  With advanced micro- and nano-photonic structures, the vacuum photon-photon
coupling rate is anticipated to approach the intrinsic loss rate and lead to
unconventional quantum effects. Here, we investigate the classical-to-quantum
transition of such photonic nonlinear systems using the quantum
cluster-expansion method, which addresses the computational challenge in
tracking large photon number states of the fundamental and harmonic optical
fields involved in the second harmonic generation process. Compared to the
mean-field approximation used in weak coupling limit, the quantum
cluster-expansion method solves multimode dynamics efficiently and reveals the
quantum behaviors of optical parametric oscillations around the threshold. This
work presents a universal tool to study quantum dynamics of multimode systems
and explore the nonlinear photonic devices for continuous-variable quantum
information processing.
