Recently synthetic frequency dimensions constructed in ring resonator systems have attracted broad interest. Dynamic modulations from electro-optic phase modulators (EOM) connect resonant modes to create complex lattice structures. Study of 1D synthetic Su-Schrieffer-Heeger (SSH) model constructed along the frequency dimension in ring resonator systems would provide richer physics and new routes to extract topological phase information.

The synthetic lattice sites unequally spaced along the frequency dimension are constructed by symmetric and antisymmetric supermodes in two coupled ring resonators of 10.2 m long. An EOM in ring A provides bichromatic sinusoidal modulations at different amplitudes and generates alternating hopping amplitudes g_1 and g_2. By linearly scanning the frequency of the input laser, the time-resolved projected band structures of the synthetic lattice are obtained in the experiment. A data-analysis scheme is applied to calculate the geometric phase φ encoded in the signal intensity of bulk bands. With the external modulation g_1＜g_2 and g_1＞g_2, Zak phase values (~0.98π and ∼ 0) are then directly extracted in non-trivial and trivial topological phases.

This work for the first time shows directly reading the topological Zak phase from the distinguishable time-resolved transmission spectra in a synthetic SSH model, constructed with the frequency dimension in modulated ring resonators. The proposed method is universal in the synthetic frequency dimension with experimental feasibility and reconfigurability. This experiment offers new possibility to construct more complex lattice structures with nonuniform connectivities and to explore spectral non-reciprocity and potential applications in optical communications.

Figure. (a) An illustration of two identical coupled ring resonators with bichromatic dynamic modulations. (b) The system in (a) can be mapped into a synthetic 1D SSH model along the frequency dimension. For (c) non-trivial (g_1＜g_2) and (d) trivial (g_1＞g_2) phases, experimental observed band structures and measured phases.

The research was published in “Guangzhen Li, Luojia Wang, Rui Ye, Yuanlin Zheng, Da-Wei Wang, Xiong-Jun Liu, Avik Dutt, Luqi Yuan, and Xianfeng Chen, Direct extraction of topological Zak phase with the synthetic dimension, Light: Science & Applications 12, 81 (2023)”.

Link: https://www.nature.com/articles/s41377-023-01126-1