The polarized and unpolarized spin-orbit coupled quantum droplets in a harmonic trap

Spin-orbit (SO) coupled QDs are predicted in SO coupled Bose gases with Lee-Huang-Yang (LHY) correction confined in harmonic potential. Polarized and unpolarized QDs are predicted. The phase transition condition of the QDs is obtained analytically, showing that the QDs are more favorite to be in the unpolarized state. Furthermore, the collective modes of the QDs in different phases are obtained analytically. Softening of the collective modes at phase transition point occurs. LHY correction has strong impact only on the breathing dynamics, resulting the breathing dynamics of SO coupled QDs is distinct from that of dipolar dynamics. Interestingly, when LHY correction is dominated, the breathing dynamics of SO coupled QDs also has distinct properties in the two phases and the softening of the breathing mode disappears at phase transition point. We demonstrate that SO coupled QDs have rich and interesting ground states and collective dynamics, which are observable in current experiments.