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Arrays of circuit cavities offer fascinating perspectives for exploringquantum many-body systems in a driven dissipative regime where excitationlosses are continuously compensated by coherent input drives. Here weinvestigate a system consisting of three transmission line resonators, wherethe two outer ones are driven by coherent input sources and the centralresonator interacts with a superconducting qubit. Whereas a low excitationnumber regime of such a device has been considered previously with a numericalintegration, we here specifically address the high excitation density regime.We present analytical approximations to these regimes in the form of twomethods. The first method is a Bogoliubov or linear expansion in quantumfluctuations which can be understood as an approximation for weaknonlinearities. As the second method we introduce a combination of mean-fielddecoupling for the photon tunneling with an exact approach to a driven Kerrnonlinearity which can be understood as an approximation for low tunnelingrates. In contrast to the low excitation regime we find that for highexcitation numbers the anti-bunching of output photons from the central cavitydoes not monotonously disappear as the tunnel coupling between the resonatorsis increased.

The Circuit Quantum Electrodynamical Josephson Interferometer