Transient dynamics of a quantum dot embedded between two superconducting leads and a metallic reservoir

We study time-dependent subgap properties of a quantum dot (QD) embedded between two superconductors and another metallic lead, solving the Heisenberg equations of motion by the Laplace transform technique subject to the initial conditions. Focusing on the response of the system induced by a sudden coupling of the QD to external reservoirs, we analyze the transient currents and their differential conductance. We also derive analytical expressions for measurable quantities and find that they oscillate in time with the frequency governed by the QD coupling to superconducting reservoirs. Such quantum oscillations are damped due to relaxation processes caused by the normal lead, whereas their period is controlled by the phase difference phi between the order parameters of superconducting leads (except the case phi = pi, when all observables evolve to their stationary values without any oscillations). We also explore time-dependent development of the subgap quasiparticles and find their signatures measurable in the differential conductance. We evaluate (numerically and analytically) three typical time scales, characterizing the initial and large-time stages of the transient dynamics which in the asymptotic limit (t -> infinity) drives these subgap quasiparticles to the true Andreev states.