Spin textures in rotating two-component Bose-Einstein condensates

We investigate two kinds of coreless vortices with axisymmetric and nonaxisymmetric configurations in rotating two-component Bose-Einstein condensates. Starting from the Gross-Pitaevskii energy functional in a rotating frame, we derive a nonlinear sigma model generalized to the two-component condensates. In terms of a pseudospin representation, an axisymmetric vortex and a nonaxisymmetric one correspond to spin textures referred to as a “skyrmion” and a “meron-pair,” respectively. A variational method is used to investigate the dependence of the sizes of the stable spin textures on system parameters, and the optimized variational function is found to reproduce well the numerical solution. In the SU(2) symmetric case, the optimal skyrmion and meron-pair are degenerate and transform to each other by a rotation of the pseudospin. An external rf field that couples coherently the hyperfine states of two components breaks the degeneracy in favor of the meron-pair texture due to an effective transverse pseudomagnetic field. The difference between the intracomponent and intercomponent interactions yields a longitudinal pseudomagnetic field and a ferromagnetic or an antiferromagnetic pseudospin interaction, leading to a meron-pair texture with an anisotropic distribution of vorticity.

Figure 1

(Color online) (a) The density profile of the coreless vortex state consisting of the rotating ${\psi }_1$ component and the nonrotating ${\psi }_2$ component for $u_1=u_2=u_{12}=1000$ ($c_0=1000$, $c_1=c_2=0$), ${\omega }_{\mathrm{R}}=0$, and $\Omega =0.15$. In this calculation, we fix the total particle number $N=N_1+N_2$, but do not fix each particle number $N_i$. Then, the solution converges to $N_1∕N_2=2.465$. (b) The cross sections of ${\mid {\psi }_1\mid }^2$ (solid curve), ${\mid {\psi }_2\mid }^2$ (dashed curve), and the total density ${\rho }_{\mathrm{T}}$ (dotted curve) along the $x$ axis at $y=0$, where bold and thin curves represent the results obtained from the numerical calculation and the variational calculation, respectively.

Figure 2

(Color online) (a) The pesudospin density $\mathbf{S}=\overline{\chi }\mathbf{\sigma }\chi$ corresponding to the vortex state in Fig. 1. (b) The vectorial representation of the spin texture projected onto the $x\text{−}y$ plane in the region $[-6.7⩽x,y⩽+6.7]$. (c) The cross section of the spin texture along the $x$ axis at $y=0$.

Figure 3

(a) The distribution of the topological charge density $q\left(\mathbf{r}\right)$ of Eq. (14) and (b) the vectorial plot of the effective velocity field ${\mathbf{v}}_{\mathrm{eff}}$ of Eq. (9), corresponding to the solution shown in Fig. 1. The insets represent the cross sections of $q\left(\mathbf{r}\right)$ and $\mid {\mathbf{v}}_{\mathrm{eff}}\mid$ along the $y=0$ line within the range $-5⩽x⩽5$.

Figure 4

(a) The effective potential $V_{\mathrm{eff}}$ of Eq. (18) with $\alpha =0$ (for $\Omega =0$ and ${\omega }_{\mathrm{R}}=0$) for several values of $\lambda$. A pure harmonic potential (obtained for $\lambda \to \infty$) is shown by a dotted curve. The inset shows the corresponding total density profiles ${\rho }_{\mathrm{T}}$ for $c_0=1000$.

Figure 5

The total energy as a function of the skyrmion size $\lambda$ for several values of the rotation frequency $\Omega$ $(\Omega =0,0.55,0.1,0.15)$ for $c_0={10}^3$ and $c_1=c_2=0$. The energy of the nonvortex state is 11.894. The inset shows the optimized value of $\alpha$ as a function of $\lambda$.

Figure 6

The optimized value of the variational parameter $\lambda$ (solid curve) and that of $\alpha$ (dashed curve) as a function of rotation frequency $\Omega$ for $c_0={10}^3$, $2.5\times {10}^3$, ${10}^4$ and $c_1=c_2=0$.

Figure 7

The bending angle $\beta \left(r\right)={\cos }^{-1}{S}_z$ at the Thomas-Fermi boundary $r=R_{\mathrm{TF}}$ at which the total density ${\rho }_{\mathrm{T}}$ vanishes, as a function of $\Omega$. The optimized values of the variational parameters $\lambda$ and $\alpha$ in Fig. 6 are used.

Figure 8

(Color online) The optimized values of the variational parameters $\lambda$ (solid curve) and $\alpha$ (dashed curve) as a function of the longitudinal (pseudo)magnetic field $c_1$ for $c_0={10}^3$, $c_2=0$, and $\Omega =0.15$. Insets show the cross sections of the variational function ${\mid {\psi }_1\mid }^2$ (solid curve), ${\mid {\psi }_2\mid }^2$ (dashed curve), and the total density ${\rho }_{\mathrm{T}}$ (dotted curve) along the $y=0$ line (left: $c_1=-50$, right: $c_1=8$).

Figure 9

(Color online) The optimized values of the variational parameters $\lambda$ (solid curve) and $\alpha$ (dashed curve) as a function of the spin-spin interaction strength $c_2$ for $c_0={10}^3$, $c_1=0$, and $\Omega =0.15$. The insets show the cross sections of the variational function ${\mid {\psi }_1\mid }^2$ (solid curve), ${\mid {\psi }_2\mid }^2$ (dashed curve), and the total density ${\rho }_{\mathrm{T}}$ (dotted curve) along the $y=0$ line (left: $c_2=-50$, right: $c_2=50$).

