Turbulence in binary Bose-Einstein condensates generated by highly nonlinear Rayleigh-Taylor and Kelvin-Helmholtz instabilities

Quantum turbulence (QT) generated by the Rayleigh-Taylor instability in binary immiscible ultracold ${}^{87}\text{Rb}$ atoms at zero temperature is studied theoretically. We show that the quantum vortex tangle is qualitatively different from previously considered superfluids, which reveals deep relations between QT and classical turbulence. The present QT may be generated at arbitrarily small Mach numbers, which is a unique property not found in previously studied superfluids. By numerical solution of the coupled Gross-Pitaevskii equations we find that the Kolmogorov scaling law holds for the incompressible kinetic energy. We demonstrate that the phenomenon may be observed in the laboratory.

Figure 1

(a) The ground state of binary BEC with weakly repulsive intercomponent interaction in a 2D harmonic trap for $\gamma =0.01$ and $R_0=40$. (b) Initial stage of development of the RTI for $b=1.4$. Triggered by the initial infinitesimal perturbation of the interface with wavelength $\approx 13a_z$ which corresponds to the most unstable mode for given $\gamma$, $R_0$, and $b$, the BECs start counterflowing driven by the magnetic force $b$. The relative shear motion is generated on the sides of the growing perturbations (one of them is shown by the arrow), which leads to the KHI and mushroom-shaped bubbles. Coordinates are scaled by $a_z$, time by ${\left(\pi {\nu }_z\right)}^{-1}$, and density by $n_0$.

Figure 2

Frequency ${\omega }_{\text{RT}}$ for waves on the interface between two weakly repulsive BECs for $\gamma =0.01$, and two sets of parameters $R_0$ and $b$, as function of wave number $k_x$ of the perturbation. Dashed lines show imaginary ${\omega }_{\text{RT}}$ corresponding to the RTI, solid lines show ${\omega }_{\text{RT}}$ of the superfluid “capillary-gravitational” oscillations. Wave numbers are scaled by $a_z^{-1}$, and frequencies by $2\pi {\nu }_z$.

Figure 3

Binary quantum turbulence generated for $\gamma =0.01$, $R_0=40$, and $b=1.4$. Density of BEC 1 for subsequent instants of time showing (a,b) generation and (c,d) free decay of turbulence, which follows the initial stage shown in Fig. 1. Coordinates are scaled by $a_z$, time by ${\left(\pi {\nu }_z\right)}^{-1}$, and density by $n_0$.

Figure 4

Velocity of the bubble tip in units of $\pi a_z{\nu }_z$ as a function of the scaled strength of the driving force for $\gamma =0.01$ and $R_0=30$. Line shows the analytical prediction, Eq. (4), and markers show the results of numerical simulation.

Figure 5

Maximum integrated vorticity $\Omega$ in units of $2\pi a_z^2{\nu }_z$ developed in 2D RTI as function of the scaled driving force $b$, for $\gamma =0.01$ and $R_0=30$. Line shows the analytical prediction, Eq. (7), and markers show the results of numerical simulation.

Figure 6

Angle-averaged spectrum of incompressible kinetic energy of binary quantum turbulence for $\gamma =0.01$, $R_0=40$, and $b=1.4$, corresponding to Figs. 3 and 3 and an intermediate instant. Straight lines show analytical predictions. Wave numbers are scaled by $a_z^{-1}$, and energy density by $\pi \hslash {\nu }_z{n}_0$.

Figure 7

Angle-averaged spectra of modulus of the spin-dependent interaction energy of binary quantum turbulence for $\gamma =0.01$, $R_0=40$, and $b=1.4$, corresponding to Figs. 3 and 3 and an intermediate instant. Straight lines show analytical predictions. Wave numbers are scaled by $a_z^{-1}$, and energy density by $\pi \hslash {\nu }_z{n}_0$.

Figure 8

Development of the KHI in two-component BEC with a shear flow for $\gamma =0.01$ and $R_0=30$. The shear flow is between BEC at $z<0$ (along $x$), and BEC at $z>0$ (opposite to $x$), with the relative velocity $2\hslash q_0/m$ (see the text). Panels (a,b) show BEC 1 in a system which is twice narrower in $x$ direction than the system shown in (c,d), while the other parameters are the same. BEC 2 is located symmetrically and not shown. The unstable mode is the same for both systems, (a,c). At later times (b,d), the sheet with vorticity (the interface) rolls up transferring vorticity to the largest possible eddy and forming a vortex bundle. Coordinates are scaled by $a_z$, time by $1/\pi {\nu }_z$, and density by $n_0$.