After exactly half a century of Anderson localization, the subject is more alive than ever. Direct observation of Anderson localization of electrons was always hampered by interactions and finite temperatures. Yet, many theoretical breakthroughs were made, highlighted by finite-size scaling, the self-consistent theory and the numerical solution of the Anderson tight-binding model. Theoretical understanding is based on simplified models or approximations and comparison with experiment is crucial. Despite a wealth of new experimental data, with microwaves, light, ultrasound and cold atoms, many questions remain, especially for three dimensions. Here we report the first observation of sound localization in a random three-dimensional elastic network. We study the time-dependent transmission below the mobility edge, and report ``transverse localization'' in three dimensions, which has never been observed previously with any wave. The data are well described by the self-consistent theory of localization. The transmission reveals non-Gaussian statistics, consistent with theoretical predictions.Comment: Final published version, 5 pages, 4 figure

As discovered by Philip Anderson in 1958, strong disorder can block propagation of waves and lead to the localization of wavelike excitations in space. Anderson localization of light is particularly exciting in view of its possible applications for random lasing or quantum information processing. We show that, surprisingly, Anderson localization of light cannot be achieved in a random three-dimensional ensemble of point scattering centers that is the simplest and widespread model to study the multiple scattering of waves. Localization is recovered if the vector character of light is neglected. This shows that, at least for point scatterers, the polarization of light plays an important role in the Anderson localization problem.

Abstract. We use near-infrared dynamic multiple scattering of light ͓diffusing-wave spectroscopy ͑DWS͔͒ to detect the activation of the somato-motor cortex in 11 right-handed volunteers performing a finger opposition task separately with their right and left hands. Temporal autocorrelation functions g ͑1͒ ͑r , ͒ of the scattered light field are measured during 100-s periods of motor task alternating with 100-s resting baseline periods. From an analysis of the experimental data with an analytical theory for g ͑1͒ ͑r , ͒ from a three-layer geometry with optical and dynamical heterogeneity representing scalp, skull, and cortex, we obtain quantitative estimates of the diffusion coefficient in cortical regions. Consistent with earlier results, the measured cortical diffusion coefficient is found to be increased during the motor task, with a strong contralateral and a weaker ipsilateral increase consistent with the known brain hemispheric asymmetry for right-handed subjects. Our results support the interpretation of the increase of the cortical diffusion coefficient during finger opposition being due to the functional increase in cortical blood flow rate related to vasodilation.

We study the effect of Anderson localization on the expansion of a Bose-Einstein condensate, released from a harmonic trap, in a 3D random potential. We use scaling arguments and the selfconsistent theory of localization to show that the long-time behavior of the condensate density is controlled by a single parameter equal to the ratio of the mobility edge and the chemical potential of the condensate. We find that the two critical exponents of the localization transition determine the evolution of the condensate density in time and space. Recently there has been much interest in the possibility of observing Anderson localization of Bose-Einstein condensates obtained by trapping and cooling bosonic atoms [5,6,7,8]. A Bose-Einstein condensate is characterized by a macroscopic occupation of a single quantum state [9] and hence exhibits quantum, wave-like behavior despite its macroscopic size. Atomic Bose-Einstein condensates subjected to random external (optical) potentials are potentially good candidates for observing Anderson localization of matter waves. Up to now, the experiments have focused on 1D configurations [5], where all single-particle eigenstates are localized. In a typical experiment, the condensate is created in an optical or magneto-optical trap. The trap is then turned off and the condensate is allowed to expand.In this Letter we study the expansion of the BoseEinstein condensate in a 3D random potential. Unlike in 1D, a critical energy (the mobility edge ǫ c ) exists in 3D which separates extended and localized states. An eigenstate is extended (localized) if the corresponding energy is larger (smaller) than ǫ c . When the condensate is released from the trap, the atoms achieve kinetic energies up to the chemical potential µ of the trapped condensate. For weak disorder ǫ c < µ, and a fraction of atoms diffuses away, whereas the remainder is localized, as was pointed out in [7]. We show here that, surprisingly, even for strong disorder ǫ c > µ only a fraction of the condensate will be localized. We study the full dynamics of the condensate expansion by accounting for weak localization at energies ǫ > ǫ c , strong localization at ǫ < ǫ c , and critical behavior around the mobility edge. Our main result is that the effect of disorder on the expansion of the condensate is controlled by a single parameter ǫ c /µ, and that Anderson localization plays an important role even when the chemical potential of the condensate µ is much larger than the mobility edge ǫ c . We show that the behavior of the average condensate densityn(r, t) at large distances r and long times t is governed by the critical exponents ν and s of the localization transition. This could provide a direct way to measure these exponents. The density of the localized part of the condensaten(r, ∞) does not decay exponentially with r, as one could have expected, but follows a power law.Consider a Bose-Einstein condensate of N ≫ 1 atoms of mass m trapped in a 3D spherically-symmetric harmonic potential V ω (r), characterized by the trap freque...

