Classical, semiclassical, and quantum-optical models for an x-ray planar cavity with electronic resonance

Two theoretical models of semiclassical matrix method and quantum Green's function are extended to the system of x-ray thin-film planar cavity with inner-shell electronic resonances. The semiclassical model is based on the matrix formalism by treating each layer as the propagating matrix. The crucial idea is to expand the propagating matrix of the resonant atomic layer under ultrathin-film approximation, and then derive the analytical expression of the spectral observation, i.e., the cavity reflectance. Typical cavity-enhanced decay rate and cavity-induced energy shift as well as the Fano interference which were observed in recent experiments could be phenomenologically interpreted. The quantum model employs analytical Green's function to calculate the cavity system. The system Hamiltonian and the effective energy-level are derived. The effective energy-level scheme indicates that the cavity effect can control intermediate core-hole state. To test the validity of the semiclassical matrix and quantum Green's function models, the classical Parratt's formalism and the dispersion correction of the atomic refractive index are also recalled. Very good agreements in the reflectivity spectra between the semiclassical and quantum models with Parratt's results are observed. The equivalence between the matrix and Green's function models is discussed analytically and numerically. Based on these two theoretical models, the similarities and particularities of inner-shell electronic system are discussed and compared with the nuclear system. Several examples, including the weaker dipole moment, distorted Fano lineshape, and negligible collective effect, are given. The present semiclassical matrix and quantum Green's function models will be useful to predict new phenomena, to optimize cavity structure for future experiments, and to promote the emerging of x-ray quantum optical effects with modern x-ray spectroscopy techniques.