Vortex structures of rotating spin-orbit-coupled Bose-Einstein condensates

We consider the quasi-two-dimensional two-component Bose-Einstein condensates with Rashba spin-orbit (SO) coupling in a rotating trap. The rotation angular velocity couples to the mechanical angular momentum, which contains a noncanonical part arising from SO coupling. The effects of an external Zeeman term favoring spin polarization along the radial direction is also considered, which has the same form as the noncanonical part of the mechanical angular momentum. The rotating condensate exhibits a variety of rich structures by varying the strengths of the trapping potential and interaction. With a strong trapping potential, the condensate exhibits a half-quantum vortex-lattice configuration. Such a configuration is driven to the normal one by introducing the external radial Zeeman field. In the case of a weak trap potential, the condensate exhibits a multidomain pattern of plane-wave states under the external radial Zeeman field.

Figure 1

Canonical angular momenta $m$ of the single-particle ground states described in Eq. (11) vs $\rho$ for $\gamma =-0.1$, $0.0$, and $0.15$, respectively.

Figure 2

Density profiles of spin-up and -down components for the single-particle ground state ${\phi }_{j_z=\frac{1}{2}}$ with the parameter values of $\alpha =4$ and $\rho =\gamma =0$. Only the spin-down component carries a vortex. The corresponding density profile for ${\phi }_{j_z=-\frac{1}{2}}$ is obtained by interchanging the spin indices. Here we use the length unit defined by the harmonic trap.

Figure 3

Spin-density vector distributions along the $x$ axis lie in the $xz$ plane as shown in (a) $j_z=\frac{1}{2}$ and (b) $j_z=-\frac{1}{2}$ with $\alpha =4$ and $\rho =\gamma =0$. They are time-reversal counterparts to each other, and both exhibit the Skyrmion-type texture configuration.

Figure 4

From left to right: the density and phase profiles of spin-up and -down components with parameter values of $\alpha =0.5$, $\beta =10$, $\rho =0.97$, and $c=1$. From (a) to (g), $\gamma$ is taken as $0.5$, $0.25$, $0.1$, $0.0$, $-0.1$, $-0.25$, and $-0.5$, respectively. At small values of $\left|\gamma \right|$ in (c)–(e), a half-quantum vortex lattice is formed near the trap center. The spin-up component breaks into several density peaks, and the low-density region is connected. By increasing the magnitude of $\left|\gamma \right|$ [(b) and (f)], the half-quantum vortex lattice evolves to the normal vortex lattice. For the large value of $\left|\gamma \right|=0.5$ [(a) and (g)], the condensates show a lattice configuration around a ring. The black circle with an arrow indicates the direction of the circulation around the vortex core. The unit of length for the figures is $l$.

Figure 5

Ground-state spin-density vector of Fig. 4d with parameter values of $\alpha =0.5$, $\beta =10$, $\rho =0.97$, $c=1$, and $\gamma =0$. The projection of $\langle \vec{\sigma }\rangle$ in the $xy$ plane is shown as black vectors. A color map is used to illustrate the $\langle {\sigma }_z\rangle$ component. The unit of length for the figure is $l$.

Figure 6

From left to right: the density and phase profiles of spin-up and -down components with the parameter values of $\alpha =2.5$, $\beta =40$, and $\gamma =0$. (a) $c=0.6$ and $\rho =0.2$; a plane-wave-like state is obtained with a distorted phase pattern; (b) $c=1.2$ and $\rho =0.2$; the spin-spiral condensate is favored; (c) $c=1.2$ and $\rho =0.5$; the condensate exhibits an intermediate configuration between those of (a) and (b). The color scales for the density and phase distributions are the same as those in Fig. 4. The unit of length for the figures is $l$.

Figure 7

From left to right: the density and phase profiles of spin-up and -down components with parameter values of $\alpha =4$, $\beta =20$, $c=1$, and $\rho =0.1$. From (a) to (h), $\gamma$ is taken as $0.5$, $0.3$, $0.1$, $-0.05$, $-0.25$, $-0.35$, $-0.6$, and $-0.7$, respectively. The black arrow in each domain represents the local wave-vector direction of the corresponding plane-wave state, which shows a clockwise or counter-clockwise configuration depending on the sign of $\gamma$. For sufficiently large values of $\left|\gamma \right|$, condensates distribute around a ring in space forming a giant vortex. The color scales for the density and phase distributions are the same as that in Fig. 4. The black circle with an arrow indicates the direction of the circulation around the vortex core. The unit of length for the figures is $l$.