  Single-mode Josephson junction-based parametric amplifiers are often modeled
as perfect amplifiers and squeezers. We show that, in practice, the gain,
quantum efficiency, and output field squeezing of these devices are limited by
usually neglected higher-order corrections to the idealized model. To arrive at
this result, we derive the leading corrections to the lumped-element Josephson
parametric amplifier of three common pumping schemes: monochromatic current
pump, bichromatic current pump, and monochromatic flux pump. We show that the
leading correction for the last two schemes is a single Kerr-type quartic term,
while the first scheme contains additional cubic terms. In all cases, we find
that the corrections are detrimental to squeezing. In addition, we show that
the Kerr correction leads to a strongly phase-dependent reduction of the
quantum efficiency of a phase-sensitive measurement. Finally, we quantify the
departure from ideal Gaussian character of the filtered output field from
numerical calculation of third and fourth order cumulants. Our results show
that, while a Gaussian output field is expected for an ideal Josephson
parametric amplifier, higher-order corrections lead to non-Gaussian effects
which increase with both gain and nonlinearity strength. This theoretical study
is complemented by experimental characterization of the output field of a
flux-driven Josephson parametric amplifier. In addition to a measurement of the
squeezing level of the filtered output field, the Husimi Q-function of the
output field is imaged by the use of a deconvolution technique and compared to
numerical results. This work establishes nonlinear corrections to the standard
degenerate parametric amplifier model as an important contribution to Josephson
parametric amplifier's squeezing and noise performance.
