Magnetic graphs for cavity quantum electrodynamics

Strengthening light-matter coupling has become a central challenge in cavity quantum electrodynamics (QED), enabling ultrafast gate operations, qubit protection, and deterministic nonlinear optics. As the coupling increases, even the simplest configuration-an atom interacting with a quantized field-requires careful treatment, as exemplified by ongoing debates on quantum Rabi models (QRMs). Here we propose a magnetic graph model for single-atom cavity QED that offers a unified description of quantum dynamics from the weak- to the ultrastrong-coupling regime via graph connectivity. We demonstrate that the generalized QRM maps onto a complex bipartite graph of identical sites under Floquet boundary conditions. This framework captures the crossover from weak to deep-strong coupling via a single metric-the cost of disconnecting a nonmagnetic subgraph. We examine the mechanism underlying this connectivity transition, establishing the pivotal role of phase frustration. Scalable to many-body systems, this approach bridges graph theory and cavity QED, revealing complex-graph dynamics even in the simplest setting.