Magnus Expansion Approach to Parametric Oscillator Systems in a Thermal Bath

Abstract

Abstract We develop a Magnus formalism for periodically driven systems which provides an expansion both in the driving term and in the inverse driving frequency, applicable to isolated and dissipative systems. We derive explicit formulas for a driving term with a cosine dependence on time, up to fourth order. We apply these to the steady state of a classical parametric oscillator coupled to a thermal bath, which we solve numerically for comparison. Beyond dynamical stabilisation at second order, we find that the higher orders further renormalise the oscillator frequency, and additionally create a weakly renormalised effective temperature. The renormalised oscillator frequency is quantitatively accurate almost up to the parametric instability, as we confirm numerically. Additionally, a cut-off dependent term is generated, which indicates the break down of the hierarchy of time scales of the system, as a precursor to the instability. Finally, we apply this formalism to a parametrically driven chain, as an example for the control of the dispersion of a many-body system.