Introduction

This chapter introduces the concept of diffusion and other associated forms of translational motion such as flow, together with their physical significance. Measurements of translational motion and their interpretation are necessarily tied to a mathematical framework. Consequently, a detailed coverage of the mathematics, including the partial differential equation known as the diffusion equation, is presented. Finally, the common techniques for measuring diffusion are discussed.

Types of translational motion – physical interpretation and significance

‘Diffusion’ is used in the scientific literature with imprecision and ambiguity as there are a number of types of diffusion. With respect to molecular motion, diffusion is used to denote self-diffusion, mutual diffusion and ‘distinct’ (not in the sense of individual to a species) diffusion coefficients. Confusion arises since, although related and having the same units (i.e., m2s−1), these phenomena are physically distinct. The confusion is exacerbated in the NMR literature with the term ‘spin-diffusion’ which is a distinct NMR cross relaxation – based phenomenon involving the random migration of magnetisation via mutual spin flips in neighbouring nuclei, even though it can be measured using techniques related to those outlined in this book. In this book ‘diffusion’ signifies self-diffusion, which will also be referred to as translational diffusion, although some consideration will be given to mutual diffusion since many of the alternative methods for measuring diffusion, especially those based on scattering, provide information on mutual diffusion which is often compared with the results of NMR measurements of translational diffusion.

Introduction

In the previous chapter we considered the various methods for relating echo attenuation with diffusion in the case of free isotropic diffusion for a single diffusing species. It was observed that the echo signal attenuation was single exponential with respect to q2 and the correct value of the diffusion coefficient was determined irrespective of the measuring time (i.e., Δ). Due to the relatively long timescale of the diffusion measurement (i.e., Δ), gradient-based measurements are sensitive to the enclosing geometry (or pore) in which the diffusion occurs (i.e., restricted diffusion) and an appropriate model must be used to account for the effects of restricted diffusion when analysing the data. The effects of the restriction can be used to provide structural information for pores with characterisitc distances (a) in the range of 0.01–100 μm. Thus, gradient methods are especially suited to studying the physics of restricted diffusion and transport in porous materials.

Non-single-exponential decays can arise in a number of ways including multicomponent systems, anisotropic or restricted diffusion. These effects are the subject of the next two chapters (more complex models are studied in Chapter 4). The relevant analytical formulae for diffusion between planes and inside spheres are presented (diffusion in cylinders is presented in the following chapter). It is remarked that these are the commonly used models for benchmarking numerical approaches. We also mention that Grebenkov has recently presented a review of NMR studies of restricted Brownian motion.

Introduction

Most simplistically, mutual diffusion can be probed by imaging diffusion profiles (e.g., the ingress of a solvent into a material). However, the integration of MRI techniques with the gradient-based measurements of translational motion that we have discussed in previous chapters allows for potentially more information to be obtained – especially from spatially inhomogeneous samples. It also provides additional techniques for measuring such motions. Diffusion is extremely important in MRI, and, amongst other effects, at very high resolutions it determines the ultimate resolution limit when the distance moved by a molecule is comparable to voxel dimensions. Further, since motion is more restricted near a boundary, the spins near the boundary are less dephased (attenuated) during the application of imaging gradients in high resolution imaging, consequently a stronger signal is obtained near the boundary and this has become known as diffusive edge enhancement. Relatedly, since the length scales that can be probed with NMR diffusion measurements encompass those that restrict diffusion in cellular systems, the combination of PGSE with imaging techniques can result in MRI contrasts. Whilst there can be diffusion anisotropy at the microscopic level (e.g., diffusion in a biological cell), the MRI sampling is coarse and thus if there is too much inhomogeneity of the ordering of the microscopic anisotropic systems, the information obtained from the voxel will appear isotropic.

