Abstract In this chapter we outline the microscopic theory of an interacting Bose fluid. The aim is to provide a basis for a microscopic understanding of the excitations in superfluid and normal 4He discussed in the following chapters. The microscopic picture can be omitted on a first reading and referred to when desired.We assume that the reader is familiar with the theory of Bose-Einstein condensation in a Bose gas introduced in §7.2 and presented in many texts on statistical mechanics. In §7.2 we also discussed the generalization of Bose condensation to an interacting fluid by Penrose and Onsager (1956).Condensation was identified with macroscopic occupation of one of the eigenstates of the fluid and the appearance of an infinitely long tail in the one-body density matrix, denoted as off-diagonal long range order (ODLRO). For a homogeneous fluid it is natural to take momentum states as the eigenstates and to assume that the macroscopically occupied state is the state with momentum p = 0. In this case the criterion reduces to that for a gas but with a firm foundation.
Abstract In this chapter we outline the microscopic theory of an interacting Bose fluid. The aim is to provide a basis for a microscopic understanding of the excitations in superfluid and normal 4He discussed in the following chapters. The microscopic picture can be omitted on a first reading and referred to when desired.We assume that the reader is familiar with the theory of Bose-Einstein condensation in a Bose gas introduced in §7.2 and presented in many texts on statistical mechanics. In §7.2 we also discussed the generalization of Bose condensation to an interacting fluid by Penrose and Onsager (1956).Condensation was identified with macroscopic occupation of one of the eigenstates of the fluid and the appearance of an infinitely long tail in the one-body density matrix, denoted as off-diagonal long range order (ODLRO). For a homogeneous fluid it is natural to take momentum states as the eigenstates and to assume that the macroscopically occupied state is the state with momentum p = 0. In this case the criterion reduces to that for a gas but with a firm foundation.
Abstract In this chapter we introduce some basic concepts of superfluid 4He that we will draw on in Chapters 8-12. Liquid 4 He has a long and rich history and there are many concepts that are implicit in the field today. We begin with a sketch of the unique mass and heat flow character of superfluid 4He. The assumptions of the two-fluid model used successfully to describe these properties are noted in §7.1. We introduce Bose-Einstein condensation in a Bose gas and the extension of this to interacting superfluid 4 He in §7.2. This leads naturally to the one-body density matrix which is the intermediate scattering function in the impulse approximation,We introduce a number of basic properties such as the static structure factor, the pair correlation function, and the momentum distribution which can be calculated from first principles using Monte Carlo and path integral Monte Carlo (PIMC) methods. We conclude by noting the two current descriptions of the characteristic excitation in superfluid 4He. The first is a density excitation, based on the Landau and Feynman view, with suitable modifications at higher Q to take account of correlations among atoms. In the second, the density excitation is separated into a part due to the excitation of single particles out of the condensate to states above the condensate and a part due to regular density excitations involving only states above the condensate. In the latter picture the condensate appears explicitly in S(Q, ω).
The phenomena of superfluidity and superconductivity are intimately connected with the existence of a condensate, a macroscopically occupied quantum state. Such condensates occur in a variety of different physical systems, as described in Chapter 1. The foundation for the description of superfluidity is a picture of the system as being comprised of a condensate and elementary excitations. In Chapter 8 we have seen how physical properties such as the energy and the density of a Bose–Einstein condensed system may be expressed in terms of a contribution from the condensate, plus one from the elementary excitations, and in this chapter we shall consider further developments of this basic idea to other situations. As a first application, we determine the critical velocity for creation of an excitation in a homogeneous system (Sec. 10.1). Following that, we show how to express the momentum density in terms of the velocity of the condensate and the distribution function for excitations. This provides the basis for a two-component description, the two components being the condensate and the thermal excitations (Sec. 10.2). In the past, this framework has proved to be very effective in describing the properties of superfluids and superconductors, and in Sec. 10.3 we apply it to dynamical processes. To describe the state of a superfluid, one must specify the condensate velocity, in addition to the variables needed to characterize the state of an ordinary fluid. As a consequence, the collective behaviour of a superfluid is richer than that of an ordinary one. Collective modes are most simply examined when excitations collide frequently enough that they are in local thermodynamic equilibrium.
