  Arrays of circuit cavities offer fascinating perspectives for exploring
quantum many-body systems in a driven dissipative regime where excitation
losses are continuously compensated by coherent input drives. Here we
investigate a system consisting of three transmission line resonators, where
the two outer ones are driven by coherent input sources and the central
resonator interacts with a superconducting qubit. Whereas a low excitation
number regime of such a device has been considered previously with a numerical
integration, we here specifically address the high excitation density regime.
We present analytical approximations to these regimes in the form of two
methods. The first method is a Bogoliubov or linear expansion in quantum
fluctuations which can be understood as an approximation for weak
nonlinearities. As the second method we introduce a combination of mean-field
decoupling for the photon tunneling with an exact approach to a driven Kerr
nonlinearity which can be understood as an approximation for low tunneling
rates. In contrast to the low excitation regime we find that for high
excitation numbers the anti-bunching of output photons from the central cavity
does not monotonously disappear as the tunnel coupling between the resonators
is increased.
