Figure 1

(a) Schematic diagram of our setup for the rotating quasi-2D optical lattice. Layouts of the masks for a triangular (b) and square (c) optical lattices. (d) and (e) are pictures of triangular and square optical lattices, respectively. For details of the layouts, see 17.Reuse & Permissions

Figure 2

(a) Triangular optical lattice and (b) vortex lattice in reciprocal space. Each inset shows the corresponding original real-space CCD-camera image. (c) The difference in orientation ${\theta }_{\mathrm{OL}}\mathrm{\text{−}}{\theta }_{\mathrm{VL}}$ versus the strength of pinning $U_{\mathrm{pin}}/\mu$ (the peak of the optical potential normalized by the condensate’s chemical potential) for the rotation rates ${\Omega }_{\mathrm{OL}}=1.133{\Omega }_c=0.913{\omega }_r$. With increasing pinning strength, ${\theta }_{\mathrm{OL}}\mathrm{\text{−}}{\theta }_{\mathrm{VL}}$ tends toward its locked value [20]. (d) Minimum pinning strength needed for orientation locking between two lattices as a function of the rotation rate of the optical lattice. The dashed and dotted lines are discussed in the text.Reuse & Permissions

Figure 3

Images of rotating condensates pinned to an optical lattice at ${\Omega }_{\mathrm{OL}}={\Omega }_c=0.866{\omega }_r$ with pinning strength $U_{\mathrm{pin}}/\mu =(\mathrm{a})$ 0.049, (b) 0.084, (c) 0.143, showing the structural crossover of the vortex lattice. (a)–(c) are the absorption images of the vortex lattices after expansion. (d)–(f) are the Fourier transforms of the images in (a)–(c). $k_o$ is taken by convention to be the strongest peak; $k_{\mathrm{tr}1}$, $k_{\mathrm{sq}}$, and $k_{\mathrm{tr}2}$ are at 60°, 90°, and 120°, respectively, from $k_o$.Reuse & Permissions

Figure 4

Structure factors (a) $|S(k_{\mathrm{sq}})|$ (■), (b) $|S(k_{\mathrm{tr}})|$ (▴) [average of $|S(k_{\mathrm{tr}1})|$ and $|S(k_{\mathrm{tr}2})|$], and (c) $|S(k_o)|$ (★) versus the strength of the pinning lattice at the commensurate rotation rate ${\Omega }_c$. $|S(k_{\mathrm{sq}})|$ is fitted by [23]. The fitting leads to a maximum value 0.707 of $|S(k_{\mathrm{sq}})|$. An ideal square vortex lattice would have $|S(k_{\mathrm{sq}})|=1$.Reuse & Permissions

Figure 5

Effect of square pinning lattice. (a) Contours of $|S(k_{\mathrm{sq}})|$ are plotted, showing the effect of the rotation rate and pinning strength on the squareness of the vortex lattice. (b) The maximum observed squareness. In (a) and (b), for each rotation rate, the data points are extracted from fits such as that shown in Fig. 4 for ${\Omega }_{\mathrm{OL}}={\Omega }_c$. The vertical dotted line plus arrow shows the possible range of ${\Omega }_c$ consistent with the uncertainty in $n_{\mathrm{OL}}$.Reuse & Permissions

Figure 6

Image of a dumbbell-shape defect consisting of two vortices locked to one pinning site during the formation of the square vortex lattice. Dotted lines are to guide the eye.Reuse & Permissions