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arXiv:cond-mat/0003481 · Physical Review A

Damping of condensate collective modes due to equilibration with the noncondensate

JAMES E. JUN. WILLIAMS, Adam Griffin

Published Jan 17, 2001• 31 citations
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Abstract

We consider the damping of condensate collective modes in the collisionless regime at finite temperatures arising from lack of equilibrium between the condensate and the noncondensate atoms, an effect that is ignored in the usual discussion of the collisionless region. As a first approximation, we ignore the dynamics of the thermal cloud. Our calculations should be applicable to collective modes of the condensate that are oscillating out-of-phase with the thermal cloud. We obtain a generalized Stringari equation of motion for the condensate at finite temperatures, which includes a damping term associated with the fact that the condensate is not in diffusive equilibrium with the static thermal cloud. This intercomponent collisional damping of the condensate modes is comparable in magnitude to the Landau damping considered in the recent literature.

DOI
10.1103/physreva.63.023612

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We consider the damping of condensate collective modes in the collisionless regime at finite temperatures arising from lack of equilibrium between the condensate and the noncondensate atoms, an effect that is ignored in the usual discussion of the collisionless region. As a first approximation, we ignore the dynamics of the thermal cloud. Our calculations should be applicable to collective modes of the condensate that are oscillating out-of-phase with the thermal cloud. We obtain a generalized Stringari equation of motion for the condensate at finite temperatures, which includes a damping term associated with the fact that the condensate is not in diffusive equilibrium with the static thermal cloud. This intercomponent collisional damping of the condensate modes is comparable in magnitude to the Landau damping considered in the recent literature.
We calculate the contribution to the damping of condensate collective excitations at finite temperatures arising from the lack of equilibrium between the condensate and thermal atoms. Limiting our attention to collective modes that mainly involve the motion of the condensate, we treat the noncondensate as static as a first approximation. We derive a set of generalized Bogoliubov equations for finite temperatures that contain an explicit damping term due to the collisional exchange of atoms between the two components. We have numerically solved these Bogoliubov equations to obtain the temperature dependence of the damping of the condensate modes in a harmonic trap. These results agree quite well with our recent work based on the Thomas-Fermi approximation but predict a much larger damping near the transition temperature.
The collective oscillations of a condensate at zero temperature are well described by the solutions of the linearized time-dependent Gross–Pitaevskii (GP) equation of motion for the condensate wavefunction Φ( r , t ). At finite temperatures, however, the condensate dynamics is modified by interactions with the noncondensate atoms that comprise the thermal cloud. To account for these interactions in detail involves a sophisticated numerical analysis, which will be described in Chapter 11. However, some qualitative understanding of the effect of collisions between the condensate and noncondensate components can be gained by treating the thermal cloud within an approximation that ignores its dynamics. This approximation, referred to as the static thermal cloud approximation , is the topic of the present chapter. As explained in more detail below, it is defined by the assumption that the condensate moves in the presence of a thermal cloud that remains in a state of thermal equilibrium. Thus, if the condensate is induced to oscillate, it initially departs from equilibrium with the thermal cloud, but collisions lead to a damping of the condensate oscillation and ultimately equilibrate the two components. This collisional damping is in addition to the usual Landau and Beliaev damping, which is present even in the “collisionless” regime. The approximate version of the fully coupled ZNG equations to be discussed here provides the simplest finite-temperature extension of the theory of condensate dynamics based on the usual GP equation. The extent to which the treatment gives a reasonable first approximation will be examined in Chapter 11. It will be shown that the static thermal cloud approximation does provide a qualitative understanding of the damping of modes in which the condensate is the main participant.
We use a generalized Gross-Pitaevskii equation for the condensate and a semiclassical kinetic equation for the noncondensate atoms to discuss the scissors mode in a trapped Bose-condensed gas at finite temperatures. Both equations include the effect of collisions between the condensate and noncondensate atoms. We solve the coupled moment equations describing oscillations of the quadrupole moments of the condensate and noncondensate components to find the collective-mode frequencies and collisional damping rates as a function of temperature. Our calculations extend those of Gu\'ery-Odelin and Stringari at T=0T=0T=0 and in the normal phase. They complement the numerical results of Jackson and Zaremba, although Landau damping is left out of our approach. Our results are also used to calculate the quadrupole response function, which is related to the moment of inertia. It is shown explicitly that the moment of inertia of a trapped Bose gas at finite temperatures involves a sum of an irrotational component from the condensate and a rotational component from the thermal-cloud atoms.
In Chapter 17, we derived the Landau–Khalatnikov two-fluid hydrodynamics which describes the collision-dominated region of a trapped Bose condensate interacting with a thermal cloud. In this chapter, we use these equations to discuss the damping of the hydrodynamic collective modes in a trapped Bose gas at finite temperatures. We derive variational expressions based on these equations for both the frequency and the damping of collective modes. This extends the analysis in Chapter 16 in which a variational approach was developed to calculate the hydrodynamic two-fluid oscillation frequencies in the non-dissipative limit. A novel feature of our treatment is the introduction of frequency-dependent transport coefficients, which produce a natural cutoff eliminating the collisionless region in the low-density tail of the thermal cloud. Our expression for the damping in trapped superfluid Bose gases is a natural generalization of the approach used by Landau and Lifshitz (1959) for uniform classical fluids. This chapter is mainly based on Nikuni and Griffin (2004). In Chapters 15 and 17, we derived a closed set of two-fluid hydrodynamic equations for a trapped Bose-condensed gas starting from the simplified microscopic model describing the coupled dynamics of the condensate and noncondensate atoms given in Chapter 3. These hydrodynamic two-fluid equations include dissipative terms associated with the shear viscosity, the thermal conductivity and the four second viscosity coefficients. Explicit formulas for these transport coefficients were derived in Section 18.1. Our goal in this chapter is to find a general expression for the damping of the two-fluid modes in terms of these transport coefficients. We emphasize that the damping of hydrodynamic two-fluid oscillations is completely different in nature from the Landau and Beliaev damping of oscillations in the collisionless region which is treated in Chapters 12 and 13.
We utilize a two-gas model to simulate collective oscillations of a Bose-Einstein condensate at finite temperatures. The condensate is described using a generalized Gross-Pitaevskii equation, which is coupled to a thermal cloud modeled by a Monte Carlo algorithm. This allows us to include the collective dynamics of both the condensed and noncondensed components self-consistently. We simulate quadrupolar excitations, and measure the damping rate and frequency as a function of temperature. We also observe revivals in condensate oscillations at high temperatures, and in the thermal cloud at low temperature. Extensions of the model to include nonequilibrium effects and describe more complex phenomena are discussed.
Recently it was shown that the Landau-Khalatnikov two-fluid hydrodynamics describes the collision-dominated region of a trapped Bose condensate interacting with a thermal cloud. We use these equations to discuss the low frequency hydrodynamic collective modes in a trapped Bose gas at finite temperatures. We derive variational expressions based on these equations for both the frequency and damping of collective modes. A new feature is our use of frequency-dependent transport coefficients, which produce a natural cutoff by eliminating the collisionless low-density tail of the thermal cloud. Above the superfluid transition, our expression for the damping in trapped inhomogeneous gases is analogous to the result first obtained by Landau and Lifshitz for uniform classical fluids. We also use the moment method to discuss the crossover from the collisionless to the hydrodynamic region. Recent data for the monopole-quadrupole mode in the hydrodynamic region of a trapped gas of metastable 4He{}^{4}\mathrm{He}4He is discussed. We also present calculations for the damping of the analogous m=0m=0m=0 monopole-quadrupole condensate mode in the superfluid phase.