  We analytically derive the upper bound on the overall efficiency of
single-photon generation based on cavity quantum electrodynamics (QED), where
cavity internal loss is treated explicitly. The internal loss leads to a
tradeoff relation between the internal generation efficiency and the escape
efficiency, which results in a fundamental limit on the overall efficiency. The
corresponding lower bound on the failure probability is expressed only with an
"internal cooperativity," introduced here as the cooperativity parameter with
respect to the cavity internal loss rate. The lower bound is obtained by
optimizing the cavity external loss rate, which can be experimentally
controlled by designing or tuning the transmissivity of the output coupler. The
model used here is general enough to treat various cavity-QED effects, such as
the Purcell effect, on-resonant or off-resonant cavity-enhanced Raman
scattering, and vacuum-stimulated Raman adiabatic passage. A repumping process,
where the atom is reused after its decay to the initial ground state, is also
discussed.
