## Reconsidering resonator sensing

Changes in the frequency of a nanoscale mechanical resonator can be used for many sensing applications, provided that there is an adequate signal-to-noise ratio. Normally, this ratio is improved by creating resonators with higher quality factors that “ring” for longer times. Taking a cue from the approaches used in atomic force microscopy, Roy *et al.* show that if the thermomechanical noise of the resonator is well defined, the signal-to-noise ratio of the frequency shift can improve by lowering the quality factor. They used this approach to demonstrate temperature sensing with a double-clamped silicon beam resonator, which performed better at ambient pressures than in a vacuum.

*Science*, this issue p. eaar5220

## Structured Abstract

### INTRODUCTION

Nano-optomechanical systems (NOMS) are very small resonating mechanical devices that have extraordinary sensitivity. The coupling of mechanical motion to an optical cavity allows the motion to be tracked with femtometer precision. When using NOMS (or their electrical cousin, nanoelectromechanical systems) as a stable frequency reference, tiny force and mass changes can be distinguished by small frequency shifts. This is useful in atomic force microscopy and ultrasensitive mass measurement. For example, improvements in mass sensitivity have enabled the resolution of single molecules and have launched a prospective new paradigm of mechanical mass spectrometry. Any method to improve stability improves the performance of these sensors. If stability could remain the same or improve with more damping, NOMS ultrasensitivity could be deployed in a damping medium, like air or liquid, greatly enhancing their utility for use as biosensors or gas sensors or in the environment. Better stability could also benefit oscillator clock electronics, which could ultimately improve technologies such as GPS.

### RATIONALE

The quality factor (*Q*) is the inverse of the damping and indicates how sharp the resonance is in frequency. *Q* has been used as a proxy metric for frequency stability. However, *Q* only provides half the contribution; the other half comes from how large the resonance signal is compared to noise [the signal-to-noise ratio (*SNR*)]. This relationship is known as Robins’ model. Although traditionally *Q* and *SNR* have been assumed to be correlated, we noted that when the resonance conditions are limited only by intrinsic factors, the *SNR* should be inversely proportional to *Q*. In this case, stability should be independent of *Q*, and stable performance should be maintained in a variety of damping conditions.

### RESULTS

We measured intrinsic resonator stability in NOMS while decreasing *Q* by increasing the air pressure around the device. We found that *SNR* behaved inversely to *Q* as hypothesized; however, stability unexpectedly improved with decreasing *Q*. This improved performance with damping is diametrically opposed to conventional expectations that had been established for decades. We revisited Robins’ model to find that it was based on a high-*Q* approximation. Rederiving the model without approximation gave rise to a new flatband regime for large damping (low *Q*). In this new regime, stability tracked to *SNR* only and, correspondingly, improved with damping, explaining the measured data. We confirmed the improved performance at higher damping by monitoring temperature fluctuations at different pressures and found that the best stability occurred at highest *SNR*, consistent with the new model. Finally, there is a noise source called dephasing that is known to prevent mechanical resonators from reaching their stability limits. We confirmed that this extra noise source correlated with *Q* and therefore was mitigated at large damping and removed completely at atmospheric pressure.

### CONCLUSION

Our measurements confirm that *Q* and *SNR* behave inversely for intrinsically limited resonators, refuting long-standing assumptions about *Q* as a stability proxy. More notably, we found stability improved with damping. A low-*Q* approach was further shown to elegantly solve a vexing stability limitation caused by dephasing. We rederived Robins’ model to find a new flatband regime in which stability is linked only to *SNR* and found the new model to be consistent with the measured data. The flatband model displayed intriguing properties (in addition to stability behaving inversely to *Q*), including moderation of the trade-off between low noise and bandwidth and a route to frequency-scaling enhancement. The results offer a new paradigm for thinking about stability in mechanical resonators and suggest new pathways to improve stability in resonant sensors and crystal clock oscillators.

## Abstract

Mechanical resonances are used in a wide variety of devices, from smartphone accelerometers to computer clocks and from wireless filters to atomic force microscopes. Frequency stability, a critical performance metric, is generally assumed to be tantamount to resonance quality factor (the inverse of the linewidth and of the damping). We show that the frequency stability of resonant nanomechanical sensors can be improved by lowering the quality factor. At high bandwidths, quality-factor reduction is completely mitigated by increases in signal-to-noise ratio. At low bandwidths, notably, increased damping leads to better stability and sensor resolution, with improvement proportional to damping. We confirm the findings by demonstrating temperature resolution of 60 microkelvin at 300-hertz bandwidth. These results open the door to high-performance ultrasensitive resonators in gaseous or liquid environments, single-cell nanocalorimetry, nanoscale gas chromatography, atmospheric-pressure nanoscale mass spectrometry, and new approaches in crystal oscillator stability.

