Figure 1

Condensate fraction measured after band mapping from $V_i=10$ $E_R$. The insets show high-optical-density images where the background is resolved but the Bragg peaks are saturated. The images are shown in false color, with red (blue) indicating regions of high (low) column density. The field of view is $813\times 813$ $\mu$m. (a) $\tau =10$ ns; (b) $\tau =1$ ms.Reuse & Permissions

Figure 2

Relaxation time (${\tau }_{\mathrm{rel}}$) for the condensate fraction for ramps from $V_i$ to $V_f=4E_R$ for variable ramp times $\tau$. The range of $V_i$ spans the superfluid (SF) and Mott insulator (MI) regimes (demarcated by the vertical dotted line). The experimental data is bounded by zero-temperature Gutzwiller mean-field simulations using two different initial states (see text). The solid black line shows the relaxation time assuming initial state 2, while the dashed line shows the relaxation time assuming initial state 1. The error bars represent the uncertainty in the relaxation time from a fit to data such as that shown in Fig. 1.Reuse & Permissions

Figure 3

Relevant energy scales in deep lattices. (Left) Typical excitation spectrum of a superfluid near the Mott-insulator transition ($V=13E_R$). Two modes are present: a gapless phonon mode with a linear Bogoliubov (dashed) dispersion at low $k$ and a gapped particle-hole mode with quadratic dispersion. The gap $\Delta \sim U$, sets the diabaticity time scale for band mapping. (Right) Energy scale of the phonon (${\epsilon }_{ph}$ at $k=0.2/d$) (dashed) and gap $\Delta$ at $n=0.9$ as a function of lattice depth. For a more detailed plot of $\Delta$ and ${\epsilon }_{ph}$ for different fillings and lattice depths, we refer the reader to Ref. [26].Reuse & Permissions