Quantum-gas microscopy of the Bose-glass phase

Abstract

Disordered potentials fundamentally alter the transport properties and coherence of quantum systems. They give rise to phenomena such as Anderson localization in non-interacting systems, inhibiting transport. When interactions are introduced, the interplay with disorder becomes significantly more complex, and the conditions under which localization can be observed remain an open question. In interacting bosonic systems, a Bose glass is expected to emerge at low energies as an insulating yet compressible state without long-range phase coherence. While originally predicted to occur as a ground-state phase, more recent studies indicate that it exists at finite temperature. A key open challenge has been the direct observation of reduced phase coherence in the Bose-glass regime. In this study, we utilize ultracold bosonic atoms in a quantum-gas microscope to probe the emergence of the Bose-glass phase in a two-dimensional square lattice with a site-resolved, reproducible disordered potential. We identify the phase through in-situ distribution and particle fluctuations, via a local measurement of the Edwards-Anderson parameter. To measure the short-range phase coherence in the Bose glass, we employ Talbot interferometry in combination with single-atom-resolved detection. Finally, by driving the system in and out of the Bose-glass phase, we observe signatures for non-ergodic behavior.

Figures (10)

• Figure 1: Characterising phases and coherence in the disordered Bose-Hubbard model.a Illustrations of the phases of the disordered Bose-Hubbard model highlighting the degree of coherence between lattice sites. b Schematic to calculate the Edwards-Anderson (EA) parameter. We record ten fluorescence images for a number of disorder patterns at the same disorder strength and average the occupation on each lattice site to obtain disorder averages, see Eq. (\ref{['EAparamter']}). c We probe the coherence length of the system along one axis using an interferometry technique based on the Talbot effect. Top panel: Time evolution of the sketched one-dimensional initial state after switching off the horizontal lattice. Initially phase coherent regions show (shifted) revivals after (half) integer Talbot times, $\tau_{\rm T}$. Horizontal dashed lines indicate the positions of the lattice sites. Bottom panel: Total energy after recapturing the atoms in the original lattice after a duration $t$. The decay time of the resulting oscillations can be related to the coherence length of the initial state (Appendix).
• Figure 2: Probing the disordered Bose-Hubbard model.a Average occupations of ten time-of-flight images at $U/J = 15$ and $\Delta/U_c = 2.8$ (top), $\Delta/U_c = 0$ (bottom), where $U_c$ is the interaction energy at $(U/J)_c$, plotted on a logarithmic color scale for clarity. Orange and black boxes highlight the regions used to determine $n_{\rm max}$ and $n_{\rm min}$, respectively, used to calculate the visibility, $\mathcal{V}$. b Visibility as a function of lattice depth and disorder strength. Our analysis identifies superfluid (SF), Mott insulating (MI) and Bose glass (BG) phases. c Edwards-Anderson (EA) parameter (Eq. \ref{['EAparamter']}) averaged over the central $5 \times 5$ lattice sites (black box in bottom panel of d). The dashed line indicates $\Delta/U=1$, above which the disorder strength is larger than the energy gap in the ideal Mott insulator Pollet2009. d Site-resolved EA parameter, $q_i$, at $U/J = 55$, using four different disorder realizations with ten in-situ images each, for $\Delta/U_c = 2.8$ (top) and $\Delta/U_c = 0$ (bottom). Error bars of $\mathcal{V}$ and $q$ are shown in Figs. \ref{['fig:SI_extendedDataFig2_QMC']} and \ref{['fig:SI_ExtendedData_Fig2_EA']} (Supplemental Material).
• Figure 3: Talbot interferometry. Coherence length, $\xi$, in units of the lattice spacing, $\,a_{\rm{\ell}}$, for increasing disorder strength at constant lattice depth ($V_{x,y} = 9 \,E_{\rm{r}}$, $U/J = 11$). The coherence length is extracted from the fitted decay time of the Talbot interferometry measurements (insets) by comparing to a numerical calculation (Appendix). Error bars represent the $68\%$ confidence intervals of the fitted decay times. Each data point of the curves in the insets results from the average of five images, each containing $\sim200$ atoms. The error bars are the standard error. Black lines are fits with a damped sine, the decay time of which is used to extract $\xi$ (Appendix).
• Figure 4: Ergodicity and adiabaticity.a Approximate phase diagram of the disordered Bose-Hubbard model interpreted from the datasets in Fig. \ref{['fig:Figure2']}. The blue shaded area highlights the region where the entire cloud is in the superfluid phase and the red shaded region shows where the center of the cloud is Mott-insulating. The blue circles show the points and the arrows the trajectories used for the data sets shown in b and c. b Coherence length, $\xi$, measured using Talbot interferometry for the points and trajectories shown in a (details see text). Each data point results from fits of four repetitions with the same parameters. c EA parameter, $q$, for three points and trajectories as shown in a. Each measurement is an average over four disorder patterns with ten images each. d Cloud width after one Talbot time, $\tau_{\rm T}$, after a ramp from the Mott-insulator regime to superfluid regime for increasing disorder strength (see inset where the dashed arrow indicates the trajectory for a data point at a particular disorder strength).
• Figure 5: Numerical simulation of Talbot interference.a Density distribution after free time evolution for $t=[0,1,2.5]\ \tau_{\rm{T}}$ starting from a homogeneous lowest-band distribution over $L=13$ sites with coherence length $\xi=3\,a_{\rm{\ell}}$. b The full time evolution of this density distribution shows (shifted) revivals after (half-) integer Talbot times with decreasing contrast. c Total energy after recapturing the atoms with the lattice after duration $t$. The case from a/b (red) with $\xi=3\,a_{\rm{\ell}}$ is compared to initial conditions with $\xi=0.5\,a_{\rm{\ell}}$ (yellow) and $100\,a_{\rm{\ell}}$ (green). The solid lines are fits with exponentially damped harmonic oscillations. d Relation between coherence length $\xi$ and damping of the Talbot oscillations $\tau$ from numerics for $L=13$ (blue points, solid fillings indicate curves in c). For low $\xi$, we find the expected $\tau=\xi/2 \times \tau_{\rm{T}} / \,a_{\rm{\ell}}$ (dashed), while at large $\xi$ the Talbot signal is dominated by finite-size effects (solid line, Eq.\ref{['eq:tau_xi']}). Dashed lines show results for $L=20,10,5$ (decreasing darkness).