  We have performed a quantum mechanical analysis of travelling-wave parametric
amplifiers (TWPAs) in order to investigate five experimental phenomena related
to their operations, namely the effect of impedance mismatch, the presence of
upper idler modes, the presence of quantum and thermal noise, the generation of
squeezed states, and the preservation of pre-squeezed states during
amplification. Our analysis uses momentum operators to describe the spatial
evolution of quantised modes along a TWPA. We calculate the restriction placed
on pump amplitude as well as amplifier gain as a result of impedance mismatch
between a TWPA and its external system. We apply our analysis to upper idler
modes and demonstrate that they will result in suppressed gain. We show that an
ideal TWPA is indeed quantum-limited - i.e. it introduces a half-quantum of
zero-point fluctuation which is the minimum possible noise contribution for a
phase-preserving linear amplifier. We analyse the thermal noise associated with
a TWPA by considering the effect of distributed sources along an amplifier
transmission line. Our analysis predicts a doubling of thermal noise in the
high gain limit as a result of wave-mixing between signal and idler modes. We
study the operation of a TWPA in the presence of a DC bias current, and have
shown that highly squeezed states can in principle be generated. However,
amplifying a pre-squeezed state using a non-degenerate TWPA generally reduces
the squeezing advantage.
