Ferrofluidity in a Two-Component Dipolar Bose-Einstein Condensate

It is shown that the interface in a two-component Bose-Einstein condensate (BEC) with a dipole-dipole interaction spontaneously develops patterns similar to those formed in a ferrofluid. Hexagonal, labyrinthine, solitonlike structures, and hysteretic behavior are numerically demonstrated. Superflow is found to circulate around the hexagonal pattern at rest, offering evidence of supersolidity. The system sustains persistent current with a vortex line pinned by the hexagonal pattern. These phenomena may be realized using a ${}^{52}\mathrm{Cr}$ BEC.

Figure 1

Schematic illustration of the system. A two-component BEC is confined in a trapping potential, in which component 1 has a magnetic dipole moment and component 2 does not. The magnetic dipole is polarized in the $z$ direction by an external magnetic field. The two components are separated due to the field gradient $B^{\prime }(z)<0$.

Figure 2

Stationary states of the two-component dipolar BEC for $a_{11}=a_{22}=a_{12}=100a_B$ and $\mu =6{\mu }_B$ in an axisymmetric potential with $({\omega }_{\perp },{\omega }_z)=2\pi \times (100,800)\mathrm{Hz}$. (a) Isodensity surface of component 1 and (b) column densities $\int |{\psi }_1{|}^{2}dz$, $\int |{\psi }_2{|}^{2}dz$, and $\int (|{\psi }_1{|}^2+|{\psi }_2{|}^2)dz$ normalized by $NM{\omega }_{\perp }/\hslash$ for the field gradient $B^{\prime }=-1\mathrm{G}/\mathrm{cm}$. The color in (a) represents the $z$ coordinate in units of $(\hslash /M{\omega }_{\perp }{)}^{1/2}$. (c)–(d) Column densities of component 1 for $B^{\prime }=-1.65\mathrm{G}/\mathrm{cm}$ (bistable). The inset in (c) shows isodensity surface near the central peak. The total number of atoms is $N=4\times {10}^6$ for (a)–(d) with an equal population in each component. (e) Column density of component 1 for $N=6\times {10}^5$ and $B^{\prime }=-0.2\mathrm{G}/\mathrm{cm}$. The color gauge is 0-0.003 for (b)–(d) and 0-0.0045 for (e). The field of view for (b)–(e) is $50\times 50\mu \mathrm{m}$.

Figure 3

Isodensity surface (upper panels) and integrated column density (lower panels) of component 1 for various stationary states. The difference among (a)–(c) arises only from initial seeds in the imaginary-time propagation. The field gradient is $B^{\prime }=-0.3\mathrm{G}/\mathrm{cm}$. The other parameters are the same as those in Fig. 2a.

Figure 4

(a) Column density of component 1 and (b) $\arg {\psi }_1(z=0)$ for a stationary state, in which component 1 has a vortex line at $x=y=0$ and component 2 has no vortex. (c) Magnitude $(J_x^2+J_y^2{)}^{1/2}$ (height) and direction $\arg (J_x+i{J}_y)$ (color) of the atomic flow $\mathit{J}=a_{\perp }^3\int \mathrm{Im}{\psi }_1^{*}\mathit{\nabla }{\psi }_{1}dz$. The parameters are the same as those in Fig. 2a.

Figure 5

Column density for stationary states, in which components 1 and 2 have magnetic dipole moments $6{\mu }_B$ and $-6{\mu }_B$, respectively. The field gradient is $B^{\prime }=0$. (a) and (b) $N=4\times {10}^6$. (c) $N={10}^5$. The field of view is $50\times 50\mu \mathrm{m}$ for (a) and (b), and $30\times 30\mu \mathrm{m}$ for (c).

Figure 6

Time evolution of the isodensity surface (upper figures) and that of the column density (lower panels) of component 1. The initial state is the flat-interface state for $B^{\prime }=-1.7\mathrm{G}/\mathrm{cm}$ and the field gradient changes to $B^{\prime }=-1\mathrm{G}/\mathrm{cm}$ at $t=0$. The field of view is $50\times 50\mu \mathrm{m}$. The parameters are the same as those in Fig. 2a.