Fundamental limits and optimal estimation of the resonance frequency of a linear harmonic oscillator

• Published in: Communications physics Vol. 4; no. 1; pp. 1 - 11
• Main Authors: Wang, Mingkang, Zhang, Rui, Ilic, Robert, Liu, Yuxiang, Aksyuk, Vladimir A.
• Format: Journal Article
• Language: English
• Published: London Nature Publishing Group UK 15.09.2021; Nature Publishing Group; Nature Portfolio
• Subjects: 639/766/530/2804; 639/925/927; Bandwidths; Fisher information; Frequency analysis; Frequency measurement; Harmonic oscillators; Information theory; Lower bounds; Maximum likelihood estimators; Noise; Oscillators; Perturbation; Physics; Physics and Astronomy; Resonance; Resonant frequencies; Silicon nitride; Uncertainty
• ISSN: 2399-3650; 2399-3650
• DOI: 10.1038/s42005-021-00700-6

All physical oscillators are subject to thermodynamic and quantum perturbations, fundamentally limiting measurement of their resonance frequency. Analyses assuming specific ways of estimating frequency can underestimate the available precision and overlook unconventional measurement regimes. Here we derive a general, estimation-method-independent Cramer Rao lower bound for a linear harmonic oscillator resonance frequency measurement uncertainty, seamlessly accounting for the quantum, thermodynamic and instrumental limitations, including Fisher information from quantum backaction- and thermodynamically driven fluctuations. We provide a universal and practical maximum-likelihood frequency estimator reaching the predicted limits in all regimes, and experimentally validate it on a thermodynamically limited nanomechanical oscillator. Low relative frequency uncertainty is obtained for both very high bandwidth measurements (≈10 −5 for τ = 30 μs) and measurements using thermal fluctuations alone (<10 −6 ). Beyond nanomechanics, these results advance frequency-based metrology across physical domains. Thermodynamic and quantum fluctuations limit the accuracy with which conventional methods can measure observables, often depending on the method chosen. Here, information theory is employed to determine the minimum uncertainty in the resonant frequency of a harmonic oscillator in a method-independent way.