Fig. 1: Measurement of resonance frequency.

a A linear harmonic oscillator subject to a driving force, stochastic Langevin and quantum measurement backaction forces (QMB), and detection uncertainty. The time-varying eigenfrequency induced by a parametric interaction with an external system is extracted from the continuously measured position x [Eq.(1)] by a frequency estimator. Lower panel shows the false-colored scanning electron micrograph of the nanomechanical tuning fork with a cavity-optomechanical readout. Inset: a magnified view of the coupling gap between them. b The red bubble in the phase diagram represents the steady-state distribution of the linear harmonic oscillator (LHO) rotating-frame coordinate u = X + iY [Eq. (2)] subject to thermal and quantum fluctuations. The purple bubble represents the distribution of u_k due to diffusion around the expectation \(\hat u_k\), in a short time dt after a known state \(u_{k - 1}\). The blue bubbles show the position detection uncertainty. The red, purple, and blue distributions have a standard deviation of \(\sqrt 2 \sigma\), \(\sqrt 2 \sigma _{{{{{{{{\mathrm{d}}}}}}}}t}\), and \(\sqrt 2 \sigma _{{{{{{{\mathrm{n}}}}}}}}\), respectively, in each of the two dimensions. The distance \(\widehat u_k - u_{k - 1}\) is exaggerated for illustration. c LHO position power spectral density S_uu, when driven at a small detuning from a constant resonance frequency. The purple area denotes the contribution from the mechanical motion. The blue area represents the detection noise spectrum. d Real component X of u. Blue and purple dots schematically represent the measured positions with the detection uncertainty and actual positions without detection uncertainty, respectively.