Floquet spectrum and optical behaviors in dynamic Su–Schrieffer–Heeger modeled waveguide array

In the development of topological photonics, robust propagation behaviors of topologically protected states in photonic analog of topological insulators have attracted broad attention. In the presence of dynamic time-dependent drivings, much more interesting topological phenomena are expected. Specifically, when the driving field is time-periodic, dynamic analogs of topological insulators, Floquet topological insulators FTIs, have been theoretically proposed in various dimensions. On the experimental side, researchers have already designed FTIs in cold atoms, photonic crystals and other systems. However, clear demonstrations of dynamic behaviors of FTIs have been rare, especially in one-dimensional systems, due to the lack of a flexible platform with easily tunable frequency and amplitude of the periodic driving field.

Illustration of Floquet topological insulators 

Recently a variety of physical systems have been developed using coupled-mode equations to describe the propagation of light fields to imitate the evolution of electronic wave functions described by the Schrodinger equation, among which waveguide arrays have attracted intense interest. Notably, by treating the propagation direction as the time dimension, periodic modulations can be genuinely introduced by periodically bending the waveguides, with easily tunable structural parameters of bending frequency, bending amplitude, and waveguide spacings. This enables us to realize FTIs and demonstrate corresponding exotic dynamic topological propagation behaviors in waveguide systems.

Researchers have observed an anomalous edge localized mode exhibiting periodically oscillating behaviors at moderate frequencies, which was theoretically confirmed as the long-sought Floquet π mode (PRL 2019). However, a more detailed Floquet band gap evolution and the equivalence between high frequency driven waveguide and static waveguide have not been fully revealed in the work.

Cheng Qingqing and collaborator (Wang Huaiqiang at Nanjing University), further extended the microwave to the near-infrared frequency, and built a time-dependent driving system for silicon-based waveguide array. By adjusting the frequency of the periodic driving, a complete evolution of the Floquet band gap was observed in the whole process of opening, closing and reopening. The result was published in Chinese Optics Letters, Vol. 19, Iss. 4. (Yu Ye, et al., Floquet spectrum and optical behaviors in dynamic Su–Schrieffer–Heeger modeled waveguide array).

SSH model of the bend-modulated waveguide array

This work clearly demonstrates the equivalence between high frequency driven waveguide and static waveguide. They further built an analog of static SSH (Su–Schrieffer–Heeger) model with high frequency driven waveguides, and observed the topologically equivalent zero mode and related coupling effects. This work realizes the bend-modulated silicon waveguide array obeying the Floquet one-dimensional dynamic SSH model, which provides a versatile platform for Floquet engineering designed "on-demand".

This work was supported by the National Natural Science Foundation of China (Grant Nos. 11874266, 11604208)