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Due to their conceptual appeal and computational convenience, two-level systems (TLS) and their generalisations are often used to investigate nonlinear behavior in quantum optics, and to assess the applicability of theoretical methods. Here the focus is on second harmonic generation (SHG) and, as system of interest, on the Dicke model, which consists of several TLSs inside an optical cavity. The main aspect addressed is the scope of non-equilibrium Green's function (NEGF) to describe the effect of disorder and electron-electron (e-e) interactions on the SHG signal. For benchmarking purposes, exact diagonalization (ED) results are also presented and discussed. SHG spectra obtained with NEGF and ED are found to be in very good mutual agreement in most situations. Furthermore, inhomogeneity in the TLS and e-e interactions reduce the strength of SHG, and the reduction is stronger with inhomogeneity than with interactions. This trend is consistently noted across different (small to large) system sizes. Finally, a modified NEGF approach is proposed to account for cavity leakage, where the quantum photon fields are coupled to a bath of classical oscillators. As to be expected, within this mixed quantum-classical scheme a decrease in the intensity of the fluorescent spectra takes place depending on the entity of cavity leakage. |
