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SUPPLEMENTARY INFORMATION DOI: 10.1038/NNANO.2009.343 NATURE NANOTECHNOLOGY | www.nature.com/naturenanotechnology 1 Supplementary Information for “Nanomechanical motion measured with an imprecision below that at the standard quantum limit” J. D. Teufel, 1 T. Donner, 1 M. A. Castellanos-Beltran, 1, 2 J. W. Harlow, 1, 2 and K. W. Lehnert 1, 2, 1 JILA, National Institute of Standards and Technology and the University of Colorado, Boulder, CO 80309, USA 2 Department of Physics, University of Colorado, Boulder, CO 80309, USA SPECTRAL DENSITIES EXPRESSED AS NOISE QUANTA Throughout this paper we have expressed spectral den- sities using the single-sided convention [1]. This conven- tion gives the familiar classical result that an oscillator coupled to a thermal bath of temperature T will experi- ence a random force characterized by the force spectral density S F =4k B T mγ m , where m is the mass and γ m is the dissipation rate of the oscillator. More generally, this force is S F = 4¯ m ¯ n + 1 2 m , (1) where ω m is the oscillator’s resonance frequency and ¯ n is the Bose-Einstein occupancy factor given by ¯ n 1 = e ¯ m k B T 1. In the limit of zero temperature, the average energy of the oscillator is simply the half quantum of zero-point en- ergy and S F = 2¯ m m . This force is associated with displacement fluctuations on resonance with a spectral density S x (ω m )= S F |H(ω m )| 2 = h m γ m , (2) where the mechanical susceptibility H(ω) is given by H(ω) 1 = m(ω 2 ω 2 m iωγ m ). Independent of any spec- tral density convention, these zero-point fluctuations al- ways correspond to one half quantum of noise energy in the oscillator. Thus, it is often useful to refer to the dis- placement spectral density on resonance in terms of the noise quanta of energy. In the convention used here, one quarter quantum of noise energy is naturally associated with spectral density S x h/(m γ m ). INFERRING DISPLACEMENTS FROM INTERFEROMETER RESPONSE As described in previous work [2, 3], we infer the mo- tion of the wire from the measured frequency shift in the cavity resonance. The transmission past the cavity has the following resonant form S 21 = S 0 +2i ωωc γc 1+2i ωωc γc , (3) where S 0 is the normalized transmission on resonance, ω c is the resonance frequency of the cavity, and γ c is the linewidth of the cavity. By measuring the transmission past the cavity, we find S 0 , ω c and γ c . From these pa- rameters, we can determine the quality factor associated with loss inside the cavity Q int =(ω c c )/(S 0 ) and that associated with intensional coupling to the transmission line which passes the cavity Q ext =(ω c c )/(1 S 0 ). For this experiment, Q ext =3.0 × 10 3 and Q int has a typi- cal value of 2.6×10 4 , but varies by 20% for the range of temperatures and powers used here. The amplified microwave signal emerging from the cryostat is demodulated with an inphase-quadrature mixer. The voltage V from the quadrature component of the mixer is V = V 0 Im[S 21 ], where V 0 is found by measuring the transmission far from resonance. Thus, the spectral density of cavity frequency fluctuations can be inferred from the voltage fluctuations as S ωc (ω m )= 1+4ω 2 m 2 c (∂V/∂ω c ) 2 S V (ω m ) , (4) 10 5 0 300 200 100 0 2 2 ( /2 ) (kHz ) c δω π Temperature (mK) FIG. 1: Calibration of displacement measurement with thermal motion The mean-square fluctuations in the cav- ity’s resonance frequency due to the thermal motion of the oscillator as a function of cryostat temperature. The linear fit (solid line) determines the optomechanical coupling constant g, and serves as a calibration for accurately measuring the thermal occupancy and the measurement imprecision. The lowest three temperature points are excluded from the fit be- cause the mode is no longer in thermal equilibrium with the cryostat. © 2009 Macmillan Publishers Limited. All rights reserved.

