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Cambridge University Press eBooks

Static thermal cloud approximation

Allan Griffin, Tetsuro Nikuni, E. Zaremba

Published Feb 19, 2009• 1 citation
Publisher

Abstract

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.

DOI
10.1017/cbo9780511575150.009

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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 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.
In contrast with Chapter 2, in this chapter we include the dynamics of the thermal cloud. As noted in Chapter 1, we treat the noncondensate atoms using the simplest microscopic model approximation that captures the important physics. In particular, we consider only temperatures high enough ( T ≥ 0.4 T BEC ) that the noncondensate atoms can be described by a particle-like Hartree–Fock (HF) spectrum. To extend the analysis to very low temperatures is in principle straightforward (see Chapter 7). However, the details are more complicated since the excitations of the thermal cloud take on a collective aspect (i.e. a Bogoliubov-type quasiparticle spectrum must be used). In trapped Bose gases, the HF single-particle spectrum gives a good approximation down to much lower temperatures than in the case of uniform Bose gases, as first emphasized by Giorgini et al . (1997). In Section 3.1, we derive a generalized form of the Gross–Pitaevskii equation for the Bose order parameter Φ( r , t ) that is valid at finite temperatures. It involves terms that are coupled to the noncondensate component (the thermal cloud) and thus its solution in general requires one to know the equations of motion for the dynamics of the noncondensate atoms. In Section 3.2 we restrict ourselves to finite temperatures high enough that the noncondensate atoms can be described by a quantum kinetic equation for the single-particle distribution function f ( p, r , t ). A detailed microscopic derivation of this kind of kinetic equation is given in Chapters 6 and 7 using the Kadanoff–Baym Green's function formalism. A characteristic feature of a Bose-condensed gas is that the kinetic equation governing f ( p, r , t ) involves a collision integral C 12 [ f , Φ] describing collisions between condensate and noncondensate atoms.
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.
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.
In Chapters 11–13, we gave a detailed discussion of the dynamics of a trapped Bose gas at finite temperatures in a region where the collisions described by the C 12 and C 22 terms in the kinetic equation (3.42) do not play the central role. In this “collisionless” region, the dominant interaction effects are associated with the self-consistent fields which both the condensate and noncondensate atoms feel. Thus the dynamics can be understood to a first approximation by neglecting the C 12 and C 22 collision integrals in the kinetic equation and, at the next stage, treating them as a weak perturbation on the collisionless dynamics. In the rest of this book (Chapters 15–19), we turn to the study of the coupled ZNG equations in the opposite limit, where the C 12 and C 22 collision integrals completely determine the dynamics of the thermal cloud. Specifically, the collisions lead to the thermal cloud being in local hydrodynamic equilibrium, and hence this regime is described by the equations of collisional hydrodynamics. Its characteristic feature is that the nonequilibrium behaviour of the thermal cloud atoms can be completely described in terms of a few differential equations involving coarse-grained variables that are dependent on position and time, analogous to the condensate variables n c ( r , t ) and v c ( r , t ). In the present chapter, assuming that the thermal cloud distribution function f ( p, r , t ) is given by the Bose distribution describing partial local equilibrium, (15.16), we show how the ZNG coupled equations lead precisely to Landau's two-fluid equations, reviewed in Chapter 14. This equivalence is not obvious, mainly because Landau's equations are expressed in terms of thermodynamic variables, which are not used in a more microscopic analysis such as that used in the ZNG approach.
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.