How Hamiltonian non-linearities limit the performance of Josephson parametric amplifiers

The implementation of a Josephson Parametric Amplifier with a large saturation power is an essential ingredient in achieving efficient signal detection in superconducting quantum computing circuits. Using numerical evolution of the classical non-linear equations of motion that describe a single non-degenerate gain process in a Josephson Parametric Converter (JPC), we analyze the factors limiting its performance. We demonstrate that the 3rd order coupling between the signal, idler and pump modes effectively generates a complex cross-Kerr term coupling the signal and idler modes, which can limit device saturation. By comparing properties of the full-nonlinear description of the JPC to descriptions truncated at third, fourth, and higher orders we identify which terms limit the device performance and how to optimize them.