In-memory multilevel control of generic SO(m) holonomy in photonics

Non-Abelian geometric phases offer fault-tolerant unitary operations that are essential for holonomic quantum computation. While the "all-geometric" approach has successfully enabled robust control of atoms, ions, and electrons in various quantum systems, its photonic implementations have remained limited by a lack of tunability-an essential feature for modern classical or quantum optical information processing. Here, we demonstrate nonvolatile multilevel tunable generic special-orthogonal (SO) geometric phases on a multilayer silicon photonic platform incorporating the phase-change material (PCM) of Sb₂Se₃. By toggling Sb₂Se₃ between crystalline and amorphous states, we dynamically control the number of degenerate states involved in the holonomy, enabling switching between different holonomic paths and SO(m) geometric transformations. Our results bridge the key gap between robust geometric-phase control and in-memory reconfigurable photonics, establishing a paradigm for fault-tolerant, high-dimensional operations in integrated optical systems.