Ugulava A., Mchedlishvili G., Chkhaidzec S., Chotorlishvili L.

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(I0)  ω, that is, ?sticking? in the resonance frequency. For a particular physical problem there may be a case when I  ?h is the classical quantity, whereas its correction I  ?h is the quantum quantity. Naturally, dynamics of I is described by the quantum equation of motion. In particular, in the moderate nonlinearity approximation ε  (dω/dI)(I/ω)  1/ε, where ε 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 I 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.

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