Superfluid Density of a Spin-orbit Coupled Bose Gas

We discuss the superfluid properties of a Bose-Einstein condensed gas with spin-orbit coupling, recently realized in experiments. We find a finite normal fluid density $\rho_n$ at zero temperature which turns out to be a function of the Raman coupling. In particular, the entire fluid becomes normal at the transition point from the zero momentum to the plane wave phase, even though the condensate fraction remains finite. We emphasize the crucial role played by the gapped branch of the elementary excitations and discuss its contributions to various sum rules. Finally, we prove that an independent definition of superfluid density $\rho_s$, using the phase twist method, satisfies the equality $\rho_n+\rho_s=\rho$, the total density, despite the breaking of Galilean invariance.