Experimental Observation of Two-Dimensional Anderson Localization with the Atomic Kicked Rotor

Dimension 2 is expected to be the lower critical dimension for Anderson localization in a time-reversal-invariant disordered quantum system. Using an atomic quasiperiodic kicked rotor—equivalent to a two-dimensional Anderson-like model—we experimentally study Anderson localization in dimension 2 and we observe localized wave function dynamics. We also show that the localization length depends exponentially on the disorder strength and anisotropy and is in quantitative agreement with the predictions of the self-consistent theory for the 2D Anderson localization.

Figure 1

Experimentally recorded momentum distributions for the kicked rotor exposed to a quasiperiodic driving, Eq. (1). (a) $K=5.34$, $ꝁ=2.89$, $ϵ=0.36$, 0 to 1000 kicks (step 200). The momentum distribution diffusively broadens at short times and freezes at longer times, proving the existence of 2D Anderson localization. Time increases from top to bottom curves. (b) Momentum distribution at 200 kicks in log scale, showing the exponential shape characteristic of localization. The circles are experimental points, the blue dashed line is a Gaussian fit and the black solid line an exponential fit for $|p|>3(2\hslash k_L)$. (c) Localized momentum distributions after 1000 kicks, as a function of the anisotropy parameter $ϵ$, for $K=5.34$, $ꝁ=2.89$ as in (a) and (b). The modulation amplitude $ϵ$ increases from top to bottom curves. The rapid increase of the localization length shows the evolution from the 1D localization at $ϵ=0$ to the truly 2D Anderson localization. Note the different horizontal scales in the various plots.

Figure 2

Kinetic energy $E_{\text{kin}}$ of the quasiperiodic kicked rotor vs the modulation amplitude $ϵ$, for various values of the kicking strength $K$ and effective Planck constant $ꝁ$. The error bars indicate the typical experimental uncertainty. The four curves are straight lines in this logarithmic scale, with a slope that decreases with $ꝁ$ and increases with $K$.

Figure 3

Increase in the kinetic energy at $t=1000$ ($\propto p_{\text{loc}}^2$) of the quasiperiodic kicked rotor with respect to the purely one-dimensional situation $ϵ=0$ vs the scaling parameter ${ϵ(K/ꝁ)}^2$. The cloud of experimental points—collected at various values of $K$, $ϵ$ and $ꝁ$—is distributed around an average linear dependence in this semi-logarithmic plot, which shows the exponential dependence of the localization length, characteristic of 2D Anderson localization. The red dashed line is the prediction of Eq. (3). The spread is due in part to experimental imperfections [at large ${ϵ(K/ꝁ)}^2$, the localization time is not much shorter than the duration of the experiment] and in part to fundamental reasons: The linear dependence on $ϵK^2/{ꝁ}^2$ in the argument of the exponential, Eq. (3), is valid only at small $ϵ$, and the formula assumes that the classical diffusion constant is proportional to $K^2$, while the actual diffusion constant has oscillatory corrections. The black curves are numerical simulations corresponding to the two “extreme” values of $K/ꝁ=1.3,$, $ꝁ=3.46$ (higher curve) and $K/ꝁ=2.5$, $ꝁ=2.89$ (lower curve); they display the same spreading phenomenon.