We conclude our study of superfluidity in Bose liquids with a brief summary of the strengths, and of the inadequacies, of the theory. The cornerstone of our fundamental understanding of superfluidity has been the recognition of the central importance of the condensate, a single quantum state which is macroscopically occupied. For the Bose liquid one has a particle condensate, characterized by a non-vanishing expectation value of ψ (r). The existence of the condensate explains all major aspects of superfluid behavior; in particular, we now possess: A clear understanding of the static and the long wavelength dynamic properties of an equilibrium Bose liquid, including a microscopic basis for the two-fluid model. A precise microscopic theory at T = 0 for a specific model, the dilute Bose gas. Variational calculations of the excitation spectrum of liquid He II (and of the ground state properties as well), which are in good quantitative agreement with experiment. A satisfactory phenomenological treatment of the interactions between elementary excitations which give rise to various irreversible phenomena. A qualitative understanding of the role which is played by quantized vortices in both equilibrium and non-equilibrium phenomena.
Abstract Chapter 2 describes the basic knowledge of Bose-condensed systems in order to study the quantum hydrodynamics for both the superfluid helium and the atomic-gas BECs in the following sections. Starting from the field theoretical formulation, we will first give a self-contained explanation of the derivation of the most fundamental GP equation for weakly interacting Bose gases at zero temperature. We will also highlight the theoretical treatment for incorporating the concept of elementary excitations to describe low-energy excitations of Bose systems, the definition of normal and superfluid components, and eventually the celebrated model of two-fluid hydrodynamics. Finally, the microscopic structure and dynamics of topological excitations obtained from the GP equation are discussed by exemplifying solitons, vortex lines, and vortex rings.
The Bose-Einstein condensation mechanism is shown to be capable of accounting for the existence of separate hydrodynamic velocity fields for the normal and the superfluid, provided suitable assumptions are made with regard to the single-particle energy spectrum. We consider the effect on the microcanonical gas distribution of imposing a nonzero value of the total momentum P. For an ordinary gas the effect is trivial, the whole distribution being merely shifted in momentum space. However, in the case of a degenerate Bose gas whose single-particle energy spectrum has a sharp minimum (gap or cusp), only the excited part of the gas (normal fluid) participates in the imposed motion, the condensate (superfluid) remaining "frozen" in momentum space. This rigidity of the condensate in momentum space plays the same role as the rigidity of the superelectrons on imposition of a magnetic field in the London theory of superconductivity. P acts as an additional thermodynamic variable, states with P \ensuremath{\ne} 0 being macroscopically metastable and corresponding to the existence of a relative velocity between normal and superfluid. The basic hydrodynamic assumption of the two-fluid model is thus reduced to an assumption concerning the form of an effective single-particle energy spectrum, and the parallelism between the theories of superconductivity and superfluidity is clearly exhibited. The present theory permits, in particular, the introduction of the mathematical form of Landau's phonon and "roton" spectra within the framework of the Bose-Einstein condensation picture. The statistical-thermodynamic formulas are derived and are shown to lead to characteristic two-fluid equations derived previously from a variational principle.
We present a brief overview of crucial historical stages in creation of superfluidity theory and of the current state of the microscopic theory of superfluid 4He. We pay special attention to the role of Bose-Einstein condensates (BECs) in understanding of physical mechanisms of superfluidity and identification of quantum mechanical structure of 4He superfluid component below λ-point, in particular—the possibility that at least two types of condensates may appear and coexist simultaneously in superfluid 4He. In this context we discuss the properties of the binary mixtures of BECs and types of excitations, which may appear due to intercomponent interaction in such binary mixtures of condensates. We also discuss current status of investigations of persistent currents in toroidal optical traps and present an outlook of our recent findings on this subject.