Bénard–von Kármán Vortex Street in a Bose-Einstein Condensate

Vortex shedding from an obstacle potential moving in a Bose-Einstein condensate is investigated. Long-lived alternately aligned vortex pairs are found to form in the wake, which is similar to the Bénard–von Kármán vortex street in classical viscous fluids. Various patterns of vortex shedding are systematically studied and the drag force on the obstacle is calculated. It is shown that the phenomenon can be observed in a trapped system.

Figure 1

Density and phase distributions of a condensate past an obstacle potential. The velocity and potential width are $(\tilde{v},d/\xi )=(2.4,0.04)$ in (a), (2.6,0.05) in (b), and (3.0,0.05) in (c), where $\tilde{v}=v[{10}^{3}m/(g{n}_0){]}^{1/2}/(2\pi )$ and $\xi =\hslash [{10}^3/(mg{n}_0){]}^{1/2}$. The white arrows in (a) indicate the directions in which the vortex-antivortex pairs move. The density is normalized by $n_0$. The field of view is $6\xi \times 3\xi$. In the numerical calculation, a $32\xi \times 8\xi$ space is discretized into $4096\times 1024$.

Figure 2

Serial snapshots of the density profiles for the parameters in Fig. 1b in the frame moving with the potential. The time interval is $10\hslash /(g{n}_0)$. The arrows indicate the directions of circulations. The field of view is $2\xi \times \xi$. The color scale is the same as that in Fig. 1.

Figure 3

Dependence of the patterns of wakes on the normalized Gaussian width $d/\xi$ and velocity $\tilde{v}$ of the potential. The green, blue, and red regions correspond to the flow patterns shown in Figs. 1a, 1b, 1c, respectively. The white region corresponds to stationary laminar flow. The white arrow indicates the change of $\tilde{v}$ used in Fig. 4.

Figure 4

Snapshots of the density profiles for $d/\xi =0.05$. The velocity is $\tilde{v}=2.2$ for $\tilde{t}<1000$ and $\tilde{v}=2.2+0.4(\tilde{t}-1000)/4000$ for $\tilde{t}>1000$. The black and white circles indicate clockwise and counterclockwise circulations, respectively. The field of view is $3\xi \times 8\xi$. The color scale is the same as that in Fig. 1.

Figure 5

Time evolution of the normalized drag force $\tilde{\mathit{F}}=2\mathit{F}/(Am{n}_0{v}^2)$, where $A$ is the projected area of the obstacle potential. The solid and dashed lines show ${\tilde{F}}_x$ and ${\tilde{F}}_y$. The parameters used in (a)–(c) are the same as those in Figs. 1a, 1b, 1c, respectively.

Figure 6

A BEC of ${}^{87}\mathrm{Rb}$ atoms confined in a harmonic potential at $t=160\mathrm{ms}$, where the Gaussian potential is initially located at the marked position ($\times$) and starts to move in the $z$ direction at a constant velocity. (a) Column density normalized by $Nm{\omega }_x/\hslash$. The dashed region is magnified in the lower panels. The gray scale image shows the phase at $y=0$. The field of view is $300\times 30\mu \mathrm{m}$ for the main panel and $40\times 10\mu \mathrm{m}$ for the lower panels. (b) Translucent display of the isodensity surface for the state shown in (a).