Critical velocity for vortex shedding in a Bose-Einstein condensate

Woo Jin Kwon, Geol Moon, Sang Won Seo, and Y. Shin

Phys. Rev. A 91, 053615 – Published 15 May 2015

Abstract

We present measurements of the critical velocity for vortex shedding in a highly oblate Bose-Einstein condensate with a moving repulsive Gaussian laser beam. As a function of the barrier height $V_{0}$ , the critical velocity $v_{c}$ shows a dip structure having a minimum at $V_{0} \approx μ$ , where $μ$ is the chemical potential of the condensate. At fixed $V_{0} \approx 7 μ$ , we observe that the ratio of $v_{c}$ to the speed of sound $c_{s}$ monotonically increases for decreasing $σ / ξ$ , where $σ$ is the beam width and $ξ$ is the condensate healing length. We explain our results with the density reduction effect of the soft boundary of the Gaussian obstacle, based on the local Landau criterion for superfluidity. The measured value of $v_{c} / c_{s}$ with our stiffest obstacle is about 0.4, which is in good agreement with theoretical predictions for a two-dimensional superflow past a circular cylinder.

Received 12 February 2015

DOI:https://doi.org/10.1103/PhysRevA.91.053615

Authors & Affiliations

Woo Jin Kwon, Geol Moon, Sang Won Seo, and Y. Shin^*

Department of Physics and Astronomy and Institute of Applied Physics, Seoul National University, Seoul 151-747, Korea

^*yishin@snu.ac.kr

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Issue

Vol. 91, Iss. 5 — May 2015

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Images

Figure 1
Soft boundary of the optical obstacle formed by a repulsive Gaussian potential $V (r) = V_{0} exp (- 2 r^{2} / σ^{2})$ . (a) Normalized potential slope at the obstacle boundary $\tilde{S} = (ξ / μ) {| d V / d r |}_{r = R}$ as a function of $σ / ξ$ and $R / ξ$ , where $μ$ and $ξ$ is the chemical potential and the healing length of the condensate, respectively, and $R$ is the obstacle radius such that $V (R) = μ$ . The boundary becomes stiffer as (b) $R / ξ$ becomes larger or (c) $σ / ξ$ decreases. The corresponding trajectories are indicated by the arrows in (a).
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Figure 2
Vortex shedding in a highly oblate Bose-Einstein condensate. (a) Schematic of the experiment. An optical obstacle is formed by a repulsive Gaussian laser beam penetrating through the condensate and moves horizontally at a constant velocity in the center region. (b) An in situ image of the condensate with the optical obstacle at the initial position. (c) Probability $P (v)$ of having vortex dipoles (blue circles) and the number of vortices (red diamonds) as a function of the velocity $v$ of the optical obstacle. The barrier height $V_{0} \approx 2.1 μ$ and the Gaussian width $σ \approx 20 ξ$ . The solid line is a sigmoidal function fit to $P (v)$ . The inset shows an image of a condensate containing a vortex dipole. Each data point was obtained from 15 realizations of the same experiment and the error bars indicate the standard deviations of the measurements.
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Figure 3
(a) Normalized critical velocity $v_{c} / c_{s}$ versus the relative barrier height $V_{0} / μ$ for various beam widths $σ / ξ$ ; $(σ, ξ) = [24.5 (24), 0.46]$ (red diamonds), $[16.0 (14), 0.46]$ (black squares), $[9.1 (12), 0.46]$ (blue circles), and $[9.1 (12), 0.9] μ m$ (open squares). The inset displays $v_{c} / c_{s}$ at $V_{0} \approx μ$ as a function of $σ / ξ$ and the solid line is a line to guide the eye. (b) Same data in the nonpenetrable regime $(V_{0} > μ)$ as a function of the hole radius $R / ξ$ , together with additional data points for $V_{0} / μ > 10$ .
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Figure 4
Normalized critical velocity $v_{c} / c_{s}$ versus $σ / ξ$ for fixed $V_{0} / μ \approx 7$ . Here $σ = 9.1 (12) μ m$ for black closed squares and $ξ = 0.46 μ m$ for black closed circles. The blue open circles show the results of theoretical calculation for a 2D hard cylinder of radius $R = σ \sqrt{ln (V_{0} / μ) / 2}$ from Ref. [8]. The red open diamonds indicate the numerical results for a Gaussian potential with $V_{0} / μ = 100$ from Ref. [34], where the potential slope $\tilde{S}$ would be about two times higher than ours. The dashed line denotes the theoretically predicted value $v_{c} / c_{s} = 0.37$ for a 2D homogeneous case with a hard cylinder in the large obstacle limit [7, 8, 9].
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