A hybrid on-chip optomechanical transducer forultrasensitive force measurementsE. Gavartin1, P. Verlot1* and T. J. Kippenberg1,2*

Nanoscale mechanical oscillators1 are used as ultrasensitivedetectors of force2, mass3 and charge4. Nanomechanicaloscillators have also been coupled with optical and electronicresonators to explore the quantum properties of mechanicalsystems5. Here, we report an optomechanical transducer inwhich a Si3N4 nanomechanical beam6,7 is coupled to a disk-shaped optical resonator made of silica on a single chip. Wedemonstrate a force sensitivity of 74 aN Hz21/2 at room temp-erature with a readout stability better than 1% at the minutescale. Our system is particularly suited for the detection ofvery weak incoherent forces, which is difficult with existingapproaches because the force resolution scales with thefourth root of the averaging time8. By applying dissipative feed-back9 based on radiation pressure, we significantly relax thisconstraint and are able to detect an incoherent force with aforce spectral density of just 15 aN Hz21/2 (which is 25 timesless than the thermal noise) within 35 s of averaging time(which is 30 times less than the averaging time that would beneeded in the absence of feedback). It is envisaged that ourhybrid on-chip transducer could improve the performance ofvarious forms of force microscopy8,10–12.

It is possible to detect the motion of nanomechanical oscil-lators13,14 with an imprecision below that at the standard quantumlimit15,16. However, because of the thermal limit it is possible todetect very small forces without achieving this level of displacementsensitivity. Recent work has demonstrated that trapped ions17,18 candetect forces as weak as 5 yN (ref. 18), but this approach is technicallydemanding and stable interaction times are currently in the milli-second range17. In contrast, approaches to force sensing based onnanomechanical cantilevers are well established and have been usedto detect single spins8, to reconstruct the structure of a virus19 and,increasingly, for applications in biosensing20. A particularly promisingapproach is to parametrically couple a nanomechanical oscillator to amicrowave15 or optical7 cavity; this enhances the displacementsensitivity and also makes it possible to resolve the motion of canti-levers with dimensions below the diffraction limit. However, in thepast, this approach has been best suited to cryogenic temperatures15,and the force sensitivity was limited by relatively poor readout stabilityand low mechanical quality factors at room temperature7,21,22.

Our approach overcomes these drawbacks by coupling the out-of-plane vibrational mode of a high-stress Si3N4 nanomechanicalbeam to the near field of a whispering gallery mode of a silica micro-disk resonator23 (Fig. 1). We chose these two materials for threereasons. First, Si3N4 is arguably the best material for making nano-mechanical structures with high values of mechanical quality factorQM. Second, silica has favourable optical properties23,24. Finally, it ispossible to fabricate devices made of these two materials on a singlechip using methods amenable to the mass production of devices.

The beam of the hybrid system used in this work has dimensionsof 90 mm × 700 nm × 100 nm, and the microdisk has a diameter of

76 mm, with a distance of �250 nm between the beam and themicrodisk. We coupled laser light into the cavity using thefibre–taper technique25 (Fig. 1c). In contrast to previous resultswith silica microdisks23, there are greater scattering losses as aresult of defects being formed on the microdisk during fabrication.This gives a considerably higher optical linewidth of dn≈ 3 GHz(the optical quality factor is Q¼ n0/dn≈ 65,000, where n0 is theresonance frequency of the mode).

To measure the transduction of the mechanical motion ofthe oscillator, we kept the hybrid system under a pressure of lessthan 1 × 1024 mbar and detuned the laser to the blue side of thefringe of a critically coupled optical mode25. Launching a laser attelecom wavelengths (1,550 nm) and a power of 400 mW into thecavity, we observed the fundamental out-of-plane mode of thebeam at a mechanical resonance frequency of VM/2p¼ 2.88 MHzwith a signal-to-noise ratio (SNR) exceeding 50 dB. We determinedthe optomechanical coupling of the system to be G/2p¼ dn/dx¼ 2.9 MHz nm21 and calibrated the spectrum in terms of thefrequency noise Snn[V] and the displacement spectral densitySxx[V]¼ Snn[V]/G2 units, where V is the Fourier frequency. Theeffective mass of this mechanical mode is meff¼ 9 × 10215 kg(ref. 7). Ringdown measurements revealed a high mechanicalquality factor of QM¼ 4.8 × 105.

