ÉCOLE DOCTORALE : Ondes et Matière Laboratoire

The transport properties of a coherent wave in a disordered medium are inherently determined by interference of multiple scattering paths, which can lead to spatial localization and absence of diffusion. This effect, known as Anderson localization, was first predicted for electrons in disordered crystals and then extended to classical waves, which permitted its observation in a variety of systems. The most fundamental features of Anderson localization are therefore ubiquity and universality. However, observable features can depend on the details of the system. Here we make an introduction to weak and strong localization effects. We introduce background concepts, such as the scaling theory of localization, and briefly discuss interplay between disorder and interactions, and we show that correlated disorder can significantly alter usual features of Anderson localization. The recent advent of ultra-cold atomic systems, which are under great experimental control is reviewed. They offer new possibilities to study those problems. Résumé Les propriétés de transport d’une onde cohérente en milieu désordonné sont déterminées par l’interférence des chemins de diffusion multiple, ce qui mène à la localisation spatiale et à l’absence de diffusion dans le milieu. Ce phénomène, connu sous le nom de localisation d’Anderson, a d’abord été prédit pour des électrons dans des cristaux désordonnés, avant d’être étendu au cas des ondes classiques, ce qui a permis son observation dans différents systèmes. Les caractéristiques les plus fondamentales de la localisation d’Anderson sont donc l’ubiquité et l’universalité. Cependant, ses caractéristiques observables peuvent dépendre des détails du système. Ici, nous présentons les effets de localisation faible et forte. Nous introduisons aussi des concepts de base, comme la théorie de scaling de la localisation, nous discutons brièvement comment désordre et interactions peuvent se combiner, et nous montrons que les corrélations du désordre peuvent modifier significativement les caractéristiques usuelles de la te l-0 07 96 77 7, v er si on 1 5 M ar 2 01 3 16 Chap. 1 Waves in disorder: from condensed matter to ultra-cold atoms localisation d’Anderson. L’avènement récent des systèmes d’atomes ultrafroids, qui sont très bien contrôlés exprimentalement, est passé en revue. Ils offrent de nouvelles possibilités pour étudier ces problèmes. te l-0 07 96 77 7, v er si on 1 5 M ar 2 01 3 1.1 From condensed matter systems to waves in disordered media 17 Introduction Disorder is always present at the microscopic scale in natural media, it is for example the result of inhomogeneities or impurities. Hence, by definition, each piece of matter is different, and should be modeled differently. Disorder is therefore usually viewed as non-desirable, and may be neglected in order to deal with generic models. This approach is often successful in describing macroscopic behaviour, and the microscopic disorder is then seen as a source of uncertainty in the results of physical measurements. However, it is now well-known that disorder can have dramatic effects at the macroscopic scale, in some cases. An emblematic and fascinating example is Anderson localization, in which weak disorder can turn a piece of metal into an insulator (at least in low dimension). The concept of localization of particles by disorder has been introduced by Anderson in 1958 for electrons in solids [2]. It was later realized that it results from a subtle interference effect that concerns all types of coherent waves propagating in a random medium [11]. The concept has therefore spread to many other fields of physics: electromagnetic [29], optical [5,6], acoustic [30] and seismic [31] waves, but also disordered superconductors [32] and superfluid Helium in porous media [33], where it has been studied both theoretically and experimentally. As we will see, this problem is also very interesting to study with ultra-cold atoms, for the parameters of those systems are very well controlled. While the first examples concern classical waves, the three latter concern quantum waves (electrons, Helium atoms and ultracold atoms), which can be interacting. Depending on the situation, disorder and interactions can compete or cooperate for localization, and their interplay is a difficult and interesting problem. For studying localization, it is important to understand the non-interacting problem, which has been studied a lot in optics [5,6] and more recently in acoustics [30]. Then one has to consider the role of interactions. The recent development of ultracold atomic gases is a great asset to study both problems [12, 13, 21]: They make very manipulable matterwave systems with tunable interactions and to which controlled disorder can be applied. In this thesis, we focus on Anderson localization in noninteracting systems. The aim of this chapter is to review the above concepts, mainly about Anderson localization and briefly about the role of interactions. We first describe the link between disordered condensed matter systems and other types of waves in random media, in Sec. 1.1. We then give a few basic elements of coherent propagation of waves in random media, and an introduction to Anderson localization, in Sec. 1.2. Eventually, in Sec. 1.3, we describe ultracold atomic systems, and how they are useful for studying the effect of disorder in quantum systems. 1.1 From condensed matter systems to waves in disordered media 1.1.1 Disorder in solids In molecules or solid state systems, atoms are strongly bound to each other as a result of the sharing of electrons. In usual solids, they are fixed in space and arranged in a regular structure (crystal), as is the case for water ice or quartz (see illustration on Fig. 1.1). This regular arrangement is then the background medium for the propagation of the ’shared’ electrons. In the quantum theory of solids, the periodicity of the potential felt by the electrons is te l-0 07 96 77 7, v er si on 1 5 M ar 2 01 3 18 Chap. 1 Waves in disorder: from condensed matter to ultra-cold atoms Figure 1.1: Structures of crystalline SiO2 (’quartz’), and amorphous SiO2, which is the standard glass. The structure of quartz is periodic, whereas the structure of glasses are disordered (i.e. they show no longrange order). Image from [36]. essential [34, 35]. Their eigenstates are then the so-called ’Bloch waves’, which are extended over the whole system, and they are associated to an electronic band structure, which permitted to successfully understand the insulating or metallic behaviour of some materials, by the full or partial filling of the bands [34]. However such perfect layouts of atoms are idealized objects [37, 38], they hardly exist as such. In practice every arrangement of atoms is subject to localized defects: impurities (e.g. one atom is substituted by another species), vacancies or additional atoms, as well as dislocations of the cristalline structure. In addition, perfectly ordered lattice models are also inappropriate to describe another category of solid state systems: amorphous materials, in which the disorder is structural. In this class of systems the atoms are tightly bound in an irregular arrangement. It is the case of glasses for example, see Fig. 1.1. Disorder seen by the conduction electrons in condensed matter can therefore have multiple origins [38], which are not always well-known, or at least not under experimental control. The goal of the physicist is then to describe those systems as generally as possible by looking for properties that are common to a number of disordered materials. To do so one needs to classify the systems by type and ’amount’ of disorder, and proceed to statistical averaging. 1.1.2 Link with other waves In solid-state systems, even at room temperature, the electrons are close to quantum degeneracy. It means that they cannot be described as a gas of classical particles, and one has to take into account their quantum nature. The study of a disordered material is then the study of the propagation of the electronic wave function in a random medium. For the sake of simplicity, let us consider a random potential V (r), which will be the relevant case of the study of this thesis. This problem is therefore closely connected with the propagation of other types of waves in random media, and ’classical’ waves in particular [39–41], as we discuss now on a simple example. Consider first a particle of wave function ψ(r) whose eigenstate of energy E in a random te l-0 07 96 77 7, v er si on 1 5 M ar 2 01 3 1.1 From condensed matter systems to waves in disordered media 19 potential V (r) is described by the Schrödinger equation