Energy-resolved transport of ultracold atoms across the Anderson transition: theory and experiment

Abstract

In a recent experiment [X. Yu et al., arXiv:2602.07654], energy-resolved measurements of an atomic matter wave spreading in a speckle potential enabled the direct observation of the three-dimensional Anderson transition. In this work, we present a quantitative theoretical description of the matter-wave dynamics based on a tailored implementation of the self-consistent theory of localization, which incorporates both the spectral and spatial properties of the state prepared in the disorder. We benchmark this theoretical approach against ab initio numerical simulations, and use it to analyze the atom density profiles observed experimentally in the localized, diffusive, and critical regimes. Particular emphasis is placed on the key role of the atomic energy distribution, especially on the distinct contributions of Bose-condensed and thermal atoms to interpret the experimental profiles. Our framework provides a versatile and efficient theoretical toolbox for quantitatively describing wave-packet dynamics in three-dimensional disordered quantum systems, which remain challenging for state-of-the-art large-scale numerical simulations.

Figures (7)

• Figure 1: Experimental protocol of Ref. Josse2026: atoms from a Bose–Einstein condensate, initially prepared in a disorder-free state, are transferred to a disorder-sensitive state by means of an rf pulse. The transferred atoms acquire a narrow energy distribution (green dashed curve) centered at $E_\mathrm{f}$, which can be tuned across the mobility edge $E_\mathrm{c}$ of the disordered potential.
• Figure 2: (a, b) Two-dimensional cuts of a realization of a three-dimensional speckle potential, and (c) its autocorrelation function along the longitudinal ($x$) and a transverse ($z$) directions. The speckle parameters are: $V_\mathrm{R}/h = 416$ Hz, $\sigma_x = 1.45~\mu$m, and $\sigma_\perp = 0.30~\mu$m. Crosses: experimental data; points: numerical data; solid lines: analytical autocorrelation function, Eq. \ref{['eq:autocorr']}. See Appendix \ref{['App:speckle']} for further details.
• Figure 3: (a-c) Linear density profiles $n_\text{1d}(z,t)$ across the 3D Anderson transition in a speckle potential, obtained from numerical simulations of wave-packet propagation (discrete points) for a disorder characterized by the properties shown in Fig. \ref{['Fig:speckle']}. Three target energies are considered: (a) $E_\mathrm{f}/h=316$ Hz, (b) $E_\mathrm{f}/h=264$ Hz, and (c) $E_\mathrm{f}/h=216$ Hz, corresponding respectively to the diffusive, critical, and localized regimes. The solid lines show the long-time density profiles predicted by the SCT. In the diffusive regime (a), the colored dashed curves show the theoretical profiles including boundary corrections, Eq. (\ref{['eq:profile_finiteL']}). (d) Inverse of central density as a function of time for $E_\mathrm{f}/h=316$ Hz ($E/E_\mathrm{c}\simeq 1.2$), $E_\mathrm{f}/h=264$ Hz ($E/E_\mathrm{c}\simeq 1$), and $E_\mathrm{f}/h=216$ Hz ($E/E_\mathrm{c}\simeq 0.8$), revealing the three characteristic long-time scaling laws (\ref{['eq:longtime_scaling']}) across the mobility edge. Points are the results of numerical simulations and solid lines show the predictions of the SCT. Dashed lines are guides indicating the algebraic laws expected at long time. Here numerical results are averaged over 48 disorder realizations, and we fit $(\alpha_0,\beta)\simeq (10.7\,\text{s}^{-1/3},4.6\, \mu \text{m}.\text{s}^{-1/3})$ for the SCT parameters.
• Figure 4: Initial one-dimensional density profiles of atoms in state $|2\rangle$ immediately after transfer along the $z$ axis, $n_{\rm 1d}(z,t=0)$. The shaded orange areas show the experimental profiles on a semi-log scale for three target energies: (a) $E_\mathrm{f}/h=166$, (b) $246$, and (c) $366$ Hz. Dashed curves indicate, for comparison, the time-of-flight estimate of the BEC density profile in state $|1\rangle$ before the rf transfer, including the thermal component. Both profiles are normalized to unit integral for proper comparison. Solid curves are fits of $n_{\rm 1d}(z,t=0)$ to pseudo-Voigt profiles, which are used as input for the SCT.
• Figure 5: Estimated energy distribution $\mathcal{D}(E;E_{\rm f})$ of the atoms experimentally transferred into a random potential of amplitude $V_\mathrm{R}/h=416Hz$, for three target energies $E_\mathrm{f}/h=166$ (left), $246$ (middle), and $366~\mathrm{Hz}$ (right), indicated by vertical dashed lines. The distributions $\mathcal{D}(E;E_{\rm f})$ are calculated using Eq. (\ref{['eq:Dtotal']}), assuming a thermal fraction of 25$\%$ in state $|1\rangle$. Each distribution consists of a dominant BEC component, numerically computed from Eq. (\ref{['eq:Dtotal']}), superimposed on a small, smooth thermal background [Eq. (\ref{['eq:Dthermal']})], which exhibits a long tail toward higher energies. The inset displays the distribution for $E_{\rm f}/h = 246~\mathrm{Hz}$ on a semi-logarithmic scale, highlighting the high-energy thermal tail; the dashed line shows the result obtained for a pure BEC in state $|1\rangle$ ($f_\mathrm{c}=1$). In the main panel, the shaded area shows the spectral function $A(E,{\boldsymbol{k}}=0)$, whose numerical computation is detailed in Appendix \ref{['App:spectral']}.