The versatility of the presented hybrid transducer was enhancedby implementing a feedback scheme that introduces an effective sus-ceptibility for the mechanical resonator9,26–28. The feedback consistsof applying a force proportional to the mechanical displacementFfb[V]¼ –geiF× meffGMVMx[V] to the oscillator, where g and Fdenote the amplitude and phase of the gain, respectively, andGM¼VM/QM denotes the mechanical linewidth. Thus, in the pres-ence of feedback, the response of the oscillator to an external forceFext is given by

x[V] = x[V] Fext V[ ] + Ffb V[ ]( )

= xeff [V]Fext[V] (1)

where xeff denotes the (Lorentzian) effective susceptibility29, charac-terized by an effective mechanical resonance frequency Veff and aneffective damping rate Geff described by

V2eff = V2

M 1 + gQM

cosF

( )(2)

Geff = GM 1 + gsinF( )

(3)

We implemented the feedback scheme using radiation pressureobtained by amplitude-modulating a second laser beam (Fig. 2band Supplementary Sections S2 and S3). Instead of relying on

1Ecole Polytechnique Federale de Lausanne, EPFL, 1015 Lausanne, Switzerland, 2Max-Planck-Institut fur Quantenoptik, Hans-Kopfermann-Straße 1,85748 Garching, Germany. *e-mail: pierre.verlot@epfl.ch; tobias.kippenberg@epfl.ch

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electrical circuits27,28,30, we used a method based on a demodulation–modulation technique31, enabling accurate control of any mechanicalmode up to arbitrary high frequencies. Using this scheme we demon-strated accurate feedback control of the fundamental out-of-planemode by tuning F. Figure 2c shows the feedback induced resonancefrequency shift (Veff 2 VM)/2p (left axis, blue dots) and normalizedeffective damping Geff/GM (right axis, red dots) versus feedback

phase F with constant g. The experimental data of Fig. 2c werefitted using equations (2) and (3) and give excellent agreement. Inboth cases we used the fitting parameter g¼ 30.7, thus demonstrat-ing the reliability of our scheme.

We were particularly interested in dissipative feedback (F¼p/2),which enables cooling of the mechanical oscillator27,28,30. Usingour scheme we performed high-gain cooling of the fundamental

Teff = 3.6 KTeff = 34 K

Teff = 2.2 KTeff = 1.4 KTeff = 0.7 K

a

c

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(Ωeff

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M

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b

ECDLfeedback

ECDLreadout

ECDLsignal

PD1

PD3

PD2

Taper

Hybrid structure

Vacuum chamber

Ф

freq. lock.

LPF+DCA

To AM

Quadratures demodulation

Up-mixing tophase-controlled carrier

SG1 SG2

λ1

λ2

λ3

2.881 2.8822.878 2.879 2.880

5 × 10–14

1 × 10–13

2 × 10–13

5 × 10–13

1 × 10–12

Frequency (MHz)

8.635 8.638

105106107

Frequency (MHz)

S xx

(m H

z−1/2

)1/

2

S νν

(Hz

Hz−1

/2)

1/2

cos(Ωt + Φ)

sin(Ωt + Φ) sin(Ωt)

cos(Ωt)

Figure 2 | Feedback control of the nanomechanical transducer. a, The square root of the displacement density Sxx1/2 versus mechanical frequency V for the

fundamental out-of-plane mode. Feedback is used to cool down to effective temperatures Teff ranging between 34 K (blue line) and 0.7 K (orange line).

The light blue curve corresponds to a squashing of the background; although this curve can be fitted with our model, we do not use it to determine Teff.

