On the asymptotic solutions of the coupled quasiparticle-oscillator system

A coupled quasiparticle-oscillator system is considered for an arbitrary number of excitons. The exciton dynamics is described in terms of the second quantization (i.e. by means the bosonic operators). As a consequence a radius of a Bloch sphere is obtained different to the previous results. Some integrals of motion are obtained that allowed to reduce the system of equations of motion to a single nonlinear ordinary differential equation of the fourth order. This equation contains the energy of the system as a parameter. The fixed points are found as a functions of the energy of the system, and its stability properties are investigated. It is demonstrated that a bifurcation is presented for the energies H<−1/2p. An asymptotic quasiclassical solution around fixed point for the case H>−1/2p is obtained. The solutions around other stable fixed points can be obtained analogously. The expression for the evolution operator of the quasiparticle-oscillator system is obtained as a functional on the classical solutions.