We consider cavity-enhanced single-photon generation from stimulated two-photon Raman processes in three-level systems. We compare four fundamental system configurations, one Λ-, one Vand two ladder (Ξ-) configurations. These can be realized as subsystems of a single quantum dot or of quantum-dot molecules. For a new microscopic understanding of the Raman process, we analyze the Heisenberg equation of motion applying the cluster-expansion scheme. Within this formalism an exact and rigorous definition of a cavity-enhanced Raman photon via its corresponding Raman correlation is possible. This definition for example enables us to systematically investigate the ondemand potential of Raman-transition-based single-photon sources. The four system arrangements can be divided into two subclasses, Λ-type and V-type, which exhibit strongly different Ramanemission characteristics and Raman-emission probabilities. Moreover, our approach reveals whether the Raman path generates a single photon or just induces destructive quantum interference with other excitation paths. Based on our findings and as a first application, we gain a more detailed understanding of experimental data from the literature. Our analysis and results are also transferable to the case of atomic three-level-resonator systems, and can be extended to more complicated multi-level schemes.This approach leads to new microscopic insight into the Raman process which, in general, cannot be extracted directly from the observation of optical spectra. As an example, Fig. 2 shows the time development of the cavity-arXiv:1711.11490v2 [cond-mat.mes-hall]

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