Inset: feedback cooling of the third-order harmonic of the out-of-plane mode from 295 K (blue), through 54 K (green) and 20 K (red), to 7.1 K (orange).

b, Simplified scheme of the set-up used for the feedback and force detection experiments including the feedback scheme based on demodulation and

modulation of the incoming readout signal (PD, photodiode; ECDL, external cavity diode laser). Three lasers with different wavelengths are used to

provide readout of the mechanical motion, actuation for feedback control and the small incoherent radiation pressure force used for direct force detection.

The full set-up and details on the feedback scheme are provided in Supplementary Section S3. c, Frequency detuning (Veff2VM)/2p of the fundamental out-

of-plane mode (left axis) and ratio of the effective linewidth to the intrinsic linewidth (right axis) versus the phase difference F between the demodulation

and modulation signals.

a

1 μm

c

b d

e

1.0

0.5

0.0

2.880 2.881 2.882102

103

104

10−13

10−12

10−11

Frequency (MHz)

1/2

S νν

(Hz

Hz−1

/2) S

xx (m H

z −1/2)1/2

50 μm

Figure 1 | Hybrid nanomechanical transducer system on a chip. a, Scanning electron microscopy image (false colours) of the hybrid on-chip system

consisting of a doubly clamped, high-stress Si3N4 nanomechanical beam (blue) coupled via the near-field to a silica microdisk resonator (red). Both the

resonator and the pads holding the beam rest on silicon pedestals (green). b, Magnified image of the proximity area. c, Optical micrograph of a tapered fibre

coupled to the hybrid system. d, Finite-element simulation of the electromagnetic field distribution of the fundamental optical mode confined in the microdisk

resonator. The nanomechanical beam is coupled to the evanescent near-field of the optical mode. The colour code represents the electrical field intensity

|E|2. e, Room-temperature Brownian motion of a nanomechanical beam with a fundamental resonance frequency of 2.88 MHz and meff¼ 9 pg. Fitting the

mode gives a mechanical quality factor of QM¼ 4.3× 105. The left vertical axis is calibrated in frequency noise units Snn[V] and the right vertical axis in

displacement density units Sxx[V]. Inset: finite-element simulation of the fundamental out-of-plane mode of the beam.

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mode (Fig. 2a) and achieved a minimum effective temperatureTeff¼ (GM/Geff )T of 700 mK (where T is ambient temperature).We demonstrated the versatility of our method by feedback-cooling the third-order out-of-plane mode by means of a simplechange of the modulation/demodulation frequency (inset ofFig. 2a). In general, dissipative feedback can be used for highgains, as long as g ≤ QM is satisfied.

High-Q nanomechanical oscillators, combined with high-sensitivity readout, are ideally suited for the measurement of weakforces2,15. Their force sensitivity is mainly limited by the thermalLangevin force Fth, with the spectral density given by the fluc-tuation–dissipation theorem SFF

th [V]¼24kBT/V Im(1/x[V]), wherekB is Boltzmann’s constant. In the limit of high-QM harmonicoscillators, the spectral density reads SFF

th [V] ≈ 4meffkBTGM and isproportional to both the mass and mechanical damping. For ourintegrated system this sensitivity is SFF

th [V] ≈ (74 aN)2 Hz21, com-parable to the best values reported at room temperature for cantile-ver-based systems32, but with the benefit of an almost two orders ofmagnitude higher bandwidth26.

In principle, detecting stationary incoherent forces embedded in thethermal noise is possible by means of energy averaging. However, thisaveraging converges very inefficiently, scaling as t1/4 (ref. 8) andthereby limiting the force detection threshold. In the following, wepresent a detailed study of energy averaging, and show that its conver-gence can be drastically improved by using dissipative feedback.

