Spin-orbit-coupled Bose-Einstein condensates held under a toroidal trap

We study a quasispin-1/2 Bose-Einstein condensate with synthetically generated spin-orbit coupling in a toroidal trap and show that the system has a rich variety of ground states. As the central hole region increases, i.e., the potential changes from harmoniclike to ringlike, the condensate exhibits a variety of structures, such as a modified stripe, an alternately arranged stripe, and countercircling states. In the limit of a quasi-one-dimensional ring, the quantum many-body ground state is obtained, which is found to be the fragmented condensate.

Figure 1

Ground states in a toroidal trap for the immiscible case with $A=100,g=800$, and $g_{\uparrow \downarrow }=1000$. (a) and (b) Density profiles of up- and down-components $n_{\uparrow }={\left|{\psi }_{\uparrow }\right|}^2$ and $n_{\downarrow }={\left|{\psi }_{\downarrow }\right|}^2$ and $\mathbf{k}$-space densities ${\sum }_{j=\uparrow ,\downarrow }{\left|\int {\psi }_{j}exp(-i\mathit{k}·\mathit{r})d\mathit{r}\right|}^2$ for (a) $\kappa =0.75$ and $l=2.0$ and (b-1)–(b-3) $\kappa =2.0$ and $l=1.0,l=2.5$, and $l=4.0$. The dashed circles in the $\mathbf{k}$ space indicate $\left|\mathbf{k}\right|=\kappa$. (c) Dependence of the total energy on the width $l$ of the toroidal trap for fixed SOC $\kappa =2.0$. The stripe state (b-1) is smoothly changed to state (b-3) as $l$ is increased.

Figure 2

Ground states in a toroidal trap for the miscible case with $A=100,g=1000,g_{\uparrow \downarrow }=800$, and $\kappa =2$. (a) Density $n_{\uparrow },\mathbf{k}$-space density, and phase $\arg {\psi }_{\uparrow }$ distributions for $\kappa =2.0$ and for $l=1.0,3.0$, and 5.0, corresponding to (a-1)–(a-3), respectively. The dashed circles in the $\mathbf{k}$ space indicate $\left|\mathit{k}\right|=\kappa$. In the phase distribution, the region satisfying $n_{\uparrow }+n_{\downarrow }>{10}^{-4}$ is extracted, and the arrows represent the directions of the phase gradient. (b) Dependence of the total energy on the width $l$ for two different flow structures (a-1)–(a-3). The inset shows the energy difference between two such structures within the region marked by the double-headed arrow, which indicates that the former (latter) state is the ground state for $l\lesssim (\gtrsim )4.8$.

Figure 3

(a) Density difference $n_{\uparrow }-n_{\downarrow }$ of the state in Fig. 2(a-3). (b) Density $n_{\uparrow }$ and phase $\arg {\psi }_{\uparrow }$ of the state in Fig. 1(b-3) where the dashed square region in the left panel is magnified in the right panel. The arrows indicate the phase gradient $({\psi }_{\uparrow }^{*}\mathbf{\nabla }{\psi }_{\uparrow }-{\psi }_{\uparrow }\mathbf{\nabla }{\psi }_{\uparrow }^{*})/\left(2i\right)$. (a) and (b) justify the variational wave functions of Eqs. (5) and (7), respectively.