On the properties of a class of higher-order Mathieu equations originating from a parametric quantum oscillator

Abstract

Evolution in time of an arbitrary initial state for a parametrically driven quantum oscillator is an interesting problem since there exist regions in parameter space (defined by the amplitude and frequency of the driving) where the moments of the probability distribution can diverge in time. While the first moment satisfies a Mathieu equation, the higher-order moments follow Mathieu like equations of order greater than two. It is not very often that a physical problem gives rise to higher-order Mathieu equations. Hence, we give a detailed study of the different stability zones associated with the parametric quantum oscillator, using perturbative techniques traditionally associated with the Mathieu equation. We verify our results by numerical analysis, thus demonstrating that for the higher-order Mathieu equations, the traditional perturbation theory methods give a consistent account of the stability zones.