Modeling quantum optical phenomena using transition currents

Spontaneous light emission is central to a vast range of physical systems and is a founding pillar for the theory of light-matter interactions. In the presence of complex photonic media, the description of spontaneous light emission usually requires advanced theoretical quantum optics tools such as macroscopic quantum electrodynamics, involving quantized electromagnetic fields. Although rigorous and comprehensive, the complexity of such models can obscure the intuitive understanding of many quantum-optical phenomena. Here, we review a method for calculating spontaneous emission and other quantum-optical processes without making explicit use of quantized electromagnetic fields. Instead, we introduce the concept of transition currents, comprising charges in matter that undergo transitions between initial and final quantum states. We show how predictions that usually demand advanced methods in quantum electrodynamics or quantum optics can be reproduced by feeding these transition currents as sources to the classical Maxwell equations. One then obtains the relevant quantum observables from the resulting classical field amplitudes, without washing out quantum optical effects. We show that this procedure allows for a straightforward description of quantum phenomena, even when going beyond the dipole approximation and single emitters. As illustrative examples, we calculate emission patterns and Purcell-enhanced emission rates in both bound-electron and free-electron systems. For the latter, we derive cathodoluminescence emission and energy-loss probabilities of free electrons interacting with nanostructured samples. In addition, we calculate quantum-beat phenomena in bound-electron systems and wave function-dependent optical coherence in free-electron systems. Remarkably, the transition-current formalism captures more complex phenomena, such as many-body interference effects and super-radiance of both bound- and free-electron systems, second-order processes such as two-photon emission, and quantum recoil corrections to free-electron radiation. We review a variety of light-matter interactions in fields ranging from electron microscopy to nanophotonics and quantum optics, for which the transition-current theoretical formalism facilitates practical simulations and a deeper understanding of novel applications.