Role of Dissipation on the Stability of a Parametrically Driven Quantum Harmonic Oscillator

TL;DR

This paper investigates how dissipation influences the stability of a quantum harmonic oscillator under parametric driving, revealing that dissipation can suppress dynamical instabilities and linking quantum stability to classical Mathieu equation behavior.

Contribution

It provides a novel analysis connecting quantum dissipative dynamics with classical Mathieu equation stability, using numerical and analytical methods.

Findings

Dissipation reduces parametric instability in the quantum oscillator.

A relationship between Wigner function localization and Mathieu equation stability is established.

Quantum stability can be controlled via dissipation effects.

Abstract

We study the dissipative dynamics of a single quantum harmonic oscillator subjected to a parametric driving with in an effective Hamiltonian approach. Using Liouville von Neumann approach, we show that the time evolution of a parametrically driven dissipative quantum oscillator has a strong connection with the classical damped Mathieu equation. Based on the numerical analysis of the Monodromy matrix, we demonstrate that the dynamical instability generated by the parametric driving are reduced by the effect of dissipation. Further, we obtain a closed relationship between the localization of the Wigner function and the stability of the damped Mathieu equation.