  Recent experiments on Josephson junction arrays (JJAs) in microwave cavities
have opened up a new avenue for investigating the properties of these devices
while minimising the amount of external noise coming from the measurement
apparatus itself. These experiments have already shown promise for probing
many-body quantum effects in JJAs. In this work, we develop a general
theoretical description of such experiments by deriving a quantum phase model
for planar JJAs containing quantized vortices. The dynamical susceptibility of
this model is calculated for some simple circuits, and signatures of the
injection of additional vortices are identified. The effects of decoherence are
considered via a Lindblad master equation.
