Accurate mass measurement of a levitated nanomechanical resonator for precision force sensing

TL;DR

This paper introduces a new method for precisely measuring the mass of a levitated nanoparticle, significantly reducing uncertainties and enhancing the accuracy of nanomechanical sensors for force and mass detection.

Contribution

A novel measurement protocol using electrical driving to determine nanoparticle mass with less than 1% statistical and 2% systematic error.

Findings

Mass measurement uncertainty reduced to below 1%.

Method enables more reliable sensing in levitodynamics.

Improves accuracy for force and mass sensing applications.

Abstract

Nanomechanical resonators are widely operated as force and mass sensors with sensitivities in the zepto-Newton and yocto-gram regime, respectively. Their accuracy, however, is usually undermined by high uncertainties in the effective mass of the system, whose estimation is a non-trivial task. This critical issue can be addressed in levitodynamics, where the nanoresonator typically consists of a single silica nanoparticle of well-defined mass. Yet, current methods assess the mass of the levitated nanoparticles with uncertainties up to a few tens of percent, therefore preventing to achieve unprecedented sensing performances. Here, we present a novel measurement protocol that uses the electrical field from a surrounding plate capacitor to directly drive a charged optically levitated particle in moderate vacuum. The developed technique estimates the mass within a statistical error below 1% and a systematic error of 2%, and paves the way toward more reliable sensing and metrology applications of levitodynamics systems.

Tables

Table 1. Table 1: Uncertainties table. The different quantities z i subscript 𝑧 𝑖 z_{i} involved in the calculation of the mass are here reported together with the corresponding error σ z i subscript 𝜎 subscript 𝑧 𝑖 \sigma_{z_{i}} . Color coding indicates negligibility of the uncertainty, with gray rows implying σ z i ≃ 0 similar-to-or-equals subscript 𝜎 subscript 𝑧 𝑖 0 \sigma_{z_{i}}\simeq 0 .

Quantity	Value $z_i$	Error ${\sigma }_{z_i}/z_i$
$n_q$	$8$	$0$
$q_e$	$1.602\times {10}^{-19}C$	$6.1\times {10}^{-9}$
$E_0$	$5.305{\mathrm{kVm}}^{-1}$	$0.011$
$𝒯$	$40\mathrm{ms}$	$4\times {10}^{-5}$
$S_v\left({\omega }_{\mathrm{dr}}\right)$	$1057.8{\mathrm{bit}}^2{\mathrm{Hz}}^{-1}$	$0.005$
$S_v^{\mathrm{th}}\left({\omega }_{\mathrm{dr}}\right)$	$14.7$	$0.007$
$k_B$	$1.380\times {10}^{-23}{\mathrm{JK}}^{-}1$	$5.72\times {10}^{-7}$
$T$	$295.8K$	$0.002$
$\Gamma$	$1.998\times {10}^5{\mathrm{rads}}^{-1}$	$0.003$
$𝐦$	$4.01$ fg	$0.009(\mathrm{stat}.)\pm 0.024(\mathrm{syst}.)$

Equations

$$m\ddot{x}+m\Gamma \dot{x}+kx=F_{\mathrm{th}}(t)+F_{\mathrm{el}}(t).$$

$$S_x(\omega )=S_x^{\mathrm{th}}(\omega )+S_x^{\mathrm{el}}(\omega )$$

$$=\frac{4k_{\mathrm{B}}T\Gamma }{m\left[{\left({\omega }^2-{\Omega }_0^2\right)}^2+{\Gamma }^2{\omega }^2\right]}+\frac{F_0^2\tau {\mathrm{sinc}}^2[(\omega -{\omega }_{\mathrm{dr}})\tau ]}{m^2\left[{\left({\omega }^2-{\Omega }_0^2\right)}^2+{\Gamma }^2{\omega }^2\right]}.$$

