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.
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.
In Chapter 6, we derived a generalized Gross–Pitaevskii condensate equation which is coupled to a kinetic equation for the distribution function for the thermal atoms. However, the kinetic equation in Chapter 6 is only valid in the semiclassical limit. It involves the assumption that the thermal energy k B T is much greater than the spacing between the harmonic trap energy levels ( k B T ≫ω 0 where ω 0 is the trap frequency) and also much greater than the average interaction energy ( k B T ≫ gn ). The ZNG model, based on HF excitations, is still expected to be adequate down to quite low temperatures in trapped Bose gases, as will be shown by the results in Chapter 12. However, the ZNG model will break down at very low temperatures, where the Hartree–Fock excitations must be replaced by the Bogoliubov spectrum. To deal with this, one has to derive a kinetic equation for the Bogoliubov quasiparticle excitations. This is the goal of the present chapter. In this chapter, we use the second-order Beliaev approximation to discuss the nonequilibrium dynamics of a trapped Bose-condensed gas at finite temperatures. In doing to, we combine the second-order Beliaev self-energies with the lower-order Bogoliubov excitation spectrum, including off-diagonal single-particle propagators but still omitting the anomalous correlation functions. This last condition defines the Bogoliubov–Popov approximation. In this chapter, we consider only the damping which arises from collisions. We will not explicitly calculate corrections that are second-order in g to the quasiparticle energy spectrum or to the condensate chemical potential, both of which are associated with the real parts of the second-order Beliaev self-energies. The present chapter is a natural generalization of work presented in Chapter 6 for the simpler HF excitation spectrum assumed in the ZNG analysis.
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.
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.
In Chapter 3, we introduced a simple but reasonable approximation for the nonequilibrium dynamics of a Bose-condensed gas based on a generalized GP equation coupled to a kinetic equation. In Chapters 4–7, we turn to the question of how to derive such coupled equations for the condensate and noncondensate components in a way that gives a deeper understanding of the ZNG theory. Chapters 4–7 involve an introduction to Green's functions and field theoretic techniques for nonequilibrium problems. These provide the natural language and formalism to deal with the many subtle aspects of a Bose-condensed gas at finite temperatures. These four chapters are fairly technical. This chapter is mainly based on Kadanoff and Baym (1962) and Imamović-Tomasović (2001). Readers who are not interested in these questions can go straight to Chapter 8, which begins the discussion of applications of the ZNG coupled equations given in Chapter 3. Overview of Green's function approach To derive a microscopic theory of the nonequilibrium behaviour of a dilute weakly interacting Bose-condensed gas at finite temperatures, there are several different approaches available in the literature. We will use the wellknown Kadanoff–Baym (KB) nonequilibrium Green's function method. The generalization of this formalism to a Bose-condensed system was first considered by Kane and Kadanoff (1965), whose goal was to derive the Landau two-fluid hydrodynamic equations for a system with a Bose broken symmetry. The general problem consists of how to calculate the nonequilibrium response of a system induced by an external (space- and time-dependent) perturbation. In response to such an external perturbation, many interesting physical phenomena appear, including the excitation of collective modes and various transport processes.
Using the Kadanoff-Baym non-equilibrium Green's function formalism, we derive kinetic equations for the non-condensate atoms at finite temperatures which include the effect of binary collisions between atoms. The effect of collisions is included using the second-order self-energy given by the Beliaev (gapless) approximation. We limit our discussion to finite temperatures where we can use the single-particle Hartree-Fock spectrum for the excited atoms. In this limit, we can neglect the off-diagonal propagators (g~12 and g~21). As expected, this leads to the kinetic equations and collision integrals used in recent work by Zaremba, Nikuni, and Griffin (ZNG) [1]. We also derive a consistent equation of motion for the condensate wavefunction, involving a finite-temperature generalization of the well-known Gross-Pitaevskii equation which includes a dissipative term, as well as the mean field of the non-condensate.
Since the dramatic discovery of Bose–Einstein condensation (BEC) in trapped atomic gases in 1995 (Anderson et al. , 1995), there has been an explosion of theoretical and experimental research on the properties of Bose-condensed dilute gases. The first phase of this research was discussed in the influential review article by Dalfovo et al. (1999) and in the proceedings of the 1998 Varenna Summer School on BEC (Inguscio et al. , 1999). More recently, this research has been well documented in two monographs, by Pethick and Smith (2008, second edition) and by Pitaevskii and Stringari (2003). Most of this research, both experimental and theoretical, has concentrated on the case of low temperatures (well below the BEC transition temperature, T BEC ), where one is effectively dealing with a pure Bose condensate. The total fraction of noncondensate atoms in such experiments can be as small as 10% of the total number of atoms and, equally importantly, this low-density cloud of thermally excited atoms is spread over a much larger spatial region compared with the high-density condensate, which is localized at the centre of the trapping potential. Thus most studies of Bose-condensed gases at low temperatures have concentrated entirely on the condensate degree of freedom and its response to various perturbations. This region is well described by the famous Gross–Pitaevskii (GP) equation of motion for the condensate order parameter Φ( r , t ). As shown by research since 1995, this pure condensate domain is very rich in physics. The main goal of the present book, in contrast, is to describe the dynamics of dilute trapped atomic gases at finite temperatures such that the noncondensate atoms also play an important role.