Generation of high-winding-number superfluid circulation in Bose-Einstein condensates

We experimentally and numerically demonstrate a method to generate multiply quantized superfluid circulation about an obstacle in highly oblate Bose-Einstein condensates (BECs). We experimentally achieve pinned superflow with winding numbers as high as 11, which persists for at least 4 s. Our method conceptually involves spiraling a blue-detuned laser beam, around and towards the center of the BEC, and is experimentally implemented by moving the BEC in a spiral trajectory around a stationary laser beam. This optical potential serves first as a repulsive stirrer to initiate superflow, and then as a pinning potential to transport the superfluid circulation within the BEC. The spiral technique can be used either to generate a high-winding-number persistent current, or for controlled placement of a cluster of singly quantized vortices of the same circulation. Thus, the technique may serve as a building block in experimental architectures to create on-demand vortex distributions in BECs.

Figure 1

Spiral beam trajectory $\vec{s}\left(t\right)=\{r\left(t\right),\theta \left(t\right)\}$, with radial position $r\left(t\right)=R_0\sqrt{1-t/{\tau }_{\mathrm{s}}}$, angular position $\theta \left(t\right)=2\pi N_{\mathrm{s}}{(t/{\tau }_{\mathrm{s}})}^2$, and spiral parameters $N_{\mathrm{s}}=4, {\tau }_{\mathrm{s}}=5\mathrm{s}$, and $R_0=R_r$. See text for definitions of parameters. (a) Simulated 2D density profiles of the BEC in the presence of the beam are shown at 1-s intervals [density profiles (i)–(vi) correspond to the labeled time points in plots (b) and (c)]. (b) Radial beam position $r\left(t\right)$ as a function of time during the spiral trajectory. Red shading indicates the radial extent of the Gaussian beam. The horizontal dashed line indicates the radial Thomas-Fermi radius of the circular condensate, $R_r$. (c) Angular beam position $\theta \left(t\right)$ as a function of time during the spiral trajectory.

Figure 2

Experimental setup. (a) Schematic of the experimental setup (not to scale). [(b)–(d)] Axial absorption images showing a 200-$\mu \mathrm{m}$-square field of view in the horizontal $x\text{−}y$ plane. (b) In-trap image of the BEC with the 660-nm blue-detuned beam at its center. Also shown is the effective spiral trajectory (red line) of the 660-nm beam as the condensate is translated by time-varying magnetic fields. (c) BEC after ballistic expansion. The BEC is allowed to expand right after the pinning optical potential is ramped off ($t_{\mathrm{d}}=0\mathrm{ms}$; see text). The large hole in the center of the condensate is taken as evidence of multiply charged superflow since its size is much larger than a single vortex core. (d) BEC with 11 singly quantized vortices, imaged after ballistic expansion. Here we wait for $t_{\mathrm{d}}=160\mathrm{ms}$ between the 660-nm beam ramp-off and ballistic expansion to allow the superflow to disperse into singly quantized vortices. (e) Residuals from Thomas-Fermi fit to the image shown in (d), zoomed in to the central vortex region. Green circles indicate individual vortex cores.

Figure 3

BEC expansion images for variable hold time $t_{\mathrm{h}}$ and superflow dispersal time $t_{\mathrm{d}}$. The 200-$\mu \mathrm{m}$-square axial absorption images (upper and middle row) are taken after a period of ballistic expansion. Top row: Variable hold time, prior to 600-ms beam ramp off. Spiral parameters are $N_{\mathrm{s}}=4, {\tau }_{\mathrm{s}}=4.8\mathrm{s}$. Here the BEC is allowed to expand directly after ramping off the beam ($t_{\mathrm{d}}=0$ ms). Middle row: Variable dispersal time $t_{\mathrm{d}}$, after the 600-ms beam ramp-off. Spiral parameters are $N_{\mathrm{s}}=5, {\tau }_{\mathrm{s}}=3.5\mathrm{s}$. Here we hold the BEC for $t_{\mathrm{h}}=4\mathrm{s}$, ramp the beam off, and then hold for a variable dispersal time $t_{\mathrm{d}}$ to allow the superflow to dissociate into singly quantized vortices. Bottom row: Residuals after fitting the images in the middle row to a Thomas-Fermi density profile, zoomed in to the central vortex region.