$$m=\frac{n_q^2{q}_e^2{E}_0^2\mathcal{T}}{8k_{\mathrm{B}}T\Gamma R_S}.$$

$$m\ddot{x}+m\Gamma \dot{x}+m{\Omega }_0^2\left(1+\xi x^2\right)x={\mathcal{F}}_{\mathrm{th}}(t)+F_{\mathrm{el}}(t),$$

$${\Omega }_{\mathrm{NL}}={\Omega }_0\left(1+\frac{3}{4}\xi ⟨x^2⟩\right),$$

$${\hat{\sigma }}_E(\mathcal{T})=\sqrt{\frac{1}{2(N_{\mathcal{T}}-1)}\sum _{j=1}^{N_{\mathcal{T}}}\left({\hat{E}}_{j+1}(\mathcal{T})-{\hat{E}}_j(\mathcal{T})\right)}$$

$$\{\begin{array}{rlrlrl}& {\varphi }_{\mathrm{N}}=1,& & \text{if}& & {\varphi }_{\mathrm{LI}}>0\\ & {\varphi }_{\mathrm{N}}=1,& & \text{if}& & {\varphi }_{\mathrm{LI}}\text{is undefined}\\ & {\varphi }_{\mathrm{N}}=-1,& & \text{if}& & {\varphi }_{\mathrm{LI}}<0,\end{array}$$

$$\Sigma (x)=a_{\mathrm{high}}-\frac{a_{\mathrm{high}}-a_{\mathrm{low}}}{1+{\mathrm{e}}^{-(x-x_0)/\tau }}$$

$$\mathrm{d}\text{X}=\text{a}(t,\text{X})\mathrm{d}t+\text{b}(t,\text{X})\mathrm{d}W.$$

$$\begin{array}[]{rl}\textbf{K}_{1}=&\textbf{a}(t_{k},\textbf{X}_{k})\Delta t+(\Delta W_{k}-S_{k}\sqrt{\Delta t})\cdot\textbf{b}(t_{k},\textbf{X}_{k}),\\ \textbf{K}_{2}=&\textbf{a}(t_{k+1},\textbf{X}_{k}+\textbf{K}_{1})\Delta t\,+\\ &(\Delta W_{k}+S_{k}\sqrt{\Delta t})\cdot\textbf{b}(t_{k+1},\textbf{X}_{k}+\textbf{K}_{1}),\\ \textbf{X}_{k+1}=&\textbf{X}_{k}+\frac{1}{2}(\textbf{K}_{1}+\textbf{K}_{2}),\end{array}$$

$$\mathrm{d}{x}_t$$

$$m\mathrm{d}{v}_t$$

$$\Psi (x)=m{\Omega }_0^2\left(\frac{x^2}{2}+\frac{\xi \cdot x^4}{4}\right)$$

$$G(\omega )=\mathcal{F}\left(g(t)\right)={\int }_{-\infty }^{\infty }g(t)e^{-i\omega t}\mathrm{d}t.$$

$${\mathcal{S}}_g(\omega )=\underset{T\to \infty }{\lim }{\left|\frac{1}{\sqrt{T}}{\int }_{-T/2}^{T/2}g(t)e^{-i\omega t}\mathrm{d}t\right|}^2,$$

$$S_g(\omega )$$

$$=\frac{1}{\tau }{\left|\frac{1}{\pi }G(\omega )\ast \text{sinc}(\tau \omega )\right|}^2.$$

$$S_x(\omega )=|H(\omega ){|}^2|F_{\text{d}}(\omega ){|}^2$$

$$S_g(\omega )$$

$$=\frac{1}{\tau }{\left|\frac{1}{\pi }\left(\pi (\delta (\omega -{\omega }_{\text{d}})+\delta (\omega +{\omega }_{\text{d}})\right)\ast \text{sinc}(\tau \omega )\cdot \tau \right|}^2$$

