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 87Rb 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 may be neglected. 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.

The ability to vary the force constants of a trapping potential makes it possible to create very elongated or highly flattened clouds of atoms. This opens up the study of Bose–Einstein condensation in lower dimensions, since motion in one or more directions may then effectively be frozen out at sufficiently low temperatures. In a homogeneous system Bose–Einstein condensation cannot take place at non-zero temperature in one or two dimensions, but in traps the situation is different because the trapping potential changes the energy dependence of the density of states. This introduces a wealth of new phenomena associated with lower dimensions which have been explored both theoretically and experimentally. A general review may be found in the lecture notes Ref.

For a system in thermal equilibrium, the condition for motion in a particular direction to be frozen out is that the energy difference between the ground state and the lowest excited state for the motion must be much greater than the thermal energy kT. This energy difference is ħωi if interactions are unimportant for the motion in the i direction. If the interaction energy nU0 is large compared with ħωi and the trap is harmonic, the lowest excited state is a sound mode with wavelength comparable to the spatial extent of the cloud in the i direction.

Bose–Einstein condensates of particles behave in many ways like coherent radiation fields, and the realization of Bose–Einstein condensation in dilute gases has opened up the experimental study of interactions between coherent matter waves. In addition, the existence of these dilute trapped quantum gases has prompted a re-examination of a number of theoretical issues.

In Sec. 13.1 we discuss Josephson tunnelling of a condensate between two wells and the role of fluctuations in particle number and phase. The number and phase variables play a key role in the description of the classic interference experiment, in which two clouds of atoms are allowed to expand and overlap (Sec. 13.2). Rather surprisingly, an interference pattern is produced even though initially the two clouds are completely isolated. Density fluctuations in a Bose gas are studied in Sec. 13.3, where we relate atomic clock shifts to the two-particle correlation function. The ability to manipulate gaseous Bose–Einstein condensates by lasers has made possible the study of coherent matter wave optics and in Sec. 13.4 we describe applications of these techniques to observe solitons, Bragg scattering, and non-linear mixing of matter waves. How to characterize Bose–Einstein condensation in terms of the density matrix is the subject of Sec. 13.5, where we also consider fragmented condensates.

An important feature of cold atomic vapours is that particle separations, which are typically of order 102 nm, are usually an order of magnitude larger than the length scales associated with the atom–atom interaction. Consequently, the two-body interaction between atoms dominates, and three-and higher-body interactions are unimportant. Moreover, since the atoms have low velocities, many properties of these systems may be calculated in terms of a single parameter, the scattering length.

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. 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.

The advent of the laser opened the way to the development of powerful methods for producing and manipulating cold atomic gases. 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’.

A number of atomic properties play a key role in experiments on cold atomic gases, and we shall discuss them briefly in the present chapter with particular emphasis on alkali atoms. Basic atomic structure is the subject of Sec. 3.1. Two effects exploited to trap and cool atoms are the influence of a magnetic field on atomic energy levels, and the response of an atom to radiation. In Sec. 3.2 we describe the combined influence of the hyperfine interaction and the Zeeman effect on the energy levels of an atom, and in Sec. 3.3 we review the calculation of the atomic polarizability. In Sec. 3.4 we summarize and compare some energy scales.

Atomic structure

The total spin of a Bose particle must be an integer, and therefore a boson made up of fermions must contain an even number of them. Neutral atoms contain equal numbers of electrons and protons, and therefore the statistics that an atom obeys is determined solely by the number of neutrons N: if N is even, the atom is a boson, and if it is odd, a fermion. Since the alkalis have odd atomic number Z, boson alkali atoms have odd mass numbers A. Likewise for atoms with even Z, bosonic isotopes have even A. In Table 3.1 we list N, Z, and the nuclear spin quantum number I for some alkali atoms and hydrogen.

A new facet was added to the study of dilute gases by the production and subsequent Bose–Einstein condensation of diatomic molecules from a gas of fermionic atoms. Feshbach resonances which, as we have seen in Sec. 5.4, make it possible to tune the atom–atom interaction, play a crucial role in the experiments. At the magnetic field strength for which the binding energy of the molecule vanishes, the inverse of the scattering length, which determines the low-energy effective interaction between atoms, passes through zero. In the experiments to produce molecules, one starts with a mixture of two species of fermion, most commonly different hyperfine states of the same isotope, in a magnetic field of such a strength that the molecular state has an energy higher than that of two zero-momentum atoms in the open channel. The magnetic field is then altered to a value at which the molecular state is bound with respect to two atoms in the open channel, and in this process, many of the atoms combine to form molecules. These molecules have binding energies in the 10−9 eV range, and are thus extremely weakly bound by the standards of conventional molecular physics. In addition, they are very extended, with atomic separations as large as one micron. These molecules, being bosons, can undergo Bose–Einstein condensation, just as bosonic atoms do.

