PRL 117, 140401 (2016)

PHYSICAL REVIEW LETTERS

week ending
30 SEPTEMBER 2016

Quantum Backaction Evading Measurement of Collective Mechanical Modes
C. F. Ockeloen-Korppi,1 E. Damskägg,1 J.-M. Pirkkalainen,1 A. A. Clerk,2 M. J. Woolley,3 and M. A. Sillanpää1,*
1

Department of Applied Physics, Aalto University, P.O. Box 15100, FI-00076 AALTO, Finland
Department of Physics, McGill University, 3600 rue University, Montréal, Quebec H3A 2T8, Canada
3
School of Engineering and Information Technology, UNSW Canberra, Australian Capital Territory 2600, Australia(Received 8 July 2016; published 26 September 2016)2

The standard quantum limit constrains the precision of an oscillator position measurement. It arises froma balance between the imprecision and the quantum backaction of the measurement. However, ameasurement of only a single quadrature of the oscillator can evade the backaction and be made with
arbitrary precision. Here we demonstrate quantum backaction evading measurements of a collective
quadrature of two mechanical oscillators, both coupled to a common microwave cavity. The work allows
for quantum state tomography of two mechanical oscillators, and provides a foundation for macroscopic
mechanical entanglement and force sensing beyond conventional quantum limits.
DOI: 10.1103/PhysRevLett.117.140401

The interplay of measurement imprecision and quantum
backaction limits the sensitivity with which the position of an
oscillator can be continuously monitored, to be at best equal tothe oscillator’s zero-point fluctuations [1,2]. This is known as
the standard quantum limit (SQL). However, any single
quadrature of the motion can, in principle, be measured
without limit, provided that the measurement backaction isshunted to the orthogonal quadrature. Such backaction evading (BAE) measurements [1,3,4], which are examples ofquantum nondemolition (QND) measurements, can beachieved by appropriately synchronizing the measurementwith the oscillator’s intrinsic motion. Classical analogs of
BAE measurements were demonstrated a long time ago[5–7]. In cavity optomechanical systems, where a mechanicaloscillator is dispersively coupled to a driven optical or
microwave cavity, BAE measurements that evade the quantum backaction [8–10] have recently been demonstrated [11].
In the context of generating squeezed states of mechanicalmotion [12–14] they have been used for detection [12,14], andthey have also been demonstrated in atomic spin systems [15].
BAE techniques have previously been discussed for two
coupled mechanical oscillators [16] and optomechanicalsystems [17,18], and demonstrated for two atomic spin
ensembles [19]. In a recent theoretical work [20], theconcept has been extended to the collective modes of two
uncoupled mechanical oscillators, each independently
coupled to an electromagnetic cavity. This type of measurement allows one to measure both quadratures of anarrow-band force applied to one of the oscillators without
any fundamental quantum limit [20]. Adding feedback
control [20], or perturbing the measurement slightly (i.e.,
reservoir engineering) [21], such measurements could be
used to generate steady-state entanglement between two
macroscopic mechanical oscillators.
In this Letter, we for the first time experimentally demonstrate such a mechanical two-mode BAE measurement.
We simultaneously achieve a measurement imprecision0031-9007=16=117(14)=140401(6)below the quantum zero-point fluctuations and an evasionof quantum backaction caused by microwave shot noise(below the backaction arising in a continuous position
measurement). The canonically conjugate quadrature isheated predominantly by the quantum backaction.
Our system is shown in Fig. 1(a). It consists of a microwave cavity resonator and two mechanical oscillators thathave no direct coupling. Previously, optomechanical systems
containing more than one mechanical oscillator have been
experimentally studied both in the optical [22–26] andmicrowave [27,28] regimes. Our cavity is a superconducting
on-chip LC resonator, with frequency ωa, decay rate κ, andmode operator a. The mechanical oscillators are realized asaluminum drumheads with mode operators bi , frequencies
ωi , and decay rates γ i (i ¼ 1, 2). Each mechanical oscillator
is individually coupled to the cavity via the radiationpressure interaction Hi ¼ gi a† aðb†i þ bi Þ, realizing athree-mode cavity optomechanical system [29]. The single-photon coupling rates gi arise from a position-dependentcapacitance between the cavity and each mechanicaloscillator. The total Hamiltonian is hence H ¼ ωa a† aþP†i ðωi bi bi þ H i Þ. We describe the system in a reference
frame set by the cavity frequency and the average of thetwo mechanical frequencies [30]. In thiswe definepﬃﬃframeﬃ
the position quadrature Xi ¼ ðb†i þ bi Þ= 2 and momentum
pﬃﬃﬃ
quadrature Pi ¼ iðb†i − bi Þ= 2, and then the collective
pﬃﬃﬃ
quadrature p
coordinates
X
¼
ðX