We were interested in detecting a signal xsig(t) driven by astationary incoherent force dFsig(t), which is uncorrelated withthe thermal force and for which SFF

sig[V≈VM] ≪ SFFth [V≈VM]

holds true; that is, the incoherent force is buried in the thermalnoise. In the presence of dFsig , the energy of the transducer

Em = 1/2meffV2Mkx2l can be written as the contribution of two inde-

pendent terms, Eth = 1/2meffV2Mkx2

thl and Esig = 1/2meffV2Mkx2

sigl,where k. . .l denotes the average over the statistical domain describingx as a random variable. Thus, comparing the transducer’s energywith and without dFsig being applied enables its detection,in principle.

Experimentally, statistical averages are not available and onehas to rely on estimators. Assuming dFsig to be Gaussianwhite noise for simplicity, s(t)¼ 1/t

�0t dtX2(t) is a non-biased

estimator for the measurement of the energy, with X denotingany of the motion’s quadratures. Signal detection is limitedby the estimator’s imprecision Ds(t), which is related to anequivalent force spectral resolution dF[VM,t] (also knownas EFSR, and measured in units of aN Hz21/2) by thefollowing equation:

dF[VM, t] =���������������

Ds(t)�10

dV2p

x[V]∣∣ ∣∣2

√√√√√ (4)

In the low signal-to-noise limit, Ds(t) can be shown to be dominatedby thermal fluctuations (Supplementary Section S4), with a corre-sponding EFSR of dF[VM, t] =

��24

√ �������������������H(GM, t)Sth

FF[VM]√

, whereH[GM, t] =

��2

√(e−GM − 1 + GMt)1/2/(GMt) depends only on the

linewidth of the transducer in the limit of an acquisition band-width large compared to GM. For t≫ 1/GM, the EFSR variesas (GMt)21/4, confirming previous reports8. The discussionabove shows that obtaining a given force resolution becomesfaster as the damping rate of the oscillator increases; as dissipative

b c

d f

−1.0 −0.5 0.0 0.5 1.0

−1.0

−0.5

0.0

0.5

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0.020.04

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0.40.30.20.1

0.40.30.20.1

a

e

Averaging time (s)

0.03 0.3 3

2

468

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40

0.03 0.3 3

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Averaging time (s) Averaging time (s)

Г eff

/ГM

p(X2)

p(X 1

)

X1 (10−10 m)

−1.0 −0.5 0.0 0.5 1.0X1 (10−10 m)

X 2 (1

0−10

m)

−1.0

−0.5

0.0

0.5

1.0

X 2 (1

0−10

m)

Reso

lutio

n (a

N H

z−1/2

)

Reso

lutio

n (a

N H

z−1/2

)

Reso

lutio

n (a

N H

z−1/2

)

Reso

lutio

n (a

N H

z−1/2

)

Averaging time (s)2 5 10 20 50

40

30

2010

20

40

60

0.05 0.10 0.50 1 0.05 0.1 0.5

45

50

556065

10

15

20

70

50

30

Г eff

/ГM

Гeff /ГM

Figure 3 | Force resolution enhancement via optomechanical feedback control. a, Evolution of mechanical motion in phase-space with feedback (purple

traces) and without feedback (blue). Histograms correspond to the statistical distributions of quadratures X1 and X2, which are the slowly varying in-phase

and out-of-phase components of mechanical motion. b,c, Plots showing the calculated force resolution (b) and the measured force resolution (c) as a

function of Geff/GM and averaging time t. d, Measured force resolution versus t without feedback (blue line; Geff/GM¼ 1; dashed blue line in c) and with

feedback (purple line; Geff/GM¼ 57; purple dashed line in c). In both cases the measured force scales with t1/4 (dashed grey lines). e, Measured force

resolution versus t without feedback for a short time (the framed region in d; blue circles are measured values, solid line represents predictions of a

theoretical model). f, Measured force resolution versus Geff/GM for t¼0.3 s (vertical dashed line in c). Red circles are measured values, solid line represents

predictions of a theoretical model.

