  We consider cavity-enhanced single-photon generation from stimulated
two-photon Raman processes in three-level systems. We compare four fundamental
system configurations, one $\Lambda$-, one V- and two ladder ($\Xi$-)
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 on-demand potential of Raman-transition-based
single-photon sources. The four system arrangements can be divided into two
subclasses, $\Lambda$-type and V-type, which exhibit strongly different
Raman-emission 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.
