Recurrent Scattering and Memory Effect at the Anderson Localization Transition

We report on ultrasonic measurements of the propagation operator in a strongly scattering mesoglass. The backscattered field is shown to display a deterministic spatial coherence due to a remarkably large memory effect induced by long recurrent trajectories. Investigation of the recurrent scattering contribution directly yields the probability for a wave to come back close to its starting spot. The decay of this quantity with time is shown to change dramatically near the Anderson localization transition. The singular value decomposition of the propagation operator reveals the dominance of very intense recurrent scattering paths near the mobility edge.

Figure 1

Experimental setup. Examples of a single scattering path (green dashed arrows), of a RS path (red solid arrows) and of a conventional MS path (orange dotted arrows) are drawn.

Figure 2

(a) Real part of the matrix $\mathbf{K}$ at time $t=185\mu \mathrm{s}$ and frequency $f=1.25\mathrm{MHz}$ for a given realization of disorder. (b) Real part of the RS contribution ${\mathbf{K}}_{\mathbf{R}}$. (c) Real part of the conventional MS contribution ${\mathbf{K}}_{\mathbf{M}}$. (d) Spatial mean intensity profiles at the same time and frequency.

Figure 3

(a) Time evolution of the RS ratio $I_R/I$ at $f=1.2\mathrm{MHz}$ (red circles), $f=1.225\mathrm{MHz}$ (green diamonds), and $f=1.8\mathrm{MHz}$ (blue squares). (b) Time evolution of the RS intensities $I_R(t)$, normalized by their values at time $t_0=90\mu \mathrm{s}$, at the same three frequencies, shown in a log-log scale. The corresponding fits with a power law (continuous lines) yield $I_R(t)\propto t^{-2.55\pm 0.13}$ at 1.8, $I_R(t)\propto t^{-2.04\pm 0.18}$ at 1.225 and $I_R(t)\propto t^{-0.95\pm 0.23}$ at 1.2 MHz.

Figure 4

(a) Average of the first three singular values $⟨{\lambda }_i⟩$ versus frequency at time $t=150\mu \mathrm{s}$. The experimental results (circles) are compared to the RMT predictions (lines). Error bars correspond to $\pm$ the standard error in the mean. (b) Back propagation of the first singular vector at the sample surface computed at each time for one realization of disorder at frequency $f=1.2\mathrm{MHz}$. $x$ represents the coordinate along the surface.