Observation of Interaction-Induced Mobility Edge in an Atomic Aubry-André Wire

A mobility edge, a critical energy separating localized and extended excitations, is a key concept for understanding quantum localization. The Aubry-André (AA) model, a paradigm for exploring quantum localization, does not naturally allow mobility edges due to self-duality. Using the momentum-state lattice of quantum gas of Cs atoms to synthesize a nonlinear AA model, we provide experimental evidence for a mobility edge induced by interactions. By identifying the extended-to-localized transition of different energy eigenstates, we construct a mobility-edge phase diagram. The location of a mobility edge in the low- or high-energy region is tunable via repulsive or attractive interactions. Our observation is in good agreement with the theory and supports an interpretation of such interaction-induced mobility edge via a generalized AA model. Our Letter also offers new possibilities to engineer quantum transport and phase transitions in disordered systems.

Figure 1

Illustration of interaction-induced mobility edge and experimental scheme. (a) Top: without interaction, all eigenstates in the energy spectrum are either localized (blue) or extended (red). Bottom: weak interaction can suppress or enhance localization depending on the energy, thus a mobility edge emerges in the spectrum. (b) Top: a quasi-1D ${}^{133}\mathrm{Cs}$ BEC with tunable atomic interaction is illuminated by a pair of counterpropagating laser beams, one with a frequency $\omega$ and the other with multifrequency components ${\omega }_j$ ($j=-10,\dots ,9$). Bottom: the lasers, far detuned from the atomic transition, drive a series of engineered two-photon Bragg transitions that couple 21 momentum states with the increment of $2\hslash k$ (with $k=2\pi /\lambda$). This synthesizes a nonlinear AA model with $L=21$ sites in the momentum space.

Figure 2

Theoretical prediction. (a), (b) Participation ratio $r$ of the (a) GS and (b) ES for various quasiperiodic modulation strengths $\mathrm{\Delta }/J$ and interactions $U/J$. The state is localized (extended) in the blue (red) region. The transition (white curve) is identified as where $r=0.103$, the critical value in the noninteracting case [56]. (c) Regimes in the parameter spaces $(\mathrm{\Delta }/J,U/J)$ where a ME may exist. In phases I and III, extended and localized eigenstates coexist, signaling a ME; in phase I (III), low-energy states are extended (localized). The boundaries are denoted by the green and blue curves. (d) Verification of the effective GAA model. The ${\alpha }^{*}$ defined in Eq. (3) is calculated as a function of $U/J$ (solid curve). For small $U/J$, it exhibits a linear relation ${\alpha }^{*}=-0.81U/J$ (dashed curve). In all panels, the lattice size is $L=21$.

Figure 3

Experimental realization of the nonlinear AA model in a Cs BEC with tunable quasiperiodic modulation strength $\mathrm{\Delta }/J$ and interaction $U/J$. (a) Images of momentum distribution for various $\mathrm{\Delta }/J$ in a noninteracting BEC. The $\mathrm{\Delta }/J$ is tuned via engineering the frequency difference between the pair of Bragg lasers in Fig. 1. The BEC is initialized at the momentum lattice site $j=0$. The absorption image is taken after $t=1.28\mathrm{ms}$ evolution and 18 ms TOF. (b) Measured momentum width $⟨d(t)⟩$ as a function of the evolution time $t$ in a noninteracting BEC under various $\mathrm{\Delta }/J$. Corresponding solid curves denote the numerical simulations (see Supplemental Material [58]). (c) Measured momentum width $⟨d⟩$ at $t=0.64\mathrm{ms}$ as a function of the scattering length $a$ (in units of Bohr radius $a_0$) for $\mathrm{\Delta }/J=1$. The scattering length is tuned via a Feshbach resonance. The solid curve denotes the numerical simulation [58]. Error bars denote $1\sigma$ standard deviations. In all panels, the coupling rates are $J/\hslash =2\pi \times 500\mathrm{Hz}$.

Figure 4

Construction of ME phase diagram. (a),(b) Identification of the extended-to-localized transition points of the (a) GS and (b) ES. The experimental participation ratio $r$ is shown as a function of quasiperiodic modulation $\mathrm{\Delta }/J$ in the $\mathrm{semi}{\log }_2$ plot for various interactions. By fitting (solid curves) the experimental data via an empirical relation [58], we extract the transition point. In (a) and (b), the error bar indicates the $1\sigma$ standard deviation. (c) Construction of the phase diagram and identification of ME. We superimpose the extracted transition points of the GS (green circles) and ES (blue squares) on the theoretical phase diagram in Fig. 2. Inset: we compare the data for $|U/J|<0.25$ with the predictions from the effective GAA model [green (blue) line for the GS (ES)]. The error bar in (c) shows the fitting error. In (a)–(c), the coupling rates are $J/\hslash =2\pi \times 275\mathrm{Hz}$.