  Superconducting parametric amplifiers are commonly fabricated using planar
transmission lines with a non-linear inductance provided by either Josephson
junctions or the intrinsic kinetic inductance of the thin film. However, Banys
et al. [1] reported non-linear behaviour in a niobium pillbox cavity,
hypothesising that below Tc, the pair iris-bulk resonator would act as a
superconducting contact surface exploiting a Josephson-like non-linearity. This
work investigates this effect further by applying Keysight Technologies'
Advanced Design System (ADS) to simulate the cavity using an equivalent circuit
model that includes a user defined Josephson inductance component. The
simulations show that for a resonance centred at nu0 = 30.649 GHz, when two
tones (pump and signal) are injected into the cavity, mixing and parametric
gain occur. The maximum achievable gain is explored when the resonator is taken
to its bifurcation energy. These results are compared to cryogenic measurements
where the pump and signal are provided by a Vector Network Analyzer.
