  In broadband quantum optical systems, nonlinear interactions among a large
number of frequency components induce complex dynamics that may defy heuristic
analysis. In this work we introduce a perturbative framework for factoring out
reservoir degrees of freedom and establishing a concise effective model
(effective field theory) for the remaining system. Our approach combines
approximate diagonalization of judiciously partitioned subsystems with master
equation techniques. We consider cascaded optical $\chi^{(2)}$ (quadratic)
nonlinearities as an example and show that the dynamics can be construed (to
leading order) as self-phase modulations of dressed fundamental modes plus
cross-phase modulations of dressed fundamental and second-harmonic modes. We
then formally eliminate the second-harmonic degrees of freedom and identify
emergent features of the fundamental wave dynamics, such as two-photon loss
channels, and examine conditions for accuracy of the reduced model in
dispersive and dissipative parameter regimes. Our results highlight the utility
of system-reservoir methods for deriving accurate, intuitive reduced models for
complex dynamics in broadband nonlinear quantum photonics.
