Enhanced strong interaction effect in synthetic spin-orbit coupling with mixed spin symmetry

Synthetic spin-orbit coupling in cold atoms couples the pseudo-spin and spatial degrees of freedom, and therefore the inherent spin symmetry of the system plays an important role. In systems of two pseudo-spin degrees, two particles contain symmetric states and anti-symmetric states, but the spin symmetry can be mixed for more particles. This mixed spin symmetry has been overlooked and has not been investigated thoroughly. We study the role of mixed spin symmetry in the presence of spin-orbit coupling and consider the system of three bosons with two hyper-fine states trapped in a harmonic potential. We investigate the ground state and the energy spectrum by implementing exact diagonalization. Similarly to two-particle systems, the interplay between spin-orbit coupling and repulsive interactions between anti-aligned pseudo-spins increases the population of the unaligned spin components in the ground state. Also, the emergence of the mixed spin symmetric states compensates for the rise of the interaction energy. In contrast to two-particle systems, the pair correlation of the ground state is analogous to the Tonks-Girardeau gas even for relatively small contact interactions, and this feature is enhanced by the spin-orbit coupling.