Thermodynamic Limit for Linear Harmonic Oscillator Resonance Frequency Measurement

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 ${\sigma }_f=\frac{1}{2\pi }\sqrt{\frac{\Gamma }{2\tau }}$ for averaging times $\tau$ below, as well as above, the relaxation time $\frac{1}{\Gamma }$. 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.

11 pages, 2 figures

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• Sideband Cooling Micromechanical Motion to the Quantum Ground State
• A Mechanical Mass Sensor with Yoctogram Resolution
• Fundamental Aspects of Quantum Brownian Motion
• Neutral particle Mass Spectrometry with Nanomechanical Systems
• Measurement of non-monotonic Casimir forces between silicon nanostructures

Cited by in corpus (1)

• Resolution Limits of Resonant Sensors with Duffing Non-Linearity