Observation of a Disordered Bosonic Insulator from Weak to Strong Interactions

We employ ultracold atoms with controllable disorder and interaction to study the paradigmatic problem of disordered bosons in the full disorder-interaction plane. Combining measurements of coherence, transport and excitation spectra, we get evidence of an insulating regime extending from weak to strong interaction and surrounding a superfluidlike regime, in general agreement with the theory. For strong interaction, we reveal the presence of a strongly correlated Bose glass coexisting with a Mott insulator.

Figure 1

Coherence and mobility. (a) Measured width of the momentum distribution. The diagram is built with 94 data points (crosses), with a standard deviation between 2% and 5%. $T=0$ calculations reveal a MI only on the right of the dashed line, see text. The dashed-dotted line is calculated as $\mathrm{\Delta }-2J=nU$. Stars (diamonds) refer to the measurements shown in Fig. 2 (Fig. 3). (b) Momentum acquired after an applied impulse for varying interaction strength and three disorder strengths: $\mathrm{\Delta }=0$ (triangles), $\mathrm{\Delta }=6.2J$ (squares), and $\mathrm{\Delta }=8.8J$ (circles), also shown by the arrows to the right of (a). The measured SF temperature is $k_{B}T=3.1(4)J$ or $k_{B}T=4.5(7)J$ (empty squares). The lines are a guide to the eye. The uncertainties are the standard deviation of, typically, five measurements.

Figure 2

Excitation spectrum for strong interactions. Experimental spectra for $U=26J$ [indicated by the dashed-dotted line in (a)] and $\mathrm{\Delta }=0$ (a), $\mathrm{\Delta }=6.5J$ (b), and $\mathrm{\Delta }=9.5J$ (c). These sets of parameters are also shown in Fig. 1 (stars). The arrows are at $h\nu =\mathrm{\Delta }$, the dashed line in (a) is at $h\nu =U$, and the continuous lines are fits with multiple Gaussians. (d) Comparison of the low-frequency peak for $\mathrm{\Delta }=6.5J$ with theory (continuous line). The theory includes the Gaussian tail of the Mott peak (dashed line).

Figure 3

Excitation spectrum for weak interactions. Spectra for $\mathrm{\Delta }=8.9J$, and $U=0.35J$ (circles), $U=1.4J$ (triangles), or $U=2.1J$ (squares). These sets of parameters are also shown in Fig. 1 (diamonds). Successive data sets have been shifted upwards by 0.2 for clarity. The dotted red line is the theory for noninteracting bosons, the arrow is at $h\nu =\mathrm{\Delta }$ and the continuous lines are a guide to the eye.

Figure 4

Finite-temperature effects. (a)–(d) Comparison of the experimental momentum distributions (black, continuous line) with $T=0$ DMRG calculations (blue, dashed-dotted line) supplemented by the introduction of a thermal correlation length ${\xi }_T$ (red, dashed line), for four points in the $\mathrm{\Delta }-U$ plane.