Deterministic creation, pinning, and manipulation of quantized vortices in a Bose-Einstein condensate

We experimentally and numerically demonstrate deterministic creation and manipulation of a pair of oppositely charged singly quantized vortices in a highly oblate Bose-Einstein condensate (BEC). Two identical blue-detuned, focused Gaussian laser beams that pierce the BEC serve as repulsive obstacles for the superfluid atomic gas; by controlling the positions of the beams within the plane of the BEC, superfluid flow is deterministically established around each beam such that two vortices of opposite circulation are generated by the motion of the beams, with each vortex pinned to the in situ position of a laser beam. We study the vortex creation process, and show that the vortices can be moved about within the BEC by translating the positions of the laser beams. This technique can serve as a building block in future experimental techniques to create, on-demand, deterministic arrangements of few or many vortices within a BEC for precise studies of vortex dynamics and vortex interactions.

Figure 1

(a) Diagram of the optical system that is used to translate two blue-detuned laser beams through the BEC. A 660-nm laser beam is brought into the optical system with a polarization-preserving single-mode optical fiber. The beam is collimated by lens L1 and then split into two paths by a polarizing beamsplitter cube (PBSC); the optical powers of the two beams are balanced using a half-wave plate (HWP) placed before the first PBSC. Each beam reflects off of two mirrors (M1a and M1b, or M2a and M2b). Each mirror has a piezoelectric transducer (PZT) stack attached to an adjustment screw; the two PZTs attached to the mirrors of a given beam path control the tilts of the two mirrors about orthogonal axes, enabling the two laser beams to be remotely steered. After passing through additional HWPs, allowing for further optical power adjustment, the two beams are recombined with another PBSC. Lenses L2 and L3 comprise a $6\times$ minification system in order to increase the angular deflection of the chopstick beams, prior to deflection of the beams by mirror M3 and focusing the beams at the BEC by lens L4. In the experiments reported here, the beams have $1/e^2$ radii at the BEC of ${\sigma }_c\sim 19\mu \text{m}$. During the course of an experimental run, the two laser beams move along paths $P1$ and $P2$ in the $(x,y)$ plane, as represented in the inset diagram. (b) An illustration of the BEC (large gray disk) lying in the $(x,y)$ plane. The laser beams are initially colocated and pierce the BEC (left dark-gray disk) and move along paths $P1$ and $P2$, which subtend an angle $\alpha \sim {66}^{\circ }$. The beams end up in different locations (upper and lower right dark-gray disks). The size of the laser beams, relative to the BEC Thomas-Fermi radius, and the initial and final locations of the laser beams are illustrated to scale.

Figure 2

(a) A $144-\mu \text{m-square}$ in situ image of the BEC taken along the vertical $\left(z\right)$ axis, showing the profile of the $\sim 100-\mu \text{m-diameter}$ BEC in the $(x,y)$ image plane. The hole to the left of center is created by the two colocated chopstick beams turned on at full power prior to moving them across the BEC. (b) Same as (a), but with the chopstick beams at their final locations at the ends of paths $P1$ and $P2$. The intensity minima due to the chopstick beams are $\sim 31\mu \text{m}$ apart in this image, slightly larger than the separation obtained when measuring the beam positions directly. (c) A $205-\mu \text{m-square}$ absorption image of a BEC at the end of the chopsticks procedure, after the chopstick beams were ramped off at their final positions and the BEC was released from the trap and allowed to ballistically expand a factor of $\sim 1.78$ in the radial direction. Vortex cores appear as the two small dark holes in the density distribution, and are $\sim 65\mu \text{m}$ apart in this image, compared with an expected separation of $\sim 55\mu \text{m}$ given the final positions of the chopstick beams and the expansion factor of the BEC.

Figure 3

Results of numerical simulations for parameters described in the text. Each row represents the state of the system at a time $t/t_s$ during the sweep, indicated in the left column. The final row shows the state of the system at a hold time of 0.5 s after the end of the sweep, during which time the chopstick beams' positions and powers remain unchanged from their final values. Columns one through three show (respectively) $120-\mu \text{m-square}$ regions of the density profile $n$ relative to the maximum number density $n_{\max }$, the phase profile $\varphi$, and the velocity profile relative to the maximum speed of sound $c_0=\sqrt{{\mu }_0/m}$ for the ground-state BEC without chopstick beams present. Column four shows a magnified view of the velocity profile, a $61-\mu \text{m-square}$ region centered on the point midway between the chopstick beams. Arrows indicate local fluid flow velocity vectors, rescaled for each time step; arrows are not shown for $t/t_s=0$, for which there is negligible flow. In all columns, points for which $n/n_{\max }<0.0005$ are assigned the color black; these regions within the BEC are larger at early times when the two beams are colocated and their repulsive potentials add together. The colored circles in the second column are described in the caption of Fig. 4.

