Anderson Mobility Gap Probed by Dynamic Coherent Backscattering

Abstract

We use dynamic coherent backscattering to study one of the Anderson mobility gaps in the vibrational spectrum of strongly disordered three-dimensional mesoglasses. Comparison of experimental results with the self-consistent theory of localization allows us to estimate the localization (correlation) length as a function of frequency in a wide spectral range covering bands of diffuse transport and a mobility gap delimited by two mobility edges. The results are corroborated by transmission measurements on one of our samples.A quantum particle is trapped in a three-dimensional (3D) disordered potential if its energy E is lower than the so-called mobility edge (ME) E c . As was discovered by Philip Anderson in 1958, quantum interferences may increase E c to values that are much larger than the classical percolation threshold, an energy below which a classical particle would be trapped [1,2]. The link between E c and the statistical properties of disorder has been recently studied in experiments with ultracold atoms in random optical potentials [3,4]. In contrast to quantum particles, classical waves-light or sound-may be Anderson localized by disorder only in a band of intermediate energies (or frequencies), the impact of disorder becoming weak in both high-and low-frequency limits [5,6]. One thus expects a mobility 'gap' delimited by two MEs instead of a single ME. This is due to the difference between dispersion relations of quantum and classical waves [7,8]. Resonant scattering may further complicate the spectrum by shifting the mobility gap or splitting it into several narrower ones. Mobility gaps can also exist for quantum particles when the disordered potential is superimposed on a periodic one-a common situation for electrons in crystals with impurities [9]. In the present Letter we report the first experimental observation of a mobility gap for classical waves. To this end we take full advantage of experimental techniques available for classical waves but very difficult, if not impossible, to put in practice for quantum particles and, in particular, for electrons in disordered conductors. We perform frequency-, time-, position-and angular-resolved ultrasonic reflection and transmission experiments in strongly disordered 'mesoglasses'-elastic networks of brazed aluminum beads. The results are compared with the self-consistent theory of localization to precisely locate the two MEs and to estimate the localization length ξ throughout the mobility gap. ξ diverges at the MEs, as expected.

Finite-size scaling analysis of localization transition for scalar waves in a three-dimensional ensemble of resonant point scatterers

“…The frequency range between the mobility edges roughly corresponds to a band of localized quasimodes (or, equivalently, a mobility gap) discovered in our previous work [18]. The existence of a mobility gap between two mobility edges instead of a single mobility edge separating extended states at high energies (frequencies) from localized states at low energies (frequencies) is typical for resonant scattering [9]. It may be tempting to use the numerical data of Fig.…”

Coherent backscattering and forward-scattering peaks in the quantum kicked rotor

“…At longer times, theoretical arguments predict the emergence of a second peak centered at k 0 , the coherent forward scattering (CFS) peak, on a time scale given by the Heisenberg time associated to a localization volume [24,25,26,28]. For time reversal symmetric systems, the CBS/CFS peaks settle to a twin structure in the localized regime, and both can be used to extract the critical properties of the 3D AT [27,28] as exemplified by [32] for CBS. It is worth noticing that, contrary to the CBS peak that disappears when time reversal symmetry is broken, the CFS peak is robust and exists in other symmetry classes [33].…”

Dynamic coherent backscattering of ultrasound in three-dimensional strongly-scattering media

“…Theoretical backscattering profiles I theory (ρ, D B t) were calculated as a function of parameter D B t (diffusion coefficient multiplied by time) using equations (5), (8), and (13). Then, all experimental CBS profiles were compared to all theoretical profiles, i.e., for each time t exp , I exp (ρ, t exp ) was fitted with each theoretical CBS profile I theory (ρ, D B t).…”

“…The sample has a cross-section of 230×250 mm 2 much larger than its thickness L = 25 ± 2 mm, which helps to minimize contributions from the edges of the sample when performing backscattering experiments. Other details of the sample characteristics have been described in References [12][13][14].…”

“…The result of experiments and data-processing is a large set of configurationally-averaged, time-dependent backscattered intensity profiles I(ρ, t), where ρ is the distance between source and receiver elements of the ultrasonic array, and t is time. To eliminate the effect of absorption, I(ρ, t) was normalized by I(0, t), since at time t the effect of absorption is the same for both numerator and denominator of this ratio, and therefore should cancel [13,17].…”