  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.
