Researchers have developed a new operational spectral theory to describe the dynamics of open quantum systems, specifically applied to driven Kerr resonators. This theory extends beyond Liouvillian eigenvalues, which traditionally determine decay rates and oscillation frequencies, by incorporating the excitation, propagation, and detection of quantum modes within a given protocol. The framework is based on matched left and right eigenoperators, where left operators describe excitation by an input or source, and right operators characterize the propagated density deformation and readout overlap.

For bosonic systems, coherent preparations transform left eigenoperators into phase-space excitation maps. The zeros of these maps allow for identifying mode-selective suppression, while right eigenoperators reveal the corresponding Wigner deformations. The theory enables the definition of operational coordinates in resolved slow subspaces and, under conditions of positivity and Markov admissibility, the construction of a projected routing generator. This approach provides a more comprehensive understanding of how modes are prepared and evolve within the system.

Applying this theory to driven Kerr resonators has allowed for identifying preparations that suppress a specific switching mode. It has also revealed symmetry-resolved relaxation channels and shown bias-induced crossovers in projected multichannel routing, even as the coherent-preparation partition continues to deform. The results underscore that preparation geometry and slow-sector propagation offer crucial complementary operational information, not solely obtainable from Liouvillian eigenvalues, thereby enhancing the ability to control and manipulate these quantum systems.