In the ideal Bose gas, a finite fraction of the particles occupy the same singleparticle state below a critical temperature, forming a Bose-Einstein condensate. When the particles have mutual interactions, however, single-particle states are no longer meaningful. Nevertheless, we can still define a condensate wave function ψ(r), as the quantum amplitude for removing a particle from the condensate at position r. Its complex conjugate ψ∗(r) is the amplitude for creating a particle.
In the ideal Bose gas, a finite fraction of the particles occupy the same singleparticle state below a critical temperature, forming a Bose-Einstein condensate. When the particles have mutual interactions, however, single-particle states are no longer meaningful. Nevertheless, we can still define a condensate wave function ψ(r), as the quantum amplitude for removing a particle from the condensate at position r. Its complex conjugate ψ∗(r) is the amplitude for creating a particle.
We can imagine creating a particle in the condensate by inducing a transition into it. As we saw in Section 16.2, Bose enhancement makes the rate proportional to the existing boson density. Thus, the creation amplitude ψ∗(r) should be proportional to the square root of the density. We define |ψ(r, t)|2d3r as the number of condensate particles in the volume element d3r, so that the total number of particles in the condensate is ∫ d3r |ψ(r)|2 = N0 (20.1) In Chapter 18, we were able to give a complete mathematical description of the Bose-Einstein condensation in the ideal gas. Such a treatment is not possible when there is interaction between particles. Here, we shall describe it in Ginsburg-Landau theory, with the complex field ψ as order parameter. The Landau free energy is taken to be E[ψ, ψ∗] = ∫ d3r [ h¯2 2m |∇ψ |2 + (Uext − μ)ψ∗ψ + g2 (ψ ∗ψ)2 ] (20.2) where |∇ψ |2 = ∇ψ∗ · ∇ψ , m is the mass of the particles, Uext(r) is an external potential, and μ is the chemical potential that determines N0. There are equivalent choices for independent variables: ψ and its complex conjugate ψ∗, real and imaginary parts Re ψand Im ψ, modulus and phase angle √n0 and ϕ in the polar representation ψ(x) = √ n0(x) e iϕ(x) where n0 is the density of the condensate.
At finite temperatures below the phase transition point, the Bose-Einstein condensation, the macroscopic occupation of a single quantum state by particles of integer spin, is not complete. In the language of superfluid helium, this means that the superfluid coexists with the normal fluid. Our goal is to describe this coexistence in trapped, dilute atomic gases with repulsive interactions via mean field laws that account for a {\em spatially varying} particle interaction strength. By starting with the N-body Hamiltonian, N≫1, we formally derive a system of coupled, nonlinear evolution equations in 3+1 dimensions for the following quantities: (i) the wave function of the macroscopically occupied state; and (ii) the single-particle wave functions of thermally excited states. For stationary (bound) states and a scattering length with {\em periodic microstructure} of subscale ε, we heuristically extract effective equations of motion via periodic homogenization up to second order in ε.
The properties of the many-particle Schr\"odinger wave function \ensuremath{\Psi} are examined in the presence of Bose-Einstein condensation (BEC). It is shown that it is possible to define, in terms of \ensuremath{\Psi}, a function $\ensuremath{\psi}(\stackrel{\ensuremath{\rightarrow}}{r}|\stackrel{\ensuremath{\rightarrow}}{s}),$ which can be regarded as the single-particle wave function of an arbitrary particle for a fixed configuration $\stackrel{\ensuremath{\rightarrow}}{s}$ of all other particles. It is shown that $\ensuremath{\psi}(\stackrel{\ensuremath{\rightarrow}}{r}|\stackrel{\ensuremath{\rightarrow}}{s})$ plays an analogous role to the field operator of standard field-theoretical treatments of superfluidity. It is shown that in the presence of a Bose-Einstein condensate fraction f, $\ensuremath{\psi}(\stackrel{\ensuremath{\rightarrow}}{r}|\stackrel{\ensuremath{\rightarrow}}{s})$ must be nonzero and phase coherent within at least a fraction f of the total volume of the N-particle system for essentially all $\stackrel{\ensuremath{\rightarrow}}{s}.$ Examination of the form of variational many-particle wave functions shows that in liquid 4He, $\ensuremath{\psi}(\stackrel{\ensuremath{\rightarrow}}{r}|\stackrel{\ensuremath{\rightarrow}}{s})$ extends throughout the spaces left between the hard cores of the other atoms at $\stackrel{\ensuremath{\rightarrow}}{s}.$ By contrast, in the absence of BEC, $\ensuremath{\psi}(\stackrel{\ensuremath{\rightarrow}}{r}|\stackrel{\ensuremath{\rightarrow}}{s})$ in the ground state must be nonzero only over a localized region of space. It is shown that in order for long-range phase coherence in $\ensuremath{\psi}(\stackrel{\ensuremath{\rightarrow}}{r}|\stackrel{\ensuremath{\rightarrow}}{s})$ to be maintained in the presence of velocity fields, any circulation must be quantized over macroscopic length scales. Some numerical calculations of the properties and fluctuations of liquid helium are presented. These suggest that the approach outlined in this paper may have significant advantages for the numerical calculations of the properties of Bose-Einstein condensed systems. The properties of $\ensuremath{\psi}(\stackrel{\ensuremath{\rightarrow}}{r}|\stackrel{\ensuremath{\rightarrow}}{s})$ are used to show that there is no general connection between the static structure factor and the size of the Bose-Einstein condensate fraction in a Bose fluid. It is suggested that the observed connection in liquid 4He is due to the creation of vacancies in the liquid structure, which are required so that $\ensuremath{\psi}(\stackrel{\ensuremath{\rightarrow}}{r}|\stackrel{\ensuremath{\rightarrow}}{s})$ can delocalize, in the presence of hard-core interactions. It is shown that this suggestion is quantitatively consistent with observations on liquid 4He. It is suggested that the superfluid transition in 4He occurs when the vacancies in the liquid structure form a connected network.
