  Non-linearity of the current-phase relationship of a Josephson junction is
the key resource for a Josephson parametric amplifier (JPA), the only device in
which the quantum limit has so far been achieved at microwave frequencies. A
standard approach to describe JPA takes into account only the lowest order
(cubic) non-linearity resulting in a Duffing-like oscillator equation of motion
or in a Kerr-type non-linearity term in the Hamiltonian. In this paper we
derive the quantum expression for the gain of JPA including all orders of the
Josephson junction non-linearity in the linear response regime. We then analyse
gain saturation effect for stronger signals within semi-classical approach. Our
results reveal non-linear effects of higher orders and their implications for
operation of a JPA.
