Preparation geometry and slow-sector routing in driven Kerr resonators: an operational spectral theory of Liouvillians

Abstract

Liouvillian eigenvalues determine decay rates and oscillation frequencies, but not how the corresponding modes are excited, propagated, and detected in a chosen protocol. We develop an operational spectral theory based on matched left and right eigenoperators. Left eigenoperators determine excitation by an input or source; right eigenoperators determine the propagated density deformation and readout overlap; their product is a gauge-invariant modal weight. For bosonic systems, coherent preparations turn left eigenoperators into phase-space excitation maps whose zeros identify mode-selective suppression, while right eigenoperators yield the corresponding Wigner deformations. Resolved slow subspaces define operational coordinates and, when positivity and Markov-admissibility hold, a projected routing generator. In driven Kerr resonators, the framework identifies preparations that suppress a switching mode, separates symmetry-resolved relaxation channels, and reveals bias-induced crossovers in projected multichannel routing while the coherent-preparation partition continues to deform. Preparation geometry and slow-sector propagation thus provide complementary operational information beyond Liouvillian eigenvalues alone.