Violation of classical inequalities by photon frequency filtering

The violation of the Cauchy-Schwarz and Bell inequalities ranks among the major evidence of the genuinely quantum nature of an emitter. The conventional theoretical approaches associate operators with spectral lines to study correlations between photons from real-state transitions. Instead, we use a formalism that studies directly correlations between the physical reality---the photons---with no prejudice as to their origin. This allows us to extend photon correlations to all frequencies in all the possible windows of detection and to reveal landscapes of two-photon correlations that delineate regions of quantum emission, i.e., where classical inequalities are violated. We show that quantum correlations are rooted in the joint emission of two photons involving virtual states of the emitter instead of, as previously assumed, cascaded transitions between real states. As a result, correlations can be optimized in a process akin to distillation by keeping only the emission which is quantum and filtering out that which is not.