  The underlying nonlinear mechanisms behind the operation of travelling-wave
parametric amplifiers (TWPAs) are important in determining their performance in
terms of added noise, maximum gain, and bandwidth. We describe a method of
characterising the underlying nonlinearity of a superconducting material in
terms of its dissipative-reactive ratio and the response time of the underlying
microscopic processes. We describe and calculate the different behaviour
arising from the equilibrium supercurrent nonlinearity, which has low
dissipation and fast response time, and the non-equilibrium heating
nonlinearity, which has high dissipation and slow response time. We have
fabricated TWPAs based on Al and Ti, and characterised their nonlinearities
using our analysis. For both Al and Ti, the measured dissipative-reactive
ratios and response times are quantitatively similar to predictions for the
non-equilibrium heating nonlinearity. We were able to obtain more than 20 dB of
peak power gain, although only over a narrow bandwidth of a few kilohertz. Our
method of characterising the underlying nonlinearities could also be useful in
the understanding and design of other superconducting nonlinear devices such as
parametric up-converters, kinetic inductance Fourier transform spectrometers,
and resonator parametric amplifiers.
