Density fluctuations of Bose condensates in anisotropic traps in the hydrodynamic limit

Abstract

We derive a symmetry property for the Fourier-transform of the density fluctuation modes of Bose condensates in anisotropic traps in the hydrodynamic limit connected with a somewhat hidden conservation law.. We discuss its possible observation by dispersive light scattering, its range of validity, and also experiments using inelastic scattering of light. PACS: 03.75.Fi,05.30.Jp,42.50.Gy Since the achievement of Bose-Einstein condensates in trapped alkali gases [1–3] 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 fluctuation spectrum were measured by observing condensate shape oscillations either by time of flight measurements [4–6] or by dispersive light scattering [7]. In this paper we want to report on a property of the mode functions of the low lying density fluctuations of condensates in anisotropic traps. Mode functions can be observed in dispersive light scattering experiments [7,8], but also in inelastic light scattering. We discuss how the derived property could be observed in both experiments and discuss the limits of both approaches. For a Bose gas below the Bose-Einstein transition the usual separation of the particle field 1 operator into a classical part φ0(r), the condensate wave function, and a residual operator ˆφ(r), describing fluctuations, yields for the number-density operator ˆρ(r) = |φ0(r) | 2 + ( φ ∗ 0(r) ˆ φ(r) + φ0(r) ˆ φ † (r)) + ˆ φ † (r) ˆ φ(r). (0.1) Thermal averaging of the last term gives the condensate depletion at T = 0 plus the density of thermally excited atoms. At low temperatures for a weakly interacting system the condensate density |φ0(r) | 2 contains most of the particles, and the second term in (0.1) dominates by far the last term. Now we restrict ourselves to the low-lying density fluctuations with ¯hω ≪ µ, which can be described by the hydrodynamic approach to trapped Bose systems [9,10]. In this limit the dominant density fluctuations in (0.1) take the form [10,11] φ ∗ 0 (r) ˆ φ(r) + φ0(r) ˆ φ † (r) = ∑ j