Symmetry classification of spin-orbit-coupled spinor Bose-Einstein condensates

We develop a symmetry classification scheme to find ground states of pseudo-spin-$\frac{1}{2}$, spin-1, and spin-2 spin-orbit-coupled spinor Bose-Einstein condensates and show that as the SO(2) symmetry of simultaneous spin and space rotations is broken into discrete cyclic groups, various types of lattice structures emerge in the absence of a lattice potential. Examples include two different kagome lattices for pseudo-spin-$\frac{1}{2}$ condensates and a nematic vortex lattice in which uniaxial and biaxial spin textures align alternatively for spin-2 condensates. For the pseudo-spin-$\frac{1}{2}$ system, although mean-field states always break time-reversal symmetry, there exists a time-reversal invariant many-body ground state, which is fragmented and expected to be observed in a microcondensate.

Figure 1

Schematic diagrams for states preserving the combined symmetry of spin-space rotation ${\mathcal{C}}_{nz}$, gauge transformation, and time reversal. Panels (a)–(f) correspond to $n=1$, 2, 3, 4, 6, and $\infty$, respectively, where arrows indicate the direction of spin, solid closed loops delimit regions with nonzero momentum distributions ${\left|\psi \left(k\right)\right|}^2$, and $n$ points on the circle are denoted as ${\mathcal{S}}_0,...,{\mathcal{S}}_{n-1}$, respectively. Characteristic momentum distributions of the ground state $\psi$ for a trapped system are shown in the lower right corner of each figure. Black (yellow [light gray]) color refers to the region of small (large) amplitude.

Figure 2

Ground states of trapped SO-coupled spin-1 condensates, with $v^{\prime }=15$, ${\alpha }^{\prime }=0.5$, and ${\beta }^{\prime }=0.1$. (a) Density distributions of three spin components $M_F=1,0,-1$ from left to right. Here, black (yellow [light gray]) color refers to the low- (high-) density region. (b) Spatial variation of the corresponding order parameter visualized by plotting the magnitude and phase of ${\sum }_m{\xi }_m{Y}_F^m(\theta ,\varphi )$, where $Y_F^{M_F}$ is the spherical harmonic function of degree $F$ and ${\xi }_m$ is the $m$th component of the spin wave function.

Figure 3

Ground states of trapped SO-coupled spinor BECs with $v^{\prime }=15$. (a) Density distributions of $M_F=1$ (left), 0 (middle), and $-1$ (right) of the spin-1 case with ${\alpha }^{\prime }=0.5$ and ${\beta }^{\prime }=-0.1$. (b) The corresponding order parameter, where F's indicate ferromagnetic vortex cores. (c) Density distributions of $M_F=2$ (left), 1 (middle), and 0 (right) of the spin-2 case with ${\alpha }^{\prime }=0.5$, ${\beta }^{\prime }=0.1$, and ${\gamma }^{\prime }=-0.1$. (d) The corresponding order parameter. The uniaxial and biaxial nematic vortex cores are denoted by UN and BN, respectivley. Here, black (yellow [light gray]) color refers to the low- (high-) density region.

Figure 4

Ground states of trapped SO-coupled pseudo-spin-$\frac{1}{2}$ BECs with $v^{\prime }=15$ and (a,b) ${\alpha }^{\prime }=0.3$, ${\beta }^{\prime }=0.06$, (c,d) ${\alpha }^{\prime }=0.4$, ${\beta }^{\prime }=0.08$. Here, black (yellow [light gray]) color refers to the low- (high-) density region. From left to right, we show the spin-up, spin-down, and total density distributions. Panels (a) and (c) are numerically calculated for a trapped system, while panels (b) and (d) are obtained by the superposition of six single-particle states $\mathcal{PW}\left({\varphi }_k\right)$. In panels (b) and (d), we use symbols $-$, $+$, and $+$$+$ to denote vorticities $-\hslash$, $\hslash$, and $2\hslash$, respectively. The size of panel (b) [(c)] is the same as that of panel (d) [(a)].