Long-range ordering of topological excitations in a two-dimensional superfluid far from equilibrium

We study the relaxation of a two-dimensional (2D) ultracold Bose gas from a nonequilibrium initial state containing vortex excitations in experimentally realizable square and rectangular traps. We show that the subsystem of vortex gas excitations results in the spontaneous emergence of a coherent superfluid flow with a nonzero coarse-grained vorticity field. The stream function of this emergent quasiclassical 2D flow is governed by a Poisson-Boltzmann equation. This equation reveals that maximum entropy states of a neutral vortex gas that describe the spectral condensation of energy can be classified into types of flow depending on whether or not the flow spontaneously acquires angular momentum. Numerical simulations of a neutral point vortex model and a Bose gas governed by the 2D Gross-Pitaevskii equation in a square reveal that a large-scale monopole flow field with net angular momentum emerges that is consistent with predictions of the Poisson-Boltzmann equation. The results allow us to characterize the spectral energy condensate in a 2D quantum fluid that bears striking similarity to similar flows observed in experiments of 2D classical turbulence. By deforming the square into a rectangular region, the resulting maximum entropy state switches to a dipolar flow field with zero net angular momentum.

Figure 1

Statistical weight for vortices in a square as a function of energy per vortex with minimum intervortex and vortex-boundary separations of 0.08 and 0.12 imposed on vortex distributions.

Figure 2

Entropy as a function of energy of mean flows predicted by Poisson-Boltzmann equation for a square domain (dipole, diagonal dipole, and monopole solutions) and for a rectangular domain (dipole, and monopole solutions). Insets show vorticity contours of mean flows (postitive and negative contours correspond to red solid and blue dashed lines, respectively).

Figure 3

Time sequence of locations of vortices (red circles) and antivortices (blue squares) in the square for (a) the point vortex model ($t_v=4$) and (b) the GP model ($t_v=2.09\times {10}^4$). In (b), the background contour corresponds to a plot of ${\left|\varphi \right|}^2$.

Figure 4

Contour plots in the square domain of the dominant eigenmode of the two-point correlator $\langle \psi (\mathbf{x},t)\psi ({\mathbf{x}}^{\prime },t)\rangle$ for (a) the point vortex model constructed over time intervals $t\in [45.87t_v,48.07t_v]$ for a dipole and $[64.40t_v,67.54t_v]$ for a monopole and for (b) the Gross-Pitaevskii model constructed over time intervals $t\in [1.92t_v,2.68t_v]$ for a dipole and $[3.64t_v,5.56t_v]$ for a monopole. Also shown are averaged streamlines for (c) the point vortex model and for; (d) the Gross-Pitaevskii model. Time intervals used in (c) and (d) are the same as those used in (a) and (b), respectively. In all cases, instantaneous stream functions are reconstructed from quantized vortex positions; (postitive and negative streamlines are shown as red solid and blue dashed lines, respectively).

Figure 5

Time variation of (a) total number of vortices $N^v$; (b) energy per vortex $H/N^v$, for point vortex simulations.

Figure 6

Time sequence of locations of vortices (red circles) and antivortices (blue squares) in the rectangle for (a) the point vortex model ($t_v=4$) and (b) the GP model ($t_v=2.09\times {10}^4$). In (b), the background contour corresponds to a plot of ${\left|\varphi \right|}^2$.

Figure 7

Averaged streamlines in the rectangular domain calculated from quantized vortex positions for (a) the point vortex model constructed over the time interval $t\in [75.4t_v,78.54t_v]$ and for (b) and (c) GP simulations with different parameters constructed over the time intervals $t\in [5.75t_v,6.71t_v]$ and $[8.63t_v,10.55t_v]$, respectively (postitive and negative streamlines are shown as red solid and blue dashed lines, respectively).

Figure 8

Time variation of number of vortices and antivortices (normalized with respect to their initial values at $t=0$), and vortex polarization in GP simulations; (a) square, (b) rectangle.

Figure 9

Time variation of angular momenta in (a) square and (b) rectangular geometries for the point vortex model.

Figure 10

Occupation number spectra for the GP simulation with $1025\times 1025$ grid points, with the interaction parameter set to $\tilde{g}=282920$ and starting with an initial neutral configuration consisting of 360 vortices. The phase field at corresponding time is shown in the inset with locations of vortices (red circles) and antivortices (blue squares).