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Abstract
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.
Highlights
All physical oscillators are subject to thermodynamic and quantum perturbations, fundamentally limiting measurement of their resonance frequency
We derive the Cramer Rao lower bound (CRLB) to obtain general uncertainty limits, including the fundamental quantum and thermodynamic limits, as well as the instrumental limits, for resonance frequency extracted from continuous position measurement of a linear harmonic oscillator (LHO), subject to dissipation, thermodynamic- and quantum-backaction-induced stochastic fluctuations, instrumental detection uncertainty, and external harmonic excitation
In addition to recovering the uncertainty minimum of the standard quantum limit expected for such measurement under strong coherent external excitation, we present the fundamental limits of extracting the frequency information from fluctuations driven by the quantum measurement itself solely, or in combination with thermal and external driving forces
Summary
Oscillator motion in a rotating frame and the experimental system. As shown in Fig. 1(a), we consider a LHO subject to dissipation Γ, white fluctuating force f, which includes a Langevin force coming from a thermal bath and a quantum measurement backaction force. FðtÞ þ m f ð1Þ where x is the position of the LHO, m is the effective mass, and ω0 is its resonance frequency. The fluctuating force is assumed to be frequency independent, at least over the resonator bandwidth, and effectively obeying hf ðtÞf ðt0Þi 1⁄4 fr2msδðt À t0Þ with a constant fr2ms. For thermodynamic fluctuations fr2ms 1⁄4 2ΓkbTm based on the fluctuation-dissipation theorem, kb is the Boltzmann constant, T is the effective temperature, while for quantum backaction fr2ms 1⁄4 2k_2 for position measurement strength k [Supplementary Note 8: Eq (S74)]. Due to the fluctuationdissipation theorem, low damping leads to a smaller Langevin force, reducing the thermodynamically limited frequency measurement uncertainty, as derived below. The mechanical motion of the tuning fork is measured through a near-field cavity-optomechanical readout (See Supplementary Note 1)[29] with detection noise well below the thermal fluctuation within the fork resonance linewidth. Defining fluctuating-force-induced variance of x around the harmonic response xharmonic 1⁄4
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