Simultaneous measurement for amplitude and frequency of time-harmonic force based on optomechanically induced nonlinearity

An accurate readout of the mechanical motion using optomechanical coupling is highly desired for on-chip sensing applications but it remains challenging due to the uncertainty caused by time-dependent parameters and noisy fluctuations. Here, we propose an efficient scheme to realize simultaneous measurement for both amplitude and frequency of the time-harmonic force (THF) in a hybrid optomechanical system via a nonlinear sum sideband effect. In this optomechanical system assisted by a degenerate parametric amplifier (DPA), the nonlinear optomechanical interaction between the external THF, optical, and mechanical modes is used to construct the frequency component of optical sum sidebands. Using experimentally achievable parameters, we find that the conversion efficiency of the sum sidebands has a significant enhancement when the nonlinear gain coefficient of DPA increases. In the scheme of the dual-parameter measurement, we also report that the amplitude of THF could be independently detected by observing the intensity variation of the lower sum sideband, while the frequency of THF could be separately read by monitoring the frequency of the prominent peak in this nonlinear spectrum. Benefitting from the optical cooling of a mechanical element, the theoretical results show that the minimum resolutions for detecting the amplitude and the frequency of THF are approximately 8.8×10−12N and 16Hz, respectively.