4.18) is a problem in the 2-dimensional electrostatics (because of the logarithmic potential) of n+4 point-charges on a line: The equilibrium positions θ i of the n-charges of unit strength between the 4 fixed charges of strength 3/4 at θ = 0, ( α 2 + 1 4 ) at θ = −a 2 , ( β 2 + 1 4 ) at θ = −b 2 , and ( γ 2 + 1 4 ) at θ = −c 2 determine the eigenfrequencies via eq.(4.13).…”

“…We shall analyze this case using the procedure applied to Bose-condensates in our previous paper [10]. Ellipsoidal coordinates λ, µ, ν are introduced as functions of x, y, z as the three roots of [18] with respect to ρ…”

“…The eigenfunctions of the wave-operator can now be obtained as in [10] using the method for the solution of the Lamé equation (see [18]). The polynomial ansatz (with n t = 2n + α + β + γ; α = 0, 1; β = 0, 1; γ = 0, 1)…”

“…It has turned out that within the dynamic Thomas-Fermi approximation these modes could be calculated analytically by solving the Gross-Pitaevskii equation, linearized around the static Thomas-Fermi solution. This was done first [6,7] for isotropic parabolic traps, where the symmetries are sufficient for integrability, then for anisotropic axially symmetric parabolic traps [6,8,9], where an additional hidden symmetry could be identified [8], and finally also for completely anisotropic parabolic traps [10], where the resulting wave-equation could be related to various integrable dynamical systems and the Lamé equation.…”