Two-fluid theory for a superfluid system with anisotropic effective masses

In this work, we generalize the two-fluid theory to a superfluid system with anisotropic effective masses along different principal axis directions. As a specific example, such a theory can be applied to spin-orbit coupled Bose-Einstein condensate at low temperature. The normal density from phonon excitations and the second sound velocity are obtained analytically. Near the phase transition from the plane wave to zero-momentum phases, due to the effective mass divergence, the normal density from phonon excitation increases greatly, while the second sound velocity is suppressed significantly. With quantum hydrodynamic formalism, we give a unified derivation for suppressed superfluid density and Josephson relation. At last, the momentum distribution function and fluctuation of phase for the long wavelength are also discussed.

Figure 1

Corrections of the normal density [panel (a)] and the second sound velocity [panel (b)] in spin-orbit coupled BEC (along the $x$-axis direction). Note that near the phase transition ($\mathrm{\Omega }/k_0^2\to 2$), the effective mass would diverge, i.e., $z_1\to \infty$.