X
Þ=
2 and P ¼

1
2
ﬃﬃﬃ
ðP1  P2 Þ= 2. Note that these quadratures are not QND
variables, but have (as we will see) useful dynamics. Wheneach oscillator is in equilibrium with a bath of temperature T i ,it has the thermal occupation nTi ≃ kB T i =ℏωi . In such athermal state, the variances of all collective quadratures areequal: ðΔXT Þ2 ¼ ðΔPT Þ2 ¼ ðnT1 þ nT2 þ 1Þ=2, as depicted
in Fig. 1(c).
In order to achieve two-mode BAE using only two tones,
the system is simultaneously pumped at the frequencies

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PHYSICAL REVIEW LETTERS
where S0 ½ω ¼

P

σ¼ 2γ=½γweek ending
30 SEPTEMBER 20162 þ 4ðω þ σΩÞ2  andκ2
nBA ¼ 2C 2
ð2nTc þ 1Þ
κ þ 4Ω2

FIG. 1. Two-mode BAE measurement scheme. (a) Schematic
representation of our setup. Two micromechanical oscillators (ω1 ,ω2 ) are capacitively coupled to a superconducting microwave
resonator (ωa ). Pump tones are injected to port 1, and the output
spectrum of port 2 is measured on a signal analyzer. (b) Twostrong microwave pump tones (frequencies ω− and ωþ ) realize a
BAE measurement. The mechanical spectrum appears as sidebands (pink peaks) on the thermal cavity spectrum (green line).
The backaction can be probed by a set of weak probe tones (shortarrows), slightly detuned from the pump tones. (c) In thermal
equilibrium, the fluctuations (gray circle) of all collective quadratures are equal to ΔX Tþ . (d) The backaction of the BAE
measurement heats up the Pþ quadrature, whereas X þ remains
unaffected. The red axes represent the projection on a basis
defined by the probes.

ω ¼ ωa  ðω1 þ ω2 Þ=2 with equal (real) amplitudes a, as
shown in Fig. 1(b) [20]. Assuming the system is in the
resolved-sideband limit ωi ≫ κ, we neglect terms oscillatingat 2ωi . Further assuming that g1 ≈ g2 , the effective
Hamiltonian is
H ¼ ΩðXþ X − þ Pþ P− Þ þ 2Gða† þ aÞXþ ;ð1Þ

where Ω ¼ ðω1 − ω2 Þ=2 is the effective mechanical oscillator frequency, and the effective optomechanical coupling is
G ¼ ðg1 þ g2 Þa=2 [30]. Under this Hamiltonian, Xþ and P−
act dynamically like the position and momentum of a single
harmonic oscillator, even though they commute with one
another [18,20,38]. It is an example of the recently introduced concept of a “quantum-mechanics-free subsystem”[18], previously observed in an atomic system [19]. Only theXþ collective quadrature couples to the cavity, and ismeasured by observing the cavity output.
The spectra of the mechanical collective quadrature Xþ
and its conjugate collective quadrature Pþ are given by [30]1SXþ ½ω ¼ ðnT1 þ nT2 þ 1ÞS0 ½ω;2
1
SPþ ½ω ¼ ðnT1 þ nT2 þ 1 þ 2nBA ÞS0 ½ω;
2

