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

We report the experimental observation of strong multifractality in wave functions below the Anderson localization transition in open three-dimensional elastic networks. Our results confirm the recently predicted symmetry of the multifractal exponents. We have discovered that the result of multifractal analysis of real data depends on the excitation scheme used in the experiment.PACS numbers: 72.15. Rn, 05.70.Jk, 42.25.Dd, 71.30.+h Critical phenomena are of prominent importance in condensed-matter physics. Criticality at the Anderson localization transition has been the subject of intensive theoretical research. Some important theoretical predictions have been made, among which is the remarkable aspect of multifractality of wave functions at this transition. Numerical simulations support these predictions but also raise more questions [1]. Recent experimental progress has paved the way for the direct investigation of the Anderson localization transition at the mobility edge in real samples [2,3,4].In this Letter, we report the experimental observation of strong multifractality (MF) just below the Anderson transition. This observation is based on the excitation of elastic waves in an open 3D disordered medium. The recently predicted symmetry relation of the MF exponents [5] is tested and confirmed. All results are compared with the corresponding analysis of diffusive (metallic) wave functions in the same network at a different frequency or with a light speckle pattern generated by a strongly scattering medium, showing a very clear difference between localizing and diffusive regimes. Our results not only highlight the presence of MF in wave functions close to the mobility edge, but also reveal new aspects of the MF character in real experimental systems.Before presenting the experimental results, we briefly review some general aspects of MF and their implications in the context of the Anderson transition. Multifractality quantifies the strong fluctuations of the wave function. It shows the non-trivial length-scale dependence of the moments of the intensity distribution. The dependence can be investigated by varying the system size L, or alternatively, if the system size is fixed, by dividing the system into small boxes of linear size b and varying b. This property is quantified by using the generalized Inverse Participation Ratios (gIPR)where I(r) is the normalized intensity (equal to |ψ 2 (r)|/ |ψ 2 (r)|d d r where ψ(r) is the wave function) and I Bi is the integrated probability inside a box B i of linear size b, with λ ≪ b ≪ L where λ is the wavelength. The summation is performed on the whole sample, which consists of n = (L/b) d boxes, and d is the space dimension. By definition P 1 ≡ 1 and P 0 ≡ n.At criticality, the ensemble averaged gIPR, P q , scales anomalously with the dimensionless scaling length L/b aswhere d(q − 1) and ∆ q are called the normal (Euclidean) and the anomalous dimensions, respectively. For a normal (extended) wave function, ∆ q = 0 for every q. A (single-) fractal wave function with f...

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We investigate, both experimentally and theoretically, the effect of coupling between resonant scatterers on the transmission coefficient of a model system of isotropic scatterers. The model system consists of a monodisperse layer of bubbles, which exhibit a strong monopole scattering resonance at low ultrasonic frequencies. The layer was a true 2D structure obtained by injecting very monodisperse bubbles (with radius a approximately 100 microm) into a yield-stress polymer gel. Even for a layer with a low concentration of bubbles (areal fraction, n pi a(2), of 10-20%, where n is the number of bubbles per unit area), the ultrasonic transmission was found to be significantly reduced by the presence of bubbles (-20 to -50 dB) and showed a sharp minimum at a particular frequency. Interestingly, this frequency did not correspond to the resonance frequency of the individual, isolated bubbles, but depended markedly on the concentration. This frequency shift is an indication of strong coupling between the bubbles. We propose a simple model, based on a self-consistent relation, which takes into account the coupling between the bubbles and gives good agreement with the measured transmission coefficient.

Measurements of the phase velocity and attenuation of sound in concentrated samples of bubbly gels are presented. Hair gel was used as a matrix material to obtain well characterized distributions of bubbles. Ultrasonic measurements were conducted over a large range of frequencies, including the resonance frequencies of the bubbles. Surprisingly good agreement with Foldy's prediction was found, even for monodisperse samples at resonance frequencies, up to volume fraction of 1%. Beyond this concentration, the effects of high-order multiple scattering were observed. These results support the feasability of ultrasonic techniques to investigate the size distribution of bubbles in a weak gel or liquid.