Magnus expansion approach to parametric oscillator systems in a thermal bath

TL;DR

This paper introduces a Magnus expansion method for periodically driven systems, including dissipative ones, and applies it to analyze a classical parametric oscillator in a thermal bath, revealing frequency renormalization and effective temperature changes.

Contribution

The paper develops a Magnus formalism for driven systems with explicit formulas for cosine driving, extending analysis to dissipative systems and many-body chains.

Findings

Higher-order terms further renormalize oscillator frequency.

Effective temperature is weakly renormalized by the drive.

Frequency accuracy remains near the parametric instability threshold.

Abstract

We develop a Magnus formalism for periodically driven systems which provides an expansion both in the driving term and 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 stabilization at second order, we find that the higher orders further renormalize the oscillator frequency, and additionally create a weakly renormalized effective temperature. The renormalized 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.