  We study nanowire-based Josephson junctions shunted by a capacitor and take
into account the presence of low-energy quasiparticle excitations. These are
treated by extending conventional models used to describe superconducting
qubits to include the coherent coupling between fermionic quasiparticles, in
particular the Majorana zero modes that emerge in topological superconductors,
and the plasma mode of the junction. Using accurate, unbiased matrix-product
state techniques, we compute the energy spectrum and response function of the
system across the topological phase transition. Furthermore, we develop a
perturbative approach, valid in the harmonic limit with small charging energy,
illustrating how the presence of low-energy quasiparticles affects the spectrum
and response of the junction. Our results are of direct interest to on-going
experimental investigations of nanowire-based superconducting qubits.
