Quantum states of an oscillator with periodic time-dependent frequency under quasiresonant condition: Unperturbed evolution, perturbative effects, and anharmonic effects

Abstract

In a recent paper [L. Ferrari, Phys. Rev. A 57, 2347 (1998)], one of the authors has shown that the mean energy of a quantum oscillator with periodic time-dependent frequency diverges exponentially in time, under certain conditions. In the present paper, we study the explicit form of the evolving state, and compare the results obtained with the general expressions developed by other authors for an arbitrary time-dependent frequency. Then we approach the problem of the anharmonic effects. A first-order calculation is performed in the case of a short-range perturbation. The transition rates between different states, evolving with the unperturbed Hamiltonian, are shown to vanish at long times when the unperturbed oscillator's energy diverges exponentially. A nonperturbative approach must be adopted, in the presence of anharmonic potentials of the form $V(q)\ensuremath{\propto}{q}^{j}, j>2$. In the case of weak anharmonicity $(j\ensuremath{-}2\ensuremath{\ll}1)$, a mean-field procedure can be used to show that the mean energy does actually saturate at long times, with the possible exception of periodic peaks, having nonsaturating height, that we call ``special quantum effects.''