  It is shown that a broad class of cavity quantum electrodynamics (QED)
problems - which consider the resonant propagation of a single photon
interacting with quantum emitters (QEs), such as atoms, quantum dots, or
vacancy centers - can be solved directly without application of the second
quantization formalism. In the developed approach, the Hamiltonian is expressed
through the ket-bra products of collective (photon + cavities + QEs) states.
Consequently, the S-matrix of input-output problems is determined exactly by
the Mahaux-Weidenm\"uller formula, which dramatically simplifies the analysis
of complex cavity QED systems. First, this approach is illustrated for the
problem of propagation of a photon resonantly interacting with N two-level QEs
arbitrary distributed inside the optical cavity. Solution of this problem
manifests the effect of cumulative action of QEs previously known for special
cases. Can a similar cumulative action of QEs enhance the inelastic resonant
transmission of a single photon? We solve this problem for the case of an
optical cavity having two modes resonantly coupled to electronic transitions of
N three-level QEs. It is shown that the described structure is the simplest
realistic structure which enables the down-conversion of the single photon
frequency with the amplitude approaching unity in the absence of the external
driving field and for sufficiently small cavity losses and QE dissipation.
Overall, the simplicity and generality of the developed approach suggest a
practical way to identify and describe new phenomena in cavity QED.