ð2Þ

ð3Þ

is the total measurement backaction, with nTc being thethermal occupation of the cavity. The cooperativity is
introduced as C ¼ 4G2 =ðγκÞ with γ ¼ ðγ 1 þ γ 2 Þ=2. The
backaction can be divided into classical and quantum
contributions: nBA ¼ nBA;c þ nBA;q corresponding to,respectively, the terms 2nTc and þ1 inside the parentheses
in Eq. (3). We define the occupation number by nXþ þ 12 ¼
R
ðΔXþ Þ2 ¼ ð2πÞ−1 SXþ ½ωdω, and similarly for nPþ . As isevident from Eq. (2), only the Pþ quadrature is heated bybackaction, while the measured quantity Xþ remains
unaffected. Experimentally, there can be additional technical backaction via an increase in nTi from equilibrium
values due to pump current heating.
The mechanical spectrum SXþ ½ω is faithfully reproducedin the cavity output spectrum Sout ½ω. As shown in Fig. 1(b),it appears as sidebands at frequencies ωa  jΩj, on top of anoise floor due to experimental contributions [30]. This
allows for experimental reconstruction of the Xþ quadrature
by a spectral analysis of the scattered pump microwave light.
We stress that the symmetry inherent in Eq. (1) that
protects Xþ from the backaction heating of Pþ requires
closely matched single-photon optomechanical couplings.
This thus introduces an additional experimental complication compared to the single-mode optomechanical BAE
measurement. It should be noted that the coupling asymmetry cannot be compensated by tuning the pump power
ratio. In practice, one needs to have g1 ≈ g2 within about
10% accuracy, which is experimentally challenging. A
theoretical description including deviations from the ideal
case is thoroughly discussed in Ref. [20].
As schematically depicted in Fig. 1(a), the mechanicaloscillators [39] are connected to opposite ends of a transmission line cavity resonator, nearly 1 mm apart from eachother. The experiments are carried out in a dilution refrigerator at a temperature of 27 mK (unless stated otherwise).
The cavity is probed using a transmission measurement.
The cavity has the frequency ωa ≃ 2π × 5.5 GHz, total
linewidth κ ≃ 2π × 1.22 MHz dominated by coupling tothe output line with the rate κEo ≃ 2π × 0.98 MHz. The
coupling at the input side line is κEi ≃ 2π × 60 kHz, and theinternal decay rate is κ I ≃ 2π × 180 kHz. The mechanicaloscillators have the frequencies ω1 ¼ 2π × 10.0 MHz andω2 ¼ 2π × 11.3 MHz, and linewidths γ 1 ¼ 2π × 130 Hz
and γ 2 ¼ 2π × 150 Hz, respectively. For this sample weobtained g1 =g2 ≃ 0.94  0.02, which allows for a nearly
ideal two-mode BAE measurement [30]. The two pumptones need to be equal in amplitude as well, at the highest
cooperativities presented here at 0.1% accuracy. We cancalibrate the pump amplitude ratio with about 5% accuracy,
and then use the pump power ratio as an adjustable parameterwithin the calibrated window.

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FIG. 2. Backaction evading measurement. (a) Cavity output
spectra at ωa − jΩj (left) and at ωa þ jΩj (right), providing ameasurement of the X þ collective quadrature, for C ¼ 4.4, 18, 70(bottom to top). The solid lines are Lorentzian fits. (b) Integrated
area of the spectra (purple dots) as a function of measurement
strength given by the cooperativity. The solid line is expected for
perfect BAE; the dotted line includes technical heating. The redsquares correspond to cooling [30]. (c) Effective linewidth of
the spectral peaks corresponding to the data in (b). (d) BAE
thermometry (purple dots, nXþ ) for C ¼ 0.2 as a function ofcryostat temperature. The solid line is a fit to the data, used tocalibrate the mechanical signal. Red squares show the integrated
area for cooling (arbitrary units). [(e) and (f)] Effective thermal
occupation of the (e) mechanical oscillators and (f) cavity,
measured with a single red-sideband pump. Mechanical oscillator
2 (blue diamonds) shows technical heating. Dashed lines correspond to an empirical heating model.