Nanoelectromechanical systems (NEMS) are known for extraordinary sensitivity. Mass sensing has reached the single-proton level (*1*, *2*), enabling NEMS gas chromatography (*3*–*5*) and mass spectrometry (*6*–*9*). Force sensing has produced single-spin magnetic resonance microscopy (*10*, *11*) and single-molecule force spectroscopy (*12*). Torque resonance magnetometry has been reenvisioned (*13*), with applications in spintronics and magnetic skyrmions. The mechanical quantum ground state has even become accessible (*14*–*16*) and used for absolute thermometry (*17*). The best sensitivities, however, have generally been presumed to require the highest quality factors, limiting application to vacuum environments and low temperatures. A host of new applications could result from the availability of ultrasensitivity in air and liquid: biosensing, security screening, environmental monitoring, and chemical analysis. As an example, our group aims, over the long term, to combine mass spectrometry and gas chromatography functions into one via NEMS sensing in atmospheric pressure.

Exquisite NEMS sensitivity is enabled through ultrasmall mass or stiffness combined with precise resonant frequency *f* determination. This approach allows perturbations to that frequency δ*f* (such as mass or force) to be probed (see Fig. 1A). Intuitively, the peak position (and ultimate sensitivity) is obscured by two factors: a finite linewidth and noise on the signal. The former is quantified by resonant quality factor *Q* and the latter by signal-to-noise ratio *SNR*. Frequency stability should thus relate to these two quantities. Robins’ formula (*18*), articulated in the atomic force microscope (AFM) community by Rugar and co-workers (*19*) and in NEMS by Roukes and co-workers (*20*, *21*), forms the basis for force and mass sensitivity analyses and defines this relationship as follows:

(1)where *SNR* is the ratio of driven motional amplitude to equivalent noise amplitude on resonance(2)and the dynamic range *DR* is the power level associated with this *SNR*.

The *Q* factor in the denominator of Eq. 1 has led researchers to focus primarily on achieving higher *Q* for better resolution (*22*–*26*). Strategies for *SNR* enhancement have been scarcer (*27*), with almost no consideration given to relationships between *Q* and *SNR*. There is an implicit assumption that improving *Q* will also benefit the signal fidelity. Figure 1B presents a *DR* response to changing *Q* from this traditional view. An extrinsic noise floor (e.g., readout amplifier) sets a noise that does not change with *Q*. Drive power is also assumed to be unchanged, resulting in amplitude loss, and shrinking *DR*, for lower *Q*.

There is, however, a case when *SNR* ∝ 1/*Q* that results in no sensitivity dependence on *Q*. This is not a special case. Indeed, it is the general case if the *DR* is properly maximized. When instrument noise is negligible (Fig. 1B, right), lower-*Q* resonances reveal fundamentally lower intrinsic noise floor peaks (e.g., thermomechanical noise). The intrinsic upper end of the dynamic range is associated with the end of linear response. The wider linewidth of a lower-*Q* resonance tolerates more nonlinearity and extends this linear range to larger amplitude. Combined, the two effects give 10^{–}^{DR}^{/20} ∝ *Q*.

This peculiar observation implies that frequency-fluctuation noise should not depend on *Q* in the case when thermomechanical noise is well resolved and amplitude can be driven to nonlinearity. If true, the model provides a pathway to completely mitigate sensitivity loss due to low *Q*. Even more intriguing, a detailed inspection of the phase noise model used in NEMS (*19*–*21*) reveals that Eq. 1 is a high-*Q* approximation. Removing this approximation, remarkably, implies frequency-fluctuation noise proportional to *Q* at low bandwidth: A highly damped system with full dynamic range should have better frequency stability (and sensitivity) than an equivalent lowly damped one. This is an exciting prospect with wide-ranging implications for (frequency modulated) scanning probe microscopy, mass sensing and biosensing, and inertial and timing microelectromechanical systems (gyroscopes, accelerometers, and crystal oscillators).

