Fig. 3: Zak phase reconstruction from the absorption spectra.

The experimental data of the absorption spectra for a the Semenoff insulator superradiance lattices with \({{\Omega }}_{1,2} = 120\) MHz and \({{\Delta }}_c = 174\), \(232\), and \(298\) MHz from top to bottom, and b the SSH superradiance lattices with \({{\Delta }}_c = 0\), \({{\Omega }}_1 = 118\) MHz, and \({{\Omega }}_1/{{\Omega }}_2 = 6.52\), \(4.25\), and \(2.15\) from top to bottom. We use the marked extrema \({\it{\epsilon }}_ \pm\) (\({{\Delta }}_{n,m}\)) in absorption peaks and dips to locate the band centers (anti-crossings). The measured \(R_{n,m}\) (points) compared with the normalized ratio of WCs (dashed lines) for c the Semenoff insulator and d the SSH superradiance lattices, from which we reconstruct the Zak phases \(\theta _ \pm\) in e and f. Error bars are obtained from four independent data sets (see WSLs and more absorption spectra in Supplementary Note 7)