$$=\frac{\tau }{2}{\left|\text{sinc}[(\omega -{\omega }_{\text{d}})\tau ]+\text{sinc}[(\omega +{\omega }_{\text{d}})\tau ]\right|}^2$$

$$\simeq \frac{\tau }{2}{\text{sinc}}^2[(\omega -{\omega }_{\text{d}})\tau ].$$

$${\sigma }_m=\sqrt{\sum _i{\left(\frac{\partial m}{\partial z_i}\right)}^2{\sigma }_{z_i}^2},$$

$$\frac{{\sigma }_m^{\mathrm{syst}}}{m}$$

$$\frac{{\sigma }_m^{\mathrm{stat}}}{m}$$

Full text

**Introduction

**Nanomechanical resonators play a leading role in the field of force (Moser2013Ultrasensitive, ), mass (Chaste2012ANanomechanical, ), and charge (Cleland1998ANanometer, ) sensing. Thermal noise represents the ultimate limitation in the their sensitivity (Yin2013Optomechanics, ; Norte2016Mechanical, ), and hence clamped resonators are usually operated in cryogenic environments (Purdy2012Cavity, ).

Owing to their unprecedented decoupling from the environment, levitated nanomechanical systems have recently been able to reach room temperature performances comparable to such clamped cryogenic nanoresonators(Gieseler2012Subkelvin, ; Jain2016Direct, ; Setter2018Real-Time, ), yet with a sensible reduction of the complexity of the apparatus. Moreover, the negligible mechanical stresses introduced by levitation allow to fulfill the rigid body approximation. As a result, the mass of the resonator is uniquely defined by the inertial mass of the levitated nanoparticle and does not require precise assessment of the system’s geometry, knowledge on material properties and complex flexural models for the shape of the oscillation modes, as it is the case for clamped systems.

Despite zepto-Newton resonant force sensitivities with levitated nanoparticles in vacuum have been predicted(Gieseler2013Thermal, ) and demonstrated (Ranjit2016Zeptonewton, ), and recent experiments with free falling nanoparticles enable for the detection of static forces (Hebestreit2018Sensing, ), the accuracy of these results does not outperform that of existing systems. In most of the levitation experiments, in fact, the uncertainties on the detected forces are of the order of few tens of percent (Hebestreit2018Sensing, ), sometimes even as high as $50\%$ (Hempston2017Force, ). Such large errors arise from uncertainties in the particle displacement calibration (Hebestreit2017Calibration, ), whose accuracy is critically affected by the poor knowledge on the particle’s mass. This results in severe limitations on their sensing and metrology applications, where the accuracy of a measurement is just as important as its precision.

Silica micro and nano-spheres are the most commonly used type of particle in levitated sensing experiments. Due to their fabrication process (Stober1968Controlled, ), these particles feature a finite size distribution with a $2$–$5\%$ coefficient of variation (microparticles2018Data, ). This, together with even higher uncertainties on the density of the amorphous silica used (up to $20\%$ (Parnell2016Porosity, )), leads to inaccurate values of the particle’s mass. One could avoid assumptions of the manufacturer specifications by relying on the kinetic theory of gasses to calculate the radius of the particle (Beresnev1990Motion, ). Also in this case, however, the final measurement of the mass is affected by uncertainties on the material density and on other quantities, such as pressure and molar mass (Hebestreit2018Measuring, ) of the surrounding media. A more accurate estimation of the particle’s mass is therefore highly desirable, as it would boost the accuracy of sensors based on levitated particles.

Here, we propose and experimentally demonstrate a measurement protocol that is unaffected by the above-mentioned uncertainties (density, pressure, size, etc.), and leads to an assessment of the particle’s mass within $2.2\%$ systematic error and $0.9\%$ statistical error. Our method exploits a new design of an optical trap in which a pair of electrodes is placed around the focus and is based on the analysis of the response of a trapped charged particle to an external electric field. Careful error estimation has been carried out in order to assess the final mass uncertainty, including the treatment of possible anharmonicities in the trapping potential. The technique we propose is easy to implement in any vacuum trapping set-up and improves by more than an order of magnitude the accuracy of most precision measurements.

