  Two-photon loss mechanisms often accompany a Kerr nonlinearity. The kinetic
inductance exhibited by superconducting transmission lines provides an example
of a Kerr-like nonlinearity that is accompanied by a nonlinear resistance of
the two-photon absorptive type. Such nonlinear dissipation can degrade the
performance of amplifiers and mixers employing a Kerr-like nonlinearity as the
gain or mixing medium. As an aid for parametric amplifier design, we provide a
quantum analysis of a cavity parametric amplifier employing a Kerr nonlinearity
that is accompanied by a two-photon absorptive loss. Because of their
usefulness in diagnostics, we obtain expressions for the pump amplitude within
the cavity, the reflection coefficient for the pump amplitude reflected off of
the cavity, the parametric gain, and the intermodulation gain. Expressions by
which of the degree of squeezing can be computed are also presented.
