  We develop a fully quantum theoretical approach which describes the dynamics
of Frenkel excitons and bi-excitons induced by few photon quantum light in a
quantum well or wire (atomic chain) of finite size. The eigenenergies and
eigenfunctions of the coupled exciton-photon states in a multiatomic system are
found and the role of spatial confinement as well as the energy quantization
effects in 1D and 2D cases is analyzed. Due to the spatial quantization, the
excitation process is found to consist in the Rabi-like oscillations between
the collective symmetric states characterized by discrete energy levels and
arising in the picture of the ladder bosonic operators. At the same time, the
enhanced excitation of additional states with energy close to the upper
polariton branch is revealed. The found new effect is referred to as the
formation of Rabi-shifted resonances and is analyzed in details. Such states
are shown to influence dramatically on the dynamics of excitation especially in
the limit of large times.
