Time-dependent quantum transport with superconducting leads: A discrete-basis Kohn-Sham formulation and propagation scheme

In this work we put forward an exact one-particle framework to study nanoscale 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 noninteracting Kohn-Sham (KS) system under the influence of different Peierls phases and of a pairing field. In the special case of normal systems, our result provides a formulation of time-dependent current-density-functional theory in tight-binding models. An extended Keldysh formalism for the nonequilibrium 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-de Gennes (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 waveguide 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.