  We develop a theory of parametric resonance in tunable superconducting
cavities. The nonlinearity introduced by the SQUID attached to the cavity, and
damping due to connection of the cavity to a transmission line are taken into
consideration. We study in detail the nonlinear classical dynamics of the
cavity field below and above the parametric threshold for the degenerate
parametric resonance, featuring regimes of multistability and parametric
radiation. We investigate the phase-sensitive amplification of external signals
on resonance, as well as amplification of detuned signals, and relate the
amplifier performance to that of linear parametric amplifiers. We also discuss
applications of the device for dispersive qubit readout. Beyond the classical
response of the cavity, we investigate small quantum fluctuations around the
amplified classical signals. We evaluate the noise power spectrum both for the
internal field in the cavity and the output field. Other quantum statistical
properties of the noise are addressed such as squeezing spectra, second order
coherence, and two-mode entanglement.
