Superfluid turbulence: Nonthermal fixed point in an ultracold Bose gas

Nonthermal fixed points of the dynamics of a dilute degenerate Bose gas are analyzed in two and three spatial dimensions. For such systems, universal power-law distributions, previously found within a nonperturbative quantum-field theoretic approach, are shown to be related to vortical dynamics and superfluid turbulence. The results imply an interpretation of the momentum scaling at the nonthermal fixed points in terms of independent vortex excitations of the superfluid. Long-wavelength acoustic excitations on the top of these are found to follow a nonthermal power law. The results shed light on fundamental aspects of superfluid turbulence and have strong potential implications for related phenomena, e.g., in early-universe inflation or quark-gluon plasma dynamics.

Figure 1

Phase angle $\phi (\mathbf{x},t)$ (color scale) at four times during a single run of the simulations in $d=2$. The white spots mark vortex cores where the density falls below 5% of the mean density $\overline{n}$. Parameters are as follows: $\overline{g}=2mg{a}_s^{2-d}=3\times {10}^{-5}$, $N={10}^8$, $N_s=256$, and $\overline{t}=t/\left(2m{a}_s^2\right)$. Shown times are as follows: (1) $\overline{t}=26$ (top left-hand side): Ordered phase shortly after initial preparation. (2) $\overline{t}=820$ (top right-hand side): After creation of vortex-antivortex pairs. (3) $\overline{t}=6550$ (bottom left-hand side): Critical slowing down. (4) $\overline{t}={10}^5$ (bottom right-hand side): Low density of vortex pairs before final thermalization.

Figure 2

Snapshots of a single run of the evolution in $d=3$ dimensions. The points show where the density falls below 5% of the mean density $\overline{n}$. Parameters are as follows: $\overline{g}=8\times {10}^{-4}$, $N={10}^9$, $N_s=128$. (1) $\overline{t}=0$ (top left-hand side). (2) $\overline{t}=103$ (top right-hand side). (3) $\overline{t}=410$ (bottom left-hand side). (4) $\overline{t}=1640$ (bottom right-hand side).

Figure 3

Single-particle mode occupation numbers as functions of the radial momentum $k={\left[{\sum }_{i=1}^{d}4{\sin }^2(k_i/2)\right]}^{1/2},\mathbf{k}=2\pi \mathbf{n}/N_s,\mathbf{n}=(n_1,...,n_d),n_i=-N_s/2,...,N_s/2$, for the four different times of the run in $d=2$ dimensions shown in Fig. 1. Note the double-logarithmic scale. An early development of a scaling $n\left(k\right)\sim k^{-4}$ is followed by a bimodal scaling with $n\left(k\right)\sim k^{-2}$ at larger wave numbers.

Figure 4

Single-particle occupation numbers of different fractions of the system, at time $\overline{t}={10}^5$ of the $d=2$ run shown in Fig. 3. The black points are the same as those shown in the lower right-hand panel of Fig. 3. Colors distinguish the fractions $n_{\mathrm{i}}$ (red circles), i.e., the divergence-free part of the flow field ${\mathbf{w}}_{\mathrm{v}}=\sqrt{n}\mathbf{v}$, $n_{\mathrm{c}}$ (filled blue squares), i.e., the solenoidal part of ${\mathbf{w}}_{\mathrm{v}}$, and the quantum-pressure part $n_{\mathrm{q}}$ (open gray squares). $k_{\xi }=2\pi /\xi$, where $\xi =1/\sqrt{2mgn}$ is the healing length.

Figure 5

Same as in Fig. 4, for the run in $d=3$ dimensions shown in Fig. 2, at the time $\overline{t}=1640$ (lower right-hand panel).