Isotropic vortex tangles in trapped atomic Bose-Einstein condensates via laser stirring

The generation of isotropic vortex configurations in trapped atomic Bose-Einstein condensates offers a platform to elucidate quantum turbulence on mesoscopic scales. We demonstrate that a laser-induced obstacle moving in a figure-8 path within the condensate provides a simple and effective means to generate an isotropic three-dimensional vortex tangle due to its minimal net transfer of angular momentum to the condensate. Our characterization of vortex structures and their isotropy is based on projected vortex lengths and velocity statistics obtained numerically via the Gross-Pitaevskii equation. Our methodology provides a possible experimental route for generating and characterizing vortex tangles and quantum turbulence in atomic Bose-Einstein condensates.

Figure 1

Tangle of few interacting vortices (left) exhibiting isotropic non-Gaussian flow statistics (right). Left: Density isosurface generated by stirring a spherically symmetric condensate along one plane in a figure-8 path (at time $t\approx t_{\mathrm{stir}}=17.1{\omega }^{-1}$ when the stirrer has just been removed after approximately two oscillations), with the vortex cores visualized by the dark blue dots. Right: PDFs of velocity components $v_x$, $v_y$, and $v_z$ (black, red, and blue overlapping solid lines respectively) at time $t\approx t_{\mathrm{stir}}$. The non-Gaussian nature of the turbulent velocity field is made apparent by the large deviation of the PDFs from the corresponding GPDFs, calculated by Eq. (3) (dashed lines).

Figure 2

PDFs of velocity components $v_x$, $v_y$, and $v_z$ (black, red, and blue overlapping solid lines respectively, as in Fig.1) of a single vortex line oriented in the $y\text{−}z$ direction (left) and a vortex ring in the $x\text{−}y$ plane (right). Insets show corresponding isosurfaces of the condensate density.

Figure 3

Top: Density images of the condensate at $z=0$ and $x,y=[-8,8]l_{\perp }$ at times $\omega t=9.1$ (A) and $17.1$ (B) and $24.9$ (C) [also marked on the lower graph where (B) refers to the snapshot from Fig. 1 at time $t=t_{\mathrm{stir}}$]. Regions of high (brown/red) and low (blue) density are apparent as indicated by the color bar. Density is measured in units of $l_{\perp }^{-3}$. The figure-8 path of the obstacle is shown by the white dashed line in (A). Middle: Corresponding phase snapshots. Singly charged vortices correspond to $2\pi$ phase singularities. To avoid phase artifacts at very low density, the phase is truncated beyond $R_{\mathrm{TF}}$. Bottom: Measured projected vortex lengths $L_i$, vs. time in the $x$ (black solid), $y$ (red solid with hollow circles), and $z$ (blue dashed) directions. Inset (left): Trajectory of laser in the $x$ direction, $x_l$, up to time $t_{\mathrm{stir}}$. Inset (right): The displaced density $\Delta$ vs time, where the displaced density is the difference in the norm between a condensate containing no vortices, ${\varphi }_1$, and the stirred condensate, ${\varphi }_2$, i.e., $\Delta ={\int }_{\mathcal{V}}|{\varphi }_1{|}^{2}dV-{\int }_{\mathcal{V}}{\left|{\varphi }_2\right|}^{2}dV$, where $\mathcal{V}=(4\pi /3)R_{\mathrm{cut}}^3$, with $R_{\mathrm{cut}}=0.78R_{\mathrm{TF}}$.

Figure 4

Left: Density isosurface at time $\omega t=25.7$ (as in Fig. 1) showing a few remaining vortices during the decay of the dense vortex tangle. Right: Corresponding velocity PDFs for components $v_x$, $v_y$, and $v_z$ (black, red, and blue overlapping solid lines respectively, as in Fig. 1).