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A research team led by professors Sunkyu Yu and Namkyoo Park of Seoul National University and Xianji Piao of the University of Seoul has developed a new mathematical framework to describe how a single atom interacts with light. Published in *Science Advances* on September 25, their study reveals that this simple quantum system contains an enormous, unbounded network of states. The team, supported by the National Research Foundation of Korea chaired by Won-Hwa Hong, created what they call "magnetic graph" theory to unify descriptions across different coupling regimes.

Previously, scientists used separate approximations for weak interactions and ultrastrong or deep-strong coupling, where the interaction strength exceeds the light’s intrinsic frequency. In these strong regimes, standard theories fail. The researchers translated quantum states into a graph structure similar to a subway map, where photon and atom combinations act as stations and transitions are represented by connecting lines. They found that even with just one atom, the accessible quantum states form a semi-infinite graph with unlimited long-range links.

To quantify this complexity, the team introduced a metric based on a magnetic Laplacian, which accounts for connection strengths and light wave phases. This single measure allows them to classify all interaction regimes along one continuous scale. They also discovered that phase frustration—a situation where phases cannot be made mutually compatible—drives the reorganization of quantum states in strong coupling. Professor Park noted that while graph connectivity is important, the distributed phase information fundamentally alters the system’s quantum states. This unified approach could aid in designing quantum computers and photonic neural networks.