Using nano-optomechanical systems (NOMS), we demonstrated frequency stability improving with increased damping. We changed pressure from vacuum to atmosphere to vary the extrinsic *Q* within a single nanomechanical device. We observed that the *SNR* grew inverse proportionally to *Q* while the full dynamic range was maintained. Corresponding frequency stability (Allan deviation) dropped with increased damping and approached the theoretical limit, and was better in atmosphere than in vacuum. Notably, excess intrinsic frequency-fluctuation noise [also known as decoherence (*23*, *24*, *28*–*32*)] shrank with falling *Q* and did not limit stability at atmospheric pressure. We tested implied sensitivity improvement with temperature measurements, using the optical ring as calibration, and showed 60-μK sensitivity at 300-Hz bandwidth. This performance is comparable to state-of-the-art sensitivity (*33*–*36*), even with the modest nanocalorimeter geometry of a doubly clamped beam. These results will allow proliferation of high-performance ultrasensitive resonators into gaseous and liquid environments.

## Maximizing dynamic range to minimize frequency fluctuations

Thermomechanical noise was identified early on as the primary limit for AFM force detection (*19*). In contrast to macroscale mechanical resonators (such as quartz-crystal oscillators), the smaller stiffness (*k*) and size of AFM beams result in nonnegligible fluctuations from the thermal bath. In essence, of thermal energy (where *k*_{B} is the Boltzmann constant and *T* is temperature) populates of modal energy, producing between picometer and nanometer average displacements (*<x>*). These motion levels have been resolvable since the early 1990s. Mass-detection NEMS (*20*, *21*) are stiffer and, although resolving picometer average displacements had been a challenge (*37*–*40*), new transduction techniques (*38*, *41*, *42*) have made it routine (*14*, *15*, *17*, *28*, *29*, *42*–*58*).

The use of NOMS, in particular (*14*, *15*, *17*, *24*, *29*, *42*–*46*, *48*, *49*, *51*–*62*), has resolved thermomechanical noise by orders of magnitude above the instrumentation noise background. One example is our microring cavity optomechanical system (*58*–*62*), with displacement imprecision of approximately 20 fm Hz^{–1/2}. Figure 1C shows the measured displacement noise *S*_{x}^{1/2} in an example doubly clamped beam, measured in vacuum, where *Q* is high, and at atmospheric pressure, where *Q* is low. The device is shown in Fig. 1D, with the principle of detection illustrated in Fig. 1E. As per convention, values for *S*_{x} were calibrated from voltage signals (*S*_{V}) by assuming the peak noise relation (derived via equipartition theorem):(3)where angular frequency Ω = Ω_{0} = 2π*f* and *f* is the resonance frequency, *M* is the resonator effective mass, and Γ = Ω/*Q* is the damping. We define the thermomechanical (th) noise amplitude on resonance *a*_{th} as(4)where Δ*f* is the measurement bandwidth. Details are in (*63*), section 1.2. Note that *a*_{th} is proportional to *Q*^{+1/2}; correspondingly, the high-*Q* peak of Fig. 1C is higher. In both cases, the noise is dominated by the thermomechanical term near resonance, flattening to a white background far from resonance. In this way, we have access to the bottom end of the intrinsic *DR* out to 30-kHz measurement bandwidth.

Our devices were mechanically driven with a shear piezo (fig. S1), and a nonlinear response was reached owing to large drive power. As the doubly clamped beam was driven to larger amplitudes, the stiffness became amplitude dependent, resulting in a geometric nonlinearity (*64*–*66*). This Duffing nonlinearity created sharkfin-shaped resonance traces (e.g., Fig. 2, first panel, top trace) and amplitude-dependent resonance frequency. To avoid injecting amplitude noise into phase noise (which reduces stability), the driven amplitude must remain at or below a critical value *a*_{crit} that is generally used to define the end of the linear range and the top of the intrinsic *DR* (*66*):
(5)where *L* is the beam length, *E* is the Young’s modulus, and ρ is the density [a version of the equation including tension (*66*) is in (*63*)]. It is known that *a*_{crit} has an inverse square root dependence on *Q* that comes straight out of a Duffing derivation and applies to all nonlinear resonators (*64*–*66*). Intuitively, at a given amplitude, intrinsic nonlinearity causes a defined frequency shift large enough to tilt the resonance shape for a narrow linewidth while still being hidden by a wider one. When the full dynamic range was accessed, we could equate *a*_{noise} to *a*_{th} and *a*_{driven} to *a*_{crit}, and Eqs. 2, 4, and 5 combine to produce *SNR* proportional to 1/*Q*.

