Critical dynamics at the Anderson localization mobility edge

We study the critical dynamics of matter waves at the three-dimensional Anderson mobility edge in cold-atom disorder quench experiments. General scaling arguments are supported by precision numerics for the spectral function, diffusion coefficient, and localization length in isotropic blue-detuned speckle potentials. We discuss signatures of critical slowdown in the time-dependent central column density of a spreading wave packet, and evaluate the prospects of observing anomalous diffusion right at criticality.

Figure 1

Main plot: Spectral function $A(0,E)$ of a matter wave with $\mathbf{k}=0$ inside a blue-detuned laser speckle potential of zero mean and rms strength $V_0=0.5E_{\zeta }$; the unit of energy $E_{\zeta }={\hslash }^2/m{\zeta }^2$ is set by a generic spatial Gaussian correlation length $\zeta$. The 3D Anderson mobility edge at $E_{\text{c}}\approx -0.1652E_{\zeta }$ (red vertical line) lies below the “sea level” or potential mean at $E=0$. The critical interval with half width $W=0.05E_{\zeta }$ (dashed vertical lines) contains initially more than two thirds of atoms; less than half of all atoms, with energy $E<E_{\text{c}}$, will eventually be localized. Insets: Zoom in the critical interval with plots of the smooth average density of states $\rho \left(E\right)$, critical inverse localization length ${\xi }_E^{-1}$, and critical diffusion coefficient $D_E$, in units of $l_{\text{c}}$ and $D_{\text{c}}=\hslash /3m$.

Figure 2

Energy interval around the mobility edge $E_{\text{c}}$ and different dynamical regimes. Numerical figures for the respective fractions are indicated for the showcase data of Fig. 1. The half width of the anomalous interval shrinks as $\delta E\left(t\right)=W{({\tau }_{\text{c}}/t)}^{1/3\nu }$, Eq. (7). A spatial cutoff scale ${\sigma }_0$ (width of the initial wave packet if the central column density is monitored) introduces two crossover times: (1) The upper critical Thouless time $t_1={\sigma }_0^2/D_{\text{c}}$, Eq. (8), separates regular from critical diffusion. (2) The localization time $t_2=t_1{\sigma }_0/l_{\text{c}}$, Eq. (9), separates critical from anomalous dynamics.

Figure 3

Central column density, Eq. (13), evaluated for the data of Fig. 1, as function of time. Around the upper critical Thouless time $t_1={\sigma }_0^2/D_{\text{c}}$, the fraction of regular diffusive atoms (here $f_{\text{diff}}^{\text{reg}}\approx 0.31$) has almost disappeared, and the slow critical diffusion of Eq. (17) starts to dominate. Anomalous diffusion becomes only relevant at much longer times $t\gg t_2={\sigma }_0{t}_1/l_{\text{c}}$. Especially with noise on the data, one could be tempted to extrapolate the curve to a localized fraction of 0.52 or more; in reality, it is only $f_{\text{loc}}=0.457$, indicated by the shaded baseline.