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It is shown that a broad class of cavity quantum electrodynamics (QED)problems - which consider the resonant propagation of a single photoninteracting with quantum emitters (QEs), such as atoms, quantum dots, orvacancy centers - can be solved directly without application of the secondquantization formalism. In the developed approach, the Hamiltonian is expressedthrough the ket-bra products of collective (photon + cavities + QEs) states.Consequently, the S-matrix of input-output problems is determined exactly bythe Mahaux-Weidenm\"uller formula, which dramatically simplifies the analysisof complex cavity QED systems. First, this approach is illustrated for theproblem of propagation of a photon resonantly interacting with N two-level QEsarbitrary distributed inside the optical cavity. Solution of this problemmanifests the effect of cumulative action of QEs previously known for specialcases. Can a similar cumulative action of QEs enhance the inelastic resonanttransmission of a single photon? We solve this problem for the case of anoptical cavity having two modes resonantly coupled to electronic transitions ofN three-level QEs. It is shown that the described structure is the simplestrealistic structure which enables the down-conversion of the single photonfrequency with the amplitude approaching unity in the absence of the externaldriving field and for sufficiently small cavity losses and QE dissipation.Overall, the simplicity and generality of the developed approach suggest apractical way to identify and describe new phenomena in cavity QED.

Mahaux-Weidenm\"uller approach to cavity quantum electrodynamics and complete resonant down-conversion of the single photon frequency