Equilibrium and time-dependent Josephson current in one-dimensional superconducting junctions (original) (raw)

We investigate the transport properties of a one-dimensional superconductor-normal metal-superconductor ͑S-N-S͒ system described within the tight-binding approximation. We compute the equilibrium dc Josephson current and the time-dependent oscillating current generated after the switch on of a constant bias. In the first case an exact embedding procedure to calculate the Nambu-Gorkov Keldysh Green's function is employed and used to derive the continuum and bound states contributions to the dc current. A general formalism to obtain the Andreev bound states ͑ABSs͒ of a normal chain connected to superconducting leads is also presented. We identify a regime in which all Josephson current is carried by the ABS and obtain an analytic formula for the current-phase relation in the limit of long chains. In the latter case, the condition for perfect Andreev reflections is expressed in terms of the microscopic parameters of the model, showing a limitation of the so-called wide-band-limit ͑WBL͒ approximation. When a finite bias is applied to the S-N-S junction we compute the exact time evolution of the system by solving numerically the time-dependent Bogoliubov-de Gennes equations. We provide a microscopic description of the electron dynamics not only inside the normal region but also in the superconductors, thus gaining more information with respect to WBL-based approaches. Our scheme allows us to study the ac regime as well as the transient dynamics whose characteristic time scale is dictated by the velocity of multiple Andreev reflections.