  Microwave parametric amplifiers based on Josephson junctions have become a
key component of many quantum information experiments. One key limitation which
has not been well predicted by theory is the gain saturation behavior which
determines its ability to process large amplitude signals. The typical
explanation for this behavior in phase-preserving amplifiers based on
three-wave mixing is pump depletion, in which the consumption of pump photons
to produce amplification results in a reduction in gain. However, in this work
we present experimental data and theoretical calculations showing that the
fourth-order Kerr nonlinearities inherent in the Josephson junctions are the
dominant factor in the Josephson Parametric Converter (JPC). The Kerr-based
theory has the unusual property of causing saturation to both lower and higher
gains, depending on bias conditions. This work presents a new methodology for
optimizing device performance in the presence of Kerr nonlinearities while
retaining device tunability, and points to the necessity of controlling
higher-order Hamiltonian terms to make further improvements in parametric
devices.
