We study the effect of Rashba spin-orbit interaction on the Josephson current through a double quantum dot in the presence of Coulomb repulsion. In particular, we describe the characteristic effects on the magnetic field-induced singlet-triplet transition in the molecular regime. Exploring the whole parameter space, we analyze the effects of the device asymmetry, the orientation of the applied magnetic field with respect to the spin-orbit interaction, and finite temperatures. We find that at finite temperatures the orthogonal component of the spin-orbit interaction exhibits a similar effect to the Coulomb interaction inducing the occurrence of a π-phase at particle-hole symmetry. This provides a new route to the experimental observability of the π-phase in multi-level quantum dots.

We consider a single-level quantum dot tunnel-coupled to one normal and one superconducting lead. We employ a diagrammatic real-time approach to calculate the finite-frequency current noise for subgap transport. The noise spectrum gives direct access to the internal dynamics of the dot. In particular the noise spectrum shows sharp dips at the frequency of the coherent oscillations of Cooper pairs between dot and superconductor. This feature is most pronounced when the superconducting correlation is maximal. Furthermore, in the quantum-noise regime, ω > kBT, µN, the noise spectrum exhibits steps at frequencies equal to the Andreev addition energies. The height of these steps is related to the effective coupling strength of the excitations. The finite-frequency noise spectrum hence provides a full spectroscopy of the system.