Microwave signatures of $mathbb{Z}_{2}$ and $mathbb{Z}_{4}$ fractional Josephson effects

We present a many-body exact diagonalization study of the $mathbb{Z}_2$ and $mathbb{Z}_4$ Josephson effects in circuit quantum electrodynamics architectures. Numerical simulations are conducted on Kitaev chain Josephson junctions hosting nearest-neighbor Coulomb interactions. The low-energy effective theory of highly transparent Kitaev chain junctions is shown to be identical to that of junctions created at the edge of a quantum spin-Hall insulator. By capacitively coupling the interacting junction to a microwave resonator, we predict signatures of the fractional Josephson effects on the cavity frequency and on time-resolved reflectivity measurements.