Interface dynamics of a two-component Bose-Einstein condensate driven by an external force

The dynamics of an interface in a two-component Bose-Einstein condensate driven by a spatially uniform time-dependent force is studied. Starting from the Gross-Pitaevskii Lagrangian, the dispersion relation for linear waves and instabilities at the interface is derived by means of a variational approach. A number of diverse dynamical effects for different types of driving force is demonstrated, which includes the Rayleigh-Taylor instability for a constant force, the Richtmyer-Meshkov instability for a pulse force, dynamic stabilization of the Rayleigh-Taylor instability and onset of the parametric instability for an oscillating force. Gaussian Markovian and non-Markovian stochastic forces are also considered. It is found that the Markovian stochastic force does not produce any average effect on the dynamics of the interface, while the non-Markovian force leads to exponential perturbation growth.

Figure 1

Schematic of two-component BEC in an external nonuniform magnetic field.

Figure 2

Hydrostatic profiles of two components pushed to each other by a constant force with $b=7.5\times {10}^{-5}$ and ${10}^{-4}$ for panels (a) and (b), respectively. The dashed lines show the analytical solution Eq. (11).

Figure 3

Lowest Bogoliubov modes (the RT dispersion relation) for different values of the driving force. The analytical result is obtained from Eq. (49) and presented by lines. Dashed lines show real parts of the Bogoliubov eigenvalues (waves), and solid lines correspond to the imaginary parts (instability). The markers show respective numerical solutions to the linearized GP equations, filled markers correspond to the instability regime, and empty markers correspond to waves.

Figure 4

Development of the RT instability in a trapped system of two-component BEC presented for one of the components.

Figure 5

Stabilization of the RT instability by a high-frequency force for the amplitude ratios $b_s/b_0=0,7,15,20$, for $b_0=2.5\times {10}^{-5}$, $k=k_{\max }$ and $\mathrm{\Omega }=20{\alpha }_0$.

Figure 6

Stability limits in the parameter space of the scaled wave number $k/k_c$ and the oscillation amplitude $b_s/b_0$, where $b_0=2.5\times {10}^{-5}$, $\mathrm{\Omega }=20\max ({\alpha }_0)$. Color indicates value of the instability growth rate in the unstable region; Eq. (54).

Figure 7

The RT instability growth rate, Eq. (54), in presence of external high-frequency field with $\mathrm{\Omega }=20\max ({\alpha }_0)$ vs. perturbation wave number for different values of $b_1/b_0=0,7,15,20$ and $b_0=2.5\times {10}^{-5}$.

Figure 8

Stability limits in parameter space of wave number $k$ and amplitude $b_s$ of external high-frequency force with $\mathrm{\Omega }=3.6\times {10}^{-3}$. Color indicates value of the instability growth rate in the unstable region, obtained from Eq. (57).

Figure 9

Growth rate of parametric instability, Eq. (57), as a function of $k$ for different amplitudes of external oscillating magnetic force $b_s=0.25\times {10}^{-4},1.75\times {10}^{-4},3.5\times {10}^{-4},5\times {10}^{-4}$ with $\mathrm{\Omega }=3.6\times {10}^{-3}$.

Figure 10

Development of the RM instability in a trapped system of two-component BEC presented for one of the components.

Figure 11

(a) Single realization of the external stochastic Gaussian Markovian force with $\mathrm{\Omega }=5\times {10}^{-9}$, ${\tau }_c=50$, and ${\mathrm{\Omega }}_c{\tau }_c=1.5\times {10}^{-2}$. (b) The respective numerical solution to Eq. (47). The dashed line shows the amplitude of the averaged analytical solution, Eq. (80).

Figure 12

(a) Single realization of the external stochastic Gaussian non-Markovian force with $\mathrm{\Omega }=2\times {10}^{-7}$, ${\tau }_c=1200$, and ${\mathrm{\Omega }}_c{\tau }_c=0.38$. (b) The respective numerical solution to Eq. (47) (two realizations). The dashed line shows the amplitude of the averaged analytical solution, Eq. (80).