  We theoretically study superconducting islands based on
semiconductor-nanowire Josephson junctions and take into account the presence
of subgap quasiparticle excitations in the spectrum of the junction. Our method
extends the standard model Hamiltonian for a superconducting charge qubit and
replaces the Josephson potential by the Bogoliubov--de Gennes Hamiltonian of
the nanowire junction, projected onto the relevant low-energy subgap subspace.
This allows to fully incorporate the coherent dynamics of subgap levels in the
junction. The combined effect of spin-orbit coupling and Zeeman energy in the
nanowires forming the junction triggers a topological transition, where the
subgap levels evolve from finite-energy Andreev bound states into near-zero
energy Majorana bound states. The interplay between the microscopic energy
scales governing the nanowire junction (the Josephson energy, the Majorana
coupling and the Majorana energy splitting), with the charging energy of the
superconducting island, gives rise to a great variety of physical regimes.
Based on this interplay of different energy scales, we fully characterize the
microwave response of the junction, from the Cooper pair box to the transmon
regimes, and show how the presence of Majoranas can be detected through
distinct spectroscopic features. In split-junction geometries, the plasma mode
couples to the phase-dispersing subgap levels resulting from Majorana
hybridization via a Jaynes--Cummings-like interaction. As a consequence of this
interaction, higher order plasma excitations in the junction inherit Majorana
properties, including the $4\pi$ effect.
