In this chapter we consider selected topics in the theory of trapped gases at non-zero temperature when the effects of interactions are taken into account. The task is to extend the considerations of Chapters 8 and 10 to allow for the trapping potential. In Sec. 11.1 we begin by discussing energy scales, and then calculate the transition temperature and thermodynamic properties. We show that at temperatures of the order of Tc the effect of interactions on thermodynamic properties of clouds in a harmonic trap is determined by the dimensionless parameter N1/6a/ā. Here ā, which is defined in Eq. (6.24), is the geometric mean of the oscillator lengths for the three principal axes of the trap. Generally this quantity is small, and therefore under many circumstances the effects of interactions are small. At low temperatures, thermodynamic properties may be evaluated in terms of the spectrum of elementary excitations of the cloud in its ground state, which we considered in Secs. 7.2, 7.3, and 8.2. At higher temperatures it is necessary to take into account thermal depletion of the condensate, and useful approximations for thermodynamic functions may be obtained using the Hartree–Fock theory as a starting point.

The remainder of the chapter is devoted to non-equilibrium phenomena. As we have seen in Secs. 10.3–10.5, two ingredients in the description of collective modes and other non-equilibrium properties of uniform gases are the two-component nature of condensed Bose systems, and collisions between excitations. For atoms in traps a crucial new feature is the in homogeneity of the gas. This in itself would not create difficulties if collisions between excitations were sufficiently frequent that matter remained in thermodynamic equilibrium locally.

The advent of the laser opened the way to the development of powerful new methods for manipulating and cooling atoms which were exploited in the realization of Bose–Einstein condensation in alkali atom vapours. To set the stage we describe a typical experiment, which is shown schematically in Fig. 4.1. A beam of sodium atoms emerges from an oven at a temperature of about 600 K, corresponding to a speed of about 800 m s–1, and is then passed through a so-called Zeeman slower, in which the velocity of the atoms is reduced to about 30 m s–1, corresponding to a temperature of about 1 K. In the Zeeman slower, a laser beam propagates in the direction opposite that of the atomic beam, and the radiation force produced by absorption of photons retards the atoms. Due to the Doppler effect, the frequency of the atomic transition in the laboratory frame is not generally constant, since the atomic velocity varies. However, by applying an inhomogeneous magnetic field designed so that the Doppler and Zeeman effects cancel, the frequency of the transition in the rest frame of the atom may be held fixed. On emerging from the Zeeman slower the atoms are slow enough to be captured by a magneto-optical trap (MOT), where they are further cooled by interactions with laser light to temperatures of order 100 μK. Another way of compensating for the changing Doppler shift is to increase the laser frequency in time, which is referred to as ‘chirping’. In other experiments the MOT is filled by transferring atoms from a second MOT where atoms are captured directly from the vapour.

The time-dependent behaviour of Bose–Einstein condensed clouds, such as collective modes and the expansion of a cloud when released from a trap, is an important source of information about the physical nature of the condensate. In addition, the spectrum of elementary excitations of the condensate is an essential ingredient in calculations of thermodynamic properties. In this chapter we treat the dynamics of a condensate at zero temperature starting from a time-dependent generalization of the Gross–Pitaevskii equation used in Chapter 6 to describe static properties. From this equation one may derive equations very similar to those of classical hydrodynamics, which we shall use to calculate properties of collective modes.

We begin in Sec. 7.1 by describing the time-dependent Gross–Pitaevskii equation and deriving the hydrodynamic equations. We then use the hydrodynamic equations to determine the excitation spectrum of a homogeneous Bose gas (Sec. 7.2). Subsequently, we consider modes in trapped clouds (Sec. 7.3) within the hydrodynamic approach, and also describe the method of collective coordinates and the related variational method. In Sec. 7.4 we consider surface modes of oscillation, which resemble gravity waves on a liquid surface. The variational approach is used in Sec. 7.5 to treat the free expansion of a condensate upon release from a trap. Finally, in Sec. 7.6 we discuss solitons, which are exact one-dimensional solutions of the time-dependent Gross-Pitaevskii equation.