**Experimental Set-Up

**The experimental set-up is depicted in Fig. 1a. A single silica nanoparticle ($d=143\pm 4~{}\mathrm{nm}$ in diameter; nominal value of the manufacturer) is optically trapped in vacuum with a tightly focused laser beam (wavelength $\lambda=1064~{}\mathrm{nm}$, power $P\simeq 75~{}\mathrm{mW}$, $N\!A=0.8$). The oscillation of the particle along the $x$–mode is monitored with a balanced split detection scheme that provides a signal $v(t)$ proportional to the particle displacement $x(t)=v(t)/c_{\rm cal}$, with $c_{\rm cal}$ being the linear calibration factor of the detection system(Hebestreit2017Calibration, ). Along the same axis, a pair of electrodes (see Fig. 1b and inset) form a parallel plate capacitor that we use to generate an oscillating electric field $E(t)=E_{0}\cos(\omega_{\rm dr}t)$ at the particle position, which in turn induces a harmonic force $F_{\rm el}(t)$ on the charged particle.

The equation of motion of the particle can be described by a thermally and harmonically driven damped resonator:

[TABLE]

Here, $m$ is the mass of the particle, $\Gamma$ is the damping rate and $k=m\Omega_{0}^{2}$ is the stiffness of the optical trap, with $\Omega_{0}$ being the mechanical eigenfrequency of the oscillator. The first forcing term $F_{\rm th}$ models the random collisions with residual air molecules in the chamber. It can be expressed as $F_{\rm th}=\sigma\eta(t)$, where $\eta(t)$ has a Gaussian probability distribution that satisfies $\langle\eta(t)\eta(t+t^{\prime})\rangle=\delta(t^{\prime})$, and $\sigma$ relates to the damping via the fluctuation-dissipation theorem: $\sigma=\sqrt{2k_{B}Tm\Gamma}$, with $k_{\rm B}$ being the Boltzmann constant and $T$ the bath temperature. The second forcing term $F_{\rm el}$ arises from the Coulomb interaction of the charged particle with the external electric field $E(t)$, and can be expressed as $F_{\rm el}(t)=F_{0}\cos(\omega_{\rm dr}t)$, where $F_{0}=q\cdot E_{0}$. The net charge $q=n_{q}\cdot q_{e}$, with $q_{e}$ being the elementary charge and $n_{q}$ the number of charges on the particle, can be controlled (Frimmer2017Controlling, ) by applying a high DC voltage $V_{\rm HV}\sim\pm 1~{}\mathrm{kV}$ on a bare electrode placed on a side of the vacuum chamber. Via the process of corona discharge (Fridman2004Plasma, ), this creates a plasma consisting of a mixture of positive or negative ions (depending on the $V_{\rm HV}$ polarity) and electrons that can ultimately add to, or remove from, the levitated particle one single elementary charge at a time. Positive and negative ions are accelerated towards opposite directions due to the presence of the electric field from the electrode. As a result, the ratio of positive to negative charges reaching the particle is biased by the electrode polarity, thus allowing us to fully control the final charge of the particle within positive or negative values (see Supplementary Section S3). This is a significant advantage compared to other discharging techniques that rely on shining UV light on the particle (Moore2014Search, ), where the net charge can only be diminished until reaching neutrality.

