Quantum swapping of immiscible Bose-Einstein condensates as an alternative to the Rayleigh-Taylor instability

We consider a two-component Bose-Einstein condensate in a quasi-one-dimensional harmonic trap, where the immiscible components are pressed against each other by an external magnetic force. The zero-temperature nonstationary Gross-Pitaevskii equations are solved numerically; analytical models are developed for the key steps in the process. We demonstrate that if the magnetic force is strong enough, then the condensates may swap their places in the trap due to dynamic quantum interpenetration of the nonlinear matter waves. The swapping is accompanied by development of a modulational instability leading to quasiturbulent excitations. Unlike the multidimensional Rayleigh-Taylor instability in a similar geometry of two-component quantum fluid systems, quantum interpenetration has no classical analog. In a two-dimensional geometry a crossover between the Rayleigh-Taylor instability and the dynamic quantum interpenetration is investigated.

Figure 1

Snapshots of density of the BEC components at $t=0,6,9.58,40,450$. Inset in panel (a) shows a magnified view of the interface. Inset in panel (b) presents a magnified view of the interference pattern close to the trap edge. Units are dimensionless.

Figure 2

Evolution in time of the density $n_1$ of component 1 for $R_0=20$ in driving field $b=1$. (a) Short-time development; (b) long-time development. Component 2 is situated symmetrically with respect to $z=0$; however, at long time the detailed symmetry is spontaneously broken. Units are dimensionless.

Figure 3

Ground state of the system for $R_0=20$ in the external potential $b=1$. Units are dimensionless.

Figure 4

Positions $Z_1\left(t\right)$ and $Z_2\left(t\right)$ of center-of-mass of the BEC components for $R_0=20$ and $b=0.8,0.9,1,1.5$. Units are dimensionless.

Figure 5

The c.m. coordinate $Z_1\left(t\right)$ for $R_0=20$ and different values of $b$, showing the bulk oscillation. Units are dimensionless.

Figure 6

Snapshot of the densities at time $t=2$ for $R_0=20$ and $b=5$. The inset magnifies the interference pattern. Units are dimensionless.

Figure 7

Length scale of the standing wave at the trap edge, as function of magnitude of the driving force $b$. The curves present the analytical formula Eq. (42) for $R_0=15,20,30$; the markers show the numerical result. Units are dimensionless.

Figure 8

Dispersion relation for the MI for $R_0=20$, $b=1,1.5,2$ (from bottom to top). Solid lines denote imaginary part of the frequency; dashed lines denote real part. Units are dimensionless.

Figure 9

Length scale of the multidomain structure formed during the swapping process, as function of magnitude of the driving force $b$. The curves correspond to the analytical formula Eq. (45) for $R_0=20,30$; the markers show the average numerical values obtained from the simulations for $R_0=20$; the error bars indicate the uncertainty in these values as the waves develop over a short duration of time. Units are dimensionless.

Figure 10

Time ${\tau }_0$ after which the c.m. coordinates of the two components meet at the midpoint, as a function of the external force $b$ for different system sizes $R_0=15-20$. The dashed lines are guides for eyes. Units are dimensionless.

Figure 11

Energy terms as functions of time. (a) Large energies. Full lines represent from top to bottom at the left, $E_{\mathrm{tot}}$: total energy,;$E_{\mathrm{p}1}$: pressure energy of intraspecies interactions for component 1; $E_{\mathrm{p}12}$: pressure energy of interspecies interactions; $E_{\mathrm{t}1}$: potential energy of component 1 due to the trap. Dashed line, almost coinciding with $E_{\mathrm{tot}}$, represents the sum of the large energies. (b) Small energies. Topmost line represents the sum of small energies; thereafter $E_{\mathrm{d}1}$: potential energy of component 1 due to the driving force; $E_{\mathrm{k}1}$: quasiclassical kinetic energy of component 1; $E_{\mathrm{qp}1}$: quantum pressure energy of component 1. Units are dimensionless.

Figure 12

Snapshots of density of the BEC component 1 for $L_x=0.5{\lambda }_{\mathrm{RT}}$, $R_0=30,$ and $b=5$ at $t=0,3.28,12.0$ in a 2D geometry with initially flat interface between BECs. The 2 component is located symmetrically. Note that proportions of the axes are distorted. Units are dimensionless.

Figure 13

Magnified view of the region marked by the dashed rectangle in Fig. 12b: The matter-wave interference fringes break up into droplets due to the capillary instability at $t=3.28$ (note the adjusted color shading). The two panels show density $n_1$ and phase ${\phi }_1$ of the 1 component. The 2 component is located symmetrically. Units are dimensionless.

Figure 14

Magnified view of the region marked by the dashed rectangle in Fig. 12c: a well-developed irregular pattern of 2D droplets at $t=12.0$. The two panels show the density $n_1$ and phase ${\phi }_1$ of the 1 component. The 2 component is located symmetrically. Units are dimensionless.

Figure 15

Snapshots of density $n_1$ of the BEC component 1 at $t=0,4,11$ in the dimensionless units of the paper in a 2D trap with transverse frequency ${\omega }_x=100{\omega }_z$ for $R_0=15$, $b=1.5$. The 2 component is located symmetrically. Note that proportions of the axes are distorted. Units are dimensionless.

Figure 16

Snapshots of density $n_1$ of the BEC component 1 for $L_x={\lambda }_{\mathrm{RT}}$, $R_0=30$, $b=5$, at $t=0,3.22,12$ in a 2D geometry with initially curved interface between BECs. The 2 component is located symmetrically. Note that proportions of the axes are distorted. Units are dimensionless.

Figure 17

Snapshots of density $n_1$ of the BEC component 1 at $t=2.00,2.08,3.22,12.0$ for the same parameters as in Fig. 16. Red circular arrows in (a) and (b) show velocity direction of the BEC component 1 around the quantized vortices. Units are dimensionless.