The configurations which guide the electromagnetic radiation and allow the interaction of light with matter vary considerably, with a wide variation of techniques developed over the years and employed today. We can distinguish between single-path and interferometric arrangements, and – as far as the interaction of light with the material under study is concerned – single-bounce from multiple-bounce, so-called resonant techniques. In this chapter we present a short summary.

Single-path arrangements sample the change of the electromagnetic wave if only one interaction with matter takes place; for instance if the light is reflected off the sample surface or is transmitted through the specimen in a single path. In general, part of the radiation is absorbed, and from this the optical properties of the material can be evaluated. However, only at low frequencies (i.e. at long wavelengths) does this simple configuration allow the determination of the phase change of the radiation due to the interaction; for higher frequencies only the attenuation in power is observed.

Interferometric techniques compare one part of the radiation, which undergoes the interaction with the material (i.e. reflection from or transmission through the material), with a second part of the signal, which serves as a reference. In this comparative approach – the so-called bridge configuration – the mutual coherence of the two beams is crucial. The interference of the two beams is sensitive to both the change in amplitude and in phase upon interaction, and thus allows calculation of the complex response of the specimen.

The role of electron–electron and electron–phonon interactions in renormalizing the Fermi-liquid state has been mentioned earlier. These interactions may also lead to a variety of so-called broken symmetry ground states, of which the superconducting ground state is the best known and most studied. The ground states are superpositions of electron–electron or electron–hole pairs all in the same quantum state with total momenta of zero or 2kF; these are the Cooper pairs for the superconducting case. There is an energy gap ∧, the well known BCS gap, introduced by Bardeen, Cooper, and Schrieffer [Bar57], which separates the ground state from the single-particle excitations. The states develop with decreasing temperature as the consequence of a second order phase transition.

After a short review of the various ground states, the collective modes and their response will be discussed. The order parameter is complex and can be written as ∧ exp{iφ}; the phase plays an important role in the electrodynamics of the ground state. Many aspects of the various broken symmetry states are common, but the distinct symmetries also lead to important differences in the optical properties. The absorption induced by an external probe will then be considered; it is usually discussed in terms of the so-called coherence effects, which played an important role in the early confirmation of the BCS theory. Although these effects are in general discussed in relation to the nuclear magnetic relaxation rate and ultrasonic attenuation, the electromagnetic absorption also reflects these coherence features, which are different for the various broken symmetry ground states.

With a few exceptions we have considered mainly bulk properties in the book. The physics of reduced dimensions is not only of theoretical interest, for many models can be solved analytically in one dimension only. A variety of interesting phenomena are bounded to restricted dimensions. On the other hand, fundamental models such as the theory of Fermi liquids developed for three dimensions break down in one or two dimensions. In recent years a number of possibilities have surfaced to explain how reduced dimensions can be achieved in real systems. One avenue is the study of real crystals with an extremely large anisotropy. The second approach considers artificial structures such as interfaces which might be confined further to reach the one-dimensional limit.

Dielectric response function in two dimensions

Reducing the dimension from three to two significantly changes many properties of the electron gas. If the thickness of the layer is smaller than the extension of the electronic wavefunction, the energy of the system is quantized (size quantization). We consider only the ground state to be occupied. For any practical case, just the electrons are confined to a thin sheet, while the field lines pass through the surrounding material which usually is a dielectric. A good approximation of a two-dimensional electron gas can be obtained in surfaces, semiconductor interfaces, and inversion layers; a detailed discussion which also takes the dielectric properties of the surrounding media into account can be found in [And82, Hau94].

Ever since Euclid, the interaction of light with matter has aroused interest – at least among poets, painters, and physicists. This interest stems not so much from our curiosity about materials themselves, but rather to applications, should it be the exploration of distant stars, the burning of ships of ill intent, or the discovery of new paint pigments.

It was only with the advent of solid state physics about a century ago that this interaction was used to explore the properties of materials in depth. As in the field of atomic physics, in a short period of time optics has advanced to become a major tool of condensed matter physics in achieving this goal, with distinct advantages – and some disadvantages as well – when compared with other experimental tools.

The focus of this book is on optical spectroscopy, defined here as the information gained from the absorption, reflection, or transmission of electromagnetic radiation, including models which account for, or interpret, the experimental results. Together with other spectroscopic tools, notably photoelectron and electron energy loss spectroscopy, and Raman together with Brillouin scattering, optics primarily measures charge excitations, and, because of the speed of light exceeding substantially the velocities of various excitations in solids, explores in most cases the Δq = 0 limit. While this is a disadvantage, it is amply compensated for by the enormous spectral range which can be explored; this range extends from well below to well above the energies of various single-particle and collective excitations.

