Robust Optical-Levitation-Based Metrology of Nanoparticle’s Position and Mass

Light has shown an incredible capability in precision measurement based on optomechanic interaction in high vacuum by isolating environment noises. However, there are still obstructions, such as displacement and mass estimation error, highly hampering the improvement of absolute accuracy at the nanoscale. Here, we present a nonlinearity based metrology to precisely measure the position and mass of a nanoparticle with optical levitation under ${10}^{-5}\mathrm{mbar}$. By precisely controlling the oscillation amplitude of the levitated nanoparticle at the nonlinear regime for high accuracy calibration, we realized a feasible sub-picometer-level position measurement with an uncertainty of 1.0% without the prior information of mass, which can be further applied to weigh the femtogram-level mass with an uncertainty of 2.2%. It will also pave the way to construct a fine-calibrated optomechanic platform in high vacuum for high sensitivity and accuracy measurement in force and acceleration at the nanoscale and the study in quantum superposition at the mesoscopic scale.

Figure 1

Scheme of position measurement without the information of particle mass. The measured nonlinearity induced relative natural frequency shift $\mathrm{\Delta }\mathrm{\Omega }/{\mathrm{\Omega }}_0$, which can be obtained from the spectrum analysis of the voltage signal of detector $V(t)$ is related to the actual oscillation amplitude $q_A$ as $q_A=w\sqrt{-4\mathrm{\Delta }\mathrm{\Omega }/(3{\mathrm{\Omega }}_0)}$. With the respective measured voltage signal amplitude $V_A$, the calibration constant $c=q_A/V_A$ is determined with only the information about $V(t)$. Then the position signal can be obtained as $q(t)=cV(t)$.

Figure 2

(a) Schematics of the parametric feedback locking. When the particle’s amplitude is larger (lower) than target amplitude, a feedback cooling (heating) will be applied to the particle. Solid curves represent the nonlinear Gaussian potential which deviates from the harmonic potential (dashed curves). (b) Part of the recorded $V(t)$ from an amplitude locked particle at ${10}^{-5}\mathrm{mbar}$. (c) Simplified experimental setup [38].

Figure 3

(a) Measured $V(t)$ of a trapped nanoparticle under different locked amplitudes and feedback cooling along the $Y$ axis at ${10}^{-5}\mathrm{mbar}$. (b) Dependence of the absolute (SD) and relative (RSD) standard deviations of $V_A$ when the nanoparticle was feedback amplitude locked. (c) PSDs of the voltage signals shown in (a). (d) Measured relative natural frequency shifts along the $X$ (blue) and $Y$ (green) axes as a function of voltage amplitude. The solid lines are fittings with Eq. (2). (e) The intensity distribution simulated with Debye integral in the $X\text{−}Y$ plane. (f) Simulated intensity distributions along the $X$ (blue) and $Y$ (green) axes. The dashed lines are fittings with a Gaussian function.

Figure 4

(a) Measured mass under different air pressures. The dashed line is the average of the measurement results. (b) Measured damping coefficients ${\mathrm{\Gamma }}_0$, which were averaged from the $X$, $Y$, and $Z$ axes as a function of air pressure. The solid line is a linear fitting. The error bars came from the variance of ${\mathrm{\Gamma }}_0$ in different axes.