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feedback can provide Geff ≫ GM, while keeping the thermalforce unchanged, it enables a substantial decrease of the averagingtime. Therefore, in the presence of dissipative feedback theimproved EFSR is

dF[VM,Geff , t] =����������������������H(Geff , t)/H(GM, t)

√× dF[VM, t]

≈���������GM/Geff

4√

× dF[VM, t] (5)

Figure 3a shows the trajectory of mechanical motion in its phase-space, without and in the presence of dissipative feedback (blueand purple traces, respectively). Figure 3b shows the EFSR calculatedas a function of Geff and t using equations (4) and (5). Figure 3cshows the corresponding experimental results. The contours corre-spond to plane regions with constant resolutions described by(Gefft ≈ ai)ai[R+ . The good agreement between theory and exper-iment confirms that an increase in Geff enables t to be decreasedby the same amount, while maintaining the same force resolution.Figure 3d shows the time evolution of the EFSR correspondingto Geff/GM¼ 1 and to Geff/GM¼ 57. Both curves show a scalingwith t1/4 in their asymptotic behaviour, in excellent agreement

with theoretical expectations. A curvature is present in the blueline at short averaging times, which is absent in the purple line.This is due to the mechanical memory time of the transducer,tM¼ 1/GM ≈ 26 ms, as confirmed by the excellent agreementbetween the short-term evolution of the measured EFSR in theabsence of feedback (dots in Fig. 3e) and our theoretical model(line in Fig. 3e; see Methods) with no adjustable parameter beingused. Figure 3f gives the improvement of the EFSR as a functionof Geff/GM, for a given averaging time of t¼ 0.3 s as givenalong the vertical dashed line in Fig. 3b. The straight line corre-sponds to a power-law fit with an exponent found to be 0.27,in agreement with the value of 1/4 expected from the theoreticaldiscussion.

We then used the resolution enhancement described above todetect a weak incoherent radiation pressure force, which wasapplied by sending a third laser tone with a randomly modulatedamplitude into the tapered fibre (see Fig. 2b and, for the full set-up, Supplementary Fig. S1a). It is important to emphasize that feed-back enables an improved force resolution only for signals withcoherence times that are small compared to the effective mechanicaldecay time 2p/ Geff (that is, spectrally broad signals, cf. Fig. 4a). By

a

b

d

e

f

g

(i) (ii) (iii)

δωFrequency

Pow

er s

pect

ral d

ensi

ty

0 τDelay time

Autocorrelation

0

FT

−20 0 20

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ocor

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tion

(a.u

.)

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350

350

35τ (s)

τ (s)

τ (s)

0

[s(τ

) − ε t

h]/ε

th

−0.1

0.0

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0 10 20 30τ (s)

0 10 20 30τ (s)

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[s(τ

) − ε t

h]/ε

th[s

(τ) −

ε th]

/εth

[s(τ

) − ε t

h]/ε

th[s

(τ) −

ε th]

/εth

[s(τ

) − ε t

h]/ε

th

0 5 10 15 20 25 30 350.0

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0 5 10 15 20 25 30 350.0

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0 5 10 15 20 25 30 35−0.2

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(i)

(ii)

(iii)c

−2 −1 0 1 2

10−11

10−9

10−7

τ (ms)

I(0)I(t)

(a.u

.)

Figure 4 | Feedback-assisted force detection. a, Weak random forces can be detected with the feedback-assisted approach when the quadrature spectrum

of the driving signal (blue line, left panel) is broader than that of the mechanical motion (purple) or, equivalently, when the autocorrelation function (which is

the inverse Fourier transform of the quadrature spectrum) of the driving signal (blue line, right panel) is narrower than that of the mechanical motion

