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arXiv:cond-mat/9711224 · Physical Review A

Quasi-one-dimensional character of sound propagation in elongated Bose-Einstein condensed clouds

G. M. Kavoulakis, C. J. Pethick

Published Aug 1, 1998• 50 citations
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Abstract

We show that under rather general conditions the propagation of sound pulses along the long axis of a Bose-Einstein condensed cloud in a very anisotropic trap is essentially one dimensional. In the linear regime, we demonstrate that the square of the velocity of propagation is given by the square of the local sound velocity ${c}^{2}{=nU}_{0}/m$ averaged over the cross section of the cloud. We also carry out calculations in the nonlinear regime and determine how the speed of the pulse depends on its amplitude.

DOI
10.1103/physreva.58.1563

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We show that under rather general conditions the propagation of sound pulses along the long axis of a Bose-Einstein condensed cloud in a very anisotropic trap is essentially one dimensional. In the linear regime, we demonstrate that the square of the velocity of propagation is given by the square of the local sound velocity c2=nU0/m{c}^{2}{=nU}_{0}/mc2=nU0​/m averaged over the cross section of the cloud. We also carry out calculations in the nonlinear regime and determine how the speed of the pulse depends on its amplitude.
We study sound propagation in Bose-condensed gases in a highly-elongated harmonic trap at finite temperatures. This problem is studied within the framework of Zaremba-Nikuni-Griffin (ZNG) formalism, which consistent of a generalized Gross-Pitaevskii (GP) equation for the condensate and the kinetic equation for a thermal cloud. We extend the ZNG formalism to deal with a highly-anisotropic trap potential, and use it to simulate sound propagation using the trap parameters corresponding to the experiment on sound pulse propagation at finite temperature. We focus on the high-density two-fluid hydrodynamic regime, and explore the possibility of observing first and second sound pulse propagation. The results of numerical simulation are compared with an analyitical results derived from linearized ZNG hydrodynamic equations. We show that the second sound mode makes a dominant contribution to condensate motion in relatively high temperature, while the first sound mode makes an appreciable contribution.
We study sound propagation in Bose-condensed gases in a highly elongated harmonic trap at finite temperatures. This problem is studied within the framework of the Zaremba-Nikuni-Griffin (ZNG) formalism, which consists of a generalized Gross-Pitaevskii equation for the condensate and a kinetic equation for the thermal cloud. We extend the ZNG formalism to deal with a highly anisotropic trap potential and use it to simulate sound propagation using the trap parameters corresponding to an experiment on sound pulse propagation at finite temperature. We focus on the high-density two-fluid hydrodynamic regime, and explore the possibility of observing first- and second-sound pulse propagation. The results of numerical simulation are compared with analytical results derived from linearized ZNG hydrodynamic equations. We show that the second-sound mode makes the dominant contribution to condensate motion at relatively high temperature, while the first-sound mode makes an appreciable contribution.
We use hydrodynamic equations to study sound propagation in a superfluid Fermi gas at zero temperature inside a strongly elongated cigar-shaped trap, with main attention to the transition from the BCS to the unitary regime. First, we treat the role of the radial density profile in the limit of a cylindrical geometry and then evaluate numerically the effect of the axial confinement in a configuration in which a hole is present in the gas density at the center of the trap. We find that in a strongly elongated trap the speed of sound in both the BCS and the unitary regime differs by a factor 3∕5\sqrt{3∕5}3∕5​ from that in a homogeneous three-dimensional superfluid. The predictions of the theory could be tested by measurements of sound-wave propagation in a setup such as that exploited by Andrews et al. [Phys. Rev. Lett. 79, 553 (1997)] for an atomic Bose-Einstein condensate.
A comparative study is conducted of the propagation of sound pulses in elongated Bose liquids and Bose-Einstein condensates in Gross-Pitaevskii and logarithmic models, by means of the Thomas-Fermi approximation. It is demonstrated that in the linear regime the propagation of small density fluctuations is essentially one-dimensional in both models, in the direction perpendicular to the cross section of a liquid’s lump. Under these approximations, it is demonstrated that the speed of sound scales as a square root of particle density in the case of the Gross-Pitaevskii liquid/condensate, but it is constant in a case of the homogeneous logarithmic liquid.
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.