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arXiv:2212.09449 · Fluids

Sound Propagation in Cigar-Shaped Bose Liquids in the Thomas-Fermi Approximation: A Comparative Study between Gross-Pitaevskii and Logarithmic Models

Konstantin G. Zloshchastiev

Published Nov 19, 2022• 3 citations
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Abstract

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.

DOI
10.3390/fluids7110358

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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 consider the dynamical properties of density fluctuations in the cigar-shaped Bose–Einstein condensate described by the logarithmic wave equation with a constant nonlinear coupling by using the Thomas–Fermi and linear approximations. It is shown that the propagation of small density fluctuations along the long axis of a condensed lump in a strongly anisotropic trap is essentially one-dimensional, while the trapping potential can be disregarded in the linear regime. Depending on the sign of nonlinear coupling, the fluctuations either take the form of translationally symmetric pulses and standing waves or become oscillations with varying amplitudes. We also study the condensate in an axial harmonic trap, by using elasticity theory’s notions. Linear particle density and energy also behave differently depending on the nonlinear coupling’s value. If it is negative, the density monotonously grows along with lump’s radius, while energy is a monotonous function of density. For the positive coupling, the density is bound from above, whereas energy grows monotonously as a function of density until it reaches its global maximum.
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.
The Gross-Pitaevskii equation for a Bose-Einstein condensate confined in an elongated cigar-shaped trap is reduced to an effective system of nonlinear equations depending on only one space coordinate along the trap axis. The radial distribution of the condensate density and its radial velocity are approximated by Gaussian functions with real and imaginary exponents, respectively, with parameters depending on the axial coordinate and time. The effective one-dimensional system is applied to a description of the ground state of the condensate, to dark and bright solitons, to the sound and radial compression waves propagating in a dense condensate, and to weakly nonlinear waves in repulsive condensate. In the low-density limit our results reproduce the known formulas. In the high-density case our description of solitons goes beyond the standard approach based on the nonlinear Schr\"odinger equation. The dispersion relations for the sound and radial compression waves are obtained in a wide region of values of the condensate density. The Korteweg--de Vries equation for weakly nonlinear waves is derived and the existence of bright solitons on a constant background is predicted for a dense enough condensate with a repulsive interaction between the atoms.
We investigate the dynamics of pulses in a cigar-shaped Bose-Einstein condensate with repulsive atom-atom interactions without using Thomas-Fermi approximation. In the linear level our results give the Bogoliubov excitation spectrum for sound propagation with speed c = c 0 /(2) 1/2 , where c 0 is the speed for the case without a trap. We develop a Korteweg-de Vries (KdV) description for dark soliton propagation in the system and show that it is the quantum pressure that contributes the dispersion necessary for the formation of the dark solitons.
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.
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.