Influence functional for two mirrors interacting via radiation pressure

TL;DR

This paper develops a perturbative influence functional approach to derive the effective dynamics of two mirrors in an optical cavity interacting via radiation pressure, revealing mediated interactions, stochastic noises, and quantum Langevin equations.

Contribution

It introduces a second-order effective action for the mirrors by tracing out the cavity field, highlighting interaction channels and fluctuation-dissipation relations.

Findings

Interaction mediated by pairs of field modes.

Quantum and thermal fluctuations induce colored Gaussian noise.

Derivation of quantum Langevin equations for mirror dynamics.

Abstract

We study the effective dynamics of two mirrors, forming an optical cavity, and interacting with the cavity field via radiation pressure. We pursue a perturbative influence functional approach to trace out the degrees-of-freedom of the field, and obtain the second order effective action for the system composed by the mirrors. We find that the interaction between the mirrors is mediated by pairs of field modes, which combine in such a way to give rise to two different interaction channels. We find that the quantum and thermal fluctuations of the cavity field result in coloured, Gaussian stochastic noises acting on the mirrors. To each of these noises is associated a dissipative effect, and the corresponding power spectra and susceptibilities are related via generalized fluctuation-dissipation relations. We finally demonstrate that the dynamics of the mirrors admits a stochastic interpretation, and give the relative quantum Langevin equations.