Vortex sheet in rotating two-component Bose-Einstein condensates

We investigate vortex states of immiscible two-component Bose-Einstein condensates under rotation through numerical simulations of the coupled Gross-Pitaevskii equations. For strong intercomponent repulsion, the two components undergo phase separation to form several density domains of the same component. In the presence of the rotation, the nucleated vortices are aligned between the domains to make up winding chains of singly quantized vortices, a vortex sheet, instead of periodic vortex lattices. The vortices of one component are located at the region of the density domains of the other component, which results in the serpentine domain structure. The sheet configuration is stable as long as the imbalance of the intracomponent parameter is small. We employ a planar sheet model to estimate the distance between neighboring sheets, determined by the competition between the surface tension of the domain wall and the kinetic energy of the superflow via quantized vortices. By comparing the several length scales in this system, the phase diagram of the vortex state is obtained.

Figure 1

(Color online) Vortex sheet array suggested by Landau and Lifshitz. Straight vortex lines are aligned along cylindrical planar soliton. The schematic view of the radial velocity profile is also shown.

Figure 2

(Color online) Typical structure of the vortex sheets, obtained by the numerical simulation of the coupled Gross-Pitaevskii equations. The parameter values are $u_1=u_2=4000$, $a_{12}∕a=\delta =1.1$, and $\Omega =0.85\omega$ (see the text). (a) The contour plots of the two-dimensional density profiles of the two-component condensates ${\mid {\psi }_1\mid }^2$ and ${\mid {\psi }_2\mid }^2$. The vortex sheets are made up of chains of singly quantized vortices whose positions are marked by ×. (b) Solid, dashed, and dotted curves show the radial density profiles of ${\mid {\psi }_1\mid }^2$, ${\mid {\psi }_2\mid }^2$, and the total density $n_{\mathrm{T}}={\mid {\psi }_1\mid }^2+{\mid {\psi }_2\mid }^2$, respectively. Circular and triangular dots represent the corresponding velocity profiles $\mid {\mathbf{v}}_i\mid =\sqrt{{\left({\partial }_x{\theta }_i\right)}^2+{\left({\partial }_y{\theta }_i\right)}^2}$ with $i=1$ and 2 (in unit of the length $b_{\mathrm{ho}}$ and time ${\omega }^{-1}$). The velocity of rigid body rotation $v=\Omega r$ is shown for comparison.

Figure 3

(Color online) Typical two-dimensional density profiles and locations of vortices in the vortex sheet configuration for $u=4000$ and $\delta =1.5$ (a), $\delta =2.0$ (b), $\delta =3.0$ (c). The upper and lower panels show ${\mid {\psi }_1\mid }^2$ and ${\mid {\psi }_2\mid }^2$, respectively.

Figure 4

(Color online) Typical two-dimensional density profiles and locations of vortices in the vortex states for imbalanced intracomponent parameters; $\delta =1.1$, $u_1=4000$, and $u_2=3000$ (a), 3500 (b), and 3900 (c). The upper and lower panels show ${\mid {\psi }_1\mid }^2$ and ${\mid {\psi }_2\mid }^2$, respectively.

Figure 5

(Color online) Simple model to calculate the intersheet distance $b$ for the vortex sheet in two-component BECs. The rectangular two-component domains align periodically and alternately along the $x$ axis with small domain-boundary region characterized by the penetration depth ${\lambda }_p$. The total density is uniform all over the space. The condensate velocity ${\mathbf{v}}_i=v_i\widehat{\mathbf{y}}$ has a staircaselike increase along the $x$ axis; the value jumps $2\Omega b$ across the sheet for each component and follows the rigid body velocity $2\Omega x$.

Figure 6

(Color online) The sheet distance $b$ (in unit of $b_{\mathrm{ho}}$) as a function of $\Omega$ (a) and ${\delta }^{-1}$ (b) for $u=4000$. In (b), the two results using Eqs. (12, 13) are smoothly connected at ${\delta }^{-1}=0.75$. Also, the numerically estimated values of $b$ for $\Omega =0.85$ are plotted for $\delta =1.1$ (Fig. 2), 1.5, 2.0, and 3.0 (Fig. 3); the values were taken by averaging the intersheet distance from the profile of vortex positions.

Figure 7

(Color online) $\Omega \text{−}{\delta }^{-1}$ phase diagram of the equilibrium vortex states. The borders are determined by comparing the characteristic length scales (see the text). The insets show the typical equilibrium structures of the density ${\mid {\psi }_1\mid }^2$ and the vortex positions for the parameters indicated by the star marks; (a) $(\Omega ,\delta )=(0.4,3.0)$, (b) $(\Omega ,\delta )=(0.85,1.5)$, (c) $(\Omega ,\delta )=(0.85,1.02)$. In (b), the region indicated by a circle represents the linked edges of two vortex sheets.