Searching for Supersolidity in Ultracold Atomic Bose Condensates with Rashba Spin-Orbit Coupling

We developed functional integral formulation for the stripe phase of a spinor Bose-Einstein condensates with Rashba spin-orbit coupling. The excitation spectrum is found to exhibit double gapless band structures, identified to be two Goldstone modes resulting from spontaneously broken internal gauge symmetry and translational invariance symmetry. The sound velocities display anisotropic behaviors with the lower branch vanishes in the direction perpendicular to the stripe in the x-y plane. At the transition point between the plane wave phase and the stripe phase, physical quantities such as fluctuation correction to the ground state energy and quantum depletion of the condensates exhibit discontinuity, characteristic of the first order phase transition. Despite strong quantum fluctuations induced by Rashba spin-orbit coupling, we show that the supersolid phase is stable against quantum depletion. Finally we extend our formulation to finite temperatures to account for interactions between excitations.