Control of Photon Dynamics in Non-Euclidean Polygonal Microcavities by Joint Geometric Curvatures

Non-Euclidean geometry has recently emerged as a powerful tool, offering new insights and applications in optical microcavities supporting Whispering Gallery Modes (WGMs). In this study, we extend the concept of polygonal microcavities to non-Euclidean spaces by developing a unified model that incorporates a joint geometric parameter of curvatures. This system uncovers a range of unexplored phenomena, mechanisms, and concepts that are unique to curved spaces. Notably, we observe dissipative states characterized by hyperbolic fixed points (HFPs) that appear exclusively in non-Euclidean scenarios, leading to the formation of phase diagrams within the parametric space of curvatures. Our results reveal phase transitions across geometric boundaries, marked by abrupt changes in the cavity quality factor. These transitions are strongly influenced by the wavelike nature of photon trajectories, offering intriguing insights into quantum chaos within curved spaces. Additionally, we discover that cavities with geodesic side lines exhibit a remarkable symmetry-driven avoidance of such phase transitions, highlighting the profound connection between physical dynamics and spatial geometry. Our findings establish a promising platform for optical simulations of non-Euclidean quantum chaos and open up potential applications in on-chip photonic devices.