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SUPPLEMENTARY INFORMATIONdoi: 10.1038/nnano.2009.343

nature nanotechnology | www.nature.com/naturenanotechnology 1

Supplementary Information for “Nanomechanical motion measured with animprecision below that at the standard quantum limit”

J. D. Teufel,1 T. Donner,1 M. A. Castellanos-Beltran,1, 2 J. W. Harlow,1, 2 and K. W. Lehnert1, 2, ∗1JILA, National Institute of Standards and Technology and the University of Colorado, Boulder, CO 80309, USA

2Department of Physics, University of Colorado, Boulder, CO 80309, USA

SPECTRAL DENSITIES EXPRESSED AS NOISEQUANTA

Throughout this paper we have expressed spectral den-sities using the single-sided convention [1]. This conven-tion gives the familiar classical result that an oscillatorcoupled to a thermal bath of temperature T will experi-ence a random force characterized by the force spectraldensity SF = 4kBTmγm, where m is the mass and γm isthe dissipation rate of the oscillator. More generally, thisforce is

SF = 4h̄ωm

�n̄ +

12

�mγm , (1)

where ωm is the oscillator’s resonance frequency and n̄is the Bose-Einstein occupancy factor given by n̄−1 =e

h̄ωmkBT − 1.In the limit of zero temperature, the average energy of

the oscillator is simply the half quantum of zero-point en-ergy and SF = 2h̄ωmmγm. This force is associated withdisplacement fluctuations on resonance with a spectraldensity

Sx(ωm) = SF |H(ωm)|2 =2h̄

mωmγm, (2)

where the mechanical susceptibility H(ω) is given byH(ω)−1 = m(ω2−ω2

m− iωγm). Independent of any spec-tral density convention, these zero-point fluctuations al-ways correspond to one half quantum of noise energy inthe oscillator. Thus, it is often useful to refer to the dis-placement spectral density on resonance in terms of thenoise quanta of energy. In the convention used here, onequarter quantum of noise energy is naturally associatedwith spectral density Sx = h̄/(mωmγm).

INFERRING DISPLACEMENTS FROMINTERFEROMETER RESPONSE

As described in previous work [2, 3], we infer the mo-tion of the wire from the measured frequency shift in thecavity resonance. The transmission past the cavity hasthe following resonant form

S21 =S0 + 2i

�ω−ωc

γc

1 + 2i�

ω−ωcγc

� , (3)

where S0 is the normalized transmission on resonance,ωc is the resonance frequency of the cavity, and γc is thelinewidth of the cavity. By measuring the transmissionpast the cavity, we find S0, ωc and γc. From these pa-rameters, we can determine the quality factor associatedwith loss inside the cavity Qint = (ωc/γc)/(S0) and thatassociated with intensional coupling to the transmissionline which passes the cavity Qext = (ωc/γc)/(1−S0). Forthis experiment, Qext = 3.0 × 103 and Qint has a typi-cal value of 2.6×104, but varies by 20% for the range oftemperatures and powers used here.

The amplified microwave signal emerging from thecryostat is demodulated with an inphase-quadraturemixer. The voltage V from the quadrature componentof the mixer is V = V0Im[S21], where V0 is found bymeasuring the transmission far from resonance. Thus,the spectral density of cavity frequency fluctuations canbe inferred from the voltage fluctuations as

Sωc(ωm) =1 + 4ω2

m/γ2c

(∂V/∂ωc)2 SV (ωm) , (4)

10

5

03002001000

2

2(

/2)

(kH

z)

cδω

π

Temperature (mK)

FIG. 1: Calibration of displacement measurement withthermal motion The mean-square fluctuations in the cav-ity’s resonance frequency due to the thermal motion of theoscillator as a function of cryostat temperature. The linear fit(solid line) determines the optomechanical coupling constantg, and serves as a calibration for accurately measuring thethermal occupancy and the measurement imprecision. Thelowest three temperature points are excluded from the fit be-cause the mode is no longer in thermal equilibrium with thecryostat.

© 2009 Macmillan Publishers Limited. All rights reserved.

