Synthesizing arbitrary lattice models using a single degenerate cavity

We propose a general method to simulate arbitrary lattice models by manipulating optical synthetic dimension in a single degenerate main cavity. Such a cavity supports a large number of degenerate optical modes with different angular momenta. Couplings between different optical modes can be readily controlled. These features allow us to simulate lattice models that are not convenient to realize using other systems, particularly models in high dimensions and with complicated hopping amplitudes. As a concrete example, we demonstrate how to construct two topological lattice models: the two-dimensional Haldane model and a four-dimensional time-reversal invariant model. For the latter case, we show how topological properties can be detected from the outputs of the cavity, where the second Chern number can be extracted. In the presence of open boundaries, the chirality of the Weyl edge modes can also be detected using the input-output formalism of the cavity modes.