Coherent Backscattering Reveals the Anderson Transition

We develop an accurate finite-time scaling analysis of the angular width of the coherent backscattering (CBS) peak for waves propagating in 3D random media. Applying this method to ultracold atoms in optical speckle potentials, we show how to determine both the mobility edge and the critical exponent of the Anderson transition from the temporal behavior of the CBS width. Our method could be used in experiments to fully characterize the 3D Anderson transition.

Figure 1

Contour plot of the averaged momentum distribution of a matter wave, obtained after propagation of a plane wave $|{\mathit{k}}_0⟩$ (${\mathit{k}}_0=0.6{\widehat{\mathit{e}}}_x$) in a speckle potential of strength $V_0=1$ for a duration $t=800$. The propagated state is here filtered around energy $E=-0.4$ (metallic regime). The CBS peak, of angular width $2\mathrm{\Delta }{\theta }_{\mathrm{CBS}}$, is visible at $\mathit{k}=-{\mathit{k}}_0$. Here momenta, energies, and times are, respectively, in units of ${\zeta }^{-1}$, ${\hslash }^2/(m{\zeta }^2)$, and $m{\zeta }^2/\hslash$, where $\zeta$ is the correlation length of the potential.

Figure 2

Dynamics of the CBS peak across the Anderson transition. Left panel: Angular width $\mathrm{\Delta }{\theta }_{\mathrm{CBS}}$ versus time, in the metallic regime $E=-0.4>E_c$ (green points), at the mobility edge $E=E_c\simeq -0.48$ (red points), and in the insulating regime $E=-0.56<E_c$ (blue points). Right panels: Cut along $k_x$ of the normalized CBS profile at three different energies. For each energy, profiles at three different times, $t=2000$, 4000, and 8000, are displayed, shifted with respect to each other for clarity. The CBS width rapidly saturates in the insulating regime, while it shrinks in time in the metallic and critical regimes. We find an excellent agreement with the temporal dependences predicted by Eq. (1).

Figure 3

Left panel: Diffusion coefficient (in the metallic regime $E>E_c$) and inverse of the localization length $1/\xi (E)$ (in the insulating regime $E<E_c$) versus energy $E$. Red dots are obtained from an analysis of the CBS width, blue squares from the transfer-matrix method, and green diamonds from the spatial spreading of a wave packet. Right panel: Scaling function $\mathrm{\Lambda }$ versus energy around the ME, for various times ranging from $t=1280$ to $t=7680$. The curves cross at a common point $E_c\approx -0.48$, which signals the location of the ME. Points are the results of numerical simulations of CBS, while solid curves are fits of these data using Eq. (3).

Figure 4

Left panel: Scaling function $\mathrm{\Lambda }$ constructed by fitting the data for $\mathrm{\Lambda }$ with model Eq. (3). Points are the data, and the solid black curve is the fit. All data lie on the same master curve, in agreement with the one-parameter scaling hypothesis, Eq. (2). Right panel: $1/\tilde{\xi }(E)=|{\chi }_r(E){|}^{\nu }$ versus energy $E$ (solid curve), together with the confidence interval (shadowed region, green online). $1/\tilde{\xi }(E)$ vanishes at the ME, and is proportional to $|E-E_c{|}^{\nu }$ in its vicinity. The dashed curve is the prediction obtained from an independent finite-size scaling analysis based on the transfer-matrix approach.