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SUPPLEMENTARY INFORMATION doi: 10.1038/nnano.2009.3432

300 K

4 K

Tcryo

HEMT

20 d

B

20 d

B

3dB

JPA Cavity S.C. Coax

50Ω Termination

InterferometerInput

JPAPump

I

Q ϕ

cir

cplr

sw

FIG. 2: Detailed diagram of the microwave interferometer. A microwave generator creates a tone close to the cavityresonance frequency. That tone is split into three arms. The first arm excites the cavity. The second arm provides the pumptone for the Josephson parametric amplifier. The third arm provides a phase reference for the mixer at room temperature.Most of the power in the first arm is heavily attenuated at cryogenic temperatures before it reaches the cavity. The last stage ofattenuation occurs inside a 20 dB directional coupler (cplr), which allows us to combine the deterministic sinusoidal voltage ofthe microwave tone with one of two calibrated noise sources, chosen with a switch (sw). To minimize microwave loss at unknowntemperature, we use superconducting coaxial cables to carry signals between the cryostat’s base temperature Tcryo and 4 K.The JPA is a reflection amplifier; a signal incident on the JPA is amplified and reflected. A commercial cryogenic circulator(cir), which is specified and tested to perform at low temperatures, is used to separate the incident and reflected waves, definingthe input and output ports of the JPA. The other cryogenic circulators are used to isolate the cavity from the noise emitted bythe amplifiers’ inputs. Heat sinking of the center conductors of the coaxial cables is accomplished with cryogenic attenuatorsand directional couplers. On the three superconducting cables where it is critical to minimize the attenuation, bias tees (notshown) are used. The capacitively coupled RF arm of the bias tee interrupts the heat load from 4 K to the cold stage, and theDC arm thermally anchors the center conductor to base temperature of the cryostat.

where the factor of (1 + 4ω2m/γ2

c ) accounts for the factthat the cavity field does not respond instantaneously tochanges in ωc.

In order to calibrate the displacement of the nanome-chanical oscillator, we use the thermal motion of the wire.The optomechanical coupling constant g is determined insitu by measuring the increase in the fluctuations of thecavity’s resonance frequency with the cryostat tempera-ture at sufficiently high temperatures. This calibrationis shown in Fig. 1 and yields g = 2π × (32± 3) kHz/nm.This value of g is also independently determined to beg = 2π × (30 ± 6) kHz/nm by measuring the radia-tion pressure damping from a microwave tone applied de-tuned from the cavity’s resonance [3]. Under conditionswhere it is no longer valid to assume thermal equilibrium

with the cryogenic environment (such as low tempera-ture or high power), we can use g to directly measurethe temperature of the mode from the area under theLorentzian peak in the displacement spectral density Sx.This method also accurately determines the imprecisionof the measurement. The 4% uncertainty in the slope ofthe fit line in Fig. 1 is the dominant uncertainty in ourmeasurement of Simp

x /SSQLx . The fractional uncertainty

in g is larger because it also includes the uncertainty inthe mass.

© 2009 Macmillan Publishers Limited. All rights reserved.

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SUPPLEMENTARY INFORMATIONdoi: 10.1038/nnano.2009.3433

CALIBRATING THE ADDED NOISE OF THEINTERFEROMETER

To estimate the contributions to the added noise of themicrowave interferometer, we use a calibrated and vari-able source of microwave noise power [4]. As shown inFig. 2, there is a cryogenic switch on the input of theinterferometer that can toggle between two terminationswhich serve as thermal noise sources. One terminationis held at Tcryo =15 mK and the other at TH =4 K. Thedifference in the noise at the output of the interferom-eter for the two positions of the switch provides an insitu characterization of the total noise added by the en-tire measurement chain. To accurately refer this noiseto the output of the cavity, one must also measure themicrowave loss of the components held at Tcryo. Thisdetermination is accomplished by measuring the changein the output noise of the interferometer as a functionof Tcryo with the switch connected to the termination atTH =4 K. In the presence of loss, the noise from the ter-mination at TH will be partially absorbed. Furthermore,this loss will emit noise with a spectrum given by thetemperature of the loss. The noise in units of microwavequanta (methods ref. [4]) emerging from loss with powerattenuation A is

n =12

+A

eh̄ω

kBTH − 1+

1 − A

eh̄ω

kBTcryo − 1. (5)

This allows us to determine that there is 6 dB of totalloss between the switch and the JPA, of which 2 dB isbetween the cavity and the JPA.

From these measurements, we infer that the JPA adds0.3 microwave quanta of noise and that the HEMT am-plifier alone adds 24.5 microwave quanta of noise to ourmeasurement. When operated with the JPA, the contri-bution of the HEMT amplifier’s added noise is dividedby the JPA’s gain, typically 20 dB. Including the loss,we estimate that the interferometer adds 1.16 microwavequanta of noise to the measurement. In addition to thenoise added by the interferometer, there is also a smallamount of thermal noise from the input signal. The inputsignal is heavily attenuated by more than 50 dB at cryo-genic temperatures to reduce its thermal noise to about0.17 microwave quanta of noise. In total, we estimate thenumber of added quanta to be nadd = 1.3.

