Measuring the Edwards-Anderson order parameter of the Bose glass: A quantum gas microscope approach

With the advent of spatially resolved fluorescence imaging in quantum gas microscopes, it is now possible to directly image glassy phases and probe the local effects of disorder in a highly controllable setup. Here we present numerical calculations using a spatially resolved local mean-field theory, show that it captures the essential physics of the disordered system, and use it to simulate the density distributions seen in single-shot fluorescence microscopy. From these simulated images we extract local properties of the phases which are measurable by a quantum gas microscope and show that unambiguous detection of the Bose glass is possible. In particular, we show that experimental determination of the Edwards-Anderson order parameter is possible in a strongly correlated quantum system using existing experiments. We also suggest modifications to the experiments which will allow further properties of the Bose glass to be measured.

Figure 1

Phase diagram of the disordered Bose-Hubbard model, showing values of the Edwards-Anderson order parameter $q$, sampled at a resolution of $50\times 50$ points. Each point was calculated using ten thermal configurations and ten disorder realisations on a $25\times 25$ lattice. The crosses indicate the boundary of the Mott lobes at $q=0$ and the guide-to-the-eye solid line is a fit of these points. The cross size shows the uncertainty due to our sampling resolution. The same color scale is used in Figs. 2, 3, 4.

Figure 2

In situ measurement of $q$, for $\delta =0.3,{\mu }_0/U=2.1$, and $J/U=1/300$. (a) Simulated snapshot of the parity-sensitive lattice occupation, where orange (black) denotes a site occupied by an odd (even) number of atoms. The white scale bar denotes five lattice sites. (b) $q$ extracted from site-by-site evaluation of Eq. (2) with ten snapshots at each of ten disorder realizations. (c) $q$ can also be approximated using ten repetitions of a single disorder realization by averaging along contours of constant $\mu$.

Figure 3

The site occupations of (a) a clean optical lattice at finite temperature and (b) a disordered lattice at zero temperature are cosmetically similar. However, (c) for the clean case $q\approx 0$, while (d) in the disordered case there are Bose glass regions with large values of $q$. The white scale bar shown in (a) denotes five lattice sites.

Figure 4

Melting of the Bose glass. The visibility of the Edwards-Anderson order parameter decreases with temperature but remains finite over a wide range of experimentally relevant temperatures. (a) Histogram showing the local average value of $q$ with error bars showing standard error. (b)–(d) The full distribution of $q$ in the zero-tunneling regime for $\delta =0.3,{\mu }_0/U=2.1$, and $k_{B}T/U=0.01,0.05,0.13$, and 0.25, respectively.