# Manifestation of classical nonlinear dynamics in optomechanical entanglement with a parametric amplifier

## TL;DR

This paper explores how an optical parametric amplifier in a cavity optomechanical system reveals classical nonlinear dynamics and quantum entanglement transitions, highlighting regimes with classical-quantum correspondence and unique quantum features.

## Contribution

It demonstrates the manifestation of classical nonlinear dynamics in optomechanical entanglement and identifies regimes with and without classical counterparts for quantum features.

## Key findings

- Classical dynamics transition from periodic to period-doubling at critical parametric gain.
- Optomechanical entanglement exhibits a second-order transition at the critical gain.
- Normal mode splitting in entanglement arises from mode squeezing without classical analogy.

## Abstract

Cavity optomechanical system involving an optical parametric amplifier (OPA) can exhibit rich classical and quantum dynamical behaviors. By simply modulating the frequency of the laser pumping the OPA, we find two interesting parameter regimes, with one of them enabling to study quantum-classical correspondence in system dynamics, while there exist no classical counterparts of the quantum features for the other. For the former regime, as the parametric gain of OPA increases to a critical value, the classical dynamics of the optical or mechanical modes can experience a transition from the regular periodic oscillation to period-doubling motion, in which cases the light-mechanical entanglement can be well studied by the logarithm negativity and can manifest the dynamical transition in the classical nonlinear dynamics. Moreover, the optomechanical entanglement shows a second-order transition characteristic at the critical parametric gain. For the latter regime, the kind of normal mode splitting comes up in the laser detuning dependence of optomechanical entanglement, which is induced by the squeezing of the optical and mechanical hybrid modes and finds no classical correspondence. The OPA assisted optomechanical systems therefore offer a simple way to study and exploit quantum manifestations of classical nonlinear dynamics.