General formulation

In the previous chapter we saw that the equilibrium structure of the condensate is described by a time-independent Schrödinger equation with a nonlinear contribution to the potential to take into account interactions between particles.

The topic of Bose–Einstein condensation in a uniform, non-interacting gas of bosons is treated in most textbooks on statistical mechanics. In the present chapter we discuss the properties of a non-interacting Bose gas in a trap. We shall calculate equilibrium properties of systems in a semi-classical approximation, in which the energy spectrum is treated as a continuum. For this approach to be valid the temperature must be large compared with Δ∈/k, where Δ∈ denotes the separation between neighbouring energy levels. As is well known, at temperatures below the Bose–Einstein condensation temperature, the lowest energy state is not properly accounted for if one simply replaces sums by integrals, and it must be included explicitly.

The statistical distribution function is discussed in Sec. 2.1, as is the single-particle density of states, which is a key ingredient in the calculations of thermo dynamic properties. Calculations of the transition temperature and the fraction of particles in the condensate are described in Sec. 2.2. In Sec. 2.3 the semi-classical distribution function is introduced, and from this we determine the density profile and the velocity distribution of particles. Thermodynamic properties of Bose gases are calculated as functions of the temperature in Sec. 2.4. The final two sections are devoted to effects not captured by the simplest version of the semi-classical approximation: corrections to the transition temperature due to a finite particle number (Sec. 2.5), and thermodynamic properties of gases in lower dimensions (Sec. 2.6).

In preceding chapters we have explored properties of Bose–Einstein condensates with a single macroscopically-occupied quantum state, and spin degrees of freedom of the atoms were assumed to play no role. In the present chapter we extend the theory to systems in which two or more quantum states are macroscopically occupied.

The simplest example of such a multi-component system is a mixture of two different species of bosons, for example two isotopes of the same element, or two different atoms. The theory of such systems can be developed along the same lines as that for one-component systems developed in earlier chapters, and we do this in Sec. 12.1.

Since alkali atoms have spin, it is also possible to make mixtures of the same isotope, but in different internal spin states. This was first done experimentally by the JILA group, who made a mixture of atoms in hyperfine states F = 2, mF = 2 and F = 1, mF = –1. Mixtures of hyperfine states of the same isotope differ from mixtures of distinct isotopes because atoms can undergo transitions between hyperfine states, while transitions that convert one isotope into another do not occur under most circumstances. Transitions between different hyperfine states can influence equilibrium properties markedly if the interaction energy per particle is comparable with or larger than the energy difference between hyperfine levels. In magnetic traps it is difficult to achieve such conditions, since the trapping potential depends on the particular hyperfine state. However, in optical traps (see Sec. 4.2.2) the potential is independent of the hyperfine state, and the dynamics of the spin can be investigated, as has been done experimentally.

In the present chapter we consider the structure of the Bose–Einstein condensed state in the presence of interactions. Our discussion is based on the Gross–Pitaevskii equation, which describes the zero-temperature properties of the non-uniform Bose gas when the scattering length a is much less than the mean interparticle spacing. We shall first derive the Gross–Pitaevskii equation at zero temperature by treating the interaction between particles in a mean-field approximation (Sec. 6.1). Following that, in Sec. 6.2 we discuss the ground state of atomic clouds in a harmonic-oscillator potential. We compare results obtained by variational methods with those derived in the Thomas-Fermi approximation, in which the kinetic energy operator is neglected in the Gross–Pitaevskii equation. The Thomas–Fermi approximation fails near the surface of a cloud, and in Sec. 6.3 we calculate the surface structure using the Gross–Pitaevskii equation. Finally, in Sec. 6.4 we determine how the condensate wave function ‘heals’ when subjected to a localized disturbance.

The Gross–Pitaevskii equation

In the previous chapter we have shown that the effective interaction between two particles at low energies is a constant in the momentum representation, U0 = 4πħ2a/m. In coordinate space this corresponds to a contact interaction U0δ(r – r′), where r and r′ are the positions of the two particles. To investigate the energy of man y-body states we adopt a Hartree or mean-field approach, and assume that the wave function is a symmetrized product of single-particle wave functions.

