Observation of the Anderson metal-insulator transition with atomic matter waves: Theory and experiment

Using a cold atomic gas exposed to laser pulses—a realization of the chaotic quasiperiodic kicked rotor with three incommensurate frequencies—we study experimentally and theoretically the Anderson metal-insulator transition in three dimensions. Sensitive measurements of the atomic wave function and the use of finite-size scaling techniques make it possible to unambiguously demonstrate the existence of a quantum phase transition and to measure its critical exponents. By taking proper account of systematic corrections to one-parameter scaling, we show the universality of the critical exponent $\nu =1.59\pm 0.01$, which is found to be equal to the one previously computed for the Anderson model.

Figure 1

(Color online) Classical diffusive motion for the 3D kicked rotor (22). The initial state is localized around the origin. After 1000 kicks, the classical momentum distribution (noisy black lower and red upper curves) has the Gaussian shape characteristic of a diffusive motion. The blue lower and green upper curves are fits by a Gaussian, which do not show any statistically significant deviation. The noisy black lower (respectively, red upper) curve is the momentum distribution along $p_1$ (respectively, $p_2$). The distribution along $p_3$ is identical to that along $p_2$. The anisotropic diffusion happens because the hopping along the directions “2” and “3” is diminished by a factor $\epsilon$ compared to hopping along direction “1.” Parameters are $K=10$, $k^{–}=2.85$, $\epsilon =0.8$, ${\omega }_2/2\pi =\sqrt{5}$, and ${\omega }_3/2\pi =\sqrt{13}$.

Figure 2

(Color online) Gravity effects on a slightly inclined kicked rotor (34). The deviation of the standing wave from horizontality is $\alpha =0°$ (black lower curve), $\alpha =0.1°$ (red lower middle curve), $\alpha =0.4°$ (green upper middle curve), and $\alpha =1°$ (blue upper curve). The stochasticity parameter is taken as $K=5$ and the effective Planck constant is $k^{–}=2.85$. The dynamics of an initial thermal state is simulated and the corresponding mean kinetic energy is plotted versus time. For angles larger than $0.1°$, the slow drift of momentum induces a diffusive behavior clearly visible on the time scale of the experiment.

Figure 3

(Color online) Classical chaotic diffusion for the quasiperiodic kicked rotor (22). The dynamics of an initial thermal distribution of classical particles is simulated and the corresponding mean kinetic energy is plotted versus time (number of kicks). The stochasticity parameter $K$ (the modulation amplitude $ϵ$) varies linearly between 4 and 9 (0.1 and 0.8), following the experimental path (Fig. 6). The dashed line of slope 1 demonstrates the linear increase of $⟨{\mathfrak{p}}^2⟩$ vs time $t$. No classical localization effects are observed. The chaotic diffusion is characteristic of the presence of pseudodisorder in the quasiperiodic kicked rotor, leading to a pseudorandom walk in momentum space.

Figure 4

(Color online) Experimentally measured momentum distributions after 150 kicks, exponentially localized in the insulator region (blue peaked) and Gaussian in the diffusive (metallic) region (red broad). (a) Linear scale; (b) log scale. For both curves $k^{–}=2.89$, for the localized distribution (blue peaked) $K=5.0$ and $ϵ=0.24$, for the Gaussian distribution (red broad) $K=9.0$ and $ϵ=0.8$.

Figure 5

(Color online) Temporal dynamics of the quasiperiodic kicked rotor. We experimentally measure the population ${\Pi }_0\left(t\right)$ of the zero-momentum class as a function of time (number of kicks) and plot the quantity ${\Pi }_0^{-2}\left(t\right)\propto ⟨{\mathfrak{p}}^2⟩\left(t\right)$. Clearly, it tends to a constant in the localized regime (blue lower curve corresponding to $K=4$ and $\epsilon =0.1$) and increases linearly with time in the diffusive regime (red upper curve corresponding to $K=9$ and $\epsilon =0.8$). The inset shows the behavior close to the localization time. $k^{–}=2.89$.

Figure 6

(Color online) Phase diagram of the quasiperiodic kicked rotor, from numerical simulations. The localized (insulator) region is shown in blue (left bottom corner); the diffusive (metallic) region is shown in red (right top corner). The experimental parameters are swept along the diagonal dash-dotted line.

Figure 7

(Color online) Numerically simulated time evolution of $⟨{\mathfrak{p}}^2⟩$ for the quasiperiodic kicked rotor. At the critical point $K=K_c\approx 6.4$ (purple middle curve), anomalous diffusion $⟨{\mathfrak{p}}^2⟩\sim t^{2/3}$ is clearly observed, as expected from theoretical arguments (cf. text). The log-log plot of the critical curve is very well fitted by a straight line of slope 0.664 (black dashed line). In the vicinity of the transition, the dynamics departs from the anomalous diffusion to tend gradually either to a diffusive dynamics (red upper curves corresponding to $K=>K_c$ bending upward for large $t$) or to a localized dynamics (blue lower curves corresponding to $K<K_c$ bending downwards for large $t$). Other parameters are $k^{–}=2.85$, ${\omega }_2=2\pi \sqrt{5}$, and ${\omega }_3=2\pi \sqrt{13}$.

