  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.
