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arXiv:cond-mat/9806112 · Physical Review A

Hydrodynamic modes and pulse propagation in a cylindrical Bose gas above the Bose-Einstein transition

Tetsuro Nikuni, Adam Griffin

Published Nov 1, 1998• 11 citations
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Abstract

We study hydrodynamic oscillations of a cylindrical Bose gas above the Bose-Einstein transition temperature using the hydrodynamic equations derived by Griffin, Wu, and Stringari. This extends recent studies of a cylindrical Bose-condensed gas at $T=0.$ Explicit normal mode solutions are obtained for non-propagating solutions. In the classical limit, the sound velocity is shown to be the same as a uniform classical gas. We use a variational formulation of the hydrodynamic equations to discuss the propagating modes in the degenerate Bose-gas limit and show there is little difference from the classical results. We discuss the propagation of sound pulses above and below ${T}_{\mathrm{BEC}}.$

DOI
10.1103/physreva.58.4044

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We study hydrodynamic oscillations of a cylindrical Bose gas above the Bose-Einstein transition temperature using the hydrodynamic equations derived by Griffin, Wu, and Stringari. This extends recent studies of a cylindrical Bose-condensed gas at T=0.T=0.T=0. Explicit normal mode solutions are obtained for non-propagating solutions. In the classical limit, the sound velocity is shown to be the same as a uniform classical gas. We use a variational formulation of the hydrodynamic equations to discuss the propagating modes in the degenerate Bose-gas limit and show there is little difference from the classical results. We discuss the propagation of sound pulses above and below TBEC.{T}_{\mathrm{BEC}}.TBEC​.
Griffin, Wu and Stringari have derived the hydrodynamic equations of a trapped dilute Bose gas above the Bose-Einstein transition temperature. We give the extension which includes hydrodynamic damping, following the classic work of Uehling and Uhlenbeck based on the Chapman-Enskog procedure. Our final result is a closed equation for the velocity fluctuations δvδvδv which includes the hydrodynamic damping due to the shear viscosity ηηη and the thermal conductivity κκκ. Following Kavoulakis, Pethick and Smith, we introduce a spatial cutoff in our linearized equations when the density is so low that the hydrodynamic description breaks down. Explicit expressions are given for ηηη and κκκ, which are position-dependent through dependence on the local fugacity when one includes the effect of quantum degeneracy of the trapped gas. We also discuss a trapped Bose-condensed gas, generalizing the work of Zaremba, Griffin and Nikuni to include hydrodynamic damping due to the (non-condensate) normal fluid.
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.
Griffin, Wu and Stringari have derived the hydrodynamic equations of a trapped dilute Bose gas above the Bose-Einstein transition temperature. We give the extension which includes hydrodynamic damping, following the classic work of Uehling and Uhlenbeck based on the Chapman-Enskog procedure. Our final result is a closed equation for the velocity fluctuations δv which includes the hydrodynamic damping due to the shear viscosity η and the thermal conductivity κ. Following Kavoulakis, Pethick and Smith, we introduce a spatial cutoff in our linearized equations when the density is so low that the hydrodynamic description breaks down. Explicit expressions are given for η and κ, which are position-dependent through dependence on the local fugacity when one includes the effect of quantum degeneracy of the trapped gas. We also discuss a trapped Bose-condensed gas, generalizing the work of Zaremba, Griffin and Nikuni to include hydrodynamic damping due to the (non-condensate) normal fluid.
We study the normal modes of a cylindrical Bose condensate at T=0T=0T=0 using the linearized time-dependent Gross-Pitaevskii equation in the Thomas-Fermi limit. These modes are relevant to the recent observation of pulse propagation in long, cigar-shaped traps. We find that pulses generated in a cylindrical condensate propagate with little spread at a speed $c=\sqrt{g\mathrm{n\ifmmode \bar{}\else \={}\fi{}}/m},$ where $\mathrm{n\ifmmode \bar{}\else \={}\fi{}}$ is the average density of the condensate over its cross-sectional area.
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 extend our recent work on the two-fluid hydrodynamics of the condensate and non-condensate in a trapped Bose gas by including the dissipation associated with viscosity and thermal conduction. For purposes of illustration, we consider the hydrodynamic modes in the case of a uniform Bose gas. A finite thermal conductivity and shear viscosity give rise to a damping of the first and second sound modes in addition to that found previously due to the lack of diffusive equilibrium between the condensate and non-condensate. The relaxational mode associated with this equilibration process is strongly coupled to thermal fluctuations and reduces to the usual thermal diffusion mode above the Bose-Einstein transition. In contrast to the standard Landau two-fluid hydrodynamics, we predict a damped mode centered at zero frequency, in addition to the usual second sound doublet.