Trapped Bose-condensed atomic gases are remarkable because, in spite of the fact that these are very dilute systems, they exhibit robust coherent dynamic behaviour when perturbed. These quantum “wisps of matter” are a new phase of highly coherent matter. While binary collisions are very infrequent, the large coherent mean field associated with the Bose condensate ensures that interactions play a crucial role in determining the collective response of these trapped superfluid gases.

In our discussion of the theory of collective oscillations of atomic condensates, the macroscopic Bose wavefunction Φ(r, t) plays a central role. This wavefunction is the BEC order parameter. As discussed in Chapter 1, the initial attempts at defining this order parameter began with the pioneering work of London (1938a), were further developed by Bogoliubov (1947) and finally extended to deal with any Bose superfluid using the general quantum field theoretic formalism developed by Beliaev (1958a). Almost all this early theoretical work was limited to T = 0 where, in a dilute weakly interacting Bose gas, all the atoms are in the condensate. The first extension of these ideas to nonuniform Bose condensates was by Pitaevskii (1961) and, independently, by Gross (1961), which led to the now famous Gross–Pitaevskii (GP) equation of motion for Φ(r, t). Before the discovery of BEC in trapped gases, the time-dependent GP equation was mainly used to study vortices in Bose superfluids, which involve a spatially nonuniform ground state. Apart from this application, the GP equation was largely unknown. The situation changed overnight in 1995 with the creation of trapped nonuniform Bose condensates in atomic gases.

We have seen in the previous chapter that crystals are common in nature. In this chapter we will investigate in more detail how to think about such three-dimensional periodic structures. Then we will turn to the interaction of waves with such structures. This will lead us to a discussion of correlation functions in condensed matter.

Crystals

In the previous chapter we defined a crystal as a structure which repeats periodically in space. There is a mathematical framework for dealing with physical quantities in perfect crystals; it is the science of crystallography. We will review some of the elementary concepts from this subject.

Of course, any real material is an imperfect realization of a perfect crystal; real materials always have impurities and defects. Even if a crystalline solid is very close to being strictly periodic in bulk, all materials have a surface where the periodicity fails. However, consider a large chunk of matter, say a cube of edge L where the distance between the atoms is a. The number of atoms in the bulk is of the order of (L/a), but the number on the surface is of order (L/a). If L > > a the fraction on the surface is negligible.

Lattices

The first step to defining a crystal is to define a lattice. This is a set of points in d dimensions which are generated by taking linear combinations of d linearly independent vectors called generators: ak, k = 1, … d with integer coefficients.

In Chapter 4, we introduced the Kadanoff–Baym equations of motion for the imaginary-time nonequilibrium Green's functions for a Bose gas, as given by (4.59) and (4.60). In this chapter, we will use the generalization of these equations of motion to find the equivalent equations of motion for the real-time Green's functions. These can be written in a natural way in the form of a kinetic equation. Using a simple Hartree–Fock approximation, we show how the coupled equations for the condensate and thermal cloud given in Chapter 3 emerge naturally from the Kadanoff–Baym (KB) formalism. This chapter is based on Imamović-Tomasović and Griffin (2001) and Imamović-Tomasović (2001), building on the pioneering work of Kane and Kadanoff (1965).

In this chapter and Chapter 7 we review the KB formalism. However, we also encourage the reader to read the original account given by Kadanoff and Baym (1962). The goals and accomplishments of their seminal book are beautifully captured by the following quote from p. 138:

A closely related way of treating the nonequilibrium dynamics of a Bose-condensed gas is based on the two-particle irreducible (2PI) effective action together with the Schwinger–Keldysh closed-time path formalism. Berges (2004) gives a detailed review of this approach, which allows one to derive the nonequilibrium action on the basis of controllable approximations.

This book is intended as a textbook for a graduate course in condensed matter physics. It is based on many years' experience in teaching in the Physics department at The University of Michigan. The material here is more than enough for a one-semester course. Usually I teach two semesters, and in the second, I add material such as the renormalization group.

In this book advanced techniques such as Green's functions are not used. I have tried to introduce as many of the concepts of modern condensed matter physics as I could without them. As a result, some topics that are of central importance in modern research do not appear.

The problems are an integral part of the book. Some concepts that are used in later chapters are introduced as problems.

Students are expected to have a good background in statistical physics, non-relativistic quantum theory, and, ideally, know undergraduate Solid State physics at the level of Kittel (2005).

I decided to write this book as a result of coming back to teaching Condensed Matter after a number of years covering other subjects. I had hoped to find a substitute for the grand old standards like Ziman (1972) or Ashcroft & Mermin (1976) which I used at the beginning of my teaching career. Though there are newer texts that are interesting in many ways, I found that none of them quite fit my needs as an instructor.

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 Chapter 15, we showed that in the limit of short collision times the coupled equations of motion for the condensate and noncondensate atoms lead to Landau's non-dissipative two-fluid hydrodynamics. However the approach used in Chapter 15 was not based on a small expansion parameter, in contrast with the more systematic Chapman–Enskog procedure used to derive hydrodynamic damping in the kinetic theory of classical gases. In the present chapter, we generalize the procedure of Chapter 15 to trapped Bose-condensed gases, in order to derive two-fluid hydrodynamic equations that include dissipation due to transport processes. We solve the kinetic equation by expanding the nonequilibrium single-particle distribution function f(p, r, t) around the distribution function f(0)(p, r, t) that describes complete local equilibrium between the condensate and the noncondensate components. All hydrodynamic damping effects are included by taking into account deviations from the local equilibrium distribution function f(0). Our discussion for a trapped Bose gas is a natural extension of the pioneering work of Kirkpatrick and Dorfman (1983, 1985a) for a uniform Bose-condensed gas. This chapter is mainly based on their work as well as on Nikuni and Griffin (2001a,b).

