Since the creation of Bose–Einstein condensation (BEC) in trapped atomic gases in 1995, there has been an enormous amount of research on ultracold quantum gases. However, most theoretical studies have ignored the dynamical effect of the thermally excited atoms. In this book, we try to give a clear development of the key ideas and theoretical techniques needed to deal with the dynamics and nonequilibrium behaviour of trapped Bose gases at finite temperatures. By limiting ourselves from the beginning to a relatively simple microscopic model, we can concentrate on the new physics which arises when dealing with the correlated motions of both the condensate and noncondensate degrees of freedom. This book also sets the stage for the future generalizations that will be needed to understand the coupled dynamics of the superfluid and normal fluid components in strongly interacting Bose gases, where there is significant depletion of the condensate even at T = 0.

The core of this book is based on a long paper published by the authors (Zaremba, Nikuni and Griffin, 1999). In the last decade, together with our coworkers, we have extended and applied this work in many additional papers. The starting point of our approach is not original, in that it consists of combining the Gross–Pitaevskii equation for the condensate with a Boltzmann equation for the noncondensate atoms. The kinetic equation for trapped superfluid Bose gases we use was first developed and studied in a pioneering series of papers by Kirkpatrick and Dorfman in 1985 on a uniform Bose gas at finite temperatures.

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.

In the previous chapter we considered perfect crystals which are periodic and thus infinite in extent. There is one deviation from perfect periodicity that is always present: all real crystals have surfaces. We argued above that the surface has a small effect on bulk properties. This may be true, but the growth of crystals is the process of adding surface layers. We will discuss a number of aspects of surfaces and growth here.

Observing surfaces: scanning tunneling microscopy

In the previous chapter we showed how that structure of a surface could be revealed by using LEED; this technique is useful for periodic surfaces. As we will see, crystal surfaces have interesting deviations from periodicity such as steps. An enormous advance in studying this aspect of surfaces was the development of direct atomic resolution microscopy.

A remarkable instrument of this type is the scanning tunneling microscope (STM) invented by G. Binnig and H. Rohrer (Binnig, Rohrer, Gerber & Weibel 1982). The idea of this device is that quantum mechanical tunneling is very sensitive to the distance through which the particles tunnel.

Suppose electrons tunnel through a classically forbidden region of length x. The current will be of order j ∼ e−x/a where a is the attenuation length of the electron wave function in the forbidden region. The STM consists of a sharp metal tip which is brought into near contact with a surface by manipulating piezo-electric drivers.

When atoms are assembled into a condensed state, it is often the case that the outer valence electrons become delocalized, and are no longer associated with a given atom. The most obvious such case is a metal where the electrons are free to move, and can conduct electricity. An ionized classical plasma is another system of this sort, but, as we have seen, the electrons in solids must be treated with quantum theory. That is the subject of this chapter. Metallic liquids and glasses exist, but we will concentrate on metallic crystals. As we will see, the theory that we will develop will also apply to the valence electrons in semiconductors and insulators.

Since electrons in metals are free to move it is natural to think of them as a gas; the term electron gas is often used. The most extreme version of this idea is surprisingly useful, namely the idea of the free electron gas. In this idealization the electrons don't see the ions that they were detached from except in an average way, to neutralize their charge. Also, in this model the electrons are non-interacting, and act as if their Coulomb repulsion is not present.

We should say from the outset that both these assumptions appear to be totally unreasonable. The strength of the electron-electron interaction in Cu was estimated in Chapter 1. It turned out to be about 3 eV, which is the same order of magnitude as the energies that we will find in the next section.

The transport theory of the previous chapter dealt mostly with the response to static fields. Here we allow the fields to vary with time. In order to be consistent, we describe the electrons with Maxwell's equations. Since electrons are charged we are taking some of the electron-electron interaction into account. As we will see later, only the long-wavelength properties can be got at this way. First we will deal with the so-called longitudinal response, namely the response to the introduction of an external charge density which may be time and space-dependent. Later we will turn to the transverse response to external fields such as those in a light wave.

Dielectric functions

We start with the simple observation that a fixed positive charge in a system of mobile electrons polarizes the environment. It attracts the electrons so that there is an induced negative charge. The net field produced by the positive charge plus the induced charge is smaller than the positive charge would produce alone: this is called screening. This is familiar in electrostatics; thus we try to think about these effects as a generalization of electrostatics.

We start by being quite general. Suppose we have a medium and we introduce an external charge density, ρe. We will suppose that the external charge is small, and that we can assume the medium is linear. What we are doing here is an example of a general method called linear response theory.

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 C12 and C22 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 C12 and C22 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 C12 and C22 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 nc(r, t) and vc(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 17, we derived two-fluid hydrodynamic equations that include damping related to transport coefficients. Our entire analysis was based on the coupled ZNG equations for the condensate and in the thermal cloud. These involved a generalized GP equation for the condensate and a kinetic equation for the thermal atoms. A crucial role is played by the C12 collision term in the kinetic equation, which describes the interactions between atoms in the condensate and in the thermal cloud.

