Quantum Dot Parity Effects in Trivial and Topological Josephson Junctions

An odd-occupied quantum dot in a Josephson junction can flip transmission phase, creating a $\ensuremath{\pi}$ junction. When the junction couples topological superconductors, no phase flip is expected. We investigate this and related effects in a full-shell hybrid interferometer, using gate voltage to control dot-junction parity and axial magnetic flux to control the transition from trivial to topological superconductivity. Enhanced zero-bias conductance and critical current for odd parity in the topological phase reflects hybridization of the confined spin with zero-energy modes in the leads.