  The phases of a Bose-Einstein condensate (BEC) with light-induced spin-orbit
coupling (SOC) are studied within the mean-field approximation. The mixed BEC
phase, in which the system condenses in a superposition of two plane wave
states, is found to be stable for sufficiently small light-atom coupling,
becoming unstable in a continuous fashion with increasing light-atom coupling.
The structure of the phase diagram at fixed chemical potential for bosons with
SOC is shown to imply an unusual density dependence for a trapped mixed BEC
phase, with the density of one dressed spin state increasing with increasing
radius, providing a unique experimental signature of this state. The collective
Bogoliubov sound mode is shown to also provide a signature of the mixed BEC
state, vanishing as the boundary to the regime of phase separation is
approached.
