The dynamics of a harmonic oscillator with time-dependent force constant and perturbed by weak quartic anharmonicity

Abstract

A harmonic oscillator with time-dependent force constant when perturbed by a weak static quartic anharmonicity λx⁴, λ small, is shown to generate new features in the dynamics. For weak coupling, both time-dependent perturbation theoretical analysis and near-exact numerical calculations predict that the excitation probability to an accessible excited state is a parabolic function of λ for a short time when all other parameters of the system are fixed. This is approximately independent of the form of the time-dependence of the harmonic force constant. Bound state populations are shown to oscillate in time when anharmonic coupling is present, due to mutual interference of the relaxing force constant and anharmonicity.