From a theoretical point of view, one of the appealing features of clouds of alkali atom vapours is that particle separations, which are typically of order 102 nm, are large compared with the scattering length a which characterizes the strength of interactions. Scattering lengths for alkali atoms are of the order of 100a0, where a0, is the Bohr radius, and therefore alkali atom vapours are dilute, in the sense that the dominant effects of interaction are due to two-body encounters. It is therefore possible to calculate properties of the gas reliably from a knowledge of two-body scattering at low energies, which implies that information about atomic scattering is a key ingredient in work on Bose–Einstein condensates.

An alkali atom in its electronic ground state has several different hyperfine states, as we have seen in Secs. 3.1 and 3.2. Interatomic interactions give rise to transitions between these states and, as we described in Sec. 4.6, such processes are a major mechanism for loss of trapped atoms. In a scattering process, the internal states of the particles in the initial or final states are described by a set of quantum numbers, such as those for the spin, the atomic species, and their state of excitation. We shall refer to a possible choice of these quantum numbers as a channel.1 At the temperatures of interest for Bose-Einstein condensation, atoms are in their electronic ground states, and the only relevant internal states are therefore the hyperfine states. Because of the existence of several hyperfine states for a single atom, the scattering of cold alkali atoms is a multi-channel problem.

Pervasive role of phonons in modern solid-state devices

As illustrated throughout this book, phonon effects are pervasive in modern solidstate devices. As is illustrated by the many examples for Chapters 7–10, the importance of these effects is usually at least as great for dimensionally confined structures as for bulk structures. Indeed, in Chapter 7 the effects of dimensional confinement were seen to be important even for biological structures! In this case, a cylindrical shell immersed in a fluid (Sirenko et al., 1996b) was used to model the vibrational behavior of microtubules (MTs) immersed in water. In addition, the examples of Chapters 7 and 9 illustrate that the elastic continuum model provides an accurate description of acoustic phonons in dimensionally confined structures of many geometries including thin films, nanowires with rectangular and circular cross sections, and a variety of dot-like structures. These structures will inevitably be pervasive as elements of nanoscale structures mimicking the well known and larger microelectromechanical structures. Indeed, Cleland and Roukes (1996) reported a technique for fabricating nanometer-scale mechanical structures from bulk, single-crystal Si substrates. As another example of acoustic phonon effects in dimensionally confined structures, it was recently predicted theoretically that Cerenkov-like effects lead to the generation of high-frequency confined acoustic phonons in quantum wells (Komirenko et al., 2000b); see Section 10.6.

In Chapter 8, values of carrier–optical-phonon scattering rates calculated for a variety of dimensionally confined semiconductor structures were found to exceed 1013 s–1.

Phonon effects: fundamental limits on carrier mobilities and dynamical processes

The importance of phonons and their interactions in bulk materials is well known to those working in the fields of solid-state physics, solid-state electronics, optoelectronics, heat transport, quantum electronics, and superconductivity.

As an example, carrier mobilities and dynamical processes in polar semiconductors, such as gallium arsenide, are in many cases determined by the interaction of longitudinal optical (LO) phonons with charge carriers. Consider carrier transport in gallium arsenide. For gallium arsenide crystals with low densities of impurities and defects, steady state electron velocities in the presence of an external electric field are determined predominantly by the rate at which the electrons emit LO phonons. More specifically, an electron in such a polar semiconductor will accelerate in response to the external electric field until the electron's energy is large enough for the electron to emit an LO phonon. When the electron's energy reaches the threshold for LO phonon emission – 36 meV in the case of gallium arsenide – there is a significant probability that it will emit an LO phonon as a result of its interaction with LO phonons. Of course, the electron will continue to gain energy from the electric field.

In the steady state, the processes of electron energy loss by LO phonon emission and electron energy gain from the electric field will come into balance and the electron will propagate through the semiconductor with a velocity known as the saturation velocity.