Figure 8

(Color online) Experimentally observed time evolution of ${\Pi }_0^{-2}\sim ⟨{\mathfrak{p}}^2⟩$ for the quasiperiodic kicked rotor. Close to the critical point $K=K_c\approx 6.4$ (purple middle curve), anomalous diffusion ${\Pi }_0^{-2}\left(t\right)\sim t^{2/3}$ is clearly observed. (a) The critical anomalous curve is well fitted by ${\Pi }_0^{-2}\left(t\right)=A+B{t}^{2/3}$ (black dashed line). The red upper curve evidences the far-above-criticality diffusive behavior $\left(K=9.0\right)$ and the blue lower curve evidences the far-below-criticality $\left(K=4.0\right)$ localized behavior. (b) These experimental results show a clear algebraic behavior, with exponent $\approx 0$ (blue lower curve, localized regime), 2/3 (purple middle curve, critical regime), and 1 (red upper curve, diffusive regime), slightly perturbed by decoherence processes responsible for the residual increase in the localized regime. Other parameters are the same as in Fig. 7.

Figure 9

(Color online) Numerical simulation showing the evolution of the dynamics from the critical behavior toward either a diffusive dynamics or a localized state. Plotting the quantity $\text{ln}\Lambda =\text{ln}\left(⟨{\mathfrak{p}}^2⟩t^{-2/3}\right)$ vs $\text{ln}t$ allows to easily distinguish the critical behavior from diffusive or localized behavior: the critical curve (corresponding to $K=K_C\approx 6.4$) has a zero slope; whereas the far localized $\left(K=4.0\right)$ one has a slope $-2/3$ and the far diffusive $\left(K=9.0\right)$ one has a slope 1/3. Other parameters are the same as in Fig. 7.

Figure 10

(Color online) Raw numerical data displayed in the form $\text{ln}\Lambda =\text{ln}\left(⟨{\mathfrak{p}}^2⟩t^{-2/3}\right)$ vs $\text{ln}{t}^{-1/3}$ (on the left). Each curve corresponds to a different stochasticity parameter $K$. The finite-time scaling procedure consists in shifting horizontally each curve by a quantity $\text{ln}\xi \left(K\right)$ so that the curves overlap. This allows one to determine both the scaling function $f$ (on the right) and the scaling parameter $\xi \left(K\right)$.

Figure 11

(Color online) Finite-time scaling applied to the results of numerical simulations of the quasiperiodic kicked rotor. The time evolution of $⟨p^2⟩$ is computed as a function of time, from 30 to ${10}^4$ kicks, for several values of $K$ between $K=4$ and $K=9$. The finite-time scaling procedure allows us to determine both the scaling function $f$ (a), clearly displaying an upper branch (red) associated with the diffusive regime, and a lower branch (blue) associated with the localized regime. The dependence of the scaling parameter $\xi$ on $K$ (b) displays a divergent behavior around the critical point $K_c=6.4$, which is the signature of the Anderson phase transition. The dashed line is a fit using Eq. (48). The resulting critical exponent is $\nu =1.6\pm 0.1$. Other parameters are $k^{–}=2.85$, ${\omega }_2=2\pi \sqrt{5}$, and ${\omega }_3=2\pi \sqrt{13}$.

Figure 12

(Color online) Finite-time scaling applied to the experimental results (from 30 to 150 kicks). The scaling procedure is the same as in Fig. 11. (a) The fact that all experimental points lie on a single curve, with a diffusive (red upper) and a localized (blue lower) branch, is a proof of the relevance of the one-parameter scaling hypothesis. (b) The maximum displayed by the scaling parameter $\xi$ in the vicinity of $K_c=6.4$ is a clear-cut proof of the Anderson transition. Phase-breaking mechanisms (cf. text) smooth the divergence at the critical point. When these effects are properly taken into account, one obtains a critical exponent $\nu =1.4\pm 0.3$, [the dashed line is a fit with Eq. (48)] compatible with the numerical results. This plot corresponds to 48 experimental runs. Other parameters are as in Fig. 11.

Figure 13

(Color online) Dynamics of the quasiperiodic kicked rotor in the vicinity of the critical regime. The rescaled quantity $\text{ln}\Lambda \left(K,t\right)$ vs $K$ is plotted from $t=30$ to $t=40000$. All curves intersect, to a very good approximation, at a single point $\left(K_c\simeq 6.4,\text{ln}{\Lambda }_c\simeq 1.6\right)$. This multiple crossing indicates the occurrence of the metal-insulator transition. Small deviations from crossing are due to the existence of an irrelevant scaling parameter at finite time and residual correlations in the disordered potential (see text). $K$ and $ϵ$ are swept along the straight line drawn in Fig. 6. Parameters are $k^{–}=2.85$, ${\omega }_2=2\pi \sqrt{5}$, and ${\omega }_3=2\pi \sqrt{13}$.

Figure 14

(Color online) Linear regression of ${\left(\text{ln}\Lambda \right)}^{\prime }\left(K_c,t\right)$ [Eq. (51)] vs $\text{ln}t$ for $t=30$ to $t=40000$ permits to extract the critical exponent $\nu$ from the slope $1/3\nu$, which is $\nu =1.61$. It is difficult to assess the uncertainty associated with this measurement as it depends crucially on the interval of $K$, where the behavior of $\text{ln}\Lambda$ vs $K$ can be assumed to be “linear.” The parameters are the same as in Fig. 13.

Figure 15

(Color online) $\text{ln}{\Lambda }_s$, after subtraction of corrections due to the irrelevant scaling variable, plotted versus $\text{ln}\left(\xi /t^{1/3}\right)$ and the scaling function deduced from the model (black curve). The parameters are that of the set $\mathcal{D}$ (see Table ). The best-fit estimates of the critical stochasticity and the critical exponent are in this case: $K_c=8.09\pm 0.01$, $\text{ln}{\Lambda }_c=1.64\pm 0.03$, and $\nu =1.59\pm 0.01$.