We derive a symmetry property for the Fourier-transform of the collisionless sound modes of Bose condensates in anisotropic traps connected with a somewhat hidden conservation law. We discuss its possible observation by dispersive light scattering. : 03.75.Fi,05.30.Jp,42.50.Gy Since the achievement of Bose-Einstein condensates in trapped alkali gases [1][2][3] at ultralow temperatures numerous aspects of these systems have been investigated both from the experimental and the theoretical side. Adding a time dependent perturbation to the trap potential the lowest frequencies of the density oscillation spectrum were measured by observing condensate shape oscillations either by time of flight measurements [4][5][6] or by dispersive light scattering [7]. In this paper we want to investigate these mode functions in detail and show how they reflect a special symmetry of the system. We disuss how this symmetry property could be observed in phase-contrast imaging measurements [7,8] and comment on the feasibility of its observation in inelastic light sacttering.
PACSWe restrict ourselves to the density fluctuations of energies much smaller than the chemical potential (hω ≪ µ) of a trapped Bose condensate at temperature T = 0. The equations governing these collisionless sound modes in a weakly interacting system in mean field approximation can be written in the form of hydrodynamic equations [9,10]. For isotropic [9], axially symmetric [14] and even fully anisotropic harmonic traps [16] the solutions of these equations can be classified in terms of three quantum numbers. We consider these mode functions in Fourier space where we scale all momenta by k i =k i mω 2 i /2µ for i = x, y, z with chemical potential µ and trap frequency ω i in direction i.Before going into any technical details we state the central result of this paper and discuss how it may be tested experimentally: We show in this paper that the Fourier transformφ(k) of the mode functions for traps of arbitrary anisotropy factorizes into a radial part depending only on the modulus of the scaled wave vectortimes an angular part g(θ k , φ k ) depending only on the two angles θ k , φ k of spherical coordinates in scaled Fourier space. Furthermore the radial part does not depend on the trap anisotropy anymore and can be determined explicitly. We simply have 1