Interplay between Coulomb blockade and Josephson effect in a topological superconductor-quantum dot device

We study the behavior of a topological Josephson junction in which two topological superconductors are coupled through a quantum dot. We focus on the case with the bulk superconducting gap being the largest energy scale. Two parameter regimes are investigated: a weak tunneling between the dot and the superconductors, with the dot near its charge degeneracy point, and a strong-tunneling regime in which the transmission between the dot and the superconductors is nearly perfect. We show that in the former situation, the Andreev spectrum for each sector with fixed fermion parity consists of only two levels, which gives rise to the nontrivial current-phase relation. Moreover, we study the Rabi oscillation between the two levels and indicate that the corresponding frequency is a 4 pi-periodic function of the phase difference between the two superconductors, which is immune to the quasiparticle poisoning. In the latter case, we find that the Coulomb charging energy enhances the effect of backscattering at the interfaces between the dot and the superconductors. Both the temperature and the gate-voltage dependence of the critical Josephson current are examined.