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arXiv:cond-mat/9805338 · Physical Review A

Hidden symmetry of collisionless sound of Bose condensates in anisotropic traps

Martin Fliesser, Robert Graham

Published Jan 1, 1999• 6 citations
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Abstract

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.
DOI
10.1103/physreva.59.r27

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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
We find scaling solutions for the evolution of a Bose gas in time-dependent anisotropic traps and reveal the phenomenon of stochastization in the evolution of a condensate. In contrast to collisionless gases, the evolution of the thermal gas in the hydrodynamic regime proves to be in many aspects similar to that of the condensate. We emphasize that clear signatures of Bose-Einstein condensation and the effects of the loss of coherence can be found in local correlation properties of the evolving gas through the measurement of the rates of intrinsic or light-induced inelastic collisional processes.
We provide a unified investigation of normal modes and sound propagation at zero temperature in Bose-Einstein condensed gases confined in highly asymmetric harmonic traps and interacting with repulsive forces. By using hydrodynamic theory for superfluids we obtain explicit analytic results for the dispersion law of the low-energy discretized modes for both cigar- and disk-shaped geometries, including the regime of large quantum numbers where discrete modes can be identified with phonons. The correspondence with sound propagation in cylindrical traps and the one-dimensional nature of cigar-type configurations are explicitly discussed.
We experimentally and theoretically investigate the anisotropic speed of sound of an atomic superfluid (SF) Bose-Einstein condensate in a 1D optical lattice. Because the speed of sound derives from the SF density, this implies that the SF density is itself anisotropic. We find that the speed of sound is decreased by the optical lattice, and the SF density is concomitantly reduced. This reduction is accompanied by the appearance of a zero entropy normal fluid in the purely Bose condensed phase. The reduction in SF density-first predicted [A. J. Leggett, Phys. Rev. Lett. 25, 1543 (1970).PRLTAO0031-900710.1103/PhysRevLett.25.1543] in the context of supersolidity-results from the coexistence of superfluidity and density modulations, but is agnostic about the origin of the modulations. We additionally measure the moment of inertia of the system in a scissors mode experiment, demonstrating the existence of rotational flow. As such we shed light on some supersolid properties using imposed, rather than spontaneously formed, density order.
We have calculated the first and second sound modes of a dilute interacting Bose gas in a spherical trap for temperatures (0.6<T/Tc<1.2)(0.6<{T/T}_{c}<1.2)(0.6<T/Tc​<1.2) and for systems with 104{10}^{4}104 to 108{10}^{8}108 particles. The second sound modes (which exist only below Tc{T}_{c}Tc​) generally have a stronger temperature dependence than the first sound modes. The puzzling temperature variations of the sound modes near Tc{T}_{c}Tc​ recently observed at JILA in systems with 103{10}^{3}103 particles match surprisingly well with those of the first and second sound modes of much larger systems.