Figure 2 shows the BAE measurement results. In Fig. 2(a)the measured cavity output spectra are shown for different
measurement strengths (cooperativities). Figure 2(b) shows
the integrated peak area, which corresponds to a photon
flux [30],


4γκ
1
nout ¼ κ Eo C 2nXþ þ :2
κ þ 4Ω2

ð4Þ

For cooperativity up to C ≃ 5, the data are in excellent
agreement with the expected linear behavior, indicating that
the Xþ quadrature is not perturbed by measurement backaction. At the largest measurement strengths available in our
experiment (limited by the technical requirement of equal
pump tone amplitudes), a small increase in nXþ is observed.
This is in good agreement with independently measured
technical heating of oscillator 2; see Fig. 2(e). The measured

week ending
30 SEPTEMBER 2016γ, shown in Fig. 2(c), is independent of the measurement
strength. As a comparison, we also carried out a measurement when only the red pump was switched on, characterized
by a strong backaction. Under these conditions, the backaction is associated with increased damping of each mechanical oscillator by the amount Γopt ¼ 4G2 κ=ðκ 2 þ 4Ω2 Þ, andconsequently cooling of each down to an occupation ni ¼
nTi γ i =ðγ i þ Γopt Þ [30]. As shown in Figs. 2(b) and 2(c), boththe occupation and linewidth strongly deviate from those in
the BAE scheme.
In order to calibrate the measurement results, we perform a
BAE measurement while varying the cryostat temperature T.
As shown in Fig. 2(d), for T ≳ 50 mK the measured nXþ islinear with T, indicating the system thermalizes with theenvironment. We use a linear fit to this data to calibrate themeasurements of nXþ . At the operating temperature T ¼
27 mK, the equilibrium occupation is 12 ðnT1 þ nT2 þ 1Þ ¼
62  2. The output photon flux nout is subsequently calibrated by comparing to Eq. (4) for low power (C ≲ 2).
By its nature, the BAE measurement only accesses theunperturbed collective quadrature Xþ . To fully quantify themeasurement backaction, we perform a second experiment,
where we additionally apply a second, weak BAE measurement. It is realized by two probe tones, offset δ ¼ 2π ×20 kHz below the pump tones, as shown in Fig. 1(a). The
probes perform a weak measurement, with cooperativity
Cprobe ≈ 0.3 ≪ C, such that the probe itself causes negligible
backaction. All tones are phase locked to a common
reference. By adjusting the phase of one of the probe tones
by an amount 2θ, the probes measure the generalized
collective quadrature Xθþ ¼ Xþ cos θ þ Pþ sin θ, as depicted
in figure Fig. 1(d). The probe tones cause two additional
peaks in the output spectrum, whose area corresponds tothe phase-dependent occupancy nθ as in Eq. (4). Here,nθ ¼ nXþ cos2 θ þ nPþ sin2 θ, assuming the correlations
between Xþ and Pþ are negligible. Note that other tomographic techniques for two-mode mechanical systems haverecently been demonstrated [40–42].
In Fig. 3(a) we display the total spectrum in the complete
pump-probe configuration. It consists of four peaks, twocorresponding to the strong pumps measuring Xþ andtwo corresponding to the weak probes measuring Xθþ .
In Fig. 3(b) we plot the measured occupancy nθ, showing
the strong phase dependence of the measurement backaction at large measurement strengths. Figure 3(c) shows
the quadrature occupations nXþ and nPþ , measured from asinusoidal fit to the data in panel (b), as well as nXþ
measured by the pumps, as a function of pump power.
The data are well described by a theoretical predictionthat includes quantum backaction on Pþ and technicalheating of nT2 . The quantum backaction (up to nBA;q ≈ 63)
dominates over the classical contribution (nBA;c ≈ 6).
The technical heating was independently calibrated withstandard optomechanical cooling measurements, shown in
Figs. 1(e) and 1(f), using a single pump with frequency

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FIG. 3. Backaction tomography. (a) Measured spectrum withpumps and weak probes for C ¼ 7, 22, 70 (bottom to top). Curves
are offset by a factor 10 for clarity, as indicated. The tall peaks are
due to the pumps, as was shown in Fig. 2(a). The short peaks are
the probe signal, measuring X þ (solid lines, θ ¼ 0°) and Pþ(dashed lines, θ ¼ 90°), respectively. (b) Effective occupation nθ
of the phase-dependent quadrature X θ obtained from the probe
signal. Data are shown for the same pump strengths as in (a).
Solid lines show the modeled backaction and technical heating.
(c) Effective occupations of X þ (red circles) and Pþ (bluediamonds) as functions of measurement strength. Open symbols
are extracted from the probe data in (b), while closed circles aremeasured from the pump spectra. Error bars show statisticalspread of the data (95% confidence) and thermal calibration
uncertainty. The solid line shows the occupation expected fromequilibration with the cryogenic environment, the dotted line
incorporates technical heating, and the dashed line additionally
includes the measurement backaction.

