We establish here the correspondence between the statistical models introduced in Chapter 3 and the quantum evolution of atoms undergoing subrecoil laser cooling. This enables us to establish analytical expressions connecting the parameters of the statistical models (τ0, p0, pD, Δp, pmax, τb and) to atomic and laser parameters relevant to subrecoil laser cooling.

Such a ‘dictionary’ is useful for the numerical estimation of the results derived in this book (see Chapter 8). It also leads to analytical relations between τb and, which are used for cooling optimization (see Chapter 9).

We first treat in detail Velocity Selective Coherent Population Trapping in Section A.1. Analytical expressions are given for the statistical parameters. Special attention is given to the p-dependences of the jump rates both for small p and for large p, because they control the asymptotic behaviours of the trapping and recycling times. It is thus important to include these p-dependences correctly in the simplified jump rates in order to ensure the validity of the statistical model. Raman cooling is then briefly treated in Section A.2.

Velocity Selective Coherent Population Trapping

We first present the quantum optics treatment of one-dimensional σ+/σ− VSCPT (Section A.1.1).

We now have all the mathematical tools in hand to address the important questions for the cooling process, namely: what is the proportion ftrap(θ) of ‘trapped’ atoms (i.e. those which have a very small momentum p < ptrap); what is the ‘line shape’, i.e. the momentum distribution, after an interaction time θ?

In Section 5.1, we define precisely the trapped proportion ftrap(θ) in terms of an ensemble average and compare it to a time average defined as the mean fraction of the time spent in the trap. The two averages do not always coincide, as shown by the explicit computation of Section 5.2. This reveals the non-ergodic character of the cooling process, as discussed in Section 5.3.

Ensemble averages versus time averages

We define the trapped proportion ftrap(θ) as the probability of finding the atom in the trap at time t = θ. Therefore, ftrap(θ) corresponds to an ensemble average, over many independent realizations of the stochastic process of Fig. 3.1. It is instructive to consider also a time average, by examining how a given atom shares its time between the ‘inside’ and the ‘outside’ of the trap. Because of the non-ergodic character of subrecoil laser cooling, ensemble averages and time averages do not in general coincide. In fact, we will see later on that the ensemble average ftrap(θ) and the time average only coincide when 〈τ〉 and are finite, whereas they differ when either µ or is smaller than one.

Introduction

The statistical approach presented in this book provides not only a deeper physical understanding of subrecoil cooling, but also analytical expressions for the various characteristics of the momentum distribution of the cooled atoms. A great confidence in the validity of these predictions has been obtained in the previous chapter, by comparing them with experimental and numerical results. Therefore, we are now entitled to apply the approach developed in this work to specific problems, such as the optimization of one particular feature of the cooling process, namely the height of the peak of cooled atoms. This is the subject of this chapter.

Finding empirically the optimum conditions for a subrecoil cooling experiment is a difficult task. There are a priori many parameters to be explored and each experiment with a given set of parameters is in itself lengthy. The same can also be said of numerical simulations. One needs guidelines such as those provided by the present statistical approach to reduce the size of the parameter space to be explored.

There is a variety of optimization problems that can be considered. Following usual motivations of laser cooling, like the increase of atomic beam brightness or the search for quantum degeneracy, we will concentrate here on optimizing the height h(θ) of the peak of the momentum distribution of the cooled atoms, which corresponds also to the gain in phase space provided by the cooling (see Section 6.2.3).

In the previous chapters, several important quantities characterizing the cooled atoms have been introduced and calculated. We now discuss the physical content of these results. We first show (Section 7.1) that the momentum distribution (p, θ) can be interpreted as the solution of a rate equation describing competition between rate of entry and rate of departure. This provides a new insight into the sprinkling distribution SR(t) which appears as a ‘source term’ for the trapped atoms. We then consider the tails of the momentum distribution (Section 7.2) and we show that they appear as a steady-state or ‘quasi-steady’-state solution of the rate equation describing the evolution of the momentum distribution. On the contrary, in the central part of this distribution, atoms do not have the time to reach a steady-state or a quasi-steady-state because their characteristic evolution times are longer than the observation time θ. One can understand in this way the θ-dependence of the height of the peak of the cooled atoms (Section 7.3). We also investigate (Section 7.4) the important case where the jump rate R(p) does not exactly vanish when p = 0 and we show that, when θ is increased, there is a cross-over between a regime where Lévy statistics is relevant, as in the previous case, and a regime where a true steady-state can be reached for the whole momentum distribution.

In this chapter, we introduce the main concepts and tools of Lévy statistics that will be used in subsequent chapters in the context of laser cooling. In Section 4.1, we show how statistical distributions with slowly decaying power-law tails can appear in a physical problem. Then, in Section 4.2, we introduce the generalized Central Limit Theorem enabling one to handle statistically ‘Lévy sums’, i.e. sums of independent random variables, the distributions of which have power-law tails. We also sketch, in a part that can be skipped at first reading, the proof of the theorem and present a few mathematical properties concerning distributions with power-law tails and Lévy distributions. In Section 4.3, we present some properties of Lévy sums which will turn out to be crucial for the physical discussion presented in subsequent chapters: the scaling behaviour, the hierarchy and fluctuation problems. These properties are illustrated using numerical simulations. Finally, in Section 4.4, we present the distribution S(t), called the ‘sprinkling distribution’. This distribution presents unexpected features and will play an essential role in the following chapters.

