Measuring $\mathbb{Z}_2$ invariants in dimer models and cross-coupled ladders with a programmable photonic molecule

Topological models are characterized by a quantized topological invariant and provide a description of novel phases of matter that can exhibit localized edge states, corner modes, and chiral transport. We experimentally realize two 1-D lattices supporting symmetry-protected topology - the Su-Schrieffer-Heeger (SSH) and extended SSH models using the synthetic frequency dimension of coupled fiber ring resonators. We introduce and experimentally demonstrate cascaded heterodyning as a technique for low-noise, single-shot winding number measurements through the mean chiral displacement and band structure measurements. Through our robust setup and detection techniques we can extend our capability to realizing 1-D ladder models, demonstrating a modified Creutz ladder with a staggered flux with each plaquette. This highly reconfigurable and compact fiber optics platform for Hamiltonian simulation, along with a low-noise detection scheme, provides a path forward for chip-scale realizations.