We investigate nonequilibrium transport through superconducting nanojunctions using a Liouville space approach. The formalism allows us to study finite-gap effects, and to account for both quasiparticle and Cooper-pair tunneling. With focus on the weak-Tunneling limit, we study the stationary dc and ac current up to second order (cotunneling) in the hybridization energy. For the particular case of a strongly interacting quantum dot sandwiched between two superconductors, we identify the characteristic virtual processes that yield the Andreev and Josephson current and obtain the dependence on the gate and bias voltage for the dc current, the critical current, and the phase-dependent dissipative current. In particular, the critical current is characterized by regions in the stability diagram in which its sign changes from positive to negative, resulting in a multitude of 0-π transitions. The latter signal the interplay between strong interactions and tunneling at finite bias.

We thank Julian Siegl for discussion and help on this work, and Christoph Brueckner for help with the numerical simulations. J.P.-C. and M.G. acknowledge DFG funding through Project No. B04 and A.D. through Project B02 of SFB 1277 Emerging Relativistic Phenomena in Condensed Matter. G.P. is supported by Spain's MINECO through Grant No. PID2020-117787GB-I00 and by Spanish National Research Council (CSIC) Research Platform PTI-001.

Peer reviewed