Engineering a Flux-Dependent Mobility Edge in Disordered Zigzag Chains

There has been great interest in experimental studies of charged particles in artificial gauge fields. Here, we perform the first cold-atom explorations of the combination of artificial gauge fields and disorder. Using synthetic lattice techniques based on parametrically coupled atomic momentum states, we engineer zigzag chains with a tunable homogeneous flux. The breaking of time-reversal symmetry by the applied flux leads to analogs of spin-orbit coupling and spin-momentum locking, which we observe directly through the chiral dynamics of atoms initialized to single lattice sites. We additionally introduce precisely controlled disorder in the site-energy landscape, allowing us to explore the interplay of disorder and large effective magnetic fields. The combination of correlated disorder and controlled intrarow and interrow tunneling in this system naturally supports energy-dependent localization, relating to a single-particle mobility edge. We measure the localization properties of the extremal eigenstates of this system, the ground state and the most-excited state, and demonstrate clear evidence for a flux-dependent mobility edge. These measurements constitute the first direct evidence for energy-dependent localization in a lower-dimensional system, as well as the first explorations of the combined influence of artificial gauge fields and engineered disorder. Moreover, we provide direct evidence for interaction shifts of the localization transitions for both low- and high-energy eigenstates in correlated disorder, relating to the presence of a many-body mobility edge. The unique combination of strong interactions, controlled disorder, and tunable artificial gauge fields present in this synthetic lattice system should enable myriad explorations into intriguing correlated transport phenomena.

Popular Summary

In a disordered environment, the transport of quantum particles can be arrested. This is the ultimate fate of quantum states under random disorder in one and two dimensions. However, in three dimensions, the localization behavior of a quantum state depends on its energy. This “mobility edge” separating the behavior between high- and low-energy states has never been observed in lower dimensions despite being a natural occurrence in 3D. By studying a zigzag-shaped lattice system, we report the first cold-atom observation of a one-dimensional noninteracting mobility edge by contrasting the behavior between the lowest- and highest-energy quantum states.

First, we engineer longer-range connections in a 1D lattice under quasiperiodic disorder. We slowly load atoms into the lowest- and highest-energy states of the system and observe that the highest-energy state is more likely to localize compared to the ground state. We then apply an effective magnetic field and tune its strength, causing the mobility edge to shift in energy and even causing a swap in the localization behavior of the two energy eigenstates. Our results not only show distinct signatures of a single-particle mobility edge, but they also show deviations from expected noninteracting behavior that is consistent with many-body effects from atomic interactions.

Our study is the first cold-atom work to combine synthetic gauge fields and disorder. These techniques could be extended to the study of disordered integer Hall systems, which would allow for the probing of topological robustness to disorder and disorder-induced changes in topology. With the addition of atomic interactions, even more exotic behavior of quantum Hall fluids may be open to investigation.

Figure 1

Constructing the zigzag lattice. (a) The one-dimensional multirange hopping model with NN (solid black lines, $t$) and NNN (dashed red, $t^{\prime }$) tunnelings. (b) The two-dimensional zigzag lattice representation, formed by a rearrangement of the one-dimensional picture of (a). A uniform clockwise flux $\varphi$ through each triangular plaquette is generated via NNN tunneling phases $\varphi$ with alternating sign. (c) Atomic dispersion indicating first- (black arrows) and second-order (dashed red arrows) Bragg transitions used to couple NN and NNN lattice sites, respectively, in the momentum-space lattice. The recoil energy is given by $E_R={\hslash }^2{k}^2/2M_{\mathrm{Rb}}$.

Figure 2

Chiral dynamics in the zigzag lattice. (a) Band structure for $\varphi /\pi =\pm 0.5$ considering a two-site unit cell (yellow boxes in lattice illustration), for tunneling ratio $t^{\prime }/t=0.62$. Color represents spin polarization $⟨\sigma ⟩$, or the overlap of the quasimomentum eigenstate with the top (red, spin up) or bottom (blue, spin down) row of the lattice. Dashed black curves represent the folded band structure for $t^{\prime }/t=0$. $\tilde{q}$ should be considered “quasiposition” in our momentum-space lattice and is given in terms of the unit cell lattice spacing $d=4\hslash k$. (b) Population imbalance between sites 2 and $-2$ of the 21-site lattice, measured after $180\mu \mathrm{s}$ of dynamics ($\sim 1.05\hslash /t$) with optical density (OD) images of atomic populations at $\varphi /\pi =0$, $\pm 0.5$ above. Dashed and solid curves represent an ideal simulation of the experiment using Eq. (1) and a full simulation of experimental parameters, respectively. (c),(d) Site population dynamics for applied flux (c) $\varphi /\pi =0.5$ and (d) $\varphi /\pi =-0.5$. Left to right: data, full simulation, and ideal simulation of experiment. Arrows indicate direction of chiral motion. Data for (b)–(d) were taken with averaged NN tunneling time $\hslash /t=176(2)$ $\mu \mathrm{s}$ and tunneling ratio $t^{\prime }/t=0.622(3)$. All error bars denote 1 standard error of the mean. OD images in (b) and extracted site populations in (c) and (d) are plotted with the color scale in (b).

