Non-equilibrium transport through a Josephson quantum dot

We study the electronic current through a quantum dot coupled to two superconducting leads which is driven by either a voltage $V$ or temperature $\Delta T$ bias. Finite biases beyond the linear response regime are considered. The local two-particle interaction $U$ on the dot is treated using an approximation scheme within the functional renormalization group approach set up in Keldysh-Nambu-space with $U$ being the small parameter. For $V>0$ we compare our renormalization group enhanced results for the dc-component of the current to earlier weak coupling approaches such as the Hartree-Fock approximation and second order perturbation theory in $U$. We show that in parameter regimes in which finite bias driven multiple Andreev reflections prevail small $|U|$ approaches become unreliable for interactions of appreciable strength. In the complementary regime the convergence of the current with respect to numerical parameters becomes an issue - but can eventually be achieved - and interaction effects turn out to be smaller then expected based on earlier results. For $\Delta T>0$ we find a surprising increase of the current as a function of the superconducting phase difference in the regime which at $T=0$ becomes the $\pi$ (doublet) phase.