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

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.

💡 Research Summary

Anderson localization, the complete halt of wave transport due to multiple scattering and phase coherence in a disordered medium, predicts a genuine quantum phase transition in three dimensions at a critical energy known as the mobility edge (E_c). While this transition underlies metal‑insulator behavior in electronic systems, a direct observation for matter waves has been hampered by uncontrolled energy broadening in previous ultracold‑atom experiments, which forced indirect, model‑dependent estimates of E_c.

In this work the authors introduce an energy‑resolved loading scheme that prepares a narrow‑energy slice of a 87Rb Bose‑Einstein condensate (BEC) inside a three‑dimensional laser‑speckle disorder. Two internal hyperfine states, |1⟩ (F=2, m_F=1) and |2⟩ (F=1, m_F=−1), are made magnetically insensitive at a “magic” field (B*≈3.23 G). A weak radio‑frequency (rf) pulse of duration t_rf≈40 ms transfers a small fraction (~5 %) of atoms from the disorder‑free state |1⟩ to the disorder‑sensitive state |2⟩. Because the transfer rate is in the weak‑coupling regime, the energy spread of the transferred atoms is Fourier‑limited to ΔE≈h/t_rf≈14 Hz, more than an order of magnitude narrower than in earlier studies.

The disorder is generated by superimposing two speckle fields of slightly different wavelengths (≈780 nm) that share the same diffuser. By adjusting their relative intensities and detunings, the authors cancel the speckle potential for |1⟩ while leaving a finite rms amplitude V_R for |2⟩. The speckle correlation length σ≈0.5 µm defines a correlation energy E_σ≈h·441 Hz; the dimensionless disorder strength η=V_R/E_σ controls the position of the mobility edge.

After the rf transfer the optical trap is switched off, atoms in |1⟩ are removed, and the remaining |2⟩ atoms evolve in the speckle potential for up to 5 s. In‑situ fluorescence imaging along the x‑axis yields column densities n_col(y,z,t), which are integrated over y to obtain one‑dimensional profiles n_1d(z,t). For low loading energies (E_f<h≈166 Hz) the tails of n_1d decay exponentially, a hallmark of Anderson localization. At higher energies (E_f≈366 Hz) the profiles become Gaussian and broaden diffusively. Near the critical energy the cloud size σ²(t) grows sub‑diffusively as σ²∝t^{2/3}, exactly as predicted for the 3D Anderson transition.

By measuring σ²(t) for several disorder amplitudes and loading energies, the authors locate the mobility edge at E_c/h=240(6) Hz. This value agrees quantitatively with state‑of‑the‑art numerical simulations (Ref.