ωa − ωi (i ¼ 1, 2). For the two-tone measurement
the heating is given by the total pump power. We typicallyobserve heating in varying amounts in different
samples [13,27], likely related to the presence of surface
two-level systems [43].
An important benchmark for a sensitive measurement isthat the total measurement imprecision is below thefundamental quantum limit for continuous position detection. This has been demonstrated in single-oscillator
optomechanical systems [8,44–46]. The effective quadraimp
ture occupation can be written as nXþ ¼ nTXþ þ nBA
X þ þ nX þ ,where nTXþ ¼ ðnT1 þ nT2 Þ=2 is the initial occupation to bemeasured, nBA
X þ is the residual backaction including technical heating, and nimp
X þ is the imprecision noise. The latter,assuming the cavity output is measured with a high-gain
phase-insensitive amplifier, is [30]week ending
30 SEPTEMBER 2016

FIG. 4. Mechanical spectra and measurement precision.
(a) Measured mechanical spectra around ωa  jΩj with different
cooperativities as marked. (b) Measurement imprecision nimp
Xþ(black squares), technical backaction nBA
on
X
(red
circles),andþXþ
conjugate backaction nBA
Pþ on Pþ (blue diamonds). Symbols as inFig. 3(c). Colored lines show quantum-limited imprecision(blue), quantum backaction (red), and their sum (yellow), theminimum of which is the SQL. Dotted (dashed) lines showBA
the modeled nBA
X þ (nPþ ) including technical heating. A fit to theimprecision (black solid line) corresponds to namp ¼ 28.

nimp
Xþ ¼

1 T
1 κ 2 þ 4Ω2
nc þðnamp þ 1Þ;C
8C κκ Eoð5Þ

where namp is the noise of the amplifier. The imprecisionnimp
X þ can be made arbitrarily small by increasing thecooperativity. In the case of nTc ¼ namp ¼ 0, the backaction
noise on Pþ and imprecision noise of Xþ satisfy
nBA nimp
X þ ¼ 1=4, which arises due to phase-insensitive
amplification in the measurement chain. This relation isof a similar form as the Heisenberg imprecision backaction
uncertainty relation for continuous position detection [2].
Notice that in the BAE case, however, the backaction only
affects the unmeasured quadrature (which is dynamicallydecoupled from the measured quadrature) and hence has noeffect on the spectrum of the measured quantity Xþ.
Figure 4(a) shows sample measured spectra SXþ ½ω, scaled
such that the peak height corresponds to the number ofquanta. For increasing cooperativity the imprecision (noise
floor) is reduced. In Fig. 4(b) we show the measurement
imprecision and backaction against cooperativity. The backaction is calculated from the data in Fig. 3(c) by subtracting
the initial thermal occupation nTXþ , and shown for C > 2
where the signal-to-noise ratio is larger than 1. At large
measurement strengths, the measurement imprecision is well
below the quantum zero-point fluctuations of the oscillators.

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Furthermore, the total backaction and imprecision of Xþ isbelow the quantum backaction by 3.2  1.0 dB, demonstrating that our system outperforms a perfect phase-insensitive
position measurement. The stated uncertainty is dominatedby the statistical uncertainty (95% confidence), calculated
from the spread of the data in Fig. 3(b), but also includes thecalibration uncertainty of 0.3 dB.
In summary, we have performed sensitive measurementsof the collective motion of two mechanical oscillators without disturbance from the quantum backaction of the measurement. The measurement sensitivity exceeds the standard
quantum limit, leading to a strong backaction observed in the
canonically conjugate observable. The residual backaction isdominated by technical heating of one mechanical oscillator,
which does not correspond to a fundamental limitation.
Neglecting technical heating while keeping other parameters
unchanged, the backaction reduces to a few quanta, closely
approaching the SQL. Further, with an improved imprecisionnamp ∼ 1, this can be reached at small pump power, C ∼ 1.
The two-mode backaction evading measurement can be used
for quantum state tomography and for measuring both
quadratures of a force without a quantum limit. This work
also provides a foundation for the preparation and detectionof macroscopic mechanical entanglement.
We thank F. Massel and T. T. Heikkilä for useful
discussions. This work was supported by the Academyof Finland (Grant No. 250280, CoE LTQ, 275245) and by
the European Research Council (Grant No. 615755CAVITYQPD). The work benefited from the facilities at
the Micronova Nanofabrication Center and at the Low
Temperature Laboratory infrastructure.

