Generation and decay of two-dimensional quantum turbulence in a trapped Bose-Einstein condensate

In a recent experiment, Kwon et al. [Phys. Rev. A 90, 063627 (2014)] generated a disordered state of quantum vortices by translating an oblate Bose-Einstein condensate past a laser-induced obstacle and studied the subsequent decay of vortex number. Using mean-field simulations of the Gross-Pitaevskii equation, we shed light on the various stages of the observed dynamics. We find that the flow of the superfluid past the obstacle leads initially to the formation of a classical-like wake, which later becomes disordered. Following removal of the obstacle, the vortex number decays due to vortices annihilating and drifting to the boundary. Our results are in excellent agreement with the experimental observations. Furthermore, we probe thermal effects through phenomenological dissipation.

Figure 1

Number of vortices $N_v$ in the condensate after removal of the obstacle. Shown are simulations of the GPE without dissipation (red circles) and with dissipation $\gamma =0.0003$ (blue squares) and experimental results extracted from Fig. 1 of [8] (black crosses). Each point is averaged over 20 ms once the obstacle amplitude reaches $V_0=0$. For comparison, the speed of sound in the center of the BEC is $v_c\approx 4.6$ mm/s.

Figure 2

Snapshots of the condensate density for a translational speed $v=1.4\text{mm}/\mathrm{s}$ and in the presence of dissipation ($\gamma =0.0003$). The obstacle is completely removed at 0.43 s. The field of view in each figure is of size ${(170\times {10}^{-6}\mathrm{m})}^2$ and shifted along the $x$ axis so as to best display the condensate. Vortices with positive (negative) circulation are highlighted by red circles (blue triangles). A movie of this process can be found in the Supplemental Material [14].

Figure 3

Growth of vortex number (in a single realization) at early times for a translational speed of $v=1.4\text{mm}/\mathrm{s}$. Shown are the results with no dissipation (red dashed line) and with dissipation $\gamma =0.0003$ (blue solid line).

Figure 4

Vortex decay (a) and (c) in the absence of dissipation and (b) and (d) with dissipation $\gamma =0.0003$ for a translational speed of $v=1.4\text{mm}/\mathrm{s}$. (a) and (b) Decay of the total vortex number $N_v\left(t\right)$, with the contribution of drifting and annihilation depicted by the shaded regions. (c) and (d) Drift number $N_d\left(t\right)$ and annihilation number $N_a\left(t\right)$ plus their respective fits.

Figure 5

Density (left) and phase (right) (a) just before, (b) immediately following, and (c) a later time after a vortex-antivortex annihilation event. The field of view is ${(23.5\times {10}^{-6}\mathrm{m})}^2$, centered on the vortex pair or sound pulse (highlighted by a circle in the phase).

Figure 6

Snapshots of the condensate density for a translational of speed $v=0.8\text{mm}/\mathrm{s}$ past an elliptical obstacle (ellipticity $\epsilon =3$). The field of view in each figure is of size ${(170\times {10}^{-6}\mathrm{m})}^2$ and shifted along the $x$ axis so as to best display the condensate. Compared to the corresponding snapshots in Fig. 2, the elliptical obstacle generates as many final vortices but at a lower translational speed and with reduced condensate disruption. A movie of this process can be found in the Supplemental Material [14].