Adopting the framework of single electrons or Cooper pairs where the rate of particle transfer is controlled by an external frequency, we investigate the dynamics of the coupled system, considering the single-mode cavity field prepared in a generalized geometric state. Considering a density matrix approach, we obtain a full description of the dynamics of the coupled system for a weak quantized radiation field. Using the relevant processes involved in the detection of the charge state of the box and a realistic description of the gate pulse, we are able to analyze some aspects of the information entropy in a quantitative way. We aim to clarify results from previous work, alluding to a link between the entropy squeezing and entanglement. The numerical and analytical results obtained in this work elucidate information entropy effects. We also show that the occupation probability amplitudes possess regular structures with collapses and revivals in the Rabi oscillations.