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arXiv:cond-mat/9710274 · Physical Review Letters

First and Second Sound Modes of a Bose-Einstein Condensate in a Harmonic Trap

Vivek B. Shenoy, Tin-Lun Ho

Published May 4, 1998• 27 citations
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Abstract

We have calculated the first and second sound modes of a dilute interacting Bose gas in a spherical trap for temperatures $(0.6<{T/T}_{c}<1.2)$ and for systems with ${10}^{4}$ to ${10}^{8}$ particles. The second sound modes (which exist only below ${T}_{c}$) generally have a stronger temperature dependence than the first sound modes. The puzzling temperature variations of the sound modes near ${T}_{c}$ recently observed at JILA in systems with ${10}^{3}$ particles match surprisingly well with those of the first and second sound modes of much larger systems.

DOI
10.1103/physrevlett.80.3895

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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.
We present an alternative approach for determining the sound modes in atomic Bose-Einstein condensates, based on thermodynamic global variables. The total number of trapped atoms was carefully studied as a function of temperature across the phase transition, at constant volume. It allowed us to evaluate the first and second sound independently resulting in consistent values from the quantum to classical regime, in good agreement with previous results found in the literature. We also provide some insight on the dominant mode, either thermal or superfluid, on each sound velocity depending on the temperature range
We determine the temperature (T)(T)(T) dependence of first- and second-sound-like mode frequencies for trapped Fermi gases undergoing the BCS to Bose--Einstein condensation (BEC) crossover. Our results are based on numerical solution of the two-fluid equations in conjunction with a microscopic calculation of thermodynamical variables. As in experiment and at unitarity, we show that the lowest radial breathing mode is TTT independent. At finite TTT, higher-order breathing modes strongly mix with second sound. Their complex TTT dependence should provide an alternative way of measuring the transition temperature Tc{T}_{c}Tc​.
We study the second sound dipole mode in a partially Bose-Einstein condensed gas. This mode is excited by spatially separating and releasing the center-of-mass of the Bose-Einstein condensate (BEC) with respect to the thermal cloud, after which the equilibration is observed. The oscillation frequency and the damping rate of this mode is studied for different harmonic confinements and temperatures. The measured damping rates close to the collisionless regime are found to be in good agreement with Landau damping. For increasing hydrodynamicity of the cloud we observe an increase of the damping.
Collective excitations of a dilute Bose gas were probed above and below the Bose-Einstein condensation temperature. The temperature dependencies of the frequency and damping rates of condensate oscillations indicate significant interactions between the condensate and the thermal cloud. Hydrodynamic oscillations of the thermal cloud were observed, constituting first sound. An antisymmetric dipolar oscillation of the thermal cloud and the condensate was studied, representing the bulk flow of a superfluid through the normal fluid. The excitations were observed in situ using non-destructive imaging techniques.
First and second sound speeds are calculated for a uniform superfluid gas of Fermi atoms as a function of temperature, density, and interaction strength. The second sound speed is of particular interest as it is a clear signal of a superfluid component and it determines the critical temperature. The sound modes and their dependence on density, scattering length, and temperature are calculated in the BCS, molecular Bose-Einstein condensate (BEC), and unitarity limits and a smooth crossover is extrapolated. It is found that first and second sounds undergo avoided crossing on the BEC side due to mixing. Consequently, they are detectable at crossover both as density and thermal waves in traps.
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.