Linear response theory of Josephson junction arrays in a microwave cavity

Motivated by recent experiments on Josephson junction arrays in microwave cavities, we construct a quantum phase model and calculate the susceptibility of this model in linear response. Both charge and vortex degrees of freedom are considered, as well as circuits containing either Josephson junctions or coherent quantum phase slip elements. The effects of decoherence are considered via a Lindblad master equation.