Our current design was chosen to allow us to simulta-neously measure multiple cavity optomechanical devices,at the cost of not measuring half of the microwave powerexiting the cavity. This effect is accounted for in the∂ϕ/∂x term of equation 1 in the main text. However,a simple modification of the geometry would allow us tomeasure all of the exiting power. For the same amount ofenergy stored in the resonator, this modification wouldimprove the imprecision by a factor of two.

INTRINSIC FREQUENCY NOISE OF THECAVITY

As we infer the mechanical displacement from the fre-quency shift of the microwave cavity, any other source ofresonance frequency fluctuation is indistinguishable frommotion of the wire. Intrinsic fluctuations in the reso-nance frequency of these lithographically fabricated su-perconducting microwave cavities are indeed an impor-tant source of technical noise[2, 5]. Figure 3 shows thenoise spectrum of our measurement for the measurementpower used to obtain our best imprecision; it is plottedin units of fractional frequency noise to facilitate com-parison with other work[5]. The blue trace shows theapparent frequency noise when the interferometer is readout with just the HEMT amplifier. There is a peak near1MHz from the thermal motion of the mechanical os-cillator; the noise floor comes primarily from the noiseadded by the HEMT amplifier. When we amplify withthe JPA (red), the noise floor near the mechanical res-onance frequency drops. However, at lower Fourier fre-quencies, the improvement is smaller owing to real cavityfrequency fluctuations. The black line shows the appar-ent frequency noise just due to the added noise of theamplification chain including the JPA.

FIG. 3: Cavity frequency noise. These spectra show thefractional frequency noise of the interferometer as a functionof Fourier frequency. A peak from the thermal motion ofthe oscillator is visible near 1MHz when measuring with theHEMT amplifier (blue) or the JPA (red). With the JPA, theintrinsic frequency fluctuations of the cavity are also visible,especially at low Fourier frequencies. For reference, the blackdata show the contribution from just the added noise of theinterferometer when using the JPA.

For the power shown (P = 1.4 nW) which was used tomeasure the best displacement imprecision, the frequencynoise near the mechanical resonance frequency is compa-rable to the added noise. For this reason, the displace-ment imprecision no longer improves with larger power.

© 2009 Macmillan Publishers Limited. All rights reserved.

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SUPPLEMENTARY INFORMATION doi: 10.1038/nnano.2009.3434

Our current device shows a fractional frequency noisecomparable to the lowest measured noise in aluminumresonators[5]. As the frequency noise of microwave res-onators is an active area of research, this noise sourcemay be further reduced by optimizing material parame-ters, fabrication techniques, and the resonator geometry.Even without reduction of the cavity frequency noise, itseffect on the displacement imprecision can be made lessimportant by increasing the optomechanical coupling g.

∗ Electronic address: [email protected][1] Flowers-Jacobs, N. E., Schmidt, D. R. & Lehnert, K. W.Intrinsic Noise properties of atomic point contact displace-

ment detectors. Phys. Rev. Lett. 98, 096804 (2007).[2] Regal, C. A., Teufel, J. D. & Lehnert, K. W. Measuringnanomechanical motion with a microwave cavity interfer-ometer. Nature Phys. 4, 555–560 (2008).

[3] Teufel, J. D., Harlow, J. W., Regal, C. A. & Lehnert,K. W. Dynamical backaction of microwave fields on ananomechanical oscillator. Phys. Rev. Lett. 101, 197203(2008).

[4] Castellanos-Beltran, M. A., Irwin, K. D., Hilton, G. C.,Vale, L. R. & Lehnert, K. W. Amplification and squeezingof quantum noise with a tunable Josephson metamaterial.Nature Phys. 4, 929–931 (2008).

[5] Gao, J. S., Zmuidzinas, J., Mazin, B. A., LeDuc, H. G. &Day, P. K. Noise properties of superconducting coplanarwaveguide microwave resonators. Appl. Phys. Lett. 90,102507 (2007).

© 2009 Macmillan Publishers Limited. All rights reserved.