Analysis of the parametrically periodically driven classical and quantum linear oscillator.

We study theoretically and computationally the behavior of the classical and quantum parametrically periodically driven linear oscillator. As a basic paradigm of such a Floquet system we consider the case of the harmonic oscillation of the oscillator frequency, which is convenient to handle theoretically and computationally, while keeping the general features. We derive an explicit analytic formula for the quantum propagator in terms of the classical propagator. Using this, we derive the explicit exact formula for the evolution of the expectation value of the energy starting from an arbitrary normalizable initial state. In the case of the starting pure stationary eigenstate the evolution is exactly the same as for the classical microcanonical ensemble of initial conditions of the same starting energy. We perform a rather complete computational analysis of the system's behavior inside the instability regions (lacunae), where the energy of the oscillator increases exponentially, as well as in the stability regions, and in particular in the vicinity of the (in)stability borders. We confirm also numerically with absolute certainty that the borders of (in)stability regions classically and quantally coincide exactly, in accordance with the theory, which is an important check of the numerical accuracy of computations, and we find a number of important empirical results, especially an equation of the elliptic type describing the rate of exponential energy growth inside the lacunae in terms of other systems' quantities. We believe that our approach and findings are of generic linear type, i.e., applicable in most such linear Floquet systems, and we present a strong motivation for a general theory, classically and quantally.