As we all know, single microscopic particles such as electrons or atoms do not behave according to the familiar classical (Newtonian) physics which describes the everyday world, but rather according to the prescriptions of quantum mechanics, the outcome is often bizarre and counterintuitive. When a macroscopic number of such particles (say ∼ 10) is assembled under conditions where the effects of quantum mechanics are important, the results can be even more spectacular; they include macroscopic phenomena such as superfluidity and superconductivity, which are completely impossible to explain in classical terms. Most, though not all of these exotic phenomena, are a result of a phenomenon which I shall refer to generically as “quantum condensation,” in which a macroscopic number of particles or pairs of particles—a nonzero fraction of all particles present—occupy a single quantum state; this phenomenon is traditionally referred to as Bose-Einstein condensation (BEC) when it occurs in a system of bosons such as liquid He, and as Cooper pairing when it occurs, for pairs of particles, in a system of fermions such as the electrons in a superconducting metal. As a result of quantum condensation, the characteristically quantum-mechanical behavior which we are used to seeing at the level of single electrons is as it were amplified by the cooperation of condensed particles and can manifest itself at the macroscopic level. The present essay is intended as a general introduction to the subject of quantum condensation and its consequences, as manifested in a variety of physical systems. I will try to review the fundamental physical ideas, usually without de-
In 1995 two different research groups observed for the first time the Bose-Einstein condensation (BEC) in ultracold gases. When the confining magnetic trap was turned off the gas was left free to expand, and the velocity of the particles showed a clear peak: most of the particles were occupying the same single particle state, the one of lowest energy. The Bose-Einstein condensation had been predicted in 1925 by Einstein, written by inspiration of a work on the statistic of the photons by Bose (1924). In this work Bose described the behavior of an ensemble of photons, treating them as massless particles, with no number conservation associated. Einstein extended this approach to particles with a mass and with fixed number, creating what is now called the Bose-Einstein distribution. The particles that follow such a description are called ``bosons'', as opposed to the ``fermions'' of the Fermi-Dirac statistics. Einstein predicted that in a gas of bosons - under a critical temperature - a finite fraction of the total number of particles would have been in the ground state, and act as a single entity. This amusing theoretical discovery found its utility a few years later. In the late thirties, new techniques allowed to cool Helium-4 at few Kelvins above the absolute zero. The properties of the resulting liquid were a puzzlement to the scientific community: among others, it could flow without experiencing friction. The liquid was called a ``superfluid''. A first explanation was given by London in 1938, which linked the superfluid behavior to the presence of a BEC among the bosonic Helium particles. The fermions cannot condense by themselves. On the other hand, they can form bound pairs and act as bosons, as it happens in a metal at low temperature. Using this approach, in 1957 Bardeen, Cooper and Schrieffer created a successful model of superconductivity by describing a superconductor as a superfluid in a charged system. During the course of these years we explored the superfluid properties of Bosons and Fermions in different settings. The original contributions of the thesis are described starting from the third chapter, where we speak about the generation and stability of solitons in a periodic optical lattices, both fixed or in motion. In the fourth chapter we study the generation of giant vortices in cold fermions, by using a generalized hydrodynamical approach. In chapter 5 we study the effect of a quasiperiodic lattice and the glassy phase it produces on a gas of bosons. Finally, we study the interaction of normal matter and superfluids with gravitational waves. While this interaction is seen to be extremely small, we believe that the resulting formalism is interesting by itself.
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.