We discover a transition from extended to localized quasi-modes for light in a gas of immobile two-level atoms in a magnetic field. The transition takes place either upon increasing the number density of atoms in a strong field or upon increasing the field at a high enough density. It has many characteristic features of a disorder-driven (Anderson) transition but is strongly influenced by near-field interactions between atoms and the anisotropy of the atomic medium induced by the magnetic field.The transition from extended to localized eigenstates upon increasing disorder in a quantum or wave system is called after Philip Anderson who was the first to predict it for electrons in disordered solids [1]. More recently, this transition was studied for various types of quantum particles (cold atoms [2], Bose-Einstein condensates [3]) as well as for classical waves (light [4-6], ultrasound [7,8]). In the most common case of time-reversal symmetric systems invariant under spin rotation Anderson transition takes place for a three-dimensional (3D) disorder only, eigenstates of low-dimensional systems being always localized [9,10]. Anderson localization of light may find applications in the design of future quantuminformation devices [11], miniature lasers [12] and solar cells [13]. However, no undisputable experimental observation of optical Anderson transition in 3D exists to date since alternative explanations were proposed for all published reports of it [14][15][16]. Moreover, we have recently shown that the simplest theoretical model in which light is scattered by point scatterers (atoms) does not predict Anderson localization of light at all [17].In the present Letter we show that an external magnetic field may induce a transition between extended and localized states for light in a gas of cold atoms. Magnetic field is an important and unique means of controlling wave propagation in disordered media. On the one hand, it breaks down the time-reversal invariance leading to a suppression of weak localization in electronic [18] and optical [19] systems and to metal-insulator transitions in topological insulators [20]. On the other hand, by profoundly modifying the scattering properties of individual scatterers the magnetic field can produce an enhancement of the coherent backscattering peak for light scattered by atoms with a degenerate ground state [21,22]. Our work adds a new element in the mosaic of magnetic-field-induced phenomena in disordered systems by demonstrating that the removal of degeneracy of the excited atomic state due to the Zeeman effect and the resulting reduction of the strength of resonant dipole-dipole * Sergey. We consider an ensemble of N 1 identical two-level atoms at random position {r i } inside a spherical volume V of radius R. The resonant frequency ω 0 of atoms defines the natural length scale 1/k 0 = c/ω 0 , where c is the vacuum speed of light. The ground state |g i of an isolated atom i is nondegenerate with the total angular momentum J g = 0, whereas the excited states |e i is three-fo...

We study probability distributions of eigenvalues of Hermitian and non-Hermitian Euclidean random matrices that are typically encountered in the problems of wave propagation in random media.

We develop a theory for the eigenvalue density of arbitrary non-Hermitian Euclidean matrices. Closed equations for the resolvent and the eigenvector correlator are derived. The theory is applied to the random Green's matrix relevant to wave propagation in an ensemble of pointlike scattering centers. This opens a new perspective in the study of wave diffusion, Anderson localization, and random lasing.

We develop a self-consistent theoretical approach to the dynamics of Anderson localization in open three-dimensional (3D) disordered media. The approach allows us to study time-dependent transmission and reflection, and the distribution of decay rates of quasimodes of 3D disordered slabs near the Anderson mobility edge.