Absence of supercurrent sign reversal in a topological junction with a quantum dot

Experimental techniques to verify Majoranas are of current interest. A prominent test is the effect of Majoranas on the Josephson current between two wires linked via a normal junction. Here, we study the case of a quantum dot connecting the two superconductors and the sign of the supercurrent in the trivial and topological regimes under grand-canonical equilibrium conditions, explicitly allowing for parity changes due to, e.g., quasi-particle poisoning. We find that the well-known supercurrent reversal for odd occupancy of the quantum dot (pi-junction) in the trivial case does not occur in the presence of Majoranas in the wires. However, we also find this to be a mere consequence of Majoranas being zero energy states. Therefore, the lack of supercurrent sign reversal can also be caused by trivial bound states, and is thus not a discriminating signature of Majoranas.