Emergent system-reservoir methods for broadband nonlinear quantum optics: a case study on cascaded chi(2)

In broadband quantum optical systems, nonlinear interactions among a large number of frequency components induce highly rich but complicated dynamics, for which we crucially lack intuitive understanding on the phenomenology. In this work, we introduce a perturbative framework in which to factor out effective ``reservoir'' degrees of freedom and establish concise effective models for the remaining system. Our approach relies on a partitioning of the intra-system degrees of freedom into primary and auxiliary sectors, followed by approximate diagonalization of subsystems by means of unitary transformation, as well as master equation techniques. We apply the method to the case of cascaded optical $chi^{(2)}$ nonlinearities, and show that the cascaded process can be summarized as interactions among dressed fundamental and second harmonic modes, where the former experience self-phase modulations to leading order, and these states interact via cross-phase modulations. Using this approach, we eliminate the second harmonic degree of freedom, and identify features of the fundamental wave dynamics beyond the conventional single-mode cascaded nonlinearity, such as emergent dissipative two-photon loss channels. Our results highlight the utility of system-reservoir methods to concisely understand the complicated dynamics of multimode quantum photonic systems, which we expect to be a powerful theoretical toolbox for quantum engineering with photons.