Effect of spin-orbit interaction on the critical temperature of an ideal Bose gas

We consider Bose-Einstein condensation of an ideal Bose gas with an equal mixture of 'Rashba' and 'Dresselhaus' spin-orbit interactions and study its effect on the critical temperature. In uniform Bose gas a 'cusp' and a sharp drop in the critical temperature occurs due to the change in the density of states at a critical Raman coupling where the degeneracy of the ground states is lifted. Relative drop in the critical temperature depends on the diluteness of the gas as well as on the spin-orbit coupling strength. In the presence of a harmonic trap, the cusp in the critical temperature is smoothened out and a minimum in the critical temperature appears. Both the drop in the critical temperature and lifting of 'quasi-degeneracy' of the ground states exhibit crossover phenomena and disappears for sufficiently strong trap frequency. By considering a 'Dicke'-like model we extend our calculation to bosons with large spin and observe a similar minimum in the critical temperature near the critical Raman frequency, which becomes deeper for larger spin. Finally in the limit of infinite spin, the critical temperature vanishes at the critical frequency, which is a manifestation of Dicke type quantum phase transition.