Long ago, Paul Lévy invented a strange family of random walks – where each segment has a very broad probability distribution. These flights, when they are observed on a macroscopic scale, do not follow the standard Gaussian statistics. When I was a student, Lévy's idea appeared to me as (a) amusing, (b) simple – all the statistics can be handled via Fourier transforms – and (c) somewhat baroque: where would it apply?

As often happens with new mathematical ideas, the fruits came later. For example, é. Bouchaud proved that adsorbed polymer chains often behave like Lévy flights. In a very different sector, J.P. Bouchaud showed the role of Lévy distributions in risk evaluation. Now we meet a third major example, which is described in this book: cold atoms.

The starting point is a jewel of quantum physics: we think of an atom in a state of 0 translational momentum p = 0 (zero Doppler effect), inside a suitably prescribed laser field. For instance, with an angular momentum J = 1 we can have two ground states │+〉 and │−〉, and one excited state │0〉. The particular state │+〉+│−〉 has an admirable property: it is entirely decoupled from the radiation and can live for an indefinitely long time. It is thus possible to create a trap (around p = 0 in momentum space) in which the atoms will live for very long times: this so-called ‘ subrecoil laser cooling’ has been a major advance of recent years.

This book deals with the important developments that have recently occurred in two different research fields, laser manipulation of atoms on the one hand, non-Gaussian statistics and anomalous diffusion processes on the other hand. It turns out that fruitful exchanges of ideas and concepts have taken place between these two apparently disconnected fields. This has led to cross-fertilization of each of them, providing new physical insights into the most efficient laser cooling mechanisms as well as simple and mathematically soluble examples of anomalous random walks.

We thought that it would be useful to present in this book a detailed report of these developments. Our ambition is to try to improve the dialogue between different communities of scientists and, hopefully, to stimulate new, interesting developments. This book is therefore written as a case study accessible to the non-specialist.

Our aim is also to promote, within the atomic physics and quantum optics community, a way to approach and solve problems that is less based on exact solutions, but relies more on the identification of the physically relevant features, thus allowing one to construct simplified, idealized models and qualitative (and sometimes quantitative) solutions. This approach is of course common in statistical physics, where, often, details do not matter, and only robust global features determine the relevant physical properties. Laser cooling is an ideal case study, where the power of this methodology is clearly illustrated.

Motivation

As explained in Chapter 2, the Lévy statistics treatment of subrecoil laser cooling has been introduced after a series of simplifications, where we have dropped details of the quantum microscopic description to only keep the main features of the physical process. Such a way of reasoning is standard in statistical physics. It is difficult, however, to be sure a priori that one has not dropped important features, and the validity of the statistical approach needs to be checked. An important step of this verification, although not a rigorous proof, is to compare a posteriori the predictions of the statistical approach with experimental results as well as with the predictions of microscopic theoretical approaches. This chapter presents such comparisons.

We present in Section 8.2 the approaches (theoretical and experimental) to which our statistical approach can be compared. We then proceed to compare the results obtained by the different approaches. First, in Section 8.3, we treat in detail the predictions for a global quantity, the proportion of trapped atoms. This is done for the three recycling models introduced in this work, in the one-dimensional case. Then, in Section 8.4, we study another physical quantity with a richer content, the momentum distribution of cooled atoms. In Section 8.5, we investigate the influence of the dimensionality of the problem, and the role of friction during the recycling periods – which are crucial predictions of the Lévy statistics analysis.

In this chapter, we first recall (in Section 2.1) a few properties of the most usual laser cooling schemes, which involve a friction force. In such standard situations, the motion of the atom in momentum space is a Brownian motion which reaches a steady-state, and the recoil momentum of an atom absorbing or emitting a single photon appears as a natural limit for laser cooling. We then describe in Section 2.2 some completely different laser cooling schemes, based on inhomogeneous random walks in momentum space. These schemes, which are investigated in the present study, allow the ‘recoil limit’ to be overcome. They are associated with non-ergodic statistical processes which never reach a steady-state. Section 2.3 is devoted to a brief survey of various quantum descriptions of subrecoil laser cooling, which become necessary when the ‘recoil limit’ is reached or overcome. The most fruitful one, in the context of this work, is the ‘quantum jump description’ which will allow us in Section 2.4 to replace the microscopic quantum description of subrecoil cooling by a statistical study of a related classical random walk in momentum space. It is this simpler approach that will be used in the subsequent chapters to derive some quantitative analytical predictions, in cases where the quantum microscopic approach is unable to yield precise results, in particular in the limit of very long interaction times, and/or for a momentum space of dimension D higher than 1.

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).