The unmatched performance of frequency-based sensing makes it the core of accurate scientific and cost-effective commercial measurement systems, spanning the length scales from kilometer-long LIGO 1 to mesoscopic micro- and nano-electro-mechanical systems (M/NEMs) 2, 3, 4, 5, 6, 7, 8, 9 and further to the single-atom tip of a frequency-modulation atomic force microscope (AFM) 10.ĭespite the wide applications of frequency-based sensing for scientific high-precision measurement, a general and fundamental understanding of the linear oscillator resonance frequency estimation and its uncertainty limits is currently lacking. Parametrically coupling time-varying unknown quantities to resonance frequencies of harmonic oscillators enables measurements that are insensitive to low-frequency noise sources and drifts in the detection gain and bias. Beyond nanomechanics, these results advance frequency-based metrology across physical domains. 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). 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. 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. Analyses assuming specific ways of estimating frequency can underestimate the available precision and overlook unconventional measurement regimes.
All physical oscillators are subject to thermodynamic and quantum perturbations, fundamentally limiting measurement of their resonance frequency.