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



Mingkang Wang, Rui Zhang, Robert Ilic, Yuxiang Liu, Vladimir Aksyuk


All physical oscillators, from optical cavities to mechanical cantilevers, are subject to thermodynamic and quantum perturbations and detection uncertainty, fundamentally limiting how well their resonance frequency can be measured. Many previous studies of these limits lack generality, assuming specific frequency estimators and overlooking important possibilities for high-precision frequency measurement in unconventional regimes. Here we derive an estimator- independent Cramer Rao lower bound for evaluating linear oscillators' resonance frequencies from their continuously measured position. The lower bound combines quantum and thermodynamic fundamental limits with those imposed by the experimental noise and inefficiency. It is valid for any measurement bandwidth, detection noise level, and driving strength, including frequency measurement from thermodynamically- and quantum-backaction-driven fluctuations alone, without external excitation. Additionally, we propose a universal and practical maximum-likelihood frequency estimator reaching the predicted uncertainty limits in all regimes, and experimentally validate it on a thermodynamically limited nanomechanical oscillator with an integrated cavity optomechanical readout. We demonstrate new regimes of high-bandwidth frequency monitoring and high-precision measurements using only thermodynamic fluctuations. Not limited to nanomechanics, these results advance frequency-based metrology across physical domains.
Communications Physics


frequency estimator, Cramer Rao bound, thermodynamic limit, quantum backaction, harmonic oscillator, standard quantum limit


Wang, M. , Zhang, R. , Ilic, R. , Liu, Y. and Aksyuk, V. (2021), Fundamental limits and optimal estimation of the resonance frequency of a linear harmonic oscillator, Communications Physics, [online],, (Accessed December 11, 2023)
Created September 15, 2021, Updated October 14, 2021