Universal dynamics in the expansion of vortex clusters in a dissipative two-dimensional superfluid

A large ensemble of quantum vortices in a superfluid may itself be treated as a novel kind of fluid that exhibits anomalous hydrodynamics. Here we consider the dynamics of vortex clusters under thermal friction and present an analytic solution that uncovers a new universality class in the out-of-equilibrium dynamics of dissipative superfluids. We find that the long-time dynamics of the vorticity distribution is universal in the form of an expanding Rankine vortex (i.e., top-hat distribution) independent of initial conditions. This highlights a fundamentally different decay process to classical fluids, where the Rankine vortex is forbidden by viscous diffusion. Numerical simulations of large ensembles of point vortices confirm the universal expansion dynamics and further reveal the emergence of a frustrated lattice structure marked by strong correlations. We present experimental results of expanding vortex clusters in a quasi-two-dimensional Bose-Einstein condensate that are in excellent agreement with the vortex fluid theory predictions, demonstrating that the signatures of vortex fluid theory can be observed with as few as $N\sim 11$ vortices. Our theoretical, numerical, and experimental results establish the validity of the vortex fluid theory for superfluid systems.

Figure 1

(a) Exemplar vortex distributions throughout the expansion in free space with $\gamma =0.01$. We show the (a) top-hat, (b) Gaussian, and (c) ring initial distributions as well as two nonaxisymmetric initial conditions in (d) and (e). The time in the simulation is indicated on the left. The radii of the dotted lines in the first three rows, emphasizing the expansion, have a constant radius of $r=0.2$. The $r=0.2$ dashed line is omitted in the final row as it cannot be seen. In the bottom row we have highlighted lattice dislocations (defined when vortices have either five or seven nearest neighbors) with black points.

Figure 2

Comparison between vortex fluid theory (VFT) and numerical results for (a) average radius of vortex cluster and (b) energy of the vortex cluster. In the bottom panel we plot the error between the hydrodynamic solution and the numerical simulation.

Figure 3

Normalized ensemble radial vortex density for the top-hat, Gaussian, and ring initial conditions at times (a) $t=0$, (b) $t=10\gamma \tau$, (c) $t=100\gamma \tau$, and (d) $t=1000\gamma \tau$ averaged over $n=100$ different realizations for each distribution. The black curve in (d) is the coarse-grained density for the top-hat initial distribution. (e) Normalized average density as a function of radius and time for the top-hat initial distribution.

Figure 4

Vortex-vortex correlation function $g\left(s\right)$ at times (a) $t=0$, (b) $t=100\gamma \tau$, (c) $t=1000\gamma \tau$, and (d) $t=1.3\times {10}^5\gamma \tau$. (e) Decay of the geometric disorder parameter, ${\sigma }_g$, averaged over $n=100$ runs. The shaded area represents a region where ${\sigma }_g\sim t^{-\alpha }$ with $\alpha \simeq 4/9$.

Figure 5

(a) Schematic of the procedure used to create the initial vortex cluster. An elliptical stirrer is swept around the annulus as its length is decreased over time, creating same-signed vortices (denoted by a $+$) which become pinned to the central barrier. However, some vortices can become stray and remain in the bulk of the fluid. The radius of the central barrier is then shrunk to zero, and the pinned circulation (denoted by large $+$ at the center) splits into singly charged vortices which are free to expand. (b) Experimental image after 0- and 2-s hold time and 3-ms time-of-flight expansion showing resolved vortex cores as dark holes in the atom density. White circles indicate a vortex detected by the Gaussian fitting algorithm. (c) Two-dimensional histograms of the vortex positions over time, averaged over $\sim 40\mathrm{runs}$. The red dashed circle indicates the cutoff radius for cluster analysis (see text).

Figure 6

(a–c) Radial density of vortex clusters averaged over 1 s (period given in top right) along with top-hat distribution fit. Insets: corresponding 2D histograms of vortex positions. The red dashed line represents the cluster cutoff radius. (d) Error between histograms and top-hat fit.

Figure 7

(a) Comparison of vortex fluid theory and experimental measurements of the average vortex radius $\langle r\rangle$. The dashed line is a linear fit to the late-time data. The gray region is prior to the removal of the pinning potential. (b) Energy of experiment and vortex fluid theory. Experimental energy is also shown after being scaled (hollow points) by a constant factor (see text). (c) Average measured number of vortices in the system (i.e., cluster and stray vortices) and the number of vortices measured in the cluster. Each exhibits an artificial growth due to imaging techniques. (d) Observed geometric disorder ${\sigma }_g$ of the vortex cluster compared with point vortex simulations with and without stray vortices used to calculate the dynamics. Legend in (a) applies to all four subfigures. Uncertainty in the experiment is given by the standard error.

Figure 8

Time evolution of perturbation $\delta \rho$ for different initial distributions: (a) $\delta \rho (r,0)=0.2\exp (-r^2)$; (b) $\delta \rho (r,0)=0.2/{(1+r)}^2$; (c) $\delta \rho (r,0)=0.2\exp [-{(r-1)}^2]$; (d) $\delta \rho (r,0)=0.1(sin2r+1)exp(-r)$. Here we choose $\mathrm{\Gamma }=1, {\rho }_0=1$, and $\gamma =0.01$.

Figure 9

Evolution of vortex clusters without dissipation (i.e., $\gamma =0$) for initial distributions corresponding to (a) top-hat, (b) Gaussian, (c) ring, (d) one-dimensional line, and (e) multiple random clusters. The upper and middle row are exemplar snapshots of the cluster at the beginning and end of the simulation (as labeled). The bottom row is the average radial density (averaged over $n=20$ realizations) of distributions before and after evolution.

Figure 10

Comparison of radius and energy of point vortex cluster expansion with the predictions of the vortex fluid hydrodynamics. (a) A system of $N=2$ vortices initially equidistant from origin separated by $\theta =\pi$. The black curves correspond to the left $y$ axis and are the energy of the vortex cluster, while the orange curves correspond to the right $y$ axis and indicate the average radius of the vortex cluster. Dashed lines are from simulations of the point vortex model within a disk, and the solid lines are the solution to the vortex fluid hydrodynamics. (b) $N=5$ vortices. (c) $N=10$ vortices. (d) $N=20$ vortices.