Arrays of circuit cavities offer fascinating perspectives for exploring quantum many-body systems in a driven dissipative regime where excitation losses are continuously compensatedby 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. This is of particular interest as intra cavity fields might undergo a transition from low excitation number quantum fields to high amplitude classical fields when increasing the input drives. We present analytical approximations to these regimes in the form of two methods. The first method is a Bogoliubov 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.

We introduce and study the properties of an array of QED cavities coupled by
non-linear elements, in the presence of photon leakage and driven by a coherent
source. The non-linearcouplings lead to photon hopping and to nearest-neighbor
Kerr terms. By tuning the system parameters, the steady state of the array can
exhibit a photon crystal associated to a periodic modulation of the photon
blockade. In some cases the crystalline ordering may coexist with phase
synchronisation. The class of cavity arrays we consider can be built with
superconducting circuits of existing technology.

We introduce a circuit quantum electrodynamical setup for a quantum single
photon transistor. In our approach single photons propagate in two open
transmission lines that are coupledvia two interacting transmon qubits. The
interaction is such that photons are not exchanged between the two transmission
lines but a photon in one line can completely block respectively enable the
propagation of photons in the other line. High on-off ratios can be achieved
for feasible experimental parameters. Our approach is inherently scalable as
all photon pulses can have the same pulse shape and carrier frequency such that
output signals of one transistor can be input signals for a consecutive
transistor.

We present the Josephson junction intersected superconducting transmission
line resonator. In contrast to the Josephson parametric amplifier, Josephson
bifurcation amplifier and Josephsonparametric converter we consider the regime
of few microwave photons. We review the derivation of eigenmode frequencies and
zero point fluctuations of the nonlinear transmission line resonator and the
derivation of the eigenmode Kerr nonlinearities. Remarkably these
nonlinearities can reach values comparable to Transmon qubits rendering the
device ideal for accessing the strongly correlated regime. This is particularly
interesting for investigation of quantum many-body dynamics of interacting
particles under the influence of drive and dissipation. We provide current
profiles for the device modes and investigate the coupling between resonators
in a network of nonlinear transmission line resonators.

We explore photon coincidence counting statistics in the ultrastrong-coupling
regime where the atom-cavity coupling rate becomes comparable to the cavity
resonance frequency. In thisregime usual normal order correlation functions
fail to describe the output photon statistics. By expressing the electric-field
operator in the cavity-emitter dressed basis we are able to propose correlation
functions that are valid for arbitrary degrees of light-matter interaction. Our
results show that the standard photon blockade scenario is significantly
modified for ultrastrong coupling. We observe parametric processes even for
two-level emitters and temporal oscillations of intensity correlation functions
at a frequency given by the ultrastrong photon emitter coupling. These effects
can be traced back to the presence of two-photon cascade decays induced by
counter-rotating interaction terms.

We investigate a chain of superconducting stripline resonators, each
interacting with a transmon qubit, that are capacitively coupled in a row. We
show that the dynamics of this systemcan be described by a Bose-Hubbard
Hamiltonian with attractive interactions for polaritons, superpositions of
photons and qubit excitations. This setup we envisage constitutes one of the
first platforms where all technological components that are needed to
experimentally study chains of strongly interacting polaritons have already
been realized. By driving the first stripline resonator with a microwave source
and detecting the output field of the last stripline resonator one can
spectroscopically probe properties of the system in the driven dissipative
regime. We calculate the stationary polariton density and density-density
correlations $g^{(2)}$ for the last cavity which can be measured via the output
field. Our results display a transition from a coherent to a quantum field as
the ratio of on site interactions to driving strength is increased.