**Measurement

**A single nanoparticle is loaded in the trap at ambient pressure by nebulizing a solution of ethanol and monodispersed silica particles into the chamber. The pressure is then decreased down to $P\lesssim 1~{}\mathrm{mBar}$ where the net number of charges $n_{q}$ can be set with zero uncertainty. Finally, the system is brought back up to an operating pressure of $P\simeq 50~{}\mathrm{mBar}$. At this pressure the particle is in the ballistic regime, but its dynamics is still highly damped. This condition is favorable for our experiments, as the high damping reduces the contribution of anharmonicities to the dynamics of the particle (Gieseler2013Thermal, ), allowing us to apply the fully linear harmonic oscillator model which predicts:

[TABLE]

Here, $S_{x}(\omega)$ is the single-sided Power Spectral Density (PSD) of the thermally and harmonically driven resonator, whose dynamics $x(t)$ is being observed for a time $\mathcal{T}=2\tau$. Note that $S_{x}(\omega)$ relates to the experimentally measured PSD $S_{v}(\omega)$ via the calibration factor $c_{\rm cal}$, such that $S_{v}(\omega)=c_{\rm cal}^{2}\cdot S_{x}({\omega})$ (Hebestreit2017Calibration, ).

In the absence of the electric driving, the motion of the particle in the optical trap is purely thermal and its PSD is well approximated by a typical Lorentzian function. From an experimental measurement of $S_{v}^{\rm th}(\omega)$ we can extract the value of $S_{v}^{\rm th}(\omega_{\rm dr})$ and perform maximum likelihood estimation (MLE) to obtain the values of $\Omega_{0}$ and $\Gamma$ as fitting parameters. Likewise, when the coherent driving is applied to the system, we are able to determine the magnitude of the driven resonance $S_{v}(\omega_{\rm dr})$ and to calculate from this measurement the solely electric contribution $S_{v}^{\rm el}(\omega_{\rm dr})=S_{v}(\omega_{\rm dr})-S_{v}^{\rm th}(\omega_{\rm dr})$.

Figure 2 exemplifies this process for $\omega_{\rm dr}/(2\pi)=135~{}\mathrm{kHz}$ and for a signal-to-noise ${\rm SNR}=S_{v}(\omega_{\rm dr})/S_{v}^{\rm th}(\omega_{\rm dr})\simeq 60$. The curve shown is computed with Bartlett’s method from an ensemble of $N_{\rm psd}=1000$ averages of individual PSDs, calculated from $\mathcal{T}=40~{}\mathrm{ms}$ position time traces. In Supplementary Section S2 we verify that over the whole measurement time $t=N_{\rm psd}\times\mathcal{T}=40~{}\mathrm{s}$ the system does not suffer from low frequency drifts. The electrically driven peak can be fully resolved (see inset in Fig. 2), and its shape agrees with the Fourier transform of the rectangular window function used for PSD estimation. The gray trace at the bottom of the plot represents the measurement noise, which is $\sim 40~{}\mathrm{db}$ below the thermal signal and more than $55~{}\mathrm{db}$ below the driven peak. Finally, the solid line is a MLE fit of a thermally driven Lorentzian to the experimental data. Note that, to perform the fit and to retrieve the value of $S_{v}^{\rm th}(\omega_{\rm dr})$, the electrically driven peak is numerically filtered out by applying to the time series data-set a notch filter of variable bandwidth $b$ around $\omega_{\rm dr}$. The value of $b$ depends on the driving amplitude, with typical values of the order of tens of Hertz. In Supplementary Section S7 we show how this method introduces negligible errors that remain always below $\sim 0.01~{}\%$.

The mass of the particle can ultimately be calculated considering the ratio $R_{S}=\frac{S_{v}^{\rm el}(\omega_{\rm dr})}{S_{v}^{\rm th}(\omega_{\rm dr})}=\left.\frac{S_{v}-S_{v}^{\rm th}}{S_{v}^{\rm th}}\right|_{\omega=\omega_{\rm dr}}$. In fact, note that while both $S_{v}^{\rm el}$ and $S_{v}^{\rm th}$ depend quadratically on $c_{\rm cal}$, the latter scales as $m^{-1}$ while the former scales as $m^{-2}$. Thus, from their ratio we obtain:

[TABLE]

