Invariant dynamics of a superconducting qubit strongly coupled to a cavity field without energy relaxation

Motivated by recent experiments wherein the Cooper box can be used to probe the decay of the resonator superposition state due to environmental decoherence. We theoretically investigate the dynamics of entanglement and mixedness of a superconducting qubit strongly coupled to a cavity field induced by a phase-damping reservoir governed by a master equation. With a phase-damping, the amount of the entanglement and the mixedness of the global system and of the resonator states have invariant dynamics while phase damping has no effect on the dynamic of the degree of the mixedness of the Cooper-pair state.