Thermodynamic Limit for Linear Harmonic Oscillator Resonance Frequency Measurement

TL;DR

This paper derives the fundamental thermodynamic limit for frequency measurement uncertainty in linear harmonic oscillators, providing a theoretical bound and an efficient estimator validated through numerical simulations.

Contribution

It introduces a general derivation of the Cramer-Rao lower bound and an optimal estimator for oscillator frequency under thermodynamic fluctuations, applicable to both driven and undriven resonators.

Findings

Derived the CRLB for frequency measurement uncertainty.

Validated the CRLB with numerical simulations showing agreement.

Provided an efficient maximum-likelihood estimator for practical use.

Abstract

Thermodynamic fluctuations in mechanical resonators cause uncertainty in their frequency measurement, fundamentally limiting performance of frequency-based sensors. Recently, integrating nanophotonic motion readout with micro- and nano-mechanical resonators allowed practical chip-scale sensors to routinely operate near this limit in high-bandwidth measurements. However, the exact and general expressions for either thermodynamic frequency measurement uncertainty or efficient, real-time frequency estimators are not well established, particularly for fast and weakly-driven resonators. Here, we derive, and numerically validate, the Cramer-Rao lower bound (CRLB) and an efficient maximum-likelihood estimator for the frequency of a classical linear harmonic oscillator subject to thermodynamic fluctuations. For a fluctuating oscillator without external drive, the frequency Allan deviation calculated from simulated resonator motion data agrees with the derived CRLB $\sigma_f = {1 \over 2\pi}\sqrt{\Gamma \over 2\tau}$ for averaging times $\tau$ below, as well as above, the relaxation time $1\over\Gamma$. The CRLB approach is general and can be extended to driven resonators, non-negligible motion detection imprecision, as well as backaction from a continuous linear quantum measurement.