Quantum Langevin equations for optomechanical systems

TL;DR

This paper develops a comprehensive quantum Langevin framework for optomechanical systems, enabling precise analysis of mechanical oscillators interacting with thermal environments and cavity fields at any temperature.

Contribution

It introduces a fully quantum stochastic calculus-based Langevin formulation for mechanical oscillators with thermal noise, including exact input-output relations and spectral analysis.

Findings

Exact stationary energy calculation for the oscillator

Derivation of homodyne and heterodyne spectra

Framework applicable at arbitrarily low temperatures

Abstract

We provide a fully quantum description of a mechanical oscillator in the presence of thermal environmental noise by means of a quantum Langevin formulation based on quantum stochastic calculus. The system dynamics is determined by symmetry requirements and equipartition at equilibrium, while the environment is described by quantum Bose fields in a suitable non-Fock representation which allows for the introduction of temperature. A generic spectral density of the environment can be described by introducing its state trough a suitable P-representation. Including interaction of the mechanical oscillator with a cavity mode via radiation pressure we obtain a description of a simple optomechanical system in which, besides the Langevin equations for the system, one has the exact input-output relations for the quantum noises. The whole theory is valid at arbitrarily low temperature. This allows the exact calculation of the stationary value of the mean energy of the mechanical oscillator, as well as both homodyne and heterodyne spectra. The present analysis allows in particular to study possible cooling scenarios and to obtain the exact connection between observed spectra and fluctuation spectra of the position of the mechanical oscillator.