Modeling the non-Markovian Brownian motion of an optomechanical resonator

TL;DR

This paper develops a globally consistent model for non-Markovian Brownian motion in optomechanical resonators, capturing experimental spectral features and enabling detailed environment characterization.

Contribution

It introduces a phenomenological bath spectral density that reproduces local spectral behavior while remaining well-defined globally, ensuring physical consistency.

Findings

The model reproduces observed non-Ohmic spectral features.

It encodes the environment's memory effects through a structured susceptibility.

Optical readout can reconstruct the full mechanical susceptibility.

Abstract

We propose a globally-admissible phenomenological spectral density of the bath for the non-Markovian Brownian motion of an optomechanical resonator, motivated by the near-resonance experimental observation of a non-Ohmic spectrum in [Nat. Commun. 6, 7606 (2015)]. To avoid divergences arising from a naive global extrapolation, we construct this phenomenological bath spectral density that reproduces the observed local-power-law behavior near the mechanical resonance while remaining well defined globally, ensuring the finiteness of the bath-induced renormalizations and quadrature fluctuations of the resonator. The corresponding model of the structured environment produces a nonlocal mechanical susceptibility whose analytic pole structure encodes the observed linewidth. The resulting dissipation kernel exhibits a power-law-modulated exponential decay with transient negativity, signaling strong memory effects. In the weak-coupling regime, the optical readout based on homodyne detection enables near-resonance spectroscopy and, with a calibrated drive on the resonator, permits, in principle, the reconstruction of the full mechanical susceptibility, thereby providing access to both the dissipative and dispersive bath contributions. Our results provide a consistent route from locally-inferred spectral properties to globally-admissible open-system descriptions and establish a framework for probing structured environments in cavity optomechanics.