Quantum LC circuit coupled to a vibrational network: Phonons emission dynamics

A quantum LC circuit with charge discreteness coupled to a vibrational network is studied. Considering the magnetic energy as a perturbative term (charge dominates regime), the energy spectrum is found at second order approximation. Later, the dynamics of phonons emission is studied. As a consequence of energy conservation for a particular set of emitted phonons {l _k}, forbidden and allowed values to n (charge) and l _k come out which could have consequences in quantum capacitive designs, that is, only some capacitive geometries are allowed.