To test *SNR* behavior, we measured properties of the same doubly clamped beam at different pressures (thus different *Q*s), from vacuum to atmospheric pressure. This approach kept all parameters except for *Q* identical. Results are presented in Fig. 2, with frequency sweeps for five representative pressures. At each pressure, the thermomechanical noise is plotted for a 1-Hz bandwidth along with the driven root mean square amplitude response for varying drive power. Indicated by thick red lines are traces for the drive power corresponding with Duffing critical amplitude (up to 15 torr) and, by thick purple lines, for the maximum driving power available (40 and 760 torr). For 15-torr pressures and up, the driven resonance line shape was distorted. This distortion was not caused by nonlinearity (note the conserved response shape, and see fig. S8); rather, the resonance broadened to the point where piezo drive efficiency was no longer a constant function of frequency (*67*, *68*). The distorted features are related to bulk acoustic resonances in the piezo-chip system. This distortion carries no information about the NEMS beam resonance [see (*63*), section 1.5, and fig. S7].

We note in Fig. 2 that the peak of the noise floor *a*_{th} diminished as the pressure increased and generally followed *a*_{th} ∝ *Q*^{1/2} (compare Eq. 4). This result can be conceptually understood in the following way: The area under the thermomechanical resonance curve is conserved for a given temperature (in proportion to *k*_{B}*T*); as the width of the curve increases (*Q* decreases), the peak value must fall to compensate. For the upper end of the *DR*, within the Duffing-limited pressure regime, *a*_{crit} increased in proportion to *Q*^{–1/2}, as predicted by Eq. 5. Accounting for both effects, *SNR* ∝ 1/*Q*, up to 15-torr pressure. At 40 torr and up, we no longer had sufficient drive power to reach the Duffing critical amplitude and could no longer take advantage of the full intrinsic *DR* of the system. Nonetheless, the dynamic range was still higher at atmospheric pressure than in vacuum.

Figure 3 plots the peak amplitudes *a*_{crit} and *a*_{max}, the thermal amplitude *a*_{th}, quality factor *Q*, *SNR*, and product of *Q* × *SNR* as a function of pressure. We saw that *Q* × *SNR* was conserved within the Duffing-limited regime. According to Robins’ picture (Eq. 1), the frequency fluctuations in our system should be independent of *Q*, up to 15 torr.

## Frequency-fluctuation measurements (Allan deviation)

With *Q* × *SNR* conserved, checking the fractional frequency stability δ*f*/*f* in our device remained. We did this using the two-sample Allan variance, a standard method of characterizing frequency stability (*69*) related to Robins’ formula [see (*63*), section 2.2]. The Allan deviation σ(τ), as the square root of the Allan variance, is an estimate of fractional-frequency stability for a given time τ between frequency readings. The functional form for σ(τ) is

Figure 4 presents the measured Allan deviation data for our device at the five representative pressures and *Q*s. Surprisingly, rather than staying constant, σ improved as the pressure increased, and *Q* fell (up to 40-torr pressure). Furthermore, the measured data dipped well below the theoretical minimum set by Robins’ formalism and Eq. 6 (blue lines). Frequency stability, and therefore performance, appeared to be correlated to *SNR* alone.

## Full analysis of Allan deviation from noise power

We could understand this response by revisiting the close connection between Allan deviation and phase noise (*69*). Allan variance σ^{2} is essentially an integration of close-in phase noise *S*_{ϕ}(ω) out to a measurement bandwidth Δ*f*. Here sideband angular frequency ω = 2π*f*_{mod}, where frequency-offset-from-carrier *f*_{mod} = *f* – *f*_{0}. The resulting Allan deviation σ is proportional to , where the brackets here loosely represent the integration.

