  We present the theory of a Josephson parametric amplifier employing two pump
sources. Our calculations are based on Input-Output Theory, and can easily be
generalized to any coupled system involving parametric interactions. We analyze
the operation of the device, taking into account the feedback introduced by the
reaction of the signal and noise on the pump power, and in this framework,
compute the response functions of interest - signal and idler gains, internal
gain of the amplifier, and self-oscillation signal amplitude. To account for
this back-action between signal and pump, we adopt a mean-field approach and
self-consistently explore the boundary between amplification and
self-oscillation. The coincidence of bifurcation and self-oscillation
thresholds reveals that the origin of coherent emission of the amplifier lies
in the multi-wave mixing of the noise components. Incorporation of the
back-action leads the system to exhibit hysteresis, dependent on parameters
like temperature and detuning from resonance. Our analysis also shows that the
resonance condition itself changes in the presence of back-action and this can
be understood in terms of the change in plasma frequency of the junction. The
potential of the double pump amplifier for quantum-limited measurements and as
a squeezer is also discussed.
