  We present the Josephson junction intersected superconducting transmission
line resonator. In contrast to the Josephson parametric amplifier, Josephson
bifurcation amplifier and Josephson parametric 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.
