M. Gabriela Boada G., Nicolas Dirnegger, Andrea Delgado, Prineha Narang

Superconducting circuit quantum electrodynamical (cQED) platforms present a persistent modeling challenge: the intrinsic nonlinearity of the Josephson potential couples to a dissipative electromagnetic environment in ways that resist both perturbative treatment and naive Markovian reduction. Standard approaches either scale poorly with system size or absorb undeclared approximations about the noise structure into their master equations. In this work, we generalize Garraway's pseudomode construction to accommodate nonlinearly intercoupled auxiliary modes, providing a nonperturbative and systematically reducible framework for open-system cQED dynamics. The key observation is that pseudomode elimination is not fundamentally tied to linearity but to representability: any eliminated sector whose influence on the retained subsystem admits a rational self-energy can be replaced by a finite set of damped auxiliary modes, independent of the internal nonlinear structure of the retained Hamiltonian. We develop the general theory in the Heisenberg picture via a Dyson equation for the retained-mode Green's function, then demonstrate closed-form elimination for two-, three-, and four-mode Kerr-coupled systems with bilinear exchange and three-wave mixing interactions. The resulting framework substantially reduces the computational overhead of open-system cQED modeling while remaining faithful to the underlying physics, provided the spectral description of the eliminated sector is chosen to match the experimentally measured response functions of the hardware.