Fig. 1: Schematic and experimental setup.

a Configuration of a lattice network in frequency synthetic dimensions. The lattice network is simulated by a train of modulated resonators. Instead of fixed beam splitters (BS), MZIs tunable by DC and RF signals are used to couple the adjacent resonators (purple boxes). The lattice network consequently contains three types of coupling \({J}_{i,p}^{H}\) (orange), \({J}_{i}^{V}\) (violet), \({J}_{i,p}^{C}\) (green), where i is the index of resonators (A_i) along the spatial direction, and p is the coupling range along the frequency direction. These couplings stem from RF signals on the resonators, DC and RF signals applied on the MZIs, respectively. \({\phi }_{i,p}^{H}\), \({\phi }_{i}^{V}\), \({\phi }_{i,p}^{C}\) are the corresponding phases accumulated while coupling, which stem from the initial phases of the modulation. b Schematic of the Creutz ladder consisting of two lattices A and B which serve as two pseudospins. The coupling types are of same definition in a and are abbreviated as \({J}_{A(B)}^{H}\), J^V and J^C. The phases accumulated around the rectangular or triangular plaquettes form gauge potentials such as ϕ₁, ϕ₂ and ϕ₃. c Experimental setup for realizing b with two MZI assisted race-track resonators on TFLN. Three sets of electrical signals consisting of DC and RF parts are applied on three sets of the ground-signal-ground (GSG) arranged electrodes through Bias-Tees to control the two resonators (S₁, S₃) and the MZI (S₂). For detecting band structures, a probe laser with detuning Δω is injected to excite the bands. The time- (quasi-momentum-) varying transmittance signal, which is a slice of the band structure, is detected by a photodetector (PD) followed by an oscilloscope triggered by the RF source. For obtaining mode distribution, the measurement part is replaced by scanning a Fabry–Perot (F-P) cavity together with a photodetector.