Quantum turbulence in a trapped Bose-Einstein condensate

We study quantum turbulence in trapped Bose-Einstein condensates by numerically solving the Gross-Pitaevskii equation. Combining rotations around two axes, we successfully induce quantum turbulent state in which quantized vortices are not crystallized but tangled. The obtained spectrum of the incompressible kinetic energy is consistent with the Kolmogorov law, the most important statistical law in turbulence.

Figure 1

(Color online). Schematic sketch of the rotation. The first rotation is applied along the $z$ axis and the second rotation is applied along the $x$ axis.

Figure 2

Time development of (a) $E_{\mathrm{kin}}^{c,i}\left(t\right)$ and (b) $\sigma \left(t\right)$.

Figure 3

(Color online). (a) Wave number dependence of $E_{\mathrm{kin}}^i(k,t)$ and $\Pi (k,t)$. (b) Vortex length distribution $n\left(l\right)\Delta l$ inside the Thomas-Fermi radius $R_{\mathrm{TF}}$. $E_{\mathrm{kin}}^i(k,t)$, $\Pi (k,t)$, and $n\left(l\right)\Delta l$ are obtained from an ensemble average of 25 turbulent states $(t\omega >300)$. In both figures, $a_{\text{trap}}=\sqrt{\hslash ∕m\omega }$ is the characteristic scale of the trap.

Figure 4

(Color online). Isosurface plots of 5% of the maximum condensate density (a)–(c) and configuration of quantized vortices inside the Thomas-Fermi radius $R_{\mathrm{TF}}$ (d)–(f). (a), (d) $t\omega =10$; (b), (e) $t\omega =50$; (c), (f) $t\omega =300$. The method for identifying vortices in (d)–(f) is the same as that in Fig. 7 in Ref. 15.