Understanding the frequency stability then reduces to understanding the behavior of *S*_{ϕ}. In a Robins’ formalism, displacement-derived phase noise *S*_{ϕ}^{x} is related to *S*_{x} via normalization by driven energy:

In essence, displacement fluctuations obscure the zero-crossing (phase) during the oscillation cycle by the ratio of noise amplitude to driven amplitude (fig. S13). Full details are available in (*63*), section 2, and fig. S14. Displacement noise amplitude *S*_{x}, such as in Fig. 1C, turns into a low-pass filter with 1/*f* ^{2} roll-off when zoomed in near the resonance frequency (compare Fig. 5B):
(8)Combining Eqs. 2, 4, 7, and 8 gives

The analysis follows closely to previous Robins’ analyses (*19*–*21*). At this point, the assumption is generally made that ω^{2} + (Γ/2)^{2} ≈ ω^{2}, that is, that *Q* is high. This assumption turns Eqs. 8 and 9 from low-pass filters into pure roll-offs (see Fig. 5A and inset). Knowing that *S*_{x}(0) ∝ 1/Γ (compare Eq. 3), it is concluded that *S*_{ϕ}^{x} ~ *S*_{x} ~ Γ^{+1}, and, ultimately, that σ ∝ Γ^{+1/2}. This result is generally well known in the AFM community.

When the high-*Q* assumption is not made, the proportionality of *S*_{x} and *S*_{ϕ}^{x} with Γ can vary considerably for different *f*_{mod}. Figure 5B shows our experimentally measured values of *S*_{x}(ω) fit directly with Eq. 8. At high *f*_{mod}, *S*_{x} ∝ Γ^{+1}, as in Fig. 5A. For low *f*_{mod}, however, *S*_{x} ∝ Γ^{−1}. When integrating to get σ^{2}, the high-*Q* assumption overestimates the integration, needlessly adding the area between the flat pass and the *f* ^{–2} line (blue shaded area in inset of Fig. 5A). This overestimate is negligible for large bandwidth (Δ*f*) but dominates for small bandwidth, leading to overestimated Allan variance.

The difference becomes more intriguing with increasing driven amplitude. Figure 5, C and D, shows Fig. 5B data in *S*_{ϕ}^{x}(ω) form (via Eq. 7). In Fig. 5C, for 15- and 40-torr pressures, *a*_{driven} had the same value, *S*_{x} and *S*_{ϕ}^{x} maintained the same relation, and the noise dependence on damping was the same as in Fig. 5B. On the other hand, in Fig. 5D, *a*_{driven} was Duffing limited (that is, ), causing *S*_{ϕ}^{x} to shrink more quickly with damping than *S*_{x} does. This resulted in *S*_{ϕ}^{x} being independent of Γ for large bandwidths and proportional to 1/ Γ^{2} at small bandwidths (compare Eq. 9). The right-hand portions of the data at different pressures and damping collapsed on top of each other. This correspondence is not a coincidence but rather the signature of *SNR* being inversely proportional to *Q*, resulting in no Γ dependence by Robins’ equation (Eqs. 1 and 6).

However, for Δ*f* ≤ 1 kHz, Fig. 5D implies σ^{2} ∝ Γ^{−2}, and therefore σ ∝ Γ^{−1}. That is, better stability results from more damping. The full functional form of σ for this case, which we refer to as the flatband (fb) regime, is [derived in (*63*), section 2]

This regime is not usually considered, as it would normally result in prohibitively low bandwidths. However, as devices reach higher frequencies, and as *Q* is pushed purposefully down, the corner frequency of (Γ/2)/(2π) can become very large; in the present case, it is 200 kHz for atmospheric pressure.

What Eqs. 9 and 10 explicitly show is that flatband *S*_{ϕ} and σ should correlate to *SNR* alone, not *Q*. We qualitatively confirmed this behavior with direct measurement of phase noise in fig. S19, where *S*_{ϕ} decreased for increasing *DR*. Returning to Fig. 4, Eq. 10 sets the new theoretical lower limit. Here σ was dominated by drift at τ = 0.1 s but approached Eq. 10 at τ = 2 ms. Figure 4B plots σ(2 ms) versus *Q* with both theoretical minimum floors (Eqs. 6 and 10), showing close approach to the full model. Within the Duffing-limited regime, where *SNR* ∝ 1/*Q*, the measured σ was proportional to *Q*, and stability became better in proportion to the amount of damping.

## Application of damping improved stability: Temperature sensing

We demonstrate an application of damping-enhanced sensitivity, with NEMS temperature resolution improving with increasing pressure. NEMS can be used as a thermometer owing to subtle temperature changes to Young’s modulus and device dimensions (*33*–*36*). Traditionally in the range of –50 parts per million (ppm)/K for silicon, intrinsic tension changes give our devices enhanced temperature coefficients with resonant frequency (TCRF) as high as –1200 ppm/K [see (*63*), section 1.7, table S1, and (*33*)]. The optical microring cavity itself also had a resonance dependence on temperature, allowing its use for calibration. The temperature responsivities for both microring and NEMS were simultaneously determined at each pressure tested by monitoring the change in resonant wavelength and mechanical resonant frequency for several 1-K temperature steps [see (*63*), section 1.7, and fig. S11].