Figure 3

Localization in the 1D AA model with NN tunneling. (a) Lattice site energies under the AA model of quasiperiodic disorder, ${ϵ}_n=\mathrm{\Delta }\cos (2\pi \beta n+\phi )$, for periodicity $\beta =(\sqrt{5}-1)/2$ and amplitude $\mathrm{\Delta }$. (b) Measured population outside the central three lattice sites ($P_{\mathrm{out}}$) versus disorder-to-tunneling ratio $\mathrm{\Delta }/t$, averaged over four values of the AA phase $\phi /\pi =\{0.96,0.64,1.35,1.88\}$. These $\phi$ values correspond to the energy eigenstates $\{|{\psi }_0⟩,|{\psi }_7⟩,|{\psi }_7⟩,|{\psi }_{18}⟩\}$, where $|{\psi }_0⟩$ is the ground state and $|{\psi }_{20}⟩$ is the highest excited state. Dashed and solid curves represent ideal and full simulations of experimental parameters, respectively. Arrow indicates experimental ramp from $(\mathrm{\Delta }/t{)}_i=\infty$ to some final disorder value. Inset: Integrated OD image showing site populations versus disorder strength, also averaged over the four eigenstates. (c) Integrated OD images for the individual eigenstates, labeled with the relevant AA phase value. (d) $P_{\mathrm{out}}$ versus disorder strength for eigenstates $|{\psi }_0⟩$ (red circles) and $|{\psi }_{18}⟩$ (blue squares). Gray circles are the averaged data of (c). Dashed red, dotted blue, and solid gray curves represent full simulation results including interactions of strength $U/\hslash \approx 2\pi \times 500\mathrm{Hz}$, for $|{\psi }_0⟩$, $|{\psi }_{18}⟩$, and the averaged data, respectively. Inset: Illustration depicting lattice site energies for initial sites with low (left, red) and high (right, blue) energy. Dashed lines show the effect of attractive interactions on the initially populated central site. Vertical lines in (b) and (d) indicate the critical disorder $(\mathrm{\Delta }/t{)}_c=2$ for an infinite system without interactions. Error bars in (b) and (d) denote 1 standard error of the mean. OD images in (b) and (c) are plotted with the color scale in Fig. 2.

Figure 4

Flux-dependent localization transition in the AA model with multirange tunneling. (a) Top: Ground state (GS) localization behavior for varying values of disorder $\mathrm{\Delta }/t$ and flux $\varphi /\pi$, in a zigzag lattice with NN tunneling $t/\hslash =493(2)\mathrm{Hz}$ and NN to NNN tunneling ratio $t^{\prime }/t=0.247(4)$. Colors indicate the width (standard deviation) of the site population distribution ${\sigma }_n$ [inset color scale in (d)], interpolated from the sampled data points (small solid black circles). The empirically determined transition disorder strength between delocalized and localized regions is shown for the data (white circles), a noninteracting simulation of the experiment (dashed black line), and a simulation including attractive interactions of strength $U/\hslash \approx 2\pi \times 500\mathrm{Hz}$ (dotted black line). Bottom: Localization properties as visualized by the integrated 1D density patterns for roughly 0 and $\pi$ flux. The 1D atomic distributions are interpolated from integrated OD images for $\varphi /\pi =0.05$ (left) and 0.95 (right), shown as a function of $\mathrm{\Delta }/t$. Horizontal lines show the empirically determined localization transition point for data (solid red), noninteracting simulation (dashed gray), and simulation including interactions (dotted gray). (b) Cuts as a function of $\varphi$ showing site populations interpolated from integrated OD images, shown for the GS (top) and the highest excited state (ES, bottom). These integrated OD plots are averaged from data taken in the ranges $2.75\le \mathrm{\Delta }/t\le 3$ and $3.2\le \mathrm{\Delta }/t\le 3.5$, respectively, as indicated by the shaded regions. (c) Band structure diagrams for the zigzag lattice with applied flux $\varphi /\pi =0$ (left) and $\varphi /\pi =1$ (right). As in Fig. 2, color represents spin polarization. Dashed black curves represent the folded band structure for $t^{\prime }/t=0$. (d) ES localization behavior for varying disorder and flux values, with the same format as in (a). Bottom: Localization properties of the ES as a function of $\mathrm{\Delta }/t$, also with the same format as in (a). Error bars in (a) and (d) denote 1 standard error of the mean. OD images in (a), (b), and (d) are plotted with the color scale in Fig. 2.

Figure 5

Flux-dependent mobility edge. Empirically determined critical disorder-to-tunneling strength ratios marking localization transition for ground state (GS, red circles) and highest excited state (ES, blue squares), as shown separately in Figs. 4 and 4. Noninteracting and interacting simulations ($U/\hslash =2\pi \times 500\mathrm{Hz}$ used in the latter) are shown as dashed and dotted lines, respectively. For flux values where no critical disorder is plotted, atomic population was determined to be delocalized (based on the set threshold value of the standard deviation) over the full range of disorder strengths. Vertical gray line at $\varphi /\pi =0.5$ denotes flux value at which the GS and ES curves should cross in the absence of interactions and any off-resonant coupling terms. Error bars denote 1 standard error of the mean.