We will prove that, with appropriate definitions of various thermodynamic variables, our two-fluid hydrodynamic equations including damping have precisely the structure of those first derived by Landau and Khalatnikov for superfluid He. In particular, the damping associated with the collisional exchange of atoms between the condensate and noncondensate components, which is discussed at length in Chapter 15, is now expressed in terms of frequency-dependent second viscosity coefficients. This special type of damping is a characteristic signature of a dilute Bose superfluid and exists in addition to the hydrodynamic damping associated with the shear viscosity and thermal conductivity of the normal fluid.

So far we have considered only the ground state of some condensed matter systems. If we excite such objects many interesting things can happen. One of the simplest and most interesting is that we can excite wave-like motions of the atoms and molecules. If we hit a solid with a hammer, we know that sound waves will move through the body. These acoustic waves are part of a class of motions called lattice vibrations which also can be excited by thermal effects. If temperatures are low or the wave amplitude is small, quantum mechanics must be used. The quantized version of a lattice vibration is made up of phonons.

Less familiar is the fact that if we wiggle a spin in a magnet it will cause its neighbors to move, and the motion will be transmitted in a spin wave. Once more there is a quantum version, the magnon.

Lattice vibrations and phonons

We first look at the vibrations of atoms away from their equilibrium positions in the crystal. We do this in three ways. First we briefly allude to the macroscopic theory of elasticity which does not use microscopic details at all. Then we look at classical atoms bound together by forces. For vibrations near the equilibrium positions the forces become an effective set of springs that bind the atoms. Then we quantize the motions.

We have seen in the previous chapter that chemical bonds are the glue for condensed matter. If the temperature is low enough so that thermal fluctuations do not break the bonds, it is no surprise that atoms and molecules condense, i.e. stick together, so that there are large pieces of matter.

However, the precise structure of condensed matter is often quite surprising. For example, we might guess that the typical result of attractive chemical bonds would be a disorderly mass of molecules. This does occur; such materials are called glasses. However, very commonly something else happens: at low enough temperatures the atoms or molecules form a remarkable ordered structure, a crystal. A crystal is an ordered, periodic array of atoms or molecules. In the next chapter we will give a precise definition of this concept. For our purposes, it is enough to understand that crystals are made up of identical building blocks that are repeated many times. See Figure 2.1 for an example, the face-centered cubic (fcc) crystal structure.

Chemistry tells us that atoms or ions can have a magnetic moment, either from orbital currents or unpaired spins. However, you might expect that when large numbers of such ions are stuck together that the orientation of the moments would be random. This is not always the case. For some elements, e.g. Fe, Ni, Co, and many compounds the moments line up in regular arrays of various kinds due to the exchange interaction, discussed above.

Condensed matter physics is the study of large numbers of atoms and molecules that are “stuck together.” Solids and liquids are examples. In the condensed state many molecules interact with each other. The physics of such a system is quite different from that of the individual molecules because of collective effects: qualitatively new things happen because there are many interacting particles. The behavior of most of the objects in our everyday experience is dominated by collective effects. Examples of materials where such effects are important are crystals and magnets.

This is a vast field: the subject matter could be taken to include traditional solid state physics (basically the study of the quantum mechanics of crystalline matter), magnetism, fluid dynamics, elasticity theory, the physics of materials, aspects of polymer science, and some biophysics. In fact, condensed matter is less a field than a collection of fields with some overlapping tools and techniques. Any course in this area must make choices. This is my personal choice.

In this chapter I will discuss orders of magnitude that are important, review ideas from quantum mechanics and chemistry that we will need, outline what holds condensed matter together, and discuss how order arises in condensed systems. The discussion here will be qualitative. Later chapters will fill in the details.

In this chapter we present several dynamical simulations that make use of the numerical methods discussed in the previous chapter. Some of these are model simulations that are not directly linked to experiment but are designed to investigate some aspect of the dynamical behaviour. Others are performed with the express purpose of explaining specific experimental data. Both kinds of simulation serve to illustrate the range of nonequilibrium phenomena that can be studied in ultracold Bose gases using the ZNG equations.

A dynamical simulation is typically initiated in one of two ways. Either an appropriate nonequilibrium initial state is imposed on the system, or the system, initially in equilibrium, is dynamically excited by the application of an external perturbation. The latter parallels the procedure used experimentally to study small-amplitude collective excitations and usually amounts to some parametric modulation of the trapping potential. However, this approach may not always be feasible if the excitation phase requires a prohibitively long simulation time. In this case, the best one can do is to specify some initial nonequilibrium state that represents the experimental situation as closely as possible. This is not an issue in model simulations, where we are at liberty to specify the initial state in whatever way serves our purpose.

In Sections 12.1–12.3 we present three examples of model simulations. All essentially check some aspect of the numerical procedures. By studying the equilibration of an initial nonequilibrium state in Section 12.1, we confirm that the total number of atoms is conserved to a very good approximation during the course of the evolution. This is a nontrivial result since, as explained at the end of subsection 11.3.2, the numbers of condensate and thermal atoms change in quite different ways.

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 ≥ 0.4TBEC) 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 C12[f, Φ] describing collisions between condensate and noncondensate atoms.