Capillary instability in a two-component Bose-Einstein condensate

Capillary instability and the resulting dynamics in an immiscible two-component Bose-Einstein condensate are investigated using the mean-field and Bogoliubov analyses. A long, cylindrical condensate surrounded by the other component is dynamically unstable against breakup into droplets due to the interfacial tension arising from the quantum pressure and interactions. A heteronuclear system confined in a cigar-shaped trap is proposed for realizing this phenomenon experimentally.

Figure 1

Capillary instability and resulting dynamics of a two-component BEC for $g_{12}/g=1.1$. The three-dimensional objects depict the isodensity surface of component 1. The lower panels in (a)–(c) show the density distributions $\left|{\psi }_j\right|{}^2$ at the cross section of $x=0$ (the field of view is $600\xi \times 100\xi$). The number of component 1 atoms in an initial cylinder with length $\xi$ is $N_{1\xi }=400$.

Figure 2

Time evolution of the system with a periodic initial perturbation given by Eq. (5). The parameters are the same as those in Fig. 1. The three-dimensional objects indicate the isodensity surfaces of component 1 and the lower panels show the density distributions $\left|{\psi }_j\right|{}^2$ at the cross section of $x=0$ (the field of view is $600\xi \times 100\xi$).

Figure 3

Radial density profiles of the stationary states for (a) $N_{1\xi }=400$ and (b) $N_{1\xi }=7854$ with $g_{12}/g=1.1$. (c) Imaginary part of the excitation frequencies. The circles and triangles are obtained by numerical diagonalization of Eq. (9) with the parameters used in (a) and (b), respectively. The plotted quantities are normalized using Eq. (10) with $R_0=11\xi$ for the circles and $R_0=50\xi$ for the triangles. The solid curve indicates Eq. (30).

Figure 4

Time evolution of a ${}^{41}\mathrm{K}$-${}^{87}\mathrm{Rb}$ condensate confined in a cigar-shaped trap. The initial state is the ground state, in which $N_1=5\times {10}^4$ ${}^{41}\mathrm{K}$ atoms (component 1) are confined in $V_1$ with $({\omega }_{1\perp },{\omega }_{1z})/2\pi =(582,11.6)$ Hz and $N_2=9.5\times {10}^5$ ${}^{87}\mathrm{Rb}$ atoms (component 2) are confined in $V_2$ with $({\omega }_{2\perp },{\omega }_{2z})/2\pi =(89.4,7.2)$ Hz. At $t=0$, ${\omega }_{1\perp }/(2\pi )$ is changed from $582$ to $145$ Hz. The panels in (a)–(c) show the density distributions $\left|{\psi }_j\right|{}^2$ at the cross section of $x=0$, where the unit of the density is ${10}^{20}$ ${\mathrm{m}}^{-3}$ and the field of view is $220\times 14$ $\mu \mathrm{m}$. The bottom figure shows the isodensity surface of component 1 at $t=21$ ms.

Figure 5

Time evolution for $N_1=2\times {10}^5$ and $N_2=8\times {10}^5$. The trap frequencies are $({\omega }_{1\perp },{\omega }_{1z})/2\pi =(436,35)$ and $({\omega }_{2\perp },{\omega }_{2z})/2\pi =(89.4,7.2)$. At $t=0$, ${\omega }_{1\perp }/2\pi$ is changed to 145 Hz. The panels show the column density $d_j=\int \left|{\psi }_j\right|{}^2\mathit{dx}$ and the phase ${\varphi }_j=\arg {\psi }_j(x=0)$. The unit of the column density is ${10}^{15}$ ${\mathrm{m}}^{-2}$ and the field of view is $170\times 13.6$ $\mu \mathrm{m}$. The dashed square in (c) shows an example of a position at which the phase jumps.

Figure 6

Creation of a Skyrmion train. (a) Density distribution of the initial state at the cross section of $x=0$. Component 1 has a singly quantized vortex along the $z$ axis. The initial trap frequencies for component 1 are $({\omega }_{1\perp },{\omega }_{1z})/2\pi =(796,14.5)$ Hz. At $t=0$, ${\omega }_{1\perp }$ is changed to 145 Hz and the trap center for component 1 is shifted to $z=65$ $\mu \mathrm{m}$. The other parameters are the same as those in Fig. 4. The field of view is $150\times 14$ $\mu \mathrm{m}$. (b) Density profile of component 2 at the cross section of $x=0$ at $t=9$ ms. The density and phase in the dashed square are magnified in the lower panels. The arrows indicate the directions of circulations of the quantized vortices. (c) Isodensity surface of component 1 at $t=9$ ms. The lower panels show the density and phase profiles of component 1 at the cross section of $z$ indicated by the arrow. The unit of the density in (a)–(c) is ${10}^{20}$ ${\mathrm{m}}^{-3}$.