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In this work, we generalize the two-fluid theory to a superfluid system withanisotropic effective masses along different principal axis directions. As aspecific example, such a theory can be applied to spin-orbit coupledBose-Einstein condensate (BEC) at low temperature. The normal density fromphonon excitations and the second sound velocity are obtained analytically.Near the phase transition from the plane wave to zero-momentum phases, due tothe effective mass divergence, the normal density from phonon excitationincreases greatly, while the second sound velocity is suppressed significantly.With quantum hydrodynamic formalism, we give unified derivations for suppressedsuperfluid density and Josephson relation. At last, the momentum distributionfunction and fluctuation of phase for the long wave length are also discussed.

Two-fluid theory for superfluid system with anisotropic effective masses