Fig. 3: Resonant-phase-controlled TPTs and statistical verification of topological robustness.
a, Calculated band structures in φS-controlled TPT. Starting from the weak coupling regime (θ = 0.4π), by increasing φS from 0 to π, site rings and link rings become detuned, reaching a maximum detuning at φS = π. TPT occurs at bandgap ❸ when φS = 0.58π, making it a trivial forbidden bandgap. b, Theoretical and experimental transmission spectra as functions of λ and φS. As a global phase shift is introduced by φS, TPT occurs at bandgap ❶. The mapping of bandgaps changes from {❶, ❷, ❸} to {❸, ❶, ❷} after a 2π evolution of phase φS. c, Calculated projected band structures at φS = 0.8π, plotted as a reference to demonstrate the robustness of topological edge modes. d,e, Experimental verification of topological robustness with statistical measurements by individually controlling the phase disorders in all rings. The generated random phase obeys the same uniform distribution in the range of δ[−0.5, 0.5]. d, A set of 100 samples with uniformly distributed random phases is chosen at δ = 0.1π in measurement. Measured transmission spectra for the disordered devices are shown as the grey background, and the spectrum for an ideal device without disorder is plotted as a blue line. In the topological edge modes the flat plateaus with high transmission are only slightly influenced, while in all other regimes severe broadening and small dips owing to obstruction from random local modes appear. e, The measured s.d. (normalized) of transmission spectra under different strengths of disorder. Evident windows with low fluctuations correspond exactly to the regimes of topological edge modes. δ increases from 0 to 0.2π with an interval of 0.01π. S.d. is colour coded and the key is provided at the bottom right.