(purple). The plots in both panels are schematics. b, Experimentally measured autocorrelation functions associated with the signal intensity noise (in blue)

and with the thermally driven mechanical motion (in purple). c, From top to bottom: intensity noise autocorrelation function for decreasing noise powers,

associated with the signals detected in d, e and f, respectively. d–f, Signal contribution to the mechanical energy (expressed in units of thermal noise),

measured along various averaging times and feedback gains. Each point corresponds to the signal contribution obtained in the presence of a feedback gain

given by its abscissa value and after a given averaging time related to its colour (single-shot measurement for the bluer points, 35 s of averaging for the

redder ones). The blue dashed lines correspond to the expected values of the signal contributions to the mechanical energy (5.25 Eth, 0.5 Eth and 0.04 Eth,

respectively), deduced from the levels of the signal autocorrelation functions shown in c. The pink lines correspond to the gain evolution of the energy

dispersion expected after 35 s of averaging (see Methods). The pink region in f gives the signal detection zone, which we define as corresponding to a

measurement accuracy of five standard deviations or better. g, Detailed evolution of the signal contribution with averaging time for feedback gains of 0 (top),

6.5 (middle) and 32 (bottom) for the lowest signal applied, Esig ≈ 0.04Eth. The blue dashed line corresponds to the expected measurement dispersion

(see Methods). The pink region corresponds to the detection zone.

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measuring the autocorrelation function of our signal, we deter-mined its coherence time to be more than three orders of magnitudesmaller than the intrinsic mechanical decay time 2p/GM (Fig. 4band Methods), so that feedback gains with values up to a fewhundred can be used safely.

To detect the signal we used the ‘on/off’ protocol suggestedabove. We first averaged the probe phase fluctuations while thesignal was being applied and compared these to the averagedprobe fluctuations in the absence of signal. We implemented thisprotocol for various signal powers, feedback gains and averagingtimes. Figure 4 presents the signal detection for three differentpower levels deduced from the signal’s autocorrelation functionsshown in Fig. 4c. Figure 4d–f show the estimated energy excessinduced by the presence of the signal (s(t) − Eth)/Eth (expressedin units of thermal energy Eth) as a function of averaging time tand feedback gain. The colour of the points is related to the aver-aging time, varying between 0 and 35 s from blue to red. For eachsignal power the estimated energy excess converges towards theexpected value (dashed blue lines, inferred from the level ofthe autocorrelation), with the final dispersion being in excellentagreement with our model (pink lines, see Methods). In particular,it is clearly confirmed that using high feedback gains enables asignificant decrease in the measurement imprecision for a givenaveraging time.

Finally, we focused on the detection of the signal with the lowestpower (SFF

sig[V≈VM] ≈ 0.04SFFth [V≈VM] ≈ (15 aN Hz21/2)2), with

the results shown in Fig. 4f,g. The pink region in Fig. 4f representswhat we define as the ‘detection zone’, highlighting the pointslocated within five standard deviations of the expected value ofthe measurement. The signal is considered to be detected in agiven averaging time after which the centred normalized energyestimator (s(t) − Eth)/Eth stays confined in the detection zone.Figure 4g shows the detailed time evolution of the centred normal-ized energy estimator given by the framed regions in Fig. 4f. In theabsence of feedback (Fig. 4g, top), 35 s of averaging is clearly insuf-ficient to make a conclusion about the presence of the signal. Settinga moderate feedback gain of g¼ 6.5 (Fig. 4g, middle) reduces themeasurement dispersion to two standard deviations after 35 s ofaveraging, but is still insufficient to decide on signal detection.Figure 4g, bottom, shows that using a relatively high gain ofg¼ 32 enables the signal to be detected (that is, the applied incoher-ent force) within the same 35 s of averaging duration.

Note that such a performance is possible due to the combinationof ultrahigh motion resolution and stable readout of an ultralow-dissipation, ultralow-mass transducer (equation (5)). A high dis-placement resolution minimizes the feedback noise, the effect ofwhich on the transducer has to remain below the measurementimprecision of five standard deviations (Supplementary SectionS5). This implies a gain limit of g , gmax given by

g2max ≈

15× Ssig

FF[V ≈ VM]Sth

FF[V ≈ VM]× rM (6)

where rM denotes the ratio between the thermal noise and thereadout background at frequency VM. The readout stability sets amaximum averaging time tmax, during which the apparent energychange due to sensitivity drifts remains less than the five standarddeviations measurement accuracy:

