  We consider a Hamiltonian model for a quantum dot which is placed between two
superconducting leads with a constant bias imposed between these leads. Using
the non-equilibrium Keldysh technique, we focus on the subgap current, where it
is known that multiple Andreev reflections (MAR) are responsible for charge
transfer through the dot. Attention is put on the DC current and on the first
harmonics of the supercurrent. Varying the energy and width of the resonant
level on the dot, we first investigate a cross-over from a quantum dot regime
to a quantum point contact regime when there is zero coupling to the normal
probe. We then study the effect on the supercurrent of the normal probe which
is attached to the dot. This normal probe is understood to lead to dephasing,
or alternatively to induce reverse proximity effect. We describe the full
crossover from zero dephasing to the incoherent case. We also compute the
Josephson current in the presence of the normal lead, and find it in excellent
agreement with the values of the non-equlibrium current extrapolated at zero
voltage.