*mika.sillanpaa@aalto.fi[1] C. M. Caves, K. S. Thorne, R. W. P. Drever, V. D. Sandberg,and M. Zimmermann, Rev. Mod. Phys. 52, 341 (1980).
[2] A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and
R. J. Schoelkopf, Rev. Mod. Phys. 82, 1155 (2010).
[3] V. B. Braginsky, Y. I. Vorontsov, and K. S. Thorne, Science
209, 547 (1980).
[4] A. A. Clerk, F. Marquardt, and K. Jacobs, New J. Phys. 10,095010 (2008).
[5] P. Bonifazi, C. Cinquegrana, E. Majorana, N. Pergola,
P. Puppo, P. Rapagnani, F. Ricci, S. Vaselli, and M. Visco,Phys. Lett. A 215, 141 (1996).
[6] T. Caniard, P. Verlot, T. Briant, P.-F. Cohadon, and
A. Heidmann, Phys. Rev. Lett. 99, 110801 (2007).
[7] J. B. Hertzberg, T. Rocheleau, T. Ndukum, M. Savva, A. A.
Clerk, and K. C. Schwab, Nat. Phys. 6, 213 (2010).
[8] J. D. Teufel, T. Donner, M. A. Castellanos-Beltran, J. W.
Harlow, and K. W. Lehnert, Nat. Nanotechnol. 4, 820(2009).
[9] D. W. C. Brooks, T. Botter, S. Schreppler, T. P. Purdy, N.
Brahms, and D. M. Stamper-Kurn, Nature (London) 488,476 (2012).
[10] T. P. Purdy, R. W. Peterson, and C. A. Regal, Science 339,
801 (2013).

