We consider a spin-orbit coupled configuration of spin-1/2 interacting bosons with equal Rashba and Dresselhaus couplings. The phase diagram of the system at T=0 is discussed with special emphasis on the role of the interaction, treated in the mean-field approximation. For a critical value of the density and of the Raman coupling we predict the occurrence of a characteristic tricritical point separating the stripe, the spin polarized and the zero momentum states of the Bose gas. The effect of harmonic trapping as well as the role of the breaking of spin symmetry in the interaction Hamiltonian are also discussed. Starting from the ground state phase, we deduced the compressibility and sin polarizability of the system.

 

By using the hydrodynamic formalism we identify the excitation spectrum for uniform case at zero temperature. The propagation of phonons is strongly affected by the coupling and the velocity of sound vanishes when one approaches the second order phase transition between the spin polarized and the zero momentum quantum phases. We also point out the emergence of a roton minimum in the excitation spectrum for small values of the Raman coupling, when one approaches the transition to the stripe phase. Using a sum rule approach we investigate the dipole oscillation in a harmonic trap. We show that near the second order transition between the spin polarized and the zero minimum phase the lowest frequency is vanishingly small for large condensates. We point out the crucial role played by the spin polarizability of the gas in the excitations for both uniform and trapped case.