Introduction

As soon as the spin-echo was discovered by Hahn in 1950 it was realised that it could form the basis of self-diffusion measurements. Indeed, certainly within the next decade the concept of spin-echo-based diffusion measurements using static magnetic gradients (i.e., Steady Gradient Spin-Echo or SGSE NMR) had become widespread and used in quite sophisticated measurements such as on water and 3He. Many of the experimental limitations of static gradient measurements were removed with the suggestion in 1963 by McCall, Douglass and Anderson and experimental introduction in 1965 by Stejskal and Tanner of applying the magnetic gradients as pulses in the spin-echo sequence (i.e., Pulsed Gradient Spin-Echo NMR or PGSE NMR). Carr and Purcell were the first to discuss NMR flow measurements and in 1960 NMR flow measurements were considered for the purpose of measuring sea-water motion.

Virtually all contemporary NMR diffusion (and flow experiments) are based on some form of spin-echo. Indeed, for all but the simplest cases the dependence of the observed echo amplitudes on diffusion rapidly becomes very complicated and this can be exacerbated in pulse sequences where the magnetisation is kept in a steady state. However, in the following discussions we will assume, unless otherwise noted, that all pulse sequences start with the spin system being in thermal equilibrium (i.e., M0). As the diffusing species necessarily contains a nuclear spin, the terms spin and particle will henceforth become synonymous.

Introduction

There are a number of potential problems that must be addressed in PGSE measurements if high quality data is to be obtained. The problems include: (i) rf interference, (ii) radiation damping and long-range dipolar interactions, (iii) convection, (iv) homogeneity of the applied magnetic field gradient, (v) background magnetic gradients, (vi) eddy currents and static magnetic field disturbances generated by the gradient pulses, and lastly (vii) gradient pulse mismatch and sample movement. Almost invariably these problems lead to increased signal attenuation and thus overestimates of the diffusion coefficient and misinterpretation of the experimental data, and it has been noted that all PGSE systems have thresholds below which artefactual attenuation exceeds diffusive attenuation. Here, we consider the origins of these problems, their symptoms and some methods to alleviate them.

RF problems

The addition of gradient coils to an NMR probe can have deleterious effects on probe performance. Due to the proximity of the gradient coils to the sample region, the gradient coils and leads can, without appropriate precautions, act as antennae and introduce rf interference. A related problem is the possible strong mutual inductance between the gradient and the rf coils. Thus, the quality factor Q (= ωL/R where ω, L and R are the resonance frequency, inductance and resistance, respectively) of the rf coil(s) are diminished resulting in longer pulses for the same flip angle, poorer decoupling efficiency and S/N.

The effects of non-ideal B1 pulses and B1 inhomogeneity are well-known on spin-echoes, but have not been widely considered with respect to NMR diffusion measurements.

Single-particle Green's functions, density response functions and other correlation functions are calculated in many different ways in the literature on Bose-condensed gases. An in-depth comparison and classification of different approaches was first given in the classic paper by Hohenberg and Martin (1965), with an emphasis on the various exact identities (conservation laws) that are satisfied. A key feature to be included in any theory is that a Bose broken symmetry leads to a hybridization of the single-particle excitations and the collective density fluctuations in such a way that the two excitation spectra become identical. This key feature is demonstrated in Chapter 5 of Griffin (1993).

How to relate and assess various approximations for correlation functions in a Bose superfluid has been a topic of continual interest (and some controversy) since the late 1950s. These questions were largely resolved by the early 1960s at a conceptual level but the detailed applications of the theory were limited to dilute Bose-condensed gases at T = 0. Since it was difficult to relate the theory to the properties of superfluid. He at a quantitative level, this formalism based on a Bose broken symmetry was of little interest to experimentalists. The creation of superfluid Bose condensed gases in 1995 changed all this and has given new life to the many body theory of dilute weakly interacting Bose condensed gases.

Various approximations for the Beliaev single-particle self-energies Σαβ were derived in Sections 4.3 and 4.4. The discussion in Chapter 4 was somewhat abstract.

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, TBEC), 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.

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 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.

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.