  Realization of a room-temperature ultra-fast photon-number-resolving (PNR)
quantum nondemolition (QND) measurement would have significant implications for
photonic quantum information processing (QIP), enabling, e.g., deterministic
quantum computation in discrete-variable architectures, but the requirement for
strong coupling has hampered the development of scalable implementations. In
this work, we propose and analyze a nonlinear-optical route to PNR QND using
quadratic (i.e., $\chi^{(2)}$) nonlinear interactions. We show that the
coherent pump field driving a phase-mismatched optical parametric amplifier
(OPA) experiences displacements conditioned on the number of signal Bogoliubov
excitations. A measurement of the pump displacement thus provides a QND
measurement of the signal Bogoliubov excitations, projecting the signal mode to
a squeezed photon-number state. We then show how our nonlinear OPA dynamics can
be utilized for deterministically generating Gottesman-Kitaev-Preskill states
only with additional Gaussian resources, offering an all-optical route for
fault-tolerant QIP in continuous-variable systems. Finally, we place these QND
schemes into a more traditional context by highlighting analogies between the
phase-mismatched optical parametric oscillator and multilevel atom-cavity QED
systems, by showing how continuous monitoring of the outcoupled pump quadrature
induces conditional localization of the intracavity signal mode onto squeezed
photon-number states. Our analysis suggests that our proposal may be viable in
near-term $\chi^{(2)}$ nonlinear nanophotonics, highlighting the rich potential
of OPA as a universal tool for ultrafast non-Gaussian quantum state engineering
and quantum computation.
