MotivatedbyrecentexperimentswhereintheCooperboxcanbeusedtoprobethedecayofthe
resonatorsuperpositionstateduetoenvironmentaldecoherence.Wetheoreticallyinvestigatethe
dynamicsofentanglementandmixednessofasuperconductingqubitstronglycoupledtoacavityfield
inducedbyaphase-dampingreservoirgovernedbyamasterequation.Withaphase-damping,the
amountoftheentanglementandthemixednessoftheglobalsystemandoftheresonatorstateshave
invariantdynamicswhilephasedampinghasnoeffectonthedynamicofthedegreeofthemixedness
of theCooper-pairstate.