  We study the emergence of non-Gaussian quantum features in pulsed squeezed
light generation with a mesoscopic number (i.e., dozens to hundreds) of pump
photons. Due to the strong optical nonlinearities necessarily involved in this
regime, squeezing occurs alongside significant pump depletion, compromising the
predictions made by conventional semiclassical models for squeezing.
Furthermore, nonlinear interactions among multiple frequency modes render the
system dynamics exponentially intractable in na\"ive quantum models, requiring
a more sophisticated modeling framework. To this end, we construct a nonlinear
Gaussian approximation to the squeezing dynamics, defining a "Gaussian
interaction frame" (GIF) in which non-Gaussian quantum dynamics can be isolated
and concisely described using a few dominant (i.e., principal) supermodes.
Numerical simulations of our model reveal non-Gaussian distortions of squeezing
in the mesoscopic regime, largely associated with signal-pump entanglement. We
argue that the state of the art in nonlinear nanophotonics is quickly
approaching this regime, providing an all-optical platform for experimental
studies of the semiclassical-to-quantum transition in a rich paradigm of
coherent, multimode nonlinear dynamics. Mesoscopic pulsed squeezing thus
provides an intriguing case study of the rapid rise in dynamic complexity
associated with semiclassical-to-quantum crossover, which we view as a
correlate of the emergence of new information-processing capacities in the
quantum regime.
