Quantum corrections to the classical model of the atom-field system

The nonlinear-oscillating system in action-angle variables is characterized by the dependence of frequency of oscillation ω(I) on action I. Periodic perturbation is capable of realizing in the system a stable nonlinear resonance at which the action I adapts to the resonance condition ω(I(0))≃ω, that is, "sticking" in the resonance frequency. For a particular physical problem there may be a case when I≫ℏ is the classical quantity, whereas its correction ΔI≃ℏ is the quantum quantity. Naturally, dynamics of ΔI is described by the quantum equation of motion. In particular, in the moderate nonlinearity approximation ɛ≪(dω/dI)(I/ω)≪1/ɛ, where ɛ is the small parameter, the description of quantum state is reduced to the solution of the Mathieu-Schrödinger equation. The state formed as a result of sticking in resonance is an eigenstate of the operator ΔI that does not commute with the Hamiltonian H. Expanding the eigenstate wave functions in Hamiltonian eigenfunctions, one can obtain a probability distribution of energy level population. Thus, an inverse level population for times lower than the relaxation time can be obtained.