This part extends quantum variational theory to continuum states. In particular, variational principles are developed for wave function continuity at specified energy, which is the usual context of scattering theory. Chapter 7, concerned with multiple scattering theory, lies somewhere between the theory of bound states and true scattering theory. Formalism appropriate to the latter is adapted to computing the electronic structure of large molecules and periodic solids, whose energy levels are determined by consistency conditions for wave function continuity. A variational formalism is derived for energy linearization. Chapter 8 develops variational principles and methods suitable for the true continuum problem of electron scattering at specified energy. Chapter 9 presents methodology, some very recent, that allows rotational and vibrational effects in electron–molecule scattering to be treated as a practicable extension of fixed-nuclei variational theory.

This part introduces variational principles relevant to the quantum mechanics of bound stationary states. Chapter 4 covers well-known variational theory that underlies modern computational methodology for electronic states of atoms and molecules. Extension to condensed matter is deferred until Part III, since continuum theory is part of the formal basis of the multiple scattering theory that has been developed for applications in this subfield. Chapter 5 develops the variational theory that underlies independent-electron models, now widely used to transcend the practical limitations of direct variational methods for large systems. This is extended in Chapter 6 to time-dependent variational theory in the context of independent-electron models, including linear-response theory and its relationship to excitation energies.

The idea that laws of nature should satisfy a principle of simplicity goes back at least to the Greek philosophers [436]. The anthropomorphic concept that the engineering skill of a supreme creator should result in rules of least effort or of most efficient use of resources leads directly to principles characterized by mathematical extrema. For example, Aristotle (De Caelo) concluded that planetary orbits must be perfect circles, because geometrical perfection is embodied in these curves: “… of lines that return upon themselves the line which bounds the circle is the shortest. That movement is swiftest which follows the shortest line”. Hero of Alexandria (Catoptrics) proved perhaps the first scientific minimum principle, showing that the path of a reflected ray of light is shortest if the angles of incidence and reflection are equal.

As theoretical physics and chemistry have developed since the great quantum revolution of the 1920s, there has been an explosive speciation of subfields, perhaps comparable to the late Precambrian period in biological evolution. The result is that these life-forms not only fail to interbreed, but can fail to find common ground even when placed in proximity on a university campus. And yet, the underlying intellectual DNA remains remarkably similar, in analogy to the findings of recent research in biology. The purpose of this present text is to identify common strands in the substrate of variational theory and to express them in a form that is intelligible to participants in these subfields. The goal is to make hard-won insights from each line of development accessible to others, across the barriers that separate these specialized intellectual niches.

Another great revolution was initiated in the last midcentury, with the introduction of digital computers. In many subfields, there has been a fundamental change in the attitude of practicing theoreticians toward their theory, primarily a change of practical goals. There is no longer a well-defined barrier between theory for the sake of understanding and theory for the sake of predicting quantitative data. Given modern resources of computational power and the coevolving development of efficient algorithms and widely accessible computer program tools, a formal theoretical insight can often be exploited very rapidly, and verified by quantitative implications for experiment. A growing archive records experimental controversies that have been resolved by quantitative computational theory.

In quantum electrodynamics (QED), the classical electromagnetic field Aμ of Maxwell and the electronic field ψ of Dirac are given algebraic properties (Bose–Einstein and Fermi–Dirac quantization, respectively), and through their interaction account for almost all physical phenomena that can be observed in ordinary human circumstances. The relativistic theory is derived from Hamilton's principle for an action defined by the space-time integral of a Lorentz invariant Lagrangian density [373]. This same action integral can be used to develop the diagrammatic perturbation theory of Feynman [121]. The cited references describe the formalism and methodology which demonstrate that QED is in remarkable agreement with all empirical data to which it is applicable. Classical and quantized QED will be used here to introduce the basic formalism of field theory, including the variational theory of invariance properties. This theory, especially gauge invariance, is central to recent developments of electroweak theory (EWT) and quantum chromodynamics (QCD).

Electron–molecule scattering data, observed experimentally or computed with methodology available as of 1980, was reviewed in detail by Lane [215]. If there were no nuclear motion, electron–molecule scattering would differ from electron–atom scattering only because of the loss of spherical symmetry and because of the presence of multiple Coulomb potentials due to the atomic nuclei. This is already a formidable challenge to theory, exemplified by the qualitative increase in computational difficulty and complexity between atomic theory and molecular theory for electronic bound states. While bound-state molecular computational methods have been extended to fixed-nuclei electron scattering [49, 178], an effective and computationally practicable treatment of rovibrational (rotational and vibrational) excitation requires a significant and historically challenging extension of bound-state theory.

Despite the simple and universal structure of the nonrelativistic Hamiltonian for N interacting electrons, it produces a broad spectrum of physical and chemical phenomena that are difficult to conceptualize within the full N-electron theory. Starting with the work of Hartree [162] in the early years of quantum mechanics, it was found to be very rewarding to develop a model of electrons that interact only indirectly with each other, through a self-consistent mean field. A deeper motivation lies in the fact that the relativistic quantum field theory of electrons is explicitly described by a field operator that corresponds more closely to a oneparticle model wave function than to that of the Schrödinger N-electron theory. The fundamental characterization of this electron field by Fermi–Dirac statistics is directly applicable to the mean-field theory, using concepts of statistical occupation numbers determined by effective one-electron orbital energy values. The variational theory appropriate to such independent-electron models is developed in this chapter.

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.