  One of the most studied model systems in quantum optics is a two-level atom
strongly coupled to a single mode of the electromagnetic field stored in a
cavity, a research field named cavity quantum electrodynamics or CQED. CQED has
recently received renewed attention due to its implementation with
superconducting artificial atoms and coplanar resonators in the so-called
circuit quantum electrodynamics (cQED) architecture. In cQED, the couplings can
be much stronger than in CQED due to the design flexibility of superconducting
circuits and to the enhanced field confinement in one-dimensional cavities.
This enabled the realization of fundamental quantum physics and quantum
information processing experiments with a degree of control comparable to that
obtained in CQED.
  The purpose of this chapter is to investigate the situation where the
resonator to which the atom is coupled is made nonlinear with a Kerr-type
nonlinearity, causing its energy levels to be nonequidistant. The system is
then described by a nonlinear Jaynes-Cummings Hamiltonian. This considerably
enriches the physics since a pumped nonlinear resonator displays bistability,
parametric amplification, and squeezing. The interplay of strong coupling and
these nonlinear effects constitutes a novel model system for quantum optics
that can be implemented experimentally with superconducting circuits.
  This chapter is organized as follows. In a first section we present the
system consisting of a superconducting Kerr nonlinear resonator strongly
coupled to a transmon qubit. In the second section, we describe the response of
the sole nonlinear resonator to an external drive. In the third section, we
show how the resonator bistability can be used to perform a high-fidelity
readout of the transmon qubit. In the last section, we investigate the quantum
backaction exerted by the intracavity field on the qubit.
