Engineering Dirac points with hybrid photonic synthetic dimensions in a cavity.

Synthetic dimensions provide a powerful route for engineering topological matter beyond the constraints of real-space geometry, with enhanced tunability and measurement accessibility. Here, we implement two independent and extended synthetic dimensions based on photonic frequency and orbital angular momentum (OAM) in a zero-dimensional cavity system. With polarization serving as a pseudo-spin, we realize an anisotropic Dirac Hamiltonian with fully controllable coupling strengths, enabling unprecedented engineering of Dirac points, including their creation, motion, and merging, as well as a Lifshitz transition from type-I to type-II Dirac cones. Using angle- and time-resolved spectroscopic techniques, we experimentally reconstruct the full two-dimensional band structure, identify Dirac points, and measure the associated Berry phase. Our platform provides a versatile setting for exploring Dirac physics and topological phenomena and establishes a scalable pathway toward higher-dimensional quantum simulation.