kdj2/j20ltmax

≤ 15× Ssig

FF[V ≈ VM]Sth

FF[V ≈ VM](7)

with kdj2/j02ltmax

denoting the fractional fluctuations of theoptomechanical transduction factor j(t) over time tmax. Thesetwo limits are conditioning signal detection, which has to be

performed within a time td ≤ tmax. With our system, detectingSFF

sig[V≈VM]¼ 0.04SFFth [V≈VM] without feedback assistance

would require td ≈ 1,200 s, whereas we determined tmax ≈ 60 s≪ td.Signal detection therefore requires reducing td by a factor of g≥ 20.With a measured rM of �1.3 × 105 (51 dB), we can safely usegains up to 34, explaining the detection capability of our system.

In conclusion, we have demonstrated an integrated hybrid forcetransducer system for sensing applications for ultrasmall signals.The integration of the system is of crucial importance, as it allowsstable operation over an extended period of time. Even the smallestchanges in the system, particularly regarding the evanescent coup-ling between the mechanical beam and the optical cavity, wouldprohibit the detection of small signals well below the thermalnoise. In contrast to other high-QM and ultrasensitive trans-ducers13,15, our system operates at room temperature and profitsfrom well-established readout techniques at optical frequencies.Concurrently, the integrated system could also be implemented ina cryogenic environment in a straightforward manner33. Our workhas shown that dissipative feedback can substantially improve theincoherent force resolution of nano- or micromechanical systems,featuring stable and sensitive operation12,30 and enabling a signifi-cant decrease of measurement time and thus extending the rangeof applications for small-scale transducers. This scheme has thepotential to be applied to magnetic resonance force microscopy8

and with other signals such as transitions in nitrogen-vacancy(NV) centres10,11. Our system can be readily functionalizedfor these applications, for example through the use of nano-manipulators34 or using a scanning atomic force microscope35.

MethodsFabrication. The samples were fabricated on a silicon wafer with thermally grown2 mm silica and 100 nm low-pressure chemical vapour deposited high-stressstoichometric Si3N4 thin films on top. The beam was created using electron-beamlithography followed by a timed C4F8–SF6 reactive ion etch. In a second lithographystep, the polymer mask for the microdisk resonator was defined in such a way thatthe disk partially overlapped the beam. The microdisk resonator was created using awet etch in a buffered hydrofluoric bath. Owing to the isotropic nature of this etch,material below the polymer mask was partially etched, resulting in an angledsidewall of the microdisk. Using this effect and ensuring the correct alignment of thedisk mask with respect to the beam, the release of the beam was achieved with thestructure being positioned above the wedge of the microdisk resonator. Release of thecombined system was accomplished by anisotropic silicon etching in a potassiumhydroxide bath.

Enhancement of force resolution using feedback. To determine the measurementaccuracy (Supplementary equation (S5)) corresponding to a given feedback gain, wefirst extracted the quadratures of motion by sending the detected signal to anelectronic spectrum analyser (ESA, Agilent MXA 9020A), operating in I/Q modewith a sampling rate of 1/ts ≈ 7 kHz. We recorded independent realizations of thethermal noise (X1(t), X2(t))i,j along N different acquisition times tj ranging from27 ms to 3.77 s. Each of the resulting sequences was used to determine thecorresponding averaged energy si,j(tj)¼ 1/tj

�0tj dt(X2

1,(i,j)(t)þ X22,(i,j)(t)).

