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.

所有物理振荡器都受到热力学和量子扰动的影响，从根本上限制了其共振频率的测量。假设估计频率的特定方法的分析可能会低估可用的精度并忽略非常规的测量方式。在这里，我们为线性谐振子共振频率测量不确定性推导出一般的、与估计方法无关的 Cramer Rao 下界，无缝地考虑了量子、热力学和仪器的限制，包括来自量子反作用和热力学驱动波动的 Fisher 信息。我们提供了一个通用且实用的最大似然频率估计器，可在所有状态下达到预测的极限，并在热力学受限的纳米机械振荡器上对其进行实验验证。^-5 τ = 30 μs）和单独使用热波动的测量（<10 ^-6）。除了纳米力学，这些结果推动了跨物理领域的基于频率的计量学。