Our analysis of the deviation from the diffusive local equilibrium solution of the kinetic equation was based on the Chapman–Enskog approach, extensively developed for classical gases and first applied to Bose-condensed gases by Kirkpatrick and Dorfman (1983, 1985a). This approach required a careful treatment of the novel feature relating to the C12 collisions both in the kinetic equation describing the thermal atoms and also in the source term Γ12 in the generalized GP equation for the condensate. Using the Chapman– Enskog approach to solve the kinetic equation for a trapped Bose gas, we obtained explicit expressions for the function ψ(p, r, t) that describes the deviation from diffusive local equilibrium, as defined by (17.25) and (17.39). This deviation can be related to various transport coefficients, as discussed in Chapter 17.

These transport coefficients are determined by the solutions of the three integral equations (17.40)–(17.42) for the three contributions to the deviation function ψ(p, r, t) in (17.39). In Section 18.1, we will solve these integral equations and obtain explicit expressions for the thermal conductivity k, the shear viscosity η and the four second viscosity coefficients ζi.

The goal of creating and observing quantized vortices in trapped Bose gases arose almost immediately following the first achievements of Bose–Einstein condensation. The motivation for doing so was the obvious analogy with vortices in liquid helium and in type-II superconductors, and the fact that the quantization of circulation is directly associated with superfluid flow. It was recognized that the observation of quantized vortices could be taken as indisputable evidence for the existence of superfluidity in these systems.

The review by Fetter and Svidzinsky (2001) contains a summary of the early experiments and the theoretical background for understanding the vortex state in a weakly interacting Bose gas based on the GP equation. This material will not be repeated here apart from those aspects that have a direct bearing on the focus of the present chapter, namely the properties of vortices at finite temperature. Although there have been some theoretical contributions to this subject, much remains to be done. The discussion in this chapter provides a framework for addressing the finite-temperature properties of vortex formation and vortex lattices in the context of the ZNG theory. The results in this chapter have not been published before, apart from those in subsection 9.8.1, which are based on Williams et al. (2002).

There are several issues that relate to finite temperatures. First, there is the nucleation and formation of vortices from an initial highly nonequilibrium state. Second, there is the interaction of vortices with thermal excitations, which is responsible for the dissipative dynamics of a nonequilibrium vortex state. Third, there is the question of the final equilibrium state, with respect to the condensate and noncondensate densities in the vicinity of a vortex and to the geometrical arrangement of vortices in space.

In 1911 H. Kamerlingh Onnes was investigating the electrical properties of metals such as Pt and Au in the range of a few K (Kammerlingh Onnes 1911). He observed that the resistance is determined by the number of impurities in this regime. (This is now called Mattheissen's rule, Eq. (7.24).) In order to eliminate impurities he turned to mercury which was available in a very pure form. Instead of a lower resistance, which he expected, he found a sudden jump to a vanishingly small resistance at about 4.2 K. This temperature is now called the transition temperature, Tc. His statement was: “Mercury has passed into a new state which on account of its extraordinary electrical properties may be called the superconductive state.” He also found transitions to zero resistance for lead and tin with different Tc's, also in the range of a few K. (We should note that later work showed that superconductivity is, in fact, not affected very much by impurities.)

The superconductive state is now known to occur for around forty elements and hundreds of compounds. The transition temperature is less than about 100 K for all known cases. The superconductivity of copper oxides discovered by Bednorz & Muller (1986) is of particular current interest, and gives the highest known Tc's.

Supercurrents flow without friction and thus are persistent: they have been observed to flow around a ring without decay for the better part of a year.

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 C22 and C12 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 C12 and C22 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.

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 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 kBT is much greater than the spacing between the harmonic trap energy levels (kBT ≫ω0 where ω0 is the trap frequency) and also much greater than the average interaction energy (kBT ≫ 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.

In this chapter we describe the numerical methods that can be used to solve the ZNG equations in the context of a dynamical simulation. These equations consist of a generalized GP equation (3.21) for the condensate and a Boltzmann equation (3.42) for the thermal component. The fact that the two equations are coupled makes their numerical solution more complex than when either is considered on its own. Indeed, the distinct quantum and classical aspects of the problem require specifically tailored numerical methods. Although most of these methods are well established and described elsewhere (Taha and Ablowitz, 1984; Sanz-Serna and Calvo, 1994), we provide in this chapter a detailed pedagogical discussion that will serve as a guide to those interested in carrying out such calculations for trapped Bose gases. This chapter is based on the papers of Jackson and Zaremba (2002a,b).

