We develop a Keldysh-Eliashberg-type formalism based on a functional theory for strongly correlated inhomogeneous superconductors subject to time-dependent external forces on the ions, four-vector, and pairing potentials. This theory incorporates exchange-correlation effects including a self-consistent treatment of the electromagnetic vector field as well as the ion motion on equal footing. Thus our theory not only contains but also goes beyond the scope of a recently published time-dependent density-functional theory for superconductors.
We analyze the effect of pairing on particle transport in time-dependent theories based on the Hartree-Fock-Bogoliubov (HFB) or BCS approximations. The equations of motion for the HFB density matrices are unique and the theory respects the usual conservation laws defined by commutators of the conserved quantity with the Hamiltonian. In contrast, the theories based on the BCS approximation are more problematic. In the usual formulation of time-dependent Hartree-Fock
(TDHF)+BCS, the equation of continuity is violated and one sees unphysical oscillations in particle densities. This can be ameliorated by freezing the occupation numbers during the evolution in
TDHF+BCS, but there are other problems with the BCS approximation that make it doubtful for reaction dynamics. We also compare different numerical implementations of the time-dependent HFB equations. The equations of motion for the
U and
V Bogoliubov transformations are not unique, but it appears that the usual formulation is also the most efficient. Finally, we compare the time-dependent HFB solutions with numerically exact solutions of the two-particle Schr\"odinger equation. Depending on the treatment of the initial state, the HFB dynamics produces a particle emission rate at short times similar to that of the Schr\"odinger equation. At long times, the total particle emission can be quite different, due to inherent mean-field approximation of the HFB theory.