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arXiv:cond-mat/9709259 · Physical Review A

Damping in dilute Bose gases: A mean-field approach

S. Giorgini

Published Apr 1, 1998• 168 citations
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Abstract

Damping in a dilute Bose gas is investigated using a mean-field approximation which describes the coupled oscillations of condensate and noncondensate atoms in the collisionless regime. Explicit results for both Landau and Beliaev damping rates are given for nonuniform gases. In the case of uniform systems, we obtain results for the damping of phonons both at zero and finite temperature. The isothermal compressibility of a uniform gas is also discussed.

DOI
10.1103/physreva.57.2949

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Damping in a dilute Bose gas is investigated using a mean-field approximation which describes the coupled oscillations of condensate and noncondensate atoms in the collisionless regime. Explicit results for both Landau and Beliaev damping rates are given for nonuniform gases. In the case of uniform systems, we obtain results for the damping of phonons both at zero and finite temperature. The isothermal compressibility of a uniform gas is also discussed.
We study the low-energy collective oscillations of a dilute Bose gas at finite temperature in the collisionless regime. By using a time-dependent mean-field scheme we derive for the dynamics of the condensate and noncondensate components a set of coupled equations, which we solve perturbatively to second order in the interaction coupling constant. This approach is equivalent to the finite-temperature extension of the Beliaev approximation and includes corrections to the Gross-Pitaevskii theory due to both quantum and thermal fluctuations. For a homogeneous system we explicitly calculate the temperature dependence of the velocity of propagation and damping rate of zero sound. In the case of harmonically trapped systems in the thermodynamic limit, we calculate, as a function of temperature, the frequency shift of the low-energy compressional and surface modes.
By using a mean-field approximation which describes the coupled oscillations of condensate and noncondensate atoms in the collisionless regime, Landau damping in a dilute dipolar Bose–Fermi mixture in the BEC limit where Fermi superfluid is treated as tightly bounded molecules, is investigated. In the case of a uniform quasi-two-dimensional (2D) case, the results for the Landau damping due to the Bose–Fermi interaction are obtained at low and high temperatures. It is shown that at low temperatures, the Landau damping rate is exponentially suppressed. By increasing the strength of dipolar interaction, and the energy of boson quasiparticles, Landau damping is suppressed over a broader temperature range.
With very few exceptions (such as the centre-of-mass dipole mode), collective oscillations in trapped superfluid Bose gases are damped. In the “collisionless” region the damping is second order in the interaction strength. There are three possible components. One is Beliaev damping, which is due to the decay of a single excitation into two excitations; this can occur even at T = 0. In addition, there is Landau damping, which is due to a collective mode scattering from thermally excited excitations. This process only occurs at finite temperatures but quickly becomes the dominant damping mechanism as the temperature increases. Both Landau and Beliaev damping arise naturally from the imaginary part of the Beliaev second-order self-energies, as given in (5.40) in the case of a uniform Bose gas. Finally there is the damping that arises from the C 22 and C 12 collision processes; this is discussed in Chapters 8, 12 and 19. In Chapter 12 we calculated the damping of various condensate modes at finite temperatures using direct numerical simulations of the ZNG equations. These numerical results were generally in very good agreement with the available experimental data. From a theoretical perspective, one advantage of the simulations is that the Landau damping contribution can be isolated simply by setting the C 12 and C 22 collision terms to zero. After providing an introduction to Landau damping in uniform Bose gases in Section 13.1, we present in Section 13.2 a detailed discussion of Landau damping based on a general formula in terms of Bogoliubov–Popov excitations. This discussion makes it clear that the Landau damping of condensate oscillations arises from the interaction with a thermal cloud of excitations.
Damping in two-dimensional (2D) and 3D dilute gases is investigated using both the hydrodynamical approach and the Hartree–Fock–Bogoliubov (HFB) approximation. We found that both methods are good for Beliaev damping at zero temperatures and Landau damping at very low temperatures. However, at high temperatures, the hydrodynamical approach overestimates the Landau damping and HFB gives a precise asymptotic behavior. This result shows that the comparison of the theoretical calculation using the hydrodynamical approach and the experimental data for high temperatures by Vincent Liu (1997 Phys. Rev. Lett. 79 4056) is not valid. For 2D systems, we show that the Beliaev damping rate is proportional to k 3 and the Landau damping rate is proportional to T 2 for low temperatures and to T for high temperatures. We also show that in 2D the hydrodynamical approach gives the same result for zero temperature and for low temperature as HFB, but overestimates the Landau damping for high temperatures.
We develop a microscopic theory of sound damping due to Landau mechanism in dilute gas with Bose condensate.It is based on the coupled evolution equations of the parameters describing the system.These equations have been derived in earlier works within a microscopic approach which employs the Peletminskii-Yatsenko reduced description method for quantum many-particle systems and Bogoliubov model for a weakly nonideal Bose gas with a separated condensate.The dispersion equations for sound oscillations were obtained by linearization of the mentioned evolution equations in the collisionless approximation.They were analyzed both analytically and numerically.The expressions for sound speed and decrement rate were obtained in high and low temperature limiting cases.We have shown that at low temperature the dependence of the obtained quantities on temperature significantly differs from those obtained by other authors in the semi-phenomenological approaches.Possible effects connected with non-analytic temperature dependence of dispersion characteristics of the system were also indicated.
We present a theory for the Landau damping of low-energy quasiparticles in a collisionless, quasi-two-dimensional dipolar Bose gas and produce expressions for the damping rate in uniform and nonuniform systems. Using simple energy-momentum conservation arguments, we show that in the homogeneous system, the nature of the low-energy dispersion in a dipolar Bose gas severely inhibits Landau damping of long wavelength excitations. For a gas with contact and dipolar interactions, the damping rate for phonons tends to decrease with increasing dipolar interactions; for strong dipole-dipole interactions, phonons are virtually undamped over a broad range of temperature. The damping rate for maxon-roton excitations is found to be significantly larger than the damping rate for phonons.