How to use this book xv 1 Introduction: mesoscopic physics 1 1.1 Interference and disorder 1 1.2 The Aharonov-Bohm effect 4 1.3 Phase coherence and the effect of disorder 7 1.4 Average coherence and multiple scattering 9 1.5 Phase coherence and seif-averaging: universal fluctuations 12 1.6 Spectral correlations 14 1.7 Classical probability and quantum crossings 15 1.7.1 Quantum crossings 17 1.8 Objectives 18 2 Wave equations in random media 31 2.1 Wave equations 2.1.1 Electrons in a disordered metal 31 2.1.2 Electromagnetic wave equation-Helmholtz equation 32 2.1.3 Other examples of wave equations 2.2 Models of disorder 36 2.2.1 The Gaussian model 2.2.2 Localized impurities: the Edwards model 39 2.2.3 The Anderson model Appendix A2.1: Theory of elastic collisions and single scattering A2.1.1 Asymptotic form of the solutions 44 A2.

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2014 Nous présentons une étude théorique de la rétrodiffusion cohérente de la lumière par un milieu désordonné dans diverses situations incluant les effets dépendant du temps, les milieux absorbants et les effets liés à la modulation d'amplitude de la lumière. Nous discutons tout particulièrement le cas de la diffusion anisotrope et les effets de la polarisation afin d'expliquer quantitativement les résultats expérimentaux. Nous donnons un calcul microscopique de l'albedo cohérent afin de justifier la relation heuristique précédemment établie. Nous prédisons aussi la forme de l'albedo cohérent d'un milieu fractal. Enfin, la validité des différentes approximations utilisées est discutée et quelques développements ultérieurs sont évoqués. Abstract. 2014 A theoretical study of the coherent backscattering effect of light from disordered semi-infinite media is presented for various situations including time-dependent effects as well as absorption and amplitude modulation. Particular attention is devoted to the case of anisotropic scattering and to polarization in order to explain quantitatively experimental results. A microscopic derivation of the coherent albedo is given which strongly supports the heuristic formula previously established. In addition the coherent albedo of a fractal system is predicted. The validity of the different approximations used are discussed and some further theoretical developments are presented.

We study the spectral determinant of the Laplacian on finite graphs characterized by their number of vertices V and bonds B. We present a path integral derivation which leads to two equivalent expressions of the spectral determinant of the Laplacian in terms of either a V_V vertex matrix or a 2B_2B link matrix that couples the arcs (oriented bonds) together. This latter expression allows us to rewrite the spectral determinant as an infinite product of contributions of periodic orbits on the graph. We also present a diagrammatic method that permits us to write the spectral determinant in terms of a finite number of periodic orbit contributions. These results are generalized to the case of graphs in a magnetic field. Several examples illustrating this formalism are presented and its application to the thermodynamic and transport properties of weakly disordered and coherent mesoscopic networks is discussed. 2000Academic Press

Photon propagation in a gas of N atoms is studied using an effective Hamiltonian describing photon-mediated atomic dipolar interactions. The density P(Gamma) of photon escape rates is determined from the spectrum of the NxN random matrix Gamma_{ij}=sin(x_{ij})/x_{ij}, where x_{ij} is the dimensionless random distance between any two atoms. Varying disorder and system size, a scaling behavior is observed for the escape rates. It is explained using microscopic calculations and a stochastic model which emphasizes the role of cooperative effects in photon localization and provides an interesting relation with statistical properties of "small world networks."

We study numerically the spectrum of the non-Hermitian effective Hamiltonian that describes the dipolar interaction of a gas of N » 1 atoms with the radiation field. We analyze the interplay between cooperative effects and disorder for both scalar and vectorial radiation fields. We show that for dense gases, the resonance width distribution follows, both in the scalar and vectorial cases, a power law P (F) ~ r _4/3 that originates from cooperative effects between more than two atoms. This power law is different from the P(T) ~ F _l behavior, which has been considered as a signature of Anderson locahzation of light in random systems. We show that in dilute clouds, the center of the energy distribution is described by Wigner's semicircle law in the scalar and vectorial cases. For dense gases, this law is replaced in the vectorial case by the Laplace distribution. Finally, we show that in the scalar case the degree of resonance overlap increases as a power law of the system size for dilute gases, but decays exponentially with the system size for dense clouds.

It has been realized that fractals may be characterized by complex dimensions, arising from complex poles of the corresponding zeta function, and we show here that these lead to oscillatory behavior in various physical quantities. We identify the physical origin of these complex poles as the exponentially large degeneracy of the iterated eigenvalues of the Laplacian, and discuss applications in quantum mesoscopic systems such as oscillations in the fluctuation Σ 2 (E) of the number of levels, as a correction to results obtained in Random Matrix Theory. We present explicit expressions for these oscillations for families of diamond fractals, also studied as hierarchical lattices.PACS numbers: 05.45.Df, 05.60. Gg, Fractals, such as the well-known Sierpinski gasket, have been thoroughly studied in physics and in mathematics. In addition to their own intriguing properties, they provide a useful testing ground to investigate properties of disordered classical or quantum systems [1], addressing such fundamental physical issues as Anderson localization, the renormalization group, and phase transitions [2]. In addition to condensed matter and statistical physics, fractals have been considered in other contexts such as gravitational systems [3,4], and in quantum field theory [5]. Despite the large amount of work dedicated to the study of the spectra of deterministic fractals, explicit expressions for spectral functions such as heat kernels or spectral zeta functions, from which many physical quantities can be derived, have remained elusive. It is well known that the heat kernel Z(t) and zeta function ζ(s) play central roles in various fields of physics: from mesoscopic physics [6], to black holes [7], to quantum field theory on curved spaces such as de Sitter and anti De Sitter spaces [8], to the physics of the Casimir effect [9]. This is largely due to their relation to the notion of the partition function in statistical physics [10], and to the ubiquity of Schwinger's proper-time formalism [11].An important step was to identify the leading contribution to Weyl's small time expansion of Z(t), showing that it is determined by the fractal's spectral dimension d s [12], rather than by its fractal (Hausdorff) dimension d h , as initially conjectured. The fact that fractals are characterized by a set of more than one dimension, as opposed to standard Euclidean spaces, illustrates the richness and peculiarity of self-similar structures. Spectral properties of deterministic fractals have recently been considered anew in mathematics, and the notion of complex valued fractal dimensions has been introduced [12, 13], leading to new results for the zeta function [14][15][16]. In this Let- * On leave from Department of Physics, Technion Israel Institute of Technology, 32000 Haifa, Israel.ter we use and extend these results to study the resulting (log-periodic) oscillations in the heat kernel and related physical quantities. We illustrate these ideas with a special class of fractals known as diamond fractals. These diamond fractals per...

Sequences of alternating-sign time-dependent electric field pulses lead to coherent interference effects in Schwinger vacuum pair production, producing a Ramsey interferometer, an all-optical time-domain realization of the multiple-slit interference effect, directly from the quantum vacuum. The interference, obeying fermionic quantum statistics, is manifest in the momentum dependence of the number of produced electrons and positrons along the linearly polarized electric field. The central value grows like N 2 for N pulses [i.e., N "slits"], and the functional form is well-described by a coherent multiple-slit expression. This behavior is generic for many driven quantum systems.