Figure 4

Snapshots from 2D GPE simulation of the spiral technique at representative times $t/{\tau }_{\mathrm{s}}$. Columns from left to right show snapshots of (i) density, (ii) phase, (iii) the superfluid speed profile scaled to the local speed of sound, and (iv) a zoomed-in region of the speed profile at the location of the stirring beam. The speed profile in column (iv) is scaled to the bulk speed of sound, $c_0$ (using the peak density $n_0$). White arrows indicate scaled velocity vectors of the flow. Simulation parameters are $N_{\mathrm{s}}=4, {\tau }_{\mathrm{s}}=5\mathrm{s}, R_0=R_r, w_0=18\mu \mathrm{m}$, and $U={\mu }_0$.

Figure 5

Snapshots from 2D GPE simulations for varying ${\tau }_{\mathrm{s}}$ (as labeled). Snapshots show the time point $t_{\mathrm{m}}\sim 0.66{\tau }_{\mathrm{s}}$ where the flow behind the beam merges with that ahead of the beam, resulting in continuous superflow around the beam. Columns from left to right show zoomed-in snapshots of (i) density, (ii) phase, (iii) the speed profile scaled to the local speed of sound, and (iv) the speed profile scaled to the bulk speed of sound, $c_0$ (using the peak density $n_0$). White arrows indicate scaled velocity vectors of the flow. Simulation parameters are $N_{\mathrm{s}}=4, R_0=R_r, w_0=18\mu \mathrm{m}$, and $U={\mu }_0$. Note the dark soliton appearing in the reconnecting flow for ${\tau }_{\mathrm{s}}=3$ s.

Figure 6

Representative 200-$\mu \mathrm{m}$-square axial absorption images of the BEC after ballistic expansion. Expansion of the BEC starts $\sim 160$ ms after the pinning potential ramps off. The pinned vortices start to dissociate and individual vortex cores can be resolved. [(a)–(h)] Vortex configurations for two to nine cores, respectively. Vortices are marked by green dots in the inset to guide the eye. Regular array configurations of the vortices can sometimes be observed for, e.g., (b) three, (c) four, and (e) six windings.

Figure 7

Average number of vortices observed after ramping off the pinning potential as a function of trajectory duration ${\tau }_{\mathrm{s}}$, for fixed $N_{\mathrm{s}}=4$ and beam waist $w_0=18\mu \mathrm{m}$. Black squares and green diamonds correspond to experimental data, while solid blue and red lines are numerical results from the 2D GPE simulations. Black squares show the average number of cores in the central cluster, whereas green diamonds show the average total number of vortex cores. Error bars indicate the standard deviation. Solid blue lines correspond to a spiraling potential $U={\mu }_0$, and solid red lines correspond to $U=0.75{\mu }_0$. Both experiment and numerics show that the number of vortices generated and pinned using our method decreases as the optical potential moves more slowly through the condensate.

Figure 8

GPE simulation for two beams both spiraling counterclockwise. Simulation parameters are $N_{\mathrm{s}}=3, {\tau }_{\mathrm{s}}=4\mathrm{s}, w_0=8\mu \mathrm{m}$, and $U={\mu }_0$.

Figure 9

GPE simulation for two beams, one spiraling clockwise and one spiraling counterclockwise. Simulation parameters are $N_{\mathrm{s}}=2, {\tau }_{\mathrm{s}}=3\mathrm{s}, w_0=8\mu \mathrm{m}$, and $U={\mu }_0$.

Figure 10

GPE simulation for four beams. First one pair of beams spirals counterclockwise. After 2 s of evolution, the second pair of beams begins to spiral clockwise. Simulation parameters are $N_{\mathrm{s}}=4, {\tau }_{\mathrm{s}}=5\mathrm{s}, w_0=4\mu \mathrm{m}$, and $U={\mu }_0$.