Approximate evolution for a hybrid system: An optomechanical Jaynes-Cummings model

TL;DR

This paper develops an approximate analytical method for the time evolution of a hybrid optomechanical Jaynes-Cummings system, combining algebraic techniques with numerical validation to enhance understanding of its dynamics.

Contribution

It introduces a novel approximate evolution operator for a combined optomechanical and Jaynes-Cummings system, enabling analytical insights into its dynamics.

Findings

Analytical evolution operator closely matches numerical results.

Method provides a linearized generalized interaction picture Hamiltonian.

Significant agreement between approximate and exact numerical calculations.

Abstract

In this work we start from a phenomenological Hamiltonian built from two known systems: the Hamiltonian of a pumped optomechanical system and the Jaynes Cummings Hamiltonian. Using algebraic techniques we construct an approximate time evolution operator $\hat U_{opt}$ for the forced optomechanical system (as a product of exponentials) and take the JC Hamiltonian as an interaction. We transform the later with $\hat U_{opt}$ to obtain a generalized interaction picture Hamiltonian which can be linearized and whose time evolution operator is written in a product form. The analytic results are compared with purely numerical calculations using the full Hamiltonian and the agreement between them is remarkable.