  We study the interplay of squeezing and phase randomization near the
hyperbolic instability of a two-site Bose-Hubbard model in the Josephson
interaction regime. We obtain results for the quantum Zeno suppression of
squeezing, far beyond the previously found short time behavior. More
importantly, we contrast the expected outcome with the case where randomization
is induced by erratic driving with the same fluctuations as the quantum noise
source, finding significant differences. These are related to the distribution
of the squeezing factor, which has log-normal characteristics: hence its
average is significantly different from its median due to the occurrence of
rare events.