This book describes a major aspect of the effort to understand nanostructures, namely the study of phonons and phonon-mediated effects in structures with nanoscale dimensional confinement in one or more spatial dimensions. The necessity for and the timing of this book stem from the enormous advances made in the field of nanoscience during the last few decades.

Indeed, nanoscience continues to advance at a dramatic pace and is making revolutionary contributions in diverse fields, including electronics, optoelectronics, quantum electronics, materials science, chemistry, and biology. The technologies needed to fabricate nanoscale structures and devices are advancing rapidly. These technologies have made possible the design and study of a vast array of novel devices, structures and systems confined dimensionally on the scale of 10 nanometers or less in one or more dimensions. Moreover, nanotechnology is continuing to mature rapidly and will, no doubt, lead to further revolutionary breakthroughs like those exemplified by quantum-dot semiconductor lasers operating at room temperature, intersubband multiple quantum-well semiconductor lasers, quantumwire semiconductor lasers, double-barrier quantum-well diodes operating in the terahertz frequency range, single-electron transistors, single-electron metal-oxide-semiconductor memories operating at room temperature, transistors based on carbon nanotubes, and semiconductor nanocrystals used for fluorescent biological labels, just to name a few!

The seminal works of Esaki and Tsu (1970) and others on the semiconductor superlattice stimulated a vast international research effort to understand the fabrication and electronic properties of superlattices, quantum wells, quantum wires, and quantum dots.

Phonon effects in intersubband lasers

The effects of dimensional confinement on optical phonons, phonon-assisted electron intersubband transition rates (Teng et al., 1998), and gain (Kisin et al., 1997) have been evaluated in a series of studies on semiconductor lasers. Many of the novel semiconductor lasers – such as the tunneling injection laser and the quantum cascade laser – contain quantum wells with confinement dimensions of about 50 Å or less. An example of such a semiconductor laser structure is given in Figure 10.1. In this laser, the conduction band is engineered so that the upper and lower energy levels are E3 and E2 respectively, with a third level, E1, such that phonon-assisted tunneling from level E2 to level E1 is promoted. The IF optical phonons are of special importance in such heterostructures. Two properties of the IF optical phonons account for their special significance (Stroscio, 1996) in narrow-well semiconductor lasers: (a) in narrow wells the interface phonons have appreciable interaction potentials throughout the quantum well since, as was demonstrated in Chapter 7, the interface optical phonons have potentials near the heterointerfaces of the form ce–q|z|e–iq·ρ, where, for example, q has values of very roughly 0.02 Å–1 for a typical intrasubband transition in GaAs; and (b) the energies associated with phonon-assisted processes in these heterostructures can be substantially different from those in semiconductor lasers without significant dimensional confinement, since the interface phonons may have frequencies ωq which are significantly different from those of the other phonons in the quantum well.

Basic properties of phonons in würtzite structure

The GaA1N-based semiconductor structures are of great interest in the electronics and optoelectronics communities because they possess large electronic bandgaps suitable for fabricating semiconductor lasers with wavelengths in the blue and ultraviolet as well as electronic devices designed to work at elevated operating temperatures. These III-V nitrides occur in both zincblende and würtzite structures. In this chapter, the würtzite structures will be considered rather than the zincblende structures, since the treatment of the phonons in these würtzite structures is more complicated than for the zincblendes. Throughout the remainder of this book, phonon effects in nanostructures will be considered for both the zincblendes and würtzites. This chapter focuses on the basic properties of phonons in bulk würtzite structures as a foundation for subsequent discussions on phonons in würtzite nanostructures.

The crystalline structure of a würtzite material is depicted in Figure 3.1. As in the zincblendes, the bonding is tetrahedral. The würtzite structure may be generated from the zincblende structure by rotating adjacent tetrahedra about their common bonding axis by an angle of 60 degrees with respect to each other. As illustrated in Figure 3.1, würtzite structures have four atoms per unit cell.

The total number of normal vibrational modes for a unit cell with s atoms in the basis is 3s. As for cubic materials, in the long-wavelength limit there are three acoustic modes, one longitudinal and two transverse. Thus, the total number of optical modes in the long-wavelength limit is 3s – 3.