To ensure the validity of the linear resonator model, we also considered a cubic term in the restoring force and performed Montecarlo Simulations of the resulting Duffing resonator with parameters compatible with our experimental settings and an overestimated value of the Duffing coefficient (Ricci2017Optically, ; Gieseler2014Nonlinear, ) $\xi=12~{}\mathrm{\mu m}^{-2}$. The outcome of the simulations is detailed in Supplementary Section S5, and confirms the negligibility of the nonlinear terms for pressures of $P\simeq 50~{}\mathrm{mBar}$. We stress that this assumption fails already at slightly lower pressures of $\sim 10~{}\mathrm{mBar}$ where a more complicated non-linear response model would be needed.

**Error Estimation

**In order to estimate the systematic error committed in calculating the mass, a careful study of all the sources of error has to be carried out. Table 1 summarizes the absolute values and the relative uncertainties of the quantities entering in Eq. (3). The specific case reported corresponds to point at $\omega_{\rm dr}=125~{}\mathrm{kHz}$ of the data shown in Fig. 3. For several variables and constants, we can neglect the corresponding uncertainty. Accordingly, for the error propagation we set: $\sigma_{q_{e}}=\sigma_{\mathcal{T}}=\sigma_{k_{\rm B}}=0$. Note that the specific number of charges $n_{q}=8$ chosen in our measurements is arbitrary. Other measurements have been previously carried out with different values of $n_{q}$ and have confirmed the independency of the method from $n_{q}$, provided the dynamics is maintained in the linear regime of oscillation. Concerning the other quantities, instead, we follow the arguments stated below:

1. (i)

The electric field was simulated with the finite elements method, and was mainly affected by uncertainties in the geometry of the electrodes (see Supplementary section S4 for further details). We measure a distance between electrodes of $d_{\rm el}=1410~{}\mathrm{\mu m}\pm 13~{}\mathrm{\mu m}$, and a corresponding electric field (for an applied dc potential of $1V$) $E_{0}=577\pm 6~{}\mathrm{V/m}$. 2. (ii)

The two heights of the power spectral densities $S_{v}(\omega_{\rm dr})$ and $S_{v}^{\rm th}(\omega_{\rm dr})$ from which the ratio $R_{S}$ is calculated are only affected by statistical errors since simulations confirm the validity of the linear model. $\sigma_{S_{v}}$ is thus calculated from an ensemble of $N_{\rm psd}$ measurements as the standard error of the mean, with the $1/\sqrt{N_{\rm psd}}$ trend being verified. The same applies for $S_{v}^{\rm th}$, where in this case $\sigma_{S_{v}^{\rm th}}$ is calculated in the absence of external electric driving. 3. (iii)

The thermal bath surrounding the partcle is assumed to be constantly thermalized with the set-up, and more precisely with the vacuum chamber walls. Again, the moderate-high pressure $P=50~{}\mathrm{mBar}$ ensures this assumption. Multiple temperature measurements on the surface of the vacuum chamber are carried out with a precision thermistor ($0.5^{\circ}{\rm C}$ accuracy) in order to exclude the presence of temperature gradients and significant variations during the experimental times (see Supplementary Section S8 for data and further discussion). 4. (iv)

The uncertainty of fitting parameters such as $\Omega_{0}$ and $\Gamma$ can be extracted directly from the lorentzian fits.

The variables involved in Eq. 3 can be considered uncorrelated and the standard uncertainty propagation(Ku1966Notes, ) can be performed. A detailed derivation is provided in Supplementary Section S9.

**Results

**The statistical error $\sigma_{m}^{\rm stat}$ of our measurement is calculated from the standard deviation of a set of $20$ independent measurements performed at $\omega_{\rm dr}/2\pi=130~{}\mathrm{kHz}$ and for ${\rm SNR}\simeq 60$. We find $\sigma_{m}^{\rm stat}/m=0.7\%$. This dispersion is displayed as error bars in Fig. 3, where we plot the calculated mass as a function of $\omega_{\rm dr}$, again for a ${\rm SNR}\simeq 60$. The region within green dot-dashed lines corresponds to the standard deviation $\pm\sigma_{m}^{\rm sweep}$ of the presented data.

