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arXiv:cond-mat/9801067 · Physical Review A

Dynamics of Bose-Einstein condensed gases in highly deformed traps

S. Stringari

Published Sep 1, 1998• 182 citations
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Abstract

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.

DOI
10.1103/physreva.58.2385

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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.
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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.
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 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 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.
Ultracold atoms are exceptional tools to explore the physics of quantum matter. In fact, the high degree of tunability of ultracold Bose and Fermi gases makes them ideal systems for quantum simulation and for investigating macroscopic manifestations of quantum effects, such as superfluidity. In ultracold gas research, a central role is played by collective oscillations. They can be used to study different dynamical regimes, such as superfluid, collisional, or collisionless limits or to test the equation of state of the system. In this thesis, we present a unified description of collective oscillations in low dimensions covering both Bose and Fermi statistics, different trap geometries and zero as well as finite temperature, based on the formalism of hydrodynamics and sum rules. We discuss the different behaviour exhibited by the second excited breathing mode in the collisional regime at low temperature and in the collisionless limit at high temperature in a 1D trapped Bose gas with repulsive contact interaction. We show how this mode exhibits a single-valued excitation spectrum in the collisional regime and two different frequencies in the collisionless limit. Our predictions could be important for future research related to the thermalization and damping phenomena in this low-dimensional system. We show that 1D uniform Bose gases exhibit a non-monotonic temperature dependence of the chemical potential characterized by an increasing-with-temperature behaviour at low temperature. This is due to the thermal excitation of phonons and reveals an interesting analogy with the behaviour of superfluids. Finally, we investigate a gas with a finite number N of atoms in a ring geometry at T = 0. We discuss explicitly the deviations of the thermodynamic behaviour in the ring from the one in the large N limit.