In this work we put forward an exact one-particle framework to study
nano-scale Josephson junctions out of equilibrium and propose a propagation
scheme to calculate the time-dependent current in response to an external
applied bias. Using a discrete basis set and Peierls phases for the
electromagnetic field we prove that the current and pairing densities in a
superconducting system of interacting electrons can be reproduced in a
non-interacting Kohn-Sham (KS) system under the influence of different Peierls
phases {\em and} of a pairing field. An extended Keldysh formalism for the
non-equilibrium Nambu-Green's function (NEGF) is then introduced to calculate
the short- and long-time response of the KS system. The equivalence between the
NEGF approach and a combination of the static and time-dependent
Bogoliubov-deGennes (BdG) equations is shown. For systems consisting of a
finite region coupled to N superconducting semi-infinite leads we
numerically solve the static BdG equations with a generalized wave-guide
approach and their time-dependent version with an embedded Crank-Nicholson
scheme. To demonstrate the feasibility of the propagation scheme we study two
paradigmatic models, the single-level quantum dot and a tight-binding chain,
under dc, ac and pulse biases. We provide a time-dependent picture of single
and multiple Andreev reflections, show that Andreev bound states can be
exploited to generate a zero-bias ac current of tunable frequency, and find a
long-living resonant effect induced by microwave irradiation of appropriate
frequency.Comment: 20 pages, 9 figures, published versio
