Rayleigh-Taylor, Kelvin-Helmholtz and immiscible to miscible quenching instabilities in binary Bose-Einstein condensates

TL;DR

This study analyzes RT, KH, and IMQT instabilities in a binary immiscible BEC confined to a quasi-2D circular box using a two-component GP framework. It decomposes the kinetic energy into incompressible and compressible parts and examines spectra for signs of Kolmogorov-like scaling $k^{-5/3}$ and $k^{-3}$, as well as a Bottleneck feature in IMQT, across density, vortex, and phonon dynamics. The authors find robust vortex production and phonon emission across all instability types, with KH remaining vortex-dominated over long times, RT showing an incompressible-to-compressible transition near $t\,\approx\,9$, and IMQT exhibiting a pronounced bottleneck and eventual phonon-dominated turbulence; these behaviors reflect the impact of geometry, interspecies interactions, and nonlinear quenches on energy transfer in quantum turbulence. The work advances understanding of quantum turbulence in multi-component BECs, highlighting similarities and differences with classical turbulence and providing guidance for future experiments on engineered instabilities in binary condensates. Overall, the results illuminate how energy cascades and vortex dynamics evolve in low-dimensional, immiscible quantum fluids and suggest avenues for exploring universal aspects of turbulence in quantum systems.

Abstract

We investigate three kinds of instabilities in binary immiscible homogeneous Bose-Einstein condensate, considering rubidium isotopes $^{85}$Rb and $^{87}$Rb confined in two-dimensional circular box. Rayleigh-Taylor (RT) and Kelvin-Helmholtz (KH) instability types are studied under strong perturbations. Without external perturbation, instabilities are also probed by immiscible to miscible quenching transition (IMQT), under two different initial configurations. Our numerical simulations show that all such instability dynamics are dominated by large vortex productions and sound-wave (phonon) propagations. For long-term propagation, vortex dynamics become dominant over sound waves in the KH instability, while sound wave excitations predominate in the other cases. For all the dynamical simulations, the emergence of possible scaling laws are investigated for the compressible and incompressible parts of the kinetic energy spectra, in terms of the wave number $k$. The corresponding results are compared with the classical Kolmogorov scalings, $k^{-5/3}$ and $k^{-3}$, for turbulence, which are observed in the kinetic energy spectra at some specific time intervals. Deviating from the classical scaling, a kind of ``Bottleneck effect" is noticed in the IMQT spectra.

Figures (14)

• Figure 1: (Color online) RT instability in the binary mixture $^{85}$Rb [(a$_1$)-(g$_1$)] and $^{87}$Rb [(a$_2$)-(g$_2$)], shown by sample time snapshots of the densities $|\psi_i|^2$, together with respective phases (as indicated, with $t$ given inside the panels for the densities). The immiscible condition $\delta=a_{12}/a_{ii}=1.05$ is kept along the numerical simulations. The color-bar levels for densities and phases are indicated at the top, with the units for time and length being, respectively, $\omega_\perp^{-1}$ and $l_\perp$. The corresponding full-dynamical evolution is provided in the supplemental material Suppl.
• Figure 2: (Color online) Time evolutions of incompressible and compressible kinetic energies, $K_I$ and $K_C$ (units $\hbar \omega_{\perp}$), for the $^{85}$Rb-$^{87}$Rb mixture, associated with RT instabilities, with convention as indicated inside the panel. The vertical dashed line at $t\approx 9$ identifies the approximate time instant for the energy transition from incompressible (related to the vorticity) to a compressible (sound waves) dominated fluid.
• Figure 3: (Color online) Incompressible (a$_i$) and compressible (b$_i$) kinetic energy spectra, ${\cal K}(k)$ (units of $\hbar\omega_\perp l_\perp$), for the RT dynamics, shown as functions of the dimensionless $k\xi$ for the $^{85}$Rb (upper panels) and $^{87}$Rb (lower panels) components of the mixture, considering four time instants $t$ in the evolution (as indicated). The inclined straight lines provide the $k^{-5/3}$ and $k^{-3}$ behaviors, for comparison. The two close instants 4.0 and 4.1 refer to the fast behavior transition following the panels (d$_i$) shown in Fig. \ref{['fig01']}. The vertical lines refer to the size of the box (infrared limit), $k\xi=k_L\xi=0.01\pi$ (solid line), and the starting ultraviolet region $k\xi=1$ (dotted line), where $\xi=0.4$. The allowed maximum (defined by $\Delta x=0.2$) goes to $k_{\rm max}=5\pi\sim 15.7$.
• Figure 4: (Color online) KH instability in the binary mixture $^{85}$Rb [panels (a$_1$)-(g$_1$)] and $^{87}$Rb [panels (a$_2$)-(g$_2$)], shown by sample time snapshots (with $t$ given inside the left panels) of the respective densities $|\psi_i|^2$ and phases, obtained by numerical simulations with the immiscible condition $\delta=1.05$. Here, a constant linear force $\nu_i=(-)^{i+1}0.7$ (in the $x-$direction) is applied to the components. With the color-bar levels for densities and phases indicated at the top, the units for time and length are, respectively, $\omega_\perp^{-1}$ and $l_\perp$. The corresponding full-dynamical evolution is provided in the supplemental material Suppl.
• Figure 5: (Color online) Time evolution of the incompressible (empty symbols) and compressible (filled symbols) kinetic energies $K$ (units $\hbar \omega_{\perp}$) of the two components, evidencing KH instabilities. The incompressible results become larger than the compressible ones due to the dominance of vortex emission with their interaction. As legends indicate, the solid lines are for $^{85}$Rb, with dashed ones for $^{87}$Rb results.