Figure 1

(Color online) Adiabatic spin up. Properties of cloud during adiabatic spin up with parameters $\left(t/U=0.2,{\mu }_0/U=0.3,R_{TF}=26.8\right)$. (a) Energy vs rotation rate. Sharp drops indicate vortex formation. Energy scaled by on-site interaction parameter $U$. Rotation rate quoted as $\nu$, the number of circulation quanta per optical-lattice site. [(b), (d), and (f)] Superfluid density profile at parameters labeled in (a). Light-to-dark shading corresponds to low-to-high density, and position is measured in lattice spacing. Light spots correspond to vortex cores. Red (gray) lines are guides for the eyes. [(c), (e), and (g)] Superfluid phase represented by hue. Solid white circle denotes edge of cloud. Other white lines are guides for the eyes. Black circles denote vortex locations. In (b),(c) and (f),(g) rotation speed should favor a commensurate square vortex lattice rotated by $\pi /4$ from the optical-lattice axes. (d) and (e) represents an incommensurate rotation speed.Reuse & Permissions

Figure 2

(Color online) Dependence of vortex entry on $t/U$. Plots of energy vs rotation rate during adiabatic spin up with parameters $\left({\mu }_0/U=0.3,R_{TF}=13.4\right)$ and $\left\{t/U=0.15,0.20,0.25,0.3\right\}$. Axis labels are consistent with Fig. 1a. Sharp drops indicate vortex entry. We see that as $t/U$ is decreased, vortices enter at higher rotation speeds, and the energy drops are smaller. Four vortices enter at the first energy drop, while either four (for $t/U=0.15$) or eight enter at the second drop.Reuse & Permissions

Figure 3

(Color online) Hysteresis. (a) Energy versus rotation rate during increase [solid red (gray) line] and decrease (dotted black line) of $\nu$. Energy steps in the solid (dotted) curve correspond to nucleation (expulsion) of vortices. [(b)–(e)] Order-parameter complex phase for parameters labeled in (a). Black circles are drawn around vortex cores, and white circles indicate the approximate extent of the gas.Reuse & Permissions

Figure 4

Simulation of column density seen in time-of flight absorption images. Darker shading corresponds to higher column density. The initial state corresponds to the $4\times 4$ square vortex lattice pictured in Figs. 1b, 1c. [(a)–(c)] 0, 2, and 20 ms expansion. Left: column density profile calculated using Eq. (7); right: column density convolved with a $3\text{−}\mu \text{m}$-wide Gaussian distribution to represent the finite resolution of a typical imaging system. (d) Long-time scaling limit, where $D_{\overline{t}}=\hslash \overline{t}/m\lambda ⪢R_{TF}$.Reuse & Permissions