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 param- eter space (defined by the amplitude and frequency of the driving) where the moments of the probability dis- tribution 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 associ- ated with the parametric quantum oscillator, using per- turbative 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.