  We study a nonlinear interferometer consisting of two consecutive parametric
amplifiers, where all three optical fields (pump, signal and idler) are treated
quantum mechanically, allowing for pump depletion and other quantum phenomena.
The interaction of all three fields in the final amplifier leads to an
interference pattern from which we extract the phase uncertainty. We find that
the phase uncertainty oscillates around a saturation level that decreases as
the mean number $N$ of input pump photons increases. For optimal interaction
strengths, we also find a phase uncertainty below the shot-noise level and
obtain a Heisenberg scaling $1/N$. This is in contrast to the conventional
treatment within the parametric approximation, where the Heisenberg scaling is
observed as a function of the number of down-converted photons inside the
interferometer.
