Time-dependent linear response of an inhomogeneous Bose superfluid: Microscopic theory and connection to current-density functional theory

The dynamics of a confined fluid of Bose atoms is treated within the linear response regime, with a view to establishing a current-density functional formalism for an inhomogeneous superfluid state. After evaluating in full detail a simplified case of an external coupling to the density and phase of the condensate, the theory is extended to include the coupling to the total current density. The Kohn-Sham response functions of the condensate and all the exchange-correlation kernels for the superfluid are introduced from the microscopic equations of motion and are expressed in a physically transparent way through functional derivatives of correlation functions. A microscopic formula for the superfluid density is derived and used to introduce a generalized hydrodynamic approach for a weakly inhomogeneous two-fluid model in isothermal conditions. Local-density expressions are thereby derived for the velocities of first and second sound in the weakly inhomogeneous superfluid and for visco-elastic functions describing the transition from the hydrodynamic to the collisionless regime. Landau's hydrodynamic theory and known results in Green's functions language are recovered in the limiting case of a homogeneous superfluid.