Power-law distributions. When do they occur?

Situations where broad distributions appear and where rare events play a dominant role are more and more frequently encountered in physics, as well as in many other fields, such as geology, economy and finance. The term ‘broad distributions’ usually refers to distributions decaying very slowly for large deviations, typically as a power law, implying that some moments of the distribution are formally infinite.

The experimental discovery of Bose–Einstein condensation in trapped atomic clouds opened up the exploration of quantum phenomena in a qualitatively new regime. Our aim in the present work is to provide an introduction to this rapidly developing field.

The study of Bose–Einstein condensation in dilute gases draws on many different subfields of physics. Atomic physics provides the basic methods for creating and manipulating these systems, and the physical data required to characterize them. Because interactions between atoms play a key role in the behaviour of ultra cold atomic clouds, concepts and methods from condensed matter physics are used extensively. Investigations of spatial and temporal correlations of particles provide links to quantum optics, where related studies have been made for photons. Trapped atomic clouds have some similarities to atomic nuclei, and insights from nuclear physics have been helpful in understanding their properties.

In presenting this diverse range of topics we have attempted to explain physical phenomena in terms of basic principles. In order to make the presentation self-contained, while keeping the length of the book within reasonable bounds, we have been forced to select some subjects and omit others. For similar reasons and because there now exist review articles with extensive bibliographies, the lists of references following each chapter are far from exhaustive. A valuable source for publications in the field is the archive at Georgia Southern University: http://amo.phy.gasou.edu/bec.html

This book originated in a set of lecture notes written for a graduate level one-semester course on Bose-Einstein condensation at the University of Cop enhagen. We have received much inspiration from contacts with our colleagues in both experiment and theory. In particular we thank Gordon Baym and George Kavoulakis for many stimulating and helpful discussions over the past few years.

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.

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 many aspects 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. This field is a vast one, and in this chapter we shall touch briefly on selected topics.

In Sec. 13.1 we describe the classic interference experiment, in which two clouds of atoms are allowed to expand and overlap. Rather surprisingly, an interference pattern is produced even though initially the two clouds are completely isolated. We shall analyse the reasons for this effect. The marked decrease in density fluctuations in a Bose gas when it undergoes Bose–Einstein condensation is demonstrated in Sec. 13.2. Gaseous Bose–Einstein condensates can be manipulated by lasers, and this has made possible the study of coherent matter wave optics. We describe applications of these techniques to observe solitons, Bragg scattering, and non-linear mixing of matter waves in Sec. 13.3. The atom laser and amplification of matter waves is taken up in Sec. 13.4. How to characterize Bose–Einstein condensation microscopically is the subject of Sec. 13.5, where we also consider fragmented condensates.

Interference of two condensates

One of the striking manifestations of the wave nature of Bose–Einstein condensates is the observation of an interference pattern when two condensed and initially separated clouds are allowed to overlap. An example is shown in Fig. 13.1.

Atomic properties of the alkali atoms play a key role in experiments on cold atomic gases, and we shall discuss them briefly in the present chapter. 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. In Table 3.1 we list N, Z, and the nuclear spin quantum number I for some alkali atoms and hydrogen.

Bose–Einstein condensates in dilute atomic gases, which were first realized experimentally in 1995 for rubidium, sodium, and lithium, provide unique opportunities for exploring quantum phenomena on a macroscopic scale. These systems differ from ordinary gases, liquids, and solids in a number of respects, as we shall now illustrate by giving typical values of some physical quantities.

The particle density at the centre of a Bose-Einstein condensed atomic cloud is typically 1013–1015 cm−3. By contrast, the density of molecules in air at room temperature and atmospheric pressure is about 1019 cm−3. In liquids and solids the density of atoms is of order 1022 cm−3, while the density of nucleons in atomic nuclei is about 1038 cm−3.

To observe quantum phenomena in such low-density systems, the temperature must be of order 10−5 K or less. This may be contrasted with the temperatures at which quantum phenomena occur in solids and liquids. In solids, quantum effects become strong for electrons in metals below the Fermi temperature, which is typically 104–105 K, and for phonons below the De bye temperature, which is typically of order 102 K. For the helium liquids, the temperatures required for observing quantum phenomena are of order 1 K. Due to the much higher particle density in atomic nuclei, the corresponding degeneracy temperature is about 1011 K.

The path that led in 1995 to the first realization of Bose–Einstein condensation in dilute gases exploited the powerful methods developed over the past quarter of a century for cooling alkali metal atoms by using lasers.

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.