We analyze a photon transport through an 1D open waveguide side coupled to the $N$-photon microwave cavity with embedded artificial two- level atom (qubit). The qubit state is probed by a weak signal at the fundamental frequency of the waveguide. Within the formalism of projection operators and non-Hermitian Hamiltonian approach we develop a one-photon approximation scheme to obtain the photon wavefunction which allows for the calculation of the probability amplitudes of the spontaneous transitions between the levels of two Rabi doublets in $N$- photon cavity. We obtain analytic expressions for the transmission and reflection factors of the microwave signal through a waveguide, which contain the information of the qubit parameters. We show that for small number of cavity photons the Mollow spectrum consists of four spectral lines which is a direct manifestation of quantum nature of light. The results obtained in the paper are of general nature and can be applied to any type of qubits. The specific properties of the qubit are only encoded in the two parameters: the energy $Omega$ of the qubit and its coupling $lambda$ to the cavity photons.
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