Vortex Molecules in Coherently Coupled Two-Component Bose-Einstein Condensates

A vortex molecule is predicted in rotating two-component Bose-Einstein condensates whose internal hyperfine states are coupled coherently by an external field. A vortex in one component and one in the other are connected by a domain wall of the relative phase, constituting a “vortex molecule,” which features a nonaxisymmetric (pseudo)spin texture with a pair of merons. The binding mechanism of the vortex molecule is discussed based on a generalized nonlinear sigma model and a variational ansatz. The anisotropy of vortex molecules is caused by the difference in the scattering lengths, yielding a distorted vortex-molecule lattice in fast rotating condensates.

Figure 1

(a) The profile of the density $|{\Psi }_1{|}^2$ and $|{\Psi }_2{|}^2$ (line contours) with $\Omega =0.15$ and ${\omega }_{\mathrm{R}}=0.02$. (b) The gray-scale plot of the relative phase $\varphi (\mathbf{r})={\theta }_1-{\theta }_2$. Arrows show the direction of the circulation in the space of relative phase around the vortices. (c) The vectorial representation of the pseudospin $\mathbf{S}=\overline{\chi }\mathit{\sigma }\chi /2$ projected onto the $x\mathrm{\text{−}}y$ plane. The locations of the defects are indicated by ⊙ (with $S_z=+\widehat{\mathbf{z}}/2$) and ⨂ (with $S_z=-\widehat{\mathbf{z}}/2$). (d) The contour plot of the topological charge density $q(\mathbf{r})$ (the largest value at the center) and the vectorial plot of the ${\mathbf{v}}_{\mathrm{e}\mathrm{f}\mathrm{f}}$ field.

Figure 2

(a) The cross sections of the condensate density along the $y=0$ line ($|{\Psi }_1{|}^2$, solid curve; $|{\Psi }_2{|}^2$, dashed curve; ${\rho }_{\mathrm{T}}$, dotted curve). (b) The total energy as a function of $\lambda$ (the size of the meron pair) for $\Omega =0$. The inset shows the separation $2d_{\mathrm{m}}$ between two vortices as a function of the Rabi frequency ${\omega }_{\mathrm{R}}$ for $\Omega =0.15$. The solid curve represents the result obtained by the variational analysis with Eqs. (2, 4) ($d_{\mathrm{m}}=2\lambda$) and the dots the numerical result.

Figure 3

The contour plots of the topological charge density $q(\mathbf{r})$ and the vectorial plots of the ${\mathbf{v}}_{\mathrm{e}\mathrm{f}\mathrm{f}}$ field for $\Omega =0.15$, ${\omega }_{\mathrm{R}}=0.02$, $c_0=1000$, and (a) $c_2=20$ (antiferromagnetic), and (b) $c_2=-20$ (ferromagnetic). The inset shows the corresponding density profiles.

Figure 4

The upper figures show the profiles of the condensate density $|{\Psi }_1{|}^2$ and $|{\Psi }_2{|}^2$ and the lower ones the relative phase for $\Omega =0.80$, ${\omega }_{\mathrm{R}}=0.2$, and (a) $c_2=20$ (antiferromagnetic), (b) $c_2=-20$ (ferromagnetic). The centers of mass of the molecules are linked by dashed lines. The bottom inset shows the distribution of the topological charge within [$-3<x$, $y<3$].