Inverse determination of light-matter coupling in disordered systems from transmittance spectra

We investigate quantum inverse problems in one-dimensional (1D) electronic disordered systems strongly coupled to optical cavities. More specifically, we consider the Anderson and the Aubry-Andre-Harper models connected to electronic reservoirs and embedded in a single-mode optical cavity. The light-matter interaction enables photon-assisted hopping processes that significantly modify the transmittance spectrum. Within the nonequilibrium Green's function formalism, we implement an inversion-based approach capable of accurately extracting the electron-photon coupling strength directly from transmittance spectra. While cavity coupling acts as a minor perturbation within the Anderson model, yielding broad yet precise parameter estimates, its influence is markedly different in the Aubry-Andr\'e-Harper model. The latter exhibits a sharp metal-insulator transition in 1D, thus resulting in more pronounced cavity-induced spectral changes. This renders even more accurate inverse solutions, offering unparalleled precision in the characterization of low-dimensional disordered systems. Altogether, our results demonstrate that the quantum inverse problem provides a robust diagnostic tool for quantum materials, particularly effective for systems exhibiting metal-insulator transitions.