Ultraquantum turbulence in a quenched homogeneous Bose gas

Using the classical field method, we study numerically the characteristics and decay of the turbulent tangle of superfluid vortices which is created in the evolution of a Bose gas from highly nonequilibrium initial conditions. By analyzing the vortex line density, the energy spectrum, and the velocity correlation function, we determine that the turbulence resulting from this effective thermal quench lacks the coherent structures and the Kolmogorov scaling; these properties are typical of both ordinary classical fluids and of superfluid helium when driven by grids or propellers. Instead, thermal quench turbulence has properties akin to a random flow, more similar to another turbulent regime called ultraquantum turbulence, which has been observed in superfluid helium.

Figure 1

Sample evolution of the turbulent vortex tangle (present in the condensed part of the Bose gas) as it decays to equilibrium. Here the condensate fraction of the gas is ${\rho }_0/\rho =0.22$. Shown are isosurfaces of the quasicondensate density at isosurface level $0.05\langle |\widehat{\psi }{|}^2\rangle$ at times (a) $t/\tau =0$, (b) 250, (c) 1250, and (d) 2500. At later times (not shown) all vortex lines disappear from the system. Here the quasicondensate is visualized using a cutoff of $k_c=10(2\pi /D)$. In the Supplemental Material [16], we provide a movie of this process. A similar figure and further description of the quasicondensate filtering are shown in Ref. [13].

Figure 2

Upper: Occupation numbers $n_k$ as a function of wave number $k\xi$ at different times $t/\tau$ as the Bose gas decays to equilibrium: $t/\tau =0$ (red circles), $t/\tau =25$ (orange squares), $t/\tau =75$ (magenta diamonds), $t/\tau =200$ (green pentagons), and $t/\tau =1000$ (blue triangles). Here the final condensate fraction at equilibrium is ${\rho }_0/\rho =0.48 \left(\langle H\rangle /D^3=1.33\right)$. A line proportional to the Rayleigh-Jeans distribution, $n_k\propto k^{-2}$, is shown by a black dashed line as a guide to the eye. Inset: Integral distribution of the particles, $F_k={\sum }_{k^{\prime }<k}{n}_{{\mathbf{k}}^{\prime }}$ vs $k\xi$ at the same times as in the main figure. In both the main figure and the inset $k_{c1}=10(2\pi /D)$ and $k_{c2}=20(2\pi /D)$ are labeled by red dashed lines. Lower: Condensate fraction over time for the quasicondensate filtered with a cutoff of $k_c=10(2\pi /D)$ (dashed lines) and $k_c=20(2\pi /D)$ (solid lines), for energy densities $\langle H\rangle /D^3=2.13$ (lower red/intermediate gray lines), $\langle H\rangle /D^3=1.33$ (middle green/light gray lines), and $\langle H\rangle /D^3=0.53$ (upper blue/dark gray lines). We find our choices of $k_c$ lead to only minimal variation of ${\rho }_0/\rho$ at equilibrium.

Figure 3

Incompressible kinetic energy spectrum of the quasicondensate for different temperatures, corresponding to condensate fraction ${\rho }_0/\rho =0.22$ (red/intermediate gray circles), ${\rho }_0/\rho =0.48$ (green/light gray squares), and ${\rho }_0/\rho =0.77$ (blue/dark gray diamonds) at time $t/\tau =300$, and with a cutoff of $k_c=10(2\pi /D)$ (marked by a dashed line labeled $k_{c1}$). Dashed lines also mark wave numbers $k_D,k_{\ell }$, and $k_{\xi }$ corresponding to the length scale of the box, the typical inter-vortex spacing, and the healing length, respectively. A black dashed line with $k^{-3}$ dependence is plotted as a guide to the eye. Inset: The same simulations, but with a quasicondensate cutoff of $k_c=20(2\pi /D)$ (marked by a dashed line labeled $k_{c2}$).

Figure 4

Vortex line density $L$ over time $t$ for condensate fraction ${\rho }_0/\rho =0.22$ (red/intermediate gray circles), ${\rho }_0/\rho =0.48$ (green/light gray squares), and ${\rho }_0/\rho =0.77$ (blue/dark gray diamonds). Each line is an average of five simulations. The line proportional to $t^{-1}$ (characteristic of the decay of ultraquantum turbulence) is shown (dashed black line) as a guide to the eye. Inset: The same simulations, but with a quasicondensate cutoff of $k_c=20(2\pi /D)$.

Figure 5

Integral scale $I$ as a function of time $t$ as the vortex tangle decays, for $\rho /{\rho }_0=0.22$ (red/intermediate gray circles), $\rho /{\rho }_0=0.48$ (green/light gray squares), $\rho /{\rho }_0=0.77$ (blue/dark gray diamonds), and half the typical intervortex spacing $\ell /2$ (dashed line). Inset: Velocity correlation function $f(r,t)$ at time $t=300\tau$ for equilibrium states at $\rho /{\rho }_0=0.22$ (red/intermediate gray solid line), $\rho /{\rho }_0=0.48$ (green/light gray dashed line), and $\rho /{\rho }_0=0.77$ (blue/dark gray dot-dashed line).