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

Abstract

The nonlinear-oscillating system in action-angle variables is characterized by the dependence of frequency of oscillation $\ensuremath{\omega}(I)$ 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 $\ensuremath{\omega}({I}_{0})\ensuremath{\simeq}\ensuremath{\omega}$, that is, ``sticking'' in the resonance frequency. For a particular physical problem there may be a case when $I\ensuremath{\gg}\ensuremath{\hbar}$ is the classical quantity, whereas its correction $\ensuremath{\Delta}I\ensuremath{\simeq}\ensuremath{\hbar}$ is the quantum quantity. Naturally, dynamics of $\ensuremath{\Delta}I$ is described by the quantum equation of motion. In particular, in the moderate nonlinearity approximation $\ensuremath{\varepsilon}\ensuremath{\ll}(d\ensuremath{\omega}/dI)(I/\ensuremath{\omega})\ensuremath{\ll}1/\ensuremath{\varepsilon}$, where $\ensuremath{\varepsilon}$ is the small parameter, the description of quantum state is reduced to the solution of the Mathieu-Schr\"odinger equation. The state formed as a result of sticking in resonance is an eigenstate of the operator $\ensuremath{\Delta}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{I}$ that does not commute with the Hamiltonian $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}$. 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.