Figure 6A shows the NEMS response at 3-torr pressure to a +0.3-K step change (followed later by a –0.3-K step change) in the temperature controller setting. The oscillations and long settling resulted from the PID (proportional integral differential) controller settings. The microring had a similar response (fig. S12). Zoom-ins of frequency (temperature) fluctuations for different pressures in Fig. 6B revealed a sensitivity sweet spot at 60 torr, where *DR* was maximum. Peak-to-peak fluctuations improved by one order of magnitude from vacuum to 60 torr, dropping to submillikelvin values. More formally, Fig. 6C presents the temperature resolution σ_{Δ}* _{T}* as a function of

*Q*, where σ

_{Δ}

*(τ) = σ(τ) ×*

_{T}*f*

_{0}/

*S*, and

_{f,T}*S*is the NEMS temperature responsivity (table S1). Temperature resolution improved with falling

_{f,T}*Q*to the 60-torr sweet spot, where it reaches 60 μK. This performance is comparable to that in (

*33*,

*34*).

## Discussion

That resolution could be independent of *Q* in the Robins’ picture has been hinted at (*28*, *29*), but not tested, and is not widely appreciated in the NEMS community. The further revelation that low-bandwidth sensitivity and stability actually improve with damping is a momentous development with implications in NEMS, AFM, crystal clock oscillators, and other applications. As an example, the AFM community has long known of force noise proportional to square root damping and has tried to reduce the apparent thermal-force noise off resonance by increasing *Q*. This approach works for high bandwidth (above the corner) but increases noise on resonance, which is usually truncated and ignored. Instead, by purposefully suppressing *Q*, one simultaneously suppresses close-in noise while extending the corner frequency (and bandwidth). In essence, the usually inevitable trade-off between bandwidth and low noise is eliminated.

The flatband regime stability (Eq. 10) has no explicit dependence on *Q*; our results derive from improving the *DR* with damping. The prevalent idea that stability improves with *Q* arises partly from the assumption that high *Q* and high *DR* are equivalent (compare Fig. 1B, left). Intrinsic noise–resolved and low-power resonators invert this assumption.

It is known that Eq. 6 has no explicit Ω dependence. Equation 10, on the other hand, varies inversely with Ω, opening additional paths to sensitivity improvement. Increasing the mechanical frequency should directly improve flatband sensitivity, while also extending the bandwidth available for a given *Q*. These enhancements are in addition to simultaneous sensitivity improvements coming from mass reduction.

The flatband suppression of the thermal-noise peak is reminiscent of cold damping and feedback cooling (*14*, *15*, *70*) but is distinct in that thermal noise is spread out rather than reduced. As such, feedback cooling could give cumulative benefit with the flatband technique. Similarly, techniques for using the nonlinear regime (*71*, *72*) or parametric squeezing (*73*) can be piggybacked onto flatband.

Another side benefit of low *Q* is suppression of intrinsic resonator frequency-fluctuation noise (*23*, *24*, *28*–*32*). Sansa *et al*. (*28*) recently noted this noise as ubiquitous in preventing NEMS from reaching thermal limits, though Gavartin *et al*. (*29*) were able to mitigate it with sophisticated force feedback. The transfer function responsible for conveying this intrinsic noise is proportional to *Q* (*24*), which may help explain why we did not see it at atmospheric pressure and saw clear evidence of it only at long gate times in vacuum (figs. S17 to S19).

We note that the limitations of our drive power keep us from accessing the full dynamic range at atmospheric pressure. This problem can be solved by using optomechanical drive force, which can be turned up almost with impunity. Nonlinearities in the optomechanical transduction, in both readout and excitation, could eventually limit the present technique from extending dynamic range indefinitely.

## Methods summary

A full description of the materials and methods is available in the supplementary materials (*63*).

The optomechanical system was fabricated on a silicon photonics platform from silicon-on-insulator. A doubly clamped beam (9.75 μm by 220 nm by 160 nm) was coupled to a race-track optical cavity resonator and probed with a tunable diode laser around 1550 nm. Mechanical excitations were made via shear piezo. The device chip resided in a vacuum chamber with controllable pressure. Other electronics supporting the measurement included lock-in amplification, high-power excitation amplification, temperature control, computerized data acquisition, and digitally implemented phase-lock techniques (section 1.1).

