Direct Observation of the Three-Dimensional Anderson Transition with Ultracold Atoms in a Disordered Potential

Abstract

Anderson localization of particles -- the complete halt of wave transport through multiple scattering and phase coherence -- is a paradigmatic manifestation of quantum interference in disordered media. In three dimensions, the scaling theory predicts a quantum phase transition at a critical energy, the mobility edge, separating localized from diffusive states and underpinning metal-insulator transitions in electronic systems. Despite decades of experimental efforts, a direct observation of this emblematic transition for matter waves has remained elusive. Previous attempts with ultracold atoms were hindered by strong and uncontrolled energy broadening, resulting in indirect, sometimes inaccurate, and model-dependent estimates of the mobility edge. Here we implement a novel energy-resolved scheme to prepare atomic matter waves with a narrow energy distribution and track their expansion dynamics over long timescales. This allows for a direct observation of the three-dimensional Anderson transition in a laser-speckle disordered potential, and for a precise measurement of the mobility edge that is independent of any underlying theoretical modeling. Our measurements show excellent agreement with state-of-the-art numerical predictions over a wide range of disorder strengths, resolving long-standing discrepancies between prior experiments and theory. Beyond the three-dimensional Anderson transition, our approach opens new avenues for quantitative investigations of quantum critical phenomena in spatially disordered systems, including the roles of dimensionality, symmetry class, and interactions.

Figures (12)

• Figure 1: Energy-resolved characterization of 3D Anderson transition. (A-B) Experimental scheme: a $^{87}$Rb Bose-Einstein condensate (BEC), initially prepared in the disorder-free state $|1\rangle$, is partially transferred via two-photon rf transition into the disorder-sensitive state $|2\rangle$ with a well-defined energy $E_\mathrm{f}$ and a narrow width $\Delta E=h/t_\text{rf}$, with $t_\text{rf}$ the rf pulse duration. Both states are suspended against gravity using a magnetic levitation (yellow coils). The state-dependent disorder is achieved by using two laser speckle fields, a principal (purple) and a compensating (red) one. Their wavelengths and amplitudes are finely tuned to cancel the disorder in state $|1\rangle$, and not in the state $|2\rangle$ (see Methods). Inset of (A): Schematic energy spectrum in the disordered speckle potential $V$, illustrating the position of the mobility edge $E_c$. With our setup, we probe the dynamics of the atomic sample through its cloud size $\sigma(t)$. The expansion is diffusive for $E_\text{f}>E_c$, $\sigma(t)\propto \sqrt{t}$, whereas its saturates for $E_\text{f}<E_c$, $\sigma(t)=\text{const}$. At the mobility edge $E=E_c$, finally, the dynamics is subdiffusive, with the critical behaviour $\sigma(t)\propto t^{1/3}$.
• Figure 2: Direct observation of the Anderson transition. (A) Two-dimensional column density in the $(y,z)$ plane, recorded after $5s$ of expansion in a speckle potential of amplitude $V_\mathrm{R}/h=$ 416Hz. It shows a striking change in the behavior of the central density depending on the loading energy $E_\mathrm{f}$. For proper comparison, all density profiles are normalized to a fixed number of atoms. The numerically predicted position of the mobility edge from Ref. Ec_Delande2017, $E_\mathrm{c}^\text{num}/h=240$ Hz, is indicated on the energy axis. (B) Integrated one-dimensional atomic densities along the $z$ axis, $n_\mathrm{1d}(z,t=5s)$, displayed on a semi-log scale and corresponding to the column densities shown in (A) for the low energy $E_\mathrm{f}/h=166Hz$ (blue plot) and the high energy $E_\mathrm{f}/h=366Hz$ (red plot). The light orange shading indicates the initial density profile. The wings of the profiles are independently fitted with an exponential (blue) or a Gaussian (red) function.
• Figure 3: Dynamical behaviour as a function of the energy $E_\mathrm{f}$. (A) Experimental column-density profiles as a function of time for three fixed energies $E_\mathrm{f}$ (166, 246 and 366 Hz) for the disorder amplitude $V_\mathrm{R}/h=$ 416Hz. As in Fig. \ref{['fig:Transition']}, the profiles are normalized to a fixed number of atoms. The initial cloud, which corresponds to atoms transferred from the same BEC in state $|1\rangle$ but with a different rf frequency, may slightly vary with the energy $E_\mathrm{f}$ (see Supplementary Information), see also panel B. (B) Main panel: squared size $\sigma^2_\text{exp}(t)$ of the cloud (symbols) as a function of loading energy $E_\mathrm{f}$ for different expansion times (same disorder amplitude $V_\mathrm{R}/h=$ 416Hz). Solid lines show theoretical predictions based on the 3D self-consistent theory of Anderson localization adapted to our configuration (see Methods). Inset: Time evolution, in a log-log plot, of the squared cloud size $\sigma^2_\text{exp}(t)$ at the three energies $E_\mathrm{f}$ corresponding to the profiles in (A). The dashed lines show fits of the form $\sigma^2_\text{fit}(t)=A t^\kappa$, performed on data with $t>$ 1s. The colored bands denote one-standard-deviation confidence intervals.
• Figure 4: Direct estimation of the mobility edge for $V_\mathrm{R}/h=$ 416Hz ($\eta=0.94$). Exponent $\kappa$, extracted from fits of the squared cloud width to $\sigma^2_\text{exp}(t)=A t^\kappa$ (see inset of Fig. \ref{['fig:Dynamics']}B), shown as a function of loading energy $E_\mathrm{f}$, for the disorder amplitude $V_\mathrm{R}/h=$ 416Hz. The red dotted horizontal line indicates the critical value $\kappa_\text{cri}=2/3$, corresponding to the subdiffusive dynamics expected at the mobility edge, while the two grey dotted horizontal lines mark the localized ($\kappa=0$) and diffusive ($\kappa=1$) limits. The blue solid curve represents the ad-hoc fitted function (an error function, see Methods). The shaded area represents the experimentally inferred critical region, whose half-width is defined by the rms width $W_{\rm exp}$ of the fitting function. The vertical black dashed line indicates the numerically predicted mobility edge from Ref. Ec_Delande2017, while the red point marks the experimentally estimated mobility edge $E_\mathrm{c}^{\rm exp}$.
• Figure 5: State of the art: measurements of the mobility edge and comparison with numerics. Mobility edge $E_\mathrm{c}$, relative to the disorder mean value $V_\mathrm{R}$ (horizontal dotted line), shown as a function of the normalized disorder strength $\eta = V_\mathrm{R}/E_\sigma$, where $E_{\sigma}$ is the correlation energy. The red squares show the direct measurements of $E_\mathrm{c}^\mathrm{exp}$ obtained in the present work at four disorder amplitudes, using the methodology presented in Fig. \ref{['fig:Dynamics']}. The vertical error bars indicate the half-widths $W_{\rm exp}$ of the critical regions, while the disorder amplitudes are calibrated with a 5$\%$ uncertainty (horizontal error bars) -- see supplementary Information. The thick red line correspond to the estimated scaling of the mobility edge estimated in our previous work Fred2012, resulting from a comparison between experiments and theoretical modeling. The yellow diamonds indicate the experimental estimation of the mobility edge from Ref. G.Semeghini2015. Last, the black dashed curve corresponds to the numerical prediction of Ref. Ec_Delande2017 for repulsive laser speckle disorder.