Trajectory-engineered geometric phase in two-level photonic systems

Although the geometric phase (GP) in two-level systems has been extensively studied, realizing tailored photonic-state evolution trajectories and experimentally resolving the corresponding phase responses within a controllable platform remain challenging. Here, we establish a synthetic two-level photonic system using rotated electro-optic crystals, in which two orthogonal polarization components serve as a pseudospin basis whose evolution is governed by a tunable synthetic magnetic field. Independent control of the field direction and magnitude enables tailored trajectories on the Poincaré sphere, while an explicit intensity-phase mapping provides a non-interferometric route to phase retrieval. We experimentally observe continuous GP evolution, abrupt π-phase jumps associated with antipodal crossings, and path-dependent GP accumulation along composite trajectories. These results provide a controllable electro-optic route to engineering and observing trajectory-dependent GP in synthetic photonic systems.