Observation of Vortex Dipoles in an Oblate Bose-Einstein Condensate

We report experimental observations and numerical simulations of the formation, dynamics, and lifetimes of single and multiply charged quantized vortex dipoles in highly oblate dilute-gas Bose-Einstein condensates (BECs). We nucleate pairs of vortices of opposite charge (vortex dipoles) by forcing superfluid flow around a repulsive Gaussian obstacle within the BEC. By controlling the flow velocity we determine the critical velocity for the nucleation of a single vortex dipole, with excellent agreement between experimental and numerical results. We present measurements of vortex dipole dynamics, finding that the vortex cores of opposite charge can exist for many seconds and that annihilation is inhibited in our trap geometry. For sufficiently rapid flow velocities, clusters of like-charge vortices aggregate into long-lived multiply charged dipolar flow structures.

Figure 1

The BEC initial state and experimental sequence. (a) Side-view phase-contrast image and (b) axial absorption image of a BEC in the highly oblate trap in the absence of the obstacle. Lighter shades indicate higher column densities integrated along the line of sight. Our axial and radial trapping frequencies are measured to be ${\omega }_z=2\pi \times (90\mathrm{Hz})$ and ${\omega }_r=2\pi \times (8\mathrm{Hz})$, respectively. (c) BEC initial state with the obstacle located at $x_s=-20\mu \mathrm{m}$ relative to the BEC center. (d),(e) The maximum repulsive potential energy of the obstacle is $U_0\approx 1.2{\mu }_0$ (where ${\mu }_0\sim 8\hslash {\omega }_z$ is the BEC chemical potential) and is held constant during evaporative cooling. It is ramped down linearly as the trap translates; relative to the trap center, the beam moves from position $x_s=-20\mu \mathrm{m}$ to $x_s=14\mu \mathrm{m}$ over a variable sweep time $t_s$. The BEC is then held in the harmonic trap for a variable time $t_h$ prior to expansion and imaging.

Figure 2

Average number of vortex cores observed for a range of translation velocities $v_s$. Experimental data points (black diamonds) are averages of 10 runs each, with error bars showing the standard deviation of the observations. Numerical data for $N_c=2\times {10}^6$ at a system temperature of $T=52\mathrm{nK}$, corresponding to experimental conditions, are indicated by triangles joined with dotted lines. Fewer atoms and lower temperatures reduce the critical velocity; as an example, open circles show the results of numerical simulations using $N_c=1.3\times {10}^6$ and $T=0$. For $v_s\lesssim 170\mu \mathrm{m}/\mathrm{s}$, remnant vortices arising spontaneously during condensate formation (see Ref. 11) account for the baseline of $\sim 0.3$. For $v_s\sim 190\mu \mathrm{m}/\mathrm{s}$, the inset images show a single pair of vortices having formed in the experiment (left) and simulations (right). Because the vortex core diameters are well below our imaging resolution, the BEC is expanded prior to imaging. Similarly, the vortex cores in the unexpanded numerical image are barely visible; a $10\mathrm{\text{−}}\mu \mathrm{m}$-wide inset provides a magnified scale for the core size. Above $v_s\sim 200\mu \mathrm{m}/\mathrm{s}$, multiply-charged vortex dipoles are observed. The critical velocity calculated using the methods of Ref. 23 is indicated by the vertical dashed line.

Figure 3

Sequences of images showing the first orbit of vortex dipole dynamics. (a) Back-to-back expansion images [34] from the experiment with 200 ms of successive hold time between the $180\mathrm{\text{−}}\mu \mathrm{m}$-square images. This sequence was taken using an obstacle height 3.15 times larger than that used for the data of Fig. 2, as this gave the highest degree of repeatability and the least sensitivity to beam displacement. (b) $62\mathrm{\text{−}}\mu \mathrm{m}$-square images from numerical data obtained for conditions similar to the data of sequence (a), but for a temperature of $T=0$. The orbital period is $\sim 1.2\mathrm{s}$, and the apparent vortex core size is smaller in the simulations because we show in-trap numerical data. (c) Black and dark gray small circles show average positions $x_v$ and $y_v$ (as fractions of the BEC radius) of each of the two vortices from 5 sequences of experimental data identical to that of sequence (a). The larger circle around each average position point represents the standard deviation of the vortex positions at that specific hold time, and is calculated from the 5 images obtained at that time step. A continuous dipole trajectory from sequence (b) is rescaled to the Thomas-Fermi radius of the expanded experimental images, and superimposed as solid lines on the experimental data; no further adjustments are made for this comparison. (d) Similar to sequence (b), but for a faster translation velocity with which a doubly charged vortex dipole is formed. Images are spaced by 150 ms, and the orbital period is 900 ms. (e) An experimental image of a doubly charged dipole; see text for explanation. (f),(g) Similar to (d),(e) but for a triply charged dipole, 112 ms between images, and an orbital period of 675 ms.

Figure 4

Number of vortices remaining after dipole nucleation as a function of hold time $t_h$, averaged over 17 realizations per data point. The circles show conditions for which a singly charged dipole is created, while the squares show data from a faster sweep where four cores (a doubly charged dipole) preferably occur. The error bars represent statistical uncertainty (as in Fig. 2) rather than counting uncertainty; however, for the doubly charged dipole case there is additional uncertainty in counting vortex cores because we do not always resolve 4 well-defined cores at the earlier hold times. The vortex cores can clearly persist for times much longer than the dipole orbital period of $\sim 1.2\mathrm{s}$ for a singly quantized dipole, and $\sim 0.8\mathrm{s}$ for a doubly quantized dipole. Inset figures: experimental images of singly charged dipoles obtained with identical sweeps and held for 4 sec in the trap prior to imaging. The images illustrate the variability of vortex positions after such a long hold time, precluding our ability to track consistent vortex positions beyond one orbital period.