There are two main parts to the numerical problem. The first is developing a method for solving the time-dependent GP equation for an arbitrary three-dimensional geometry. This we take up in Section 11.1. Second, a method is needed for solving the Boltzmann equation that accounts for the dynamics of the thermal component. Here one must deal both with the Hamiltonian dynamics of the thermal atoms, as they move in the self-consistent mean field of the condensate and thermal cloud, and with the collisions that take place between the thermal atoms themselves (the C22 collisions) and between the thermal atoms and the condensate (the C12 collisions). The methods used to account for these two distinct collisional processes are taken up in Section 11.3. As we shall see, collisions play an important role and cannot be neglected even when the dynamical behaviour is dominated by mean-field interactions.

In the collisional region at finite temperatures, the collective modes of superfluids are described by the Landau two-fluid hydrodynamic equations reviewed in Chapter 14. In the case of trapped Bose gases, these are coupled differential equations with position-dependent coefficients associated with the local thermodynamic functions. Building on the approach initiated by Zaremba et al. (1999) for trapped atomic Bose gases, in this chapter we develop an alternative variational formulation of two-fluid hydrodynamics. This is based on the work of Zilsel (1950), originally developed to deal with superfluid He. Assuming a simple variational ansatz for the superfluid and normal fluid velocity fields, this approach reduces the problem of finding the hydrodynamic collective mode frequencies to solving coupled algebraic equations for a few variational parameters. These equations contain constants that involve spatial integrals over various equilibrium thermodynamic derivatives. Such a variational approach is both simpler and more physical than a direct attempt to solve the Landau two-fluid equations numerically.

This chapter is mainly based on Taylor and Griffin (2005), Taylor (2008) and Zilsel (1950). In it, we discuss the normal modes of the non-dissipative Landau two-fluid equations for a trapped superfluid. In Section 16.3, we illustrate this formalism by deriving expressions for the frequencies of the dipole and breathing modes of a trapped Bose superfluid. In Chapters 17 and 18, we discuss an extended version of the two-fluid equations that includes hydrodynamic damping. The hydrodynamic damping of the collective modes is calculated in Chapter 19 using a generalized version of the variational approach developed in this chapter.

Zilsel's variational formulation

Since two-fluid hydrodynamics only describes a system in local equilibrium, all thermodynamic quantities are functions of position and time. Even in static equilibrium, in the presence of a trapping potential, most thermodynamic quantities will be position dependent.

Electrons in metals interact via the Coulomb interaction. The size of this effect is not negligible: it is of order e2/a where a is the mean distance between electrons. For a metal this ≈ 3 eV, that is, comparable to EF. An approach to the problem would be to treat the interaction as a perturbation. However, the perturbation is strong and long-ranged, and this gives rise to many difficulties, such as divergences in finite orders of perturbation theory. The problems can be overcome by an application of the methods of quantum field theory to the problem, and is treated in books on many-body theory. We will not pursue this approach here, but try to get some of the central conclusions with physical reasoning.

At first glance, we might be tempted to conclude that the sharp Fermi surface that we have discussed, and which is readily observed, should be wiped out by the interaction. The much smaller energy kBT does lead to smearing of the Fermi distribution; it is natural to conclude that this is true for interactions as well. This natural conclusion is not correct in three dimensions. The situation in one dimension is complicated, and the Fermi surface does disappear, leading to a state called the Luttinger liquid. In three dimensions the sharp Fermi surface survives the presence of interactions. This is an important phenomenon which demands explanation.

In this chapter, we review the famous Landau theory of superfluidity at finite temperatures. This theory is based on coupled hydrodynamic equations for the superfluid and normal fluid components. Landau's two-fluid description is only valid when collisions among the thermal excitations making up the normal fluid are strong enough to produce local hydrodynamic equilibrium. These two-fluid equations were originally developed for liquid. He but are thought to be generic in form, describing the collision-dominated hydrodynamic region of all Bose superfluids. In this chapter, we will consider the solutions of the two-fluid equations mainly for a uniform superfluid. We discuss the existence of second sound (involving the out-of-phase motion of the superfluid and normal fluid components) as a characteristic feature of a Bose superfluid at finite temperatures.

This chapter gives background material needed for Chapters 15–19. In Chapter 15, we will show that, in the appropriate limit, the Landau two-fluid equations can be derived from the ZNG coupled equations given in Chapter 3 for a trapped dilute Bose-condensed gas. In Chapters 17–19, we extend this discussion and derive the Landau–Khalatnikov two-fluid equations, which include hydrodynamic damping associated with various transport coefficients. Useful reviews of the two-fluid equations in the context of dilute spatially uniform Bose-condensed gases are given by Pethick and Smith (2008, Chapter 10) and Pitaevskii and Stringari (2003, Chapter 6).

History of two-fluid equations

The original discovery of superfluidity in liquid He was dramatically announced with the publication of the famous back-to-back papers of Kapitza (1938) in Moscow and Allen and Misener (1938) in Cambridge. These and subsequent experiments in the next few years showed that superfluid He could exhibit very bizarre hydrodynamic behaviour compared to classical liquids.