Graphs on Chip
Hugo Girin (C2N, Palaiseau)
A wave graph is a network of one-dimensional bonds—optical waveguides, microwave cables, quantum wires—connected at vertices where waves scatter [1]. Despite this simple structure, wave graphs are one of the cleanest playgrounds for quantum chaos: every eigenmode and scattering property can be computed exactly, while the underlying classical dynamics can be tuned from regular to fully chaotic by the topology alone. In the chaotic regime, the spectrum follows the universal statistics of random-matrix theory, and eigenfunctions spread uniformly across the graph [2]. Yet decades after these predictions, their experimental test has remained confined to microwave networks [3, 4].
We implement wave graphs on a silicon photonic chip [5], operating at telecom wavelength and room temperature (Fig. 1(a)): graph bonds are silicon waveguides, connected by 2:2 directional couplers. Injecting laser light through a lensed fiber, we exploit high-Q resonances (up to 2 × 105) to resolve ∼ 850 modes (Fig. 1(b)): a chaotic graph reproduces the predicted universal statistics, while a non-chaotic one does not—the first on-chip verification of this cornerstone of quantum chaos.
Spectral statistics, however, only probe the spectrum as a whole—they say nothing about what a single wavefunction actually looks like. To see chaos directly, we exploit silicon’s optical nonlinearity to image the intracavity field itself: near-field third-harmonic generation maps its intensity across the graph with sub-wavelength resolution (Fig. 1(c,c’)). Resonances less than a nanometer apart display starkly different intensity landscapes across the same network—a direct, real-space view of the mode-to-mode fluctuations that random-matrix theory predicts only on average.
This integrated, room-temperature platform turns wave graphs into a versatile, scalable tool. In this talk, I will discuss how it opens the door to engineering topology and disorder at will, and to controlling chaotic light transport with nonlinearity.
References
[1] T. Kottos and U. Smilansky. Quantum Chaos on Graphs. Phys. Rev. Lett. 79(24).
[2] Z. Pluhař and H. A. Weidenmüller. Universal Quantum Graphs. Phys. Rev. Lett. 112(144102).
[3] O. Hul, S. Bauch, P. Pakoński, N. Savytskyy, K. Życzkowski, and L. Sirko. Experimental Simulation of Quantum Graphs by Microwave Networks. Phys. Rev. E 69(056205).
[4] B. Dietz, T. Klaus, M. Masi, M. Miski-Oglu, A. Richter, T. Skipa, and M. Wunderle. Closed and Open Superconducting Microwave Waveguide Networks as a Model for Quantum Graphs. Phys. Rev. E 109(034201).
[5] H. Girin, X. Chécoury, B. Odouard, S. Bittner, J.-R. Coudevylle, B. Dietz, C. Lafargue, and M. Lebental. “Graphs on Chip: A Silicon Photonics Platform”. arXiv:2605.12538. 2026. arXiv: 2605.12538