Parametric resonance of capillary waves at the interface between two immiscible Bose-Einstein condensates

We study the parametric resonance of capillary waves on the interface between two immiscible Bose-Einstein condensates pushed towards each other by an oscillating force. Guided by analytical models, we solve numerically the coupled Gross-Pitaevskii equations for a two-component Bose-Einstein condensate at zero temperature. We show that, at moderate amplitudes of the driving force, the instability is stabilized due to nonlinear modifications of the oscillation frequency. When the amplitude of the driving force is large enough, we observe a detachment of droplets from the Bose-Einstein condensates, resulting in the generation of quantum vortices (skyrmions). We analytically investigate the vortex dynamics, and conditions of quantized vortex generation.

Figure 1

The system geometry: density distribution for two immiscible BECs, $n_1$ and $n_2$, for $R_0=30$, with initial perturbations at the interface. The BECs are tightly trapped in the $z$ direction (which means quasi-2D geometry), trapped with resulting TF profile in the $y$ direction, and uniform in the $x$ direction. Dimensionless units are defined in Sec. 2.

Figure 2

Scaled growth rate of the parametric instability vs the wave number of the interface perturbation between BECs for $\Omega =8.55$ and different amplitudes of the driving force, obtained in Ref. [20]. Dimensionless units are defined in Sec. 2.

Figure 3

Density distribution of BEC 1, $n_1$, for $R_0=30$, $b_c=3$ at $t=2.0$, showing the maximal interface deformation in the regime of nonlinearly stabilized parametric resonance. Dimensionless units are defined in Sec. 2.

Figure 4

Amplitude $\zeta \left(t\right)$ of the interface deviation from equilibrium for $R_0=30$ and $b_c=1,3$ and the scaled amplitude of the driving force (dots). Dimensionless units are defined in Sec. 2.

Figure 5

Phase diagram of vortex generation in the fixed-size (fixed $R_0$) system, showing possible regimes with well-defined interface. For $b_c\ge 10$ the system is not well described in the mean-field approximation because it acquires too many incoherent excitations. The mean deviation of the interface from equilibrium ${\zeta }_0$, averaged over twice as many periods of modulations, as shown in Fig. 4, for various $b_c$ (time averaged for irregular dependence), $R_0=30$. The solid line corresponds to the analytical model (7) with $\kappa =0.96$. The dashed line separates two qualitatively different nonlinear regimes. Dimensionless units are defined in Sec. 2.

Figure 6

Top: The scaled amplitude of the driving force, and amplitude of the interface deviation from equilibrium for $R_0=30$ and $b_c=6$; the markers indicate the time instants for the density snapshots (bottom). Bottom: Snapshots of density distribution of BEC 1, $n_1$. Dimensionless units are defined in Sec. 2.

Figure 7

Amplitude of the interface deviation from equilibrium for $R_0=30$ and $b_c=8$–10, and the scaled amplitude of the driving force. The time instants of vortex detachment are indicated by circles. Dimensionless units are defined in Sec. 2.

Figure 8

Snapshots of density distribution of BEC 1, $n_1$, and phase distribution for $R_0=30$ and $b_c=8$, showing vortex generation from the interface between BECs. (b) The detached droplet splits into (c) a vortex-antivortex pair, and this happens periodically (d,e) and further, due to periodic external force. Phase plots show topological excitations. Dimensionless units are defined in Sec. 2.

Figure 9

Cut of $n_1$ and $n_2$ showing dynamic AT-vortex and antivortex of component 1 (a 2D skyrmion pair) in Fig. 8d at $t=7.7$ and $z=-9.635$. Component 2 fills the vortex cores of component 1. Dimensionless units are defined in Sec. 2.

Figure 10

Snapshots of topological charge density of the two-component BEC for $R_0=30$ and $b_c=8$, for time instants the same as in Fig. 8. The topological charge of the system is generated on the interface between BECs. The direction of the $z$ axis is along $\widehat{\mathbf{x}}\times \widehat{\mathbf{y}}$. Dimensionless units are defined in Sec. 2.

Figure 11

Snapshots of location of the first generated vortex-antivortex pair in component 1 (at $y<0$), with corresponding fillings of the cores of the first vortex-antivortex pair of component 2 (at $y>0$), for $R_0=30$ and $b_c=8$. Snapshots are taken at time instants (a) $t=5.0$, (b) 5.4, (c) 5.8, (d) 6.2, (e) 6.6, (f) 7.0, (g) 7.4, (h) 7.8, (i) 8.2, and (j) 8.6, and only the area close to the pair is shown for each pair (marked by dashed borders). Note that the time instant $t=5.12$ from Fig. 8d is not shown in order to make the picture easier to understand. The second pairs, generated after the first ones, are not shown for the same reason. At the time instant $t=8.6$, shown in (j), the dynamics of the vortices is sporadic due to localization of the system and strong influence of the external field. Dimensionless units are defined in Sec. 2.