September 27, 2026 -- A new graph-theoretic framework that provides a unified description of atom-light interactions across regimes ranging from weak to deep-strong coupling has been proposed.

A research team comprising Professors Sunkyu Yu and Namkyoo Park of the Department of Electrical and Computer Engineering at Seoul National University College of Engineering and Professor Xianji Piao of the University of Seoul has discovered that even the interaction between a single atom and light—one of the simplest settings in quantum physics—contains an enormous and intricately connected network. The study appears in the journal Science Advances in September 25th.

By developing a mathematical framework that represents this hidden structure like a subway map, the researchers established a unified way to understand atom-light interactions across coupling regimes that previously required different theoretical approximations. The findings are expected to open new possibilities for quantum computing and photonic neural networks.

The National Research Foundation of Korea (NRF), chaired by Won-Hwa Hong, announced that a research team led by Professors Namkyoo Park and Sunkyu Yu of Seoul National University and Professor Xianji Piao of the University of Seoul has developed a “magnetic graph” theory that quantifies atom-light interactions using a single measure.

*Magnetic graph: A graph in which each link contains not only information about the strength of the connection but also the phase of the associated wave.

Interactions between an atom and light—theoretically described by quantum Rabi model, the most fundamental model of atom-light interactions—are a building block of quantum computing and optical technologies. When an atom absorbs a photon, its energy state changes; when it releases energy, it may emit a photon. Repeated interactions of this kind underpin quantum information processing, quantum sensing, lasers, nonlinear optics, and numerous other light-matter technologies.

When the interaction is sufficiently weak, its behavior can be described accurately using well-established approximations. However, as the coupling between the atom and light becomes extremely strong—entering the ultrastrong- or deep-strong-coupling regime—these conventional approximations begin to fail.

*Ultrastrong and deep-strong coupling: Regimes in which the strength of the atom-light interaction becomes comparable to or greater than the intrinsic frequency of light. Under these conditions, the standard approximations applied to the quantum Rabi model are no longer reliable.

As a result, researchers have traditionally used distinct approximations methods for describing different coupling regimes. The behavior across the entire regimes cannot be fully described using any single conventional theory. A new mathematical language was therefore needed—one capable of representing complex quantum states at a glance and describing the entire evolution from weak to deep-strong coupling within a unified framework.

Inspired by graph theory—interpreting subway maps, brain architecture, and social networks within a single framework—the research team devised a method for translating the quantum world into a graph. Just as a subway map represents stations as points and routes as connecting lines, the framework depicts quantum states as nodes and the possible transitions between them as links.

More specifically, combinations of the number of photons and the state of the atom are represented as “stations” on the graph. The likelihood that one quantum state will transition into another is encoded by the thickness of the connecting line, while the phase of the light wave is represented by its color.

*Phase: A value indicating the position of a wave within one complete oscillation cycle. Even waves with the same frequency and amplitude can interfere differently depending on their relative phases.

Through this representation, the researchers found that even the simplest atom-light quantum system conceals a semi-infinite graph in which two types of nodes are connected through an unlimited number of long-range links. Although the physical system may consist of only a single atom interacting with light, its accessible quantum states form a vast network whose connectivity grows without bound.

*Semi-infinite graph: A graph that has a boundary in one direction but extends indefinitely in the other direction from its starting point.

The research team then introduced a metric obtained from a magnetic Laplacian, which compresses the complexity of this network into a single number. The metric simultaneously accounts for both the strengths of the graph’s connections and the phase distributions carried by its links.

*Magnetic Laplacian: A mathematical tool used to analyze connectivity in a graph while incorporating both the strengths and the phases of its links.

Using this index, the researchers successfully distinguished all regimes of atom-light interaction—from weak coupling to deep-strong coupling—along a single continuous scale. This unified classification replaces the fragmented picture in which separate theoretical tools were required for different coupling strengths.

The team further investigated why the structure of quantum states changes so substantially under strong atom-light coupling. For each of several subgraph sizes, the researchers generated and analyzed 20,000 random configurations. The statistical results revealed that phase frustration is a key mechanism driving the dramatic reorganization of quantum states in the strong-coupling regime.

*Phase frustration: A phenomenon in which waves traveling along different paths acquire incompatible phases and consequently cancel one another. It can be compared to signals taking several different routes to the same destination but arriving out of step, thereby interfering with one another and obstructing further movement.

 The findings show that graph connectivity alone is insufficient to understand strongly interacting quantum systems. The phase information distributed throughout the network—and the frustration produced when phases cannot be made mutually compatible—can fundamentally alter the quantum states of the system.

This graph-based perspective is significant because it reveals a previously hidden structural complexity within one of the most elementary models of quantum physics. It also provides a common mathematical framework for analyzing coupling regimes that have traditionally been treated separately. The theory could ultimately contribute to the design and control of quantum computers, quantum simulators, and photonic neural networks. In particular, the vast state-space connectivity generated by even a single atom interacting with light may offer a compact physical basis for implementing artificial neural networks.

 Professor Namkyoo Park said, “We have shown that an immense connectivity structure is hidden even within the simplest quantum phenomenon.” He added, “We plan to continue investigating how artificial neural networks can be implemented by controlling the interaction between a single atom and light.”