Figure 4

For each time step $t/t_s$ represented in Fig. 3, the quantum phase $\varphi$ around the path denoted by the colored circles in the phase profiles of Fig. 3 is shown here as a function of the angle $\theta$ around the circle. The point $\theta =0$ is taken to be at the midpoint between the two laser beams, and $\theta$ increases in a clockwise direction, as indicated by the inset figure showing the phase profile at $t/t_r=1.0$. The plots of $\varphi$ vs $\theta$ in this figure are labeled with the values $t/t_s$ of the corresponding rows of Fig. 3, and the colors of the corresponding rings indicate the points for which $\varphi$ is shown. At points where the density is negligible (i.e., for black-colored values in the phase profiles of Fig. 3), values for $\varphi$ are not shown. This plot shows the gradual establishment of a clockwise $2\pi$ phase winding around the lower laser beam. At time $t/t_s=0.7$ (green line), immediately before the beams separate, a phase winding close to $2\pi$ has been established. At $t/t_s=0.8$ (pink line), just after the beams separate, a continuous loop of fluid is seen to exist around each beam (see first column Fig. 3), with a phase winding of $2\pi$ around each beam. For ease of comparison of the phase at different times $t/t_s$, each plot of $\varphi$ vs $\theta$ shown here has a constant value added such that $\varphi$ is continuous from $\theta =0$ to $\theta =2\pi$ and the minimum value of $\varphi$ over this range is zero.

Figure 5

Each column shows $61-\mu \text{m-square}$ regions of the phase profile $\varphi (x,y)$ centered on the midpoint between the chopstick beams for different times $t/t_s$, indicated above each column. Sweep times $t_s$ are indicated at the left of each row. The display colors are mapped to a range of 0 to $2\pi$, as in the second column of Fig. 3. Except for the variation of $t_s$, simulation parameters are identical in all cases to those described in the text. For 0.6 s $\le t_s\le 1.2$ s, two vortices are generated and left pinned to the beams, an indication of the robustness of the chopsticks technique. For $t_s>1.2$ s, an insufficient phase gradient is established around each beam, and vortices are not created. For $t_s<0.6$ s, too large of a phase gradient is established, and two or more vortices are nucleated for each beam. For the conditions studied, these additional vortices did not remain pinned to the laser beams, as can be seen from the singularities in the phase profiles in the last two rows.

Figure 6

Experimental demonstration of vortex position manipulation. After creating and pinning two vortices, and waiting an additional 0.5 s, one of the chopstick beams was returned to its initial position in 1.0 s while the other beam remained in place. The large images show $205-\mu \text{m-square}$ regions taken after beam ramp-down and BEC ballistic expansion. Insets show $144-\mu \text{m-square}$ in situ images taken prior to beam ramp-down, but after one beam has returned to its initial position. For the left pair of images, the upper beam (B1) was returned to its initial position, as indicated by the regions of depleted density seen in the inset, and the corresponding position of the upper vortex in the image of the expanded BEC. For the right image and inset, the lower beam (B2) was returned to its initial position, as also seen in the in situ image and the positions of the vortices in the expansion image.

Figure 7

Numerical results of 2D GPE simulations showing the return of both chopstick beams to their initial positions over a return time $t_r$, for beam and BEC parameters that are the same as those of Fig. 3. Phase profiles are shown for $61-\mu \text{m-square}$ regions centered on the midpoint between the chopstick beams, with display conditions identical to those of the second column of Fig. 3. The top row shows the return of the beams after a 0.5 s hold time, with $t_r=1.0$ s. The beams are moving from right to left, and are in the process of merging with one another. At time $t/t_r=0.07$, immediately after the return process starts (top row, leftmost image), the vortices are still colocated with each laser beam. Soon afterwards, at $t/t_r=0.11$, the vortices are clearly no longer pinned (top row, second column). Additional vortices are subsequently nucleated (top row, third column) and annihilated, leaving a state with no circulation by $t/t_r=0.38$ (top row, fourth column). For the same return sweep, but with a conjugation of the quantum phase (second row, also see text), the vortices remain pinned to the beams until the beams recombine. The return process that follows phase conjugation produces conditions that are nearly identical (aside from the phase conjugation) to the initial vortex generation stage, shown in the third row in reverse-time order for ease of comparison, for beam separations identical to those of the first two rows.