Quench dynamics of a correlated quantum dot sandwiched between normal-metal and superconducting leads

Quantum system abruptly driven from its stationary phase can reveal nontrivial dynamics upon approaching a new final state. We investigate here such dynamics for a correlated quantum dot sandwiched between the metallic and superconducting leads, considering two types of quenches feasible experimentally. The first one is related to a sudden change of the coupling between the dot and the superconducting lead, while the other one is associated with an abrupt shift of the quantum dot energy level. Using the time-dependent numerical renormalization group method, we examine and quantify the interplay between the proximity induced electron pairing with correlations caused by the on-dot Coulomb repulsion. We determine and discuss the time-dependent charge occupancy, on-dot pair correlation, transient currents, and analyze the evolution of the subgap quasiparticles, which could be empirically observed in the tunneling conductance. To get some insight into the dynamics of a biased junction, we make use of a mean-field approximation. We reveal the signatures of the time-dependent 0-$\ensuremath{\pi}$ transition and demonstrate that the evolution of local observables exhibits damped quantum oscillations with frequencies given by the energies of Andreev bound states.