Experimental Observation of the Anderson Metal-Insulator Transition with Atomic Matter Waves

We realize experimentally an atom-optics quantum-chaotic system, the quasiperiodic kicked rotor, which is equivalent to a 3D disordered system that allows us to demonstrate the Anderson metal-insulator transition. Sensitive measurements of the atomic wave function and the use of finite-size scaling techniques make it possible to extract both the critical parameters and the critical exponent of the transition, the latter being in good agreement with the value obtained in numerical simulations of the 3D Anderson model.

Figure 1

Phase diagram of the quasiperiodic kicked rotor, from numerical simulations. The localized (insulator) region is shown in blue, the diffusive (metallic) region is shown in red. The experimental parameters are swept along the diagonal dash-dotted line. The insets show the experimentally observed momentum distributions, localized in the insulator region and Gaussian in the diffusive (metallic) region.

Figure 2

Temporal dynamics of the quasiperiodic kicked rotor. We measure the population ${\Pi }_0(t)$ of the zero-momentum class as a function of time and plot the quantity ${\Pi }_0^{-2}(t)$ (proportional to $⟨p^2⟩(t)$). Clearly, it tends to saturate in the localized regime (blue curve) and increases linearly with time in the diffusive regime (red curve). Close to the critical point (purple curve), it displays anomalous diffusion ${\Pi }_0^{-2}(t)\sim t^{2/3}$, as expected from theoretical arguments [24]: the critical behavior is well fitted by ${\Pi }_0^{-2}(t)=A+B{t}^{2/3}$ (dashed line).

Figure 3

Finite-size scaling applied to the results of numerical simulations (top) and to the experimental results (bottom). Graphs (a) and (c) emphasize that all data points corresponding to the quantity $\Lambda (k,t)={\Pi }_0^{-2}(K,t)t^{-2/3}$ obtained for various values of $K$ and $t$ can actually be described by a scaling function $f(X)$ depending only on the variable $X=\xi (K)t^{-1/3}$. The finite-size technique makes it possible to determine both $f(X)$, shown in (a) and (c), and the scaling parameter $\xi (K)$, shown in (b) and (d). Close to $K_c$, the behavior of $\xi (K)$ is well fitted by Eq. (2) (dashed lines), giving $K_c=6.6\pm 0.1$ (simulation) and $K_c=6.4\pm 0.2$ (experiment). The critical exponent is $\nu =1.60\pm 0.05$ (simulation) and $\nu =1.4\pm 0.3$ (experiment).