  We investigate the quantum entropy, its power spectrum, and the excitation
inversion of a Cooper pair box interacting with a nanomechanical resonator, the
first initially prepared in its excited state, the second prepared in a
"cat"-state. The method uses the Jaynes-Cummings model with damping, with
different decay rates of the Cooper pair box and distinct detuning conditions,
including time dependent detunings. Concerning the entropy, it is found that
the time dependent detuning turns the entanglement more stable in comparison
with previous results in literature. With respect to the Cooper pair box
excitation inversion, while the presence of detuning destroys the its collapses
and revivals, it is shown that with a convenient time dependent detuning one
recovers such events in a nice way.
