Fig. 4: Using topology to study disorder.

a Study of the frequency splitting, \({\Delta }_{\,{\mbox{Topo}}\,}^{N}\), between the hybridized SSH edge modes for CCAs with different N (J₂/J₁ = 1.22). The cross and the circles show \({\Delta }_{\,{\mbox{Topo}}\,}^{N}\) extracted for CCAs with N = 14, 16, 18, 22, 26, 30, 32 and 64, respectively from spectroscopy and time resolved measurements. The continuous black line represents the expected evolution of \({\Delta }_{\,{\mbox{Topo}}\,}^{N}\)vs N for the disorderless case according to the eigenmodes of the CCAs Hamiltonian (see Methods). The continuous red line represents the median of the most likely evolution (see panel d) of \({\Delta }_{\,{\mbox{Topo}}\,}^{N}\)vs N in the presence of disorder with standard deviation \({\sigma }_{L\to z}^{C}=10.3\,{{{\rm{MHz}}}}\) applied to ω_r of the cavities in the CCAs. The shaded region represents the 1σ uncertainty on the estimation of \({\Delta }_{\,{\mbox{Topo}}\,}^{N}\). The inset shows the simulated logarithm of the Probability Density Function (PDF) of \({\Delta }_{\,{\mbox{Topo}}\,}^{64}\) as a function of σ_L→z. The red line shows the median for each disorder realization. The red dotted lines show the 1σ standard deviation. b Pulse sequence used for the time-resolved measurement. A gaussian pulse is sent at a frequency between the hybridized SSH edge modes from one of the edge of the CCA, while the signal is acquired on both side of the CCA. The orange (red) dot on the y-frequency axis on the left side, highlights the frequency of the antisymmetric (symmetric) hybridized SSH edge modes. c Time traces of the transmitted ∣S₂₁∣ and reflected ∣S₁₁∣ signals of the CCAs with N = 26 and 22 in panel (a). Each data points is averaged 20000 times. The continuous lines highlight the fit done with an exponentially decaying cosine (see Supplementary file). d Disorder likelihood extracted for different batches of devices. The dots highlight the maximum likelihood extracted for each batch. The error bars show the respective FWHM. The black line shows the combined likelihood extracted among the different batches. The shadowed purple area shows its FWHM. The color code is according to Fig. 3d,f. Batch A (C), contains the CCAs with N = 16, 32 and 64 (14, 18, 22, 26, 30) of panel (a).