The thermomechanical noise calibration was implemented in a standard way, allowing voltage noise conversion to displacement noise (section 1.2), and the instrumentation noise floor was calibrated to be around 20 fm Hz^{–1/2} (arising primarily from photodetector dark current). The onset of nonlinearity and calculation for critical amplitudes is defined in section 1.3. At high drive power, heating needed to be accounted for in calibrating the critical amplitude (section 1.3.1). Notes on high drive powers (section 1.3.2) and different drive levels (section 1.3.3) discuss the piezo technique inefficiency and the inherent energy storage advantage of a lower-*Q* device.

Optomechanical coupling was both simulated and calibrated from optomechanical spring effect (section 1.4), giving similar values of *g*_{om} = 2.8 rad GHz/nm. Section 1.5 discusses bulk acoustic interference effects of the piezo-chip system and squeeze-film effects.

A Zurich instruments HF2LI lock-in amplifier was used for phase-lock looping, and Allan deviation measurements were taken from both open- and closed-loop conditions. In closed loop, a typical setting was 4-kHz demodulation bandwidth (the integration bandwidth), allowing for a 500-Hz phase-locked loop bandwidth. Temperature calibration (section 1.7) was done using both the microring temperature tuning (found to be between 70 and 80 pm/K) and the NOMS temperature tuning (ranging from –12 to –15 kHz/K for TCRF values from –1050 to –1270 ppm/K) with a PID temperature controller. The high value for TCRF was also verified from a thermal strain model that gives –1100 ppm/K.

Section 2 provides detailed frequency stability analysis. The theorized conversion of thermal force noise to displacement noise and on to phase noise is shown in section 2.1. This derivation proceeds in typical fashion, except that no assumption is made that ω >> Γ. This results in a low-pass functional form of phase noise that reduces to the traditional form (*S*_{ϕ} α 1/ω^{2}) for large ω, but acts as frequency independent (flat) for small ω. The Allan deviation is defined with the expected stability specified based on power law phase noise (section 2.2). This produces the traditional Robins’ stability limit for large ω (Eq. 6) and the new flatband stability limit for small ω (Eq. 10). Section 2.3 confirms the power law integrations that produce Eqs. 6 and 10, and section 2.4 compares our derivation nomenclature with some existing Robins’ derivations in the literature.

Signs of frequency-fluctuation noise (in distinction from additive white noise) in data at different pressures were sought via open-loop frequency tracking, Allan deviation measurements, and quadratrue analysis (section 2.5). Clear signatures of frequency-fluctuation noise were seen at high *Q* in vacuum, and the absence of signatures of such noise was observed at low *Q* at atmospheric pressure. Finally, phase noise was measured directly in open loop at different pressures (section 2.6) and found to correlate with *DR*, consistent with behavior expected in flatband regime.

## Supplementary Materials

www.sciencemag.org/content/360/6394/eaar5220/suppl/DC1

Materials and Methods

Supplementary Text

Figs. S1 to S19

Table S1

This is an article distributed under the terms of the Science Journals Default License.

## References and Notes

**Acknowledgments:**The authors acknowledge the National Research Council’s Nanotechnology Research Centre and its fabrication, microscopy, and measurement facilities. The fabrication of the devices was facilitated through CMC Microsystems (silicon photonics services and CAD tools), and postprocessing was performed at the University of Alberta nanoFAB. We thank P. Barclay and M. Freeman for thoughtfully reviewing the manuscript.

**Funding:**Funding was provided through author institutions as well as scholarship support from Alberta Innovates Technology Futures and the Vanier Canada Graduate Scholarship program. Grant support was provided from Alberta Innovates Health Solutions through a collaborative research innovation opportunity and from the Natural Sciences and Engineering Research Council, Canada, through the Discovery Grant program.

**Author contributions:**W.K.H. conceived and designed the research. S.K.R. conducted the measurements with training and assistance from V.T.K.S., J.N.W.-B., and A.V. V.T.K.S. designed the devices for fabrication. W.K.H., S.K.R., and V.T.K.S. wrote the manuscript. All authors contributed to data interpretation and manuscript preparation.

**Competing interests:**None declared.

**Data and materials availability:**All data needed to evaluate the conclusions in the paper are present in the paper or the supplementary materials.