Analytical exploration of the optomechanical attractor diagram and of limit cycles

TL;DR

This paper provides an analytical framework for understanding the optomechanical attractor diagram and limit cycles, extending analysis beyond common approximations and exploring quantum regimes with a unified approach.

Contribution

It introduces an analytical method to evaluate the attractor diagram and investigates limit cycles beyond the resolved sideband approximation using a Fokker-Planck formalism.

Findings

Analytical expressions for the attractor diagram in elementary functions.

Characterization of limit cycles beyond the resolved sideband regime.

Quantum description of limit cycles using Fokker-Planck formalism.

Abstract

We analyse the interplay between mechanical and radiation pressure in an optomechanical cavity system. Our study is based on an analytical evaluation of the infinite Bessel summations involved, which previously had led to a numerical exploration of the so-called attractor diagram. The analytical expressions are then suitable for further asymptotic analysis in opposing regimes of the amplitude, which allows for a characterisation of the diagram in terms of elementary functions. Building on this framework, we investigate the emergence and properties of optomechanical limit cycles beyond the constraints of the resolved sideband approximation. By employing a Fokker-Planck formalism originally developed in the context of laser theory and then used in cavity optomechanics, we describe the quantum regime of these limit cycles, offering a more detailed and unified analytical perspective.