Figure 10

(Color online) The optimized values of the variational parameters $\lambda$ (solid curve) and $\alpha$ (dashed curve) as a function of (a) $c_1$ and (b) $c_2$ under the condition of a fixed particle number of each component $N_1∕N_2=1$ for $c_0={10}^3$, $\Omega =0.15$ [$c_2=0$ and $c_1=0$ for (a) and (b), respectively]. The insets show the cross sections of the variational functions ${\mid {\psi }_1\mid }^2$ (solid curve), ${\mid {\psi }_2\mid }^2$ (dashed curve), and the total density ${\rho }_{\mathrm{T}}$ (dotted curve) along the $y=0$ line [(a) left: $c_1=-50$, right: $c_1=50$ and (b) left: $c_2=-30$, right: $c_2=30$].

Figure 11

(Color online) (a) The density profile of the condensates ${\mid {\psi }_1\mid }^2$ and ${\mid {\psi }_2\mid }^2$ for $u_1=u_2=u_{12}=1000$ ($c_0={10}^3$, $c_1=c_2=0$), $\Omega =0.15$, and ${\omega }_{\mathrm{R}}=0.05$. The two components have the same particle number. (b) The cross sections of ${\mid {\psi }_1\mid }^2$ (solid curve), ${\mid {\psi }_2\mid }^2$ (dashed curve), and the total density ${\rho }_{\mathrm{T}}$ (dotted curve) along the $x$ axis at $y=0$, where bold and thin curves represent the results obtained from the numerical calculation and those obtained from the variational calculation, respectively. (c) The grayscale plot of the relative phase $\varphi ={\theta }_1-{\theta }_2$.

Figure 12

(Color online) (a) The pesudospin density $\mathbf{S}=\overline{\chi }\mathbf{\sigma }\chi$ corresponding to the vortex state in Fig. 11. (b) The vectorial representation of the spin texture projected onto the $x\text{−}y$ plane in the region $[-6.7⩽x,y⩽6.7]$. (c) The cross section of the spin texture (b) along the $x$ axis at $y=0$.

Figure 13

(Color online) (a) The optimized values of the variational parameters $\lambda$ (solid curve) and $\alpha$ (dashed curve) as a function of ${\omega }_{\mathrm{R}}$ for $c_0={10}^3$, $\Omega =0.15$, $c_1=0$, and $c_2=0$. (b) The corresponding size of the meron-pair $2d_{\mathrm{m}}$ calculated from Eq. (23) [see Fig. 11b for definition] for $c_0={10}^3$. We also show $2d_{\mathrm{m}}$ obtained from numerical simulations as filled circles. The insets show the cross sections of the variational functions ${\mid {\psi }_1\mid }^2$ (solid curve), ${\mid {\psi }_2\mid }^2$ (dashed curve), and the total density ${\rho }_{\mathrm{T}}$ (dotted curve) along the $y=0$ line (left: ${\omega }_{\mathrm{R}}=0.1$, right: ${\omega }_{\mathrm{R}}=0.5$).

Figure 14

(Color online) Equilibrium structures of the spin texture in the presence of a longitudinal magnetic field $\left(c_1\right)$ and a transverse magnetic field $\left({\omega }_{\mathrm{R}}\right)$ in the region $[-4⩽x,y⩽+4]$. The vectorial plots in the $x\text{−}y$ plane and along the $y=0$ line are shown. The values of the parameters are $\Omega =0.15$, $c_0=1000$, $c_1=-10$, $c_2=0$, and (a) ${\omega }_{\mathrm{R}}=0$, (b) ${\omega }_{\mathrm{R}}=0.02$, (c) ${\omega }_{\mathrm{R}}=0.05$, (d) ${\omega }_{\mathrm{R}}=0.10$. The cross sections of ${\mid {\psi }_1\mid }^2$ (solid curve), ${\mid {\psi }_2\mid }^2$ (dashed curve), and the total density ${\rho }_{\mathrm{T}}$ (dotted curve) along the $y=0$ line are also shown within the range $[-8⩽x⩽+8]$. (e) The distributions of the topological charge density $q\left(\mathbf{r}\right)$ along the $y=0$ line for ${\omega }_{\mathrm{R}}=0$ (solid), 0.02 (dashed), 0.05 (dashed-dotted), and 0.1 (dotted).

Figure 15

(Color online) Equilibrium structures of the spin texture in the presence of a spin-spin interaction $c_2$ and a transverse magnetic field $\left({\omega }_{\mathrm{R}}\right)$. The parameter values are $\Omega =0.15$, $c_0=1000$, $c_1=0$, ${\omega }_{\mathrm{R}}=0.05$, and (a) $c_2=20$, (b) $c_2=-20$. (top) The cross sections of ${\mid {\psi }_1\mid }^2$ (solid curve), ${\mid {\psi }_2\mid }^2$ (dashed curve), and the total density ${\rho }_{\mathrm{T}}$ (dotted curve) along the $y=0$ line are also shown. (middle) The corresponding spin texture in the region $[-4⩽x,y⩽+4]$. The vectorial plots in the $x\text{−}y$ plane and along the $y=0$ line are shown. (bottom) The spatial distributions of the topological charge density $q\left(\mathbf{r}\right)$.