Unconventional states of bosons with the synthetic spin-orbit coupling

The spin-orbit coupling with bosons gives rise to novel properties that are absent in usual bosonic systems. Under very general conditions, the conventional ground state wavefunctions of bosons are constrained by the 'no-node' theorem to be positive definite. In contrast, the linear dependence of the spin-orbit coupling leads to complex-valued condensate wavefunctions beyond this theorem. In this paper, we review the study of this class of unconventional Bose-Einstein condensations focusing on their topological properties. Both the 2D Rashba and 3D (sigma) over right arrow. (p) over right arrow -type Weyl spin-orbit couplings give rise to Landau-level-like quantization of single-particle levels in the harmonic trap. Interacting condensates develop the half-quantum vortex structure spontaneously breaking the time-reversal symmetry and exhibit topological spin textures of the skyrmion type. In particular, the 3D Weyl coupling generates topological defects in the quaternionic phase space as an SU(2) generalization of the usual U(1) vortices. Rotating spin-orbit-coupled condensates exhibit rich vortex structures due to the interplay between vorticity and spin texture. In the Mott-insulating states in optical lattices, quantum magnetism is characterized by the Dzyaloshinskii-Moriya-type exchange interactions.