week ending
30 SEPTEMBER 2016[11] J. Suh, A. J. Weinstein, C. U. Lei, E. E. Wollman, S. K.
Steinke, P. Meystre, A. A. Clerk, and K. C. Schwab, Science
344, 1262 (2014).
[12] E. E. Wollman, C. U. Lei, A. J. Weinstein, J. Suh, A.
Kronwald, F. Marquardt, A. A. Clerk, and K. C. Schwab,
Science 349, 952 (2015).
[13] J.-M. Pirkkalainen, E. Damskägg, M. Brandt, F. Massel, and
M. A. Sillanpää, Phys. Rev. Lett. 115, 243601 (2015).
[14] F. Lecocq, J. B. Clark, R. W. Simmonds, J. Aumentado, andJ. D. Teufel, Phys. Rev. X 5, 041037 (2015).
[15] G. Vasilakis, H. Shen, K. Jensen, M. Balabas, D. Salart, B.
Chen, and E. S. Polzik, Nat. Phys. 11, 389 (2015).
[16] R. Onofrio and A. Rioli, Phys. Rev. D 47, 2176 (1993).
[17] M. Tsang and C. M. Caves, Phys. Rev. Lett. 105, 123601(2010).
[18] M. Tsang and C. M. Caves, Phys. Rev. X 2, 031016 (2012).
[19] W. Wasilewski, K. Jensen, H. Krauter, J. J. Renema, M. V.
Balabas, and E. S. Polzik, Phys. Rev. Lett. 104, 133601(2010).
[20] M. J. Woolley and A. A. Clerk, Phys. Rev. A 87, 063846(2013).
[21] M. J. Woolley and A. A. Clerk, Phys. Rev. A 89, 063805(2014).
[22] Q. Lin, J. Rosenberg, D. Chang, R. Camacho, M.
Eichenfield, K. J. Vahala, and O. Painter, Nat. Photonics
4, 236 (2010).
[23] M. Zhang, G. S. Wiederhecker, S. Manipatruni, A. Barnard,
P. McEuen, and M. Lipson, Phys. Rev. Lett. 109, 233906(2012).
[24] A. B. Shkarin, N. E. Flowers-Jacobs, S. W. Hoch, A. D.
Kashkanova, C. Deutsch, J. Reichel, and J. G. E. Harris,
Phys. Rev. Lett. 112, 013602 (2014).
[25] H. Fu, T.-H. Mao, Y. Li, J.-F. Ding, J.-D. Li, and G. Cao,Appl. Phys. Lett. 105, 014108 (2014).
[26] N. Spethmann, J. Kohler, S. Schreppler, L. Buchmann, andD. M. Stamper-Kurn, Nat. Phys. 12, 27 (2016).
[27] F. Massel, S. U. Cho, J.-M. Pirkkalainen, P. J. Hakonen,
T. T. Heikkilä, and M. A. Sillanpää, Nat. Commun. 3, 987(2012).
[28] T. Faust, J. Rieger, M. J. Seitner, P. Krenn, J. P. Kotthaus,and E. M. Weig, Phys. Rev. Lett. 109, 037205 (2012).
[29] M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, Rev.
Mod. Phys. 86, 1391 (2014).
[30] See Supplemental Material http://link.aps.org/supplemental/
10.1103/PhysRevLett.117.140401,
which
includesRefs. [31–37], for experimental calibrations and theoreticalbackground.
[31] D. F. Walls and G. J. Milburn, Quantum Optics (Springer,
New York, 2008).
[32] A. J. Weinstein, C. U. Lei, E. E. Wollman, J. Suh, A.
Metelmann, A. A. Clerk, and K. C. Schwab, Phys. Rev.
X 4, 041003 (2014).
[33] C. W. Gardiner and P. Zoller, Quantum Noise (Springer,
New York, 2004).
[34] H.-P. Breuer and F. Petrucionne, The Theory of Open
Quantum Systems (Oxford University Press, Oxford, 2002).
[35] C. M. Caves, Phys. Rev. D 26, 1817 (1982).
[36] I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J.
Kippenberg, Phys. Rev. Lett. 99, 093901 (2007).
[37] F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin,
Phys. Rev. Lett. 99, 093902 (2007).

140401-5

PRL 117, 140401 (2016)

PHYSICAL REVIEW LETTERS[38] K. Hammerer, M. Aspelmeyer, E. S. Polzik, and P. Zoller,
Phys. Rev. Lett. 102, 020501 (2009).
[39] J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman,
K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, andR. W. Simmonds, Nature (London) 475, 359 (2011).
[40] I. Mahboob, H. Okamoto, K. Onomitsu, and H. Yamaguchi,
Phys. Rev. Lett. 113, 167203 (2014).
[41] Y. S. Patil, S. Chakram, L. Chang, and M. Vengalattore,
Phys. Rev. Lett. 115, 017202 (2015).
[42] A. Pontin, M. Bonaldi, A. Borrielli, L. Marconi, F. Marino,
G. Pandraud, G. A. Prodi, P. M. Sarro, E. Serra, and F.
Marin, Phys. Rev. Lett. 116, 103601 (2016).

week ending
30 SEPTEMBER 2016[43] K. B. Cooper, M. Steffen, R. McDermott, R. W. Simmonds,
S. Oh, D. A. Hite, D. P. Pappas, and J. M. Martinis, Phys.
Rev. Lett. 93, 180401 (2004).
[44] G. Anetsberger, E. Gavartin, O. Arcizet, Q. P. Unterreithmeier,E. M. Weig, M. L. Gorodetsky, J. P. Kotthaus, and T. J.
Kippenberg, Phys. Rev. A 82, 061804 (2010).
[45] T. Westphal, D. Friedrich, H. Kaufer, K. Yamamoto, S.
Goßler, H. Müller-Ebhardt, S. L. Danilishin, F. Y. Khalili, K.
Danzmann, and R. Schnabel, Phys. Rev. A 85, 063806(2012).
[46] D. J. Wilson, V. Sudhir, N. Piro, R. Schilling, A. Ghadimi,
and T. J. Kippenberg, Nature (London) 524, 325 (2015).

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