Light-matter interactions in optical cavities beyond the classical Maxwell description

In common methods for the ab-initio description of photoinduced processes typically the classical Maxwell's equations are employed to describe the propagation of light. The applicability of these equations has been demonstrated since decades. However, considering the ultimate limit of single molecules interacting with a few photons, the classical description of the electromagnetic field does not suffice anymore. In this case the quantum nature of the electromagnetic field has to be taken into account and therefore existing ab-initio approaches have to be extended.
In the present work we face the question: Whether and to what extent the analysis and simulation of photoinduced processes changes by going beyond the classical Maxwell description. Here we generalize the idea of the trajectory methods, traditionally introduced for electron-nuclear problems, to electron-photon correlated systems. We focus on the well-known semiclassical methods mean-field and path integral approach and also introduce the BBGKY-method for the fermion-boson problem as fully quantum mechanical method. In extension we also introduce the idea of exact factorization to the electron-photon problem and apply our novel approaches to spontaneous and stimulated emission for atoms and molecules in optical cavities.