The measurement resolution after averaging time tj was then determined asDs(tj)¼

p(Var[(si,j(tj))i]). We defined the equivalent force resolution

(represented in Fig. 3) as the equivalent motion resolution normalized to the areaof the mechanical amplitude response

dF(tj) =

��������������������������Ds(tj)

/∫1

0

dV2p

|x[V]|2√

the latter being determined by measuring the expected value of the energy ks(tj)l ≈1/N

∑j¼0N s(tj), using

ks(t)l = SFFth[VM]

∫1

0

dV2p

|x[V]|2

Feedback-assisted force detection. To obtain the incoherent radiation pressureforce, the output laser field of a third external cavity diode laser (ECDL) was sentinto an amplitude modulator fed with Gaussian white noise generated by a functiongenerator (Agilent 33250A). To avoid saturating the amplitude modulator, thegenerator output was filtered using a selective 3 MHz bandwidth low-pass filter. The

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optical frequency of this ECDL was tuned to match a third optical resonance of thewhispering-gallery-mode cavity, distinct from those used to probe (ECDL1) and toactuate (ECDL2) the feedback response. This third ‘noisy’ tone was sent into thetapered fibre using a fibre-based beamsplitter (C3 in Supplementary Fig. S1a) andwas later separated from the other two tones using the combination of anotherbeamsplitter (C4) and an optical filter (TOF3). A photodetector (PD3) enabled thetransmitted intensity fluctuations to be measured. To determine the autocorrelationfunctions presented in Fig. 4b, we proceeded in the following manner. The timeevolution of the signal’s quadratures was first recorded for a duration of Ts ≈ 35 s at asampling rate dt ≈ 0.2 ms using an ESA operating in I/Q mode. With theautocorrelation being quadrature-invariant for Gaussian signals, only onequadrature X was conserved for its computation, which we restricted to delay timesdt smaller than dtmax¼ 30 ms. For any dt [ [2dtmax, dtmax] > dt × N, theautocorrelation value kX(0)X(dt)l was obtained by evaluating the discrete timeintegral

kX(0)X(dt)l ≈ 1Ts − 2dtmax

∫Ts−dtmax

dtmax

dtX(t)X(t + dt)

the latter equality applying by virtue of the ergodic theorem. As both signals beingmeasured (the intensity noise and the thermal displacement of the nanomechanicaloscillator) result from white noise, their autocorrelation function is expected to bethe impulse response of the measurement apparatus used to detect them({nanomechanical oscillatorþWGM resonatorþ photodetectorþ ESA} and{WGM resonatorþ photodetectorþ ESA} for the mechanical motion and theintensity noise, respectively). For the probe signal, this impulse response is mostlydue to the mechanical response and can be approximated by a decaying exponential,because the sampling rate is dt ≪ 2p/GM. For the intensity noise, this impulseresponse prominently involves the 5 kHz wide frequency gate set by the I/Q modeparameters of the ESA, such that the autocorrelation linewidth in Fig. 4b,c is notrelated to the coherence time of the intensity noise but simply to this frequency gate.The pink dispersion lines in Fig. 4f are determined usingDE(t) = 1/2meffV

2MDsth−th(t) because the low SNR hypothesis applies. This is,

however, not the case in Fig. 4d,e, where the SNRs are kXsig2l/kXth

2l ≈ 5 andkXsig

2l/kXth2l ≈ 0.5, respectively. The corresponding dispersions are therefore

evaluated using an extended model taking into account the terms Dssig–th(t) andDssig–sig(t) (Supplementary equations (S5) and (S6)).

Received 30 January 2012; accepted 8 May 2012;published online 24 June 2012

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AcknowledgementsFabrication was carried out at the Center of MicroNanotechnology (CMi) at EPFL. Theauthors acknowledge financial support from NCCR Quantum Photonics, the DARPAOrchid programme, the SNF and an ERC starting grant (SIMP).

Author contributionsT.J.K. and E.G. conceived the hybrid transducer. E.G. performed the fabrication andmodelling. P.V. designed and conceived the incoherent force resolution enhancementscheme and performed the theoretical calculations. E.G. and P.V. performed themeasurements and analysed the data. E.G. and P.V. wrote the manuscripts with criticalcomments from T.J.K. All stages of the work were supervised by T.J.K.

Additional informationThe authors declare no competing financial interests. Supplementary informationaccompanies this paper at www.nature.com/naturenanotechnology. Reprints andpermission information is available online at http://www.nature.com/reprints. Correspondenceand requests for materials should be addressed to P.V. and T.J.K.

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