The compatibility $\sigma_{m}^{\rm stat}\simeq\sigma_{m}^{\rm sweep}$, and the reproducibility of the mass calculated at different driving frequencies reveal that the measurements are not affected by nonlinearities in the system. In fact, strong driving fields lead to anharmonic particle dynamics which in turn introduce an unphysical mass dependency on $\omega_{\rm dr}$. Fig  S6a in Supplementary Section S2 exemplifies this situation and shows how in the non-linear regime the calculated mass is affected by sever systematic errors. In our method we avoid this situation maintaining the driving field amplitude below $5.5~{}\mathrm{kV/m}$. In this regime, we have additionally verified the independency of the calculated mass from the electric field $E_{0}$ and tested the quadratic scaling of $R_{S}$ as a function of $E_{0}$. These measurements are described in fig. S3 of the Supplementary Information. The excellent agreement with the model provides a further validation of eq. (3) and of the harmonic approximation made. As a final remark, we compare the measured mass $m$ of a $d=143\pm 4~{}\mathrm{nm}$ diameter particle with the one calculated from the manufacturer specifications $m_{\rm man}$. Assuming a nominal density for Stöber silica of $\rho_{\rm p}=2200~{}{\rm kg}/{\rm m}^{3}$ and propagating the corresponding uncertainties one finds $m_{\rm man}=(3.37\pm 0.84)~{}{\rm fg}$, which shows good agreement with the value measured with our method $m=(4.01\pm 0.1)~{}{\rm fg}$.

**Conclusions

**In conclusion, we presented a novel protocol to calculate the mass of a levitated nano-sensor through its electrically driven dynamics. We stress that this method only assumes a driven damped harmonic oscillator. As such, it is suitable to measure the oscillator’s mass in a large variety of optical trapping systems and possibly also in more general mechanical resonators schemes. The level of precision and accuracy obtained establishes an improvement of more than one order of magnitude compared to the state-of-the-art methods, enabling paramount advances in the applications of levitated systems as force sensors and accelerometers. Moreover this technique leads to a much more reliable calibration of the particle’s displacement (Hebestreit2017Calibration, ), again providing an important step for the use of levitated systems for metrology and sensing applications, and towards compliance requirements of groundbreaking experiments such as MAQRO (Kaltenbaek2016Macroscopic, ).

**Corresponding authors

**francesco.ricci@icfo.eu

romain.quidant@icfo.eu

**ORCID

**Francesco Ricci: 0000-0002-5971-3369

Romain Quidant: 0000-0001-8995-8976

**Aurthor Contributions

**F.R. and A.S. conceived the experiment. F.R. designed and implemented the experimental set-up and wrote all data acquisition software. F.R. and M. T. performed the experiment and analysed the data, with inputs from G.P. Montecarlo and COMSOL simulations were performed by G.P. and M.T. respectively. All authors contributed to manuscript writing. R.Q. and A.S. supervised the work.

**Acknowledgement

**We acknowledge financial support from the ERC- QnanoMECA (Grant No. 64790), the Spanish Ministry of Economy and Competitiveness, under grant FIS2016-80293-R and through the ‘Severo Ochoa’ Programme for Centres of Excellence in R&D (SEV-2015-0522), Fundació Privada CELLEX and from the CERCA Programme/Generalitat de Catalunya. We also acknowledge N. Meyer and the rest of the PNO trapping team. F.R. acknowledges Dr. M. Frimmer and Prof. L. Novotny from ETH (Zurich)for valuable discussions and A. Bachtold (ICFO) for providing a general perspective on the accuracy of mechanically resonating nanosensors.