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. In Sec. 6.4 we determine how the condensate wave function ‘heals’ when subjected to a localized disturbance. Finally, in Sec. 6.5 we show how the magnetic dipole–dipole interaction, which is long-ranged and anisotropic, may be included in the Gross–Pitaevskii equation and determine within the Thomas–Fermi approximation its effect on the density distribution.

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.

The experimental realization in 1995 of Bose–Einstein condensation in dilute atomic gases marked the beginning of a very rapid development in the study of quantum gases. The initial experiments were performed on vapours of rubidium, sodium, and lithium. So far, the atoms 1H, 7Li, 23Na, 39K, 41K, 52Cr, 85Rb, 87Rb, 133Cs, 170Yb, 174Yb and 4He* (the helium atom in an excited state) have been demonstrated to undergo Bose–Einstein condensation. In related developments, atomic Fermi gases have been cooled to well below the degeneracy temperature, and a superfluid state with correlated pairs of fermions has been observed. Also molecules consisting of pairs of fermionic atoms such as 6Li or 40K have been observed to undergo Bose–Einstein condensation. Atoms have been put into optical lattices, thereby allowing the study of many-body systems that are realizations of models used in condensed matter physics. Although the gases are very dilute, the atoms can be made to interact strongly, thus providing new challenges for the description of strongly correlated many-body systems. In a period of less than ten years the study of dilute quantum gases has changed from an esoteric topic to an integral part of contemporary physics, with strong ties to molecular, atomic, subatomic and condensed matter physics.

The dilute quantum gases differ from ordinary gases, liquids and solids in a number of ways, as we shall now illustrate by giving values of physical quantities.

In the previous chapter we described quantized vortex lines, which are one of the characteristic features of superfluids. In a classical fluid, the circulation of vortex lines is not quantized and, in addition, vortex lines decay because of viscous processes. Another feature of a superfluid, the lack of response to rotation for a small enough angular velocity, was also demonstrated. This is analogous to the Meissner effect for superconductors. One characteristic common to superfluids and superconductors is the ability to carry currents without dissipation. Such current-carrying states are not the lowest-energy state of the system. They are metastable states, the existence of which is intimately connected to the nature of the low-lying elementary excitations. The word ‘superfluidity’ does not refer to a single property of the system, but it is used to describe a variety of different phenomena (see Ref. [1]).

Historically, the connection between superfluidity and the existence of a condensate, a macroscopically occupied quantum state, dates back to Fritz London's suggestion in 1938, as we have described in Chapter 1. However, the connection between Bose–Einstein condensation and superfluidity is a subtle one. A Bose–Einstein condensed system does not necessarily exhibit superfluidity, an example being the ideal Bose gas for which the critical velocity vanishes, as demonstrated in Sec. 10.1 below. Also lower-dimensional systems may exhibit superfluid behaviour in the absence of a true condensate, as we shall see in Chapter 15.

Dissociative recombination is part of the broader field of electron–molecule scattering, which dates back to the famous Franck–Hertz experiment in 1914 (Franck & Hertz 1914). There are also similarities between dissociative recombination and photodissociation, and being a reactive process, recombination can also be considered as branch of chemical reaction dynamics. It is not possible to give a comprehensive presentation of all these topics, or this book would take an encyclopedic format. Instead we will focus on the topics that are most closely related to dissociative recombination.

A broad overview of the entire field of atomic collisions, including electron–ion recombination, is given in McDaniel (1989) and McDaniel, Mitchell, and Rudd (1993). The theory of electron–atom and electron–molecule collisions has been covered in Khare (2002). Since the recombination of atomic ions is not covered in the present book, the reader is referred to Dunn et al. (1984), Graham et al. (1992), McDaniel, Mitchell, and Rudd (1993), Hahn (1997), and Phaneuf et al. (1999) for reviews of this topic. Edited volumes by Christophorou (1984a, b), Märk and Dunn (1985), Ehrhardt and Morgan (1994), and Becker (1998) cover well the development in electron–molecule collisions during the 1980s and 1990s. Review articles covering dissociative recombination are given in Chapter 1.