Modeling the non-Markovian Brownian motion of an optomechanical resonator

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.

Figures (3)

• Figure 1: Schematic of an optomechanical setup where a control laser enters the cavity from the left mirror and the intracavity mode $a$ couples with the micromechanical motion of the right mirror, represented by the operator $Q$ denoting the displacement of the resonator's center-of-mass position from its mean value. Assuming negligible intrinsic losses, the input and output fields are indicated with the arrows.
• Figure 2: Normalized bath spectral function $J_k(\omega)/(A_k \Omega_R^3)$ as a function of the dimensionless frequency $\omega/\Omega_R$. The solid-red curve corresponds to the experimental central estimate $k = -2.30$, while the dashed-black curve denotes $k = -1.75$. Both the profiles exhibit a super-Ohmic infrared scaling ($\propto \omega^3$) that guarantees infrared stability of the mass and stiffness renormalizations ($\delta M$ and $\delta K$). The non-monotonic profile reflects the presence of a structured environment, characterized by a redistribution of the spectral weight around a characteristic frequency scale below the mechanical resonance.
• Figure 3: Normalized dissipation kernel $\mu_k(t)/(2A_k\Omega_R^3/\sqrt{\pi})$ as a function of the dimensionless time $\Omega_R t$. The curves represent $k = -2.30$ (solid-red) and $k = -1.75$ (dashed-black). The asymptotic decay is characterized by a power-law-modulated exponential envelope $\sim t^{2-k} e^{-\Omega_R t}$. The transient sign change of the kernel reflects delayed, history-dependent friction that is characteristic of a structured non-Markovian bath.