PRL 117, 100801 (2016)

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PHYSICAL REVIEW LETTERS

Quantum Nondemolition Measurement of a Quantum Squeezed
State Beyond the 3 dB Limit
C. U. Lei,1 A. J. Weinstein,1 J. Suh,2 E. E. Wollman,1 A. Kronwald,3,4 F. Marquardt,3,4 A. A. Clerk,5 and K. C. Schwab1,*
1

Applied Physics, California Institute of Technology, Pasadena, California 91125, USA
Korea Research Institute of Standards and Science, Daejeon 305-340, Republic of Korea
3
Friedrich-Alexander-Universität Erlangen-Nürnberg, Staudtstrasse 7, D-91058 Erlangen, Germany
4
Max Planck Institute for the Science of Light Günther-Scharowsky-Straße 1/Bau 24, D-91058 Erlangen, Germany
5
Department of Physics, McGill University, Montreal, Quebec, H3A 2T8 Canada(Received 27 May 2016; published 30 August 2016)2

We use a reservoir engineering technique based on two-tone driving to generate and stabilize a quantum
squeezed state of a micron-scale mechanical oscillator in a microwave optomechanical system. Using an
independent backaction-evading measurement to directly quantify the squeezing, we observe 4.7  0.9 dBof squeezing below the zero-point level surpassing the 3 dB limit of standard parametric squeezing
techniques. Our measurements also reveal evidence for an additional mechanical parametric effect. Theinterplay between this effect and the optomechanical interaction enhances the amount of squeezing
obtained in the experiment.
DOI: 10.1103/PhysRevLett.117.100801

Generating nonclassical states of a massive object has
been a subject of considerable interest. It offers a route
toward fundamental tests of quantum mechanics in an
unexplored regime [1]. One of the most important andelementary quantum states of an oscillator is a squeezed
state [2], which is a minimum uncertainty state that has aquadrature which is smaller than the zero-point level. Suchstates have long been discussed in the context of gravitational wave detection to improve the measurement sensitivity [3–5]. It is well known that a coherent parametricdrive can be used to squeeze mechanical fluctuations [6,7],
which is essentially equivalent to the technique first used tosqueeze ground-state optical fields [8]. However, themaximum steady-state squeezing achieved by this method
is limited to 3 dB due to the onset of parametric instability.
Therefore, it is, in principle, impossible to have a steadystate where the mechanical motion is squeezed below one
half of the zero-point level using only parametric driving.
These limitations may be overcome by combining continuous quantum measurement and feedback [9–12], but itwould substantially increase the experimental complexity.
Another method to generate a robust quantum state isquantum reservoir engineering [13], which has been used togenerate quantum squeezed states and entanglement withtrapped ions [14,15] and superconducting qubits [16]. It
can also be applied to an optomechanical system togenerate strong steady-state squeezing without quantumlimited measurement and feedback [17]. By modulating theoptomechanical coupling with two imbalanced classicaldrive tones, the driven cavity acts effectively as a squeezed
reservoir. When the engineered dissipation from the cavity
dominates the dissipation from the environment, themechanical resonator relaxes to a steady squeezed state.
0031-9007=16=117(10)=100801(5)This technique has been applied recently to generatequantum squeezed states of macroscopic mechanical resonators [18–20].
In addition to being a tool for state preparation, optomechanics also provides a means to probe the quantum
behavior of macroscopic objects [21–23]. In particular, abackaction-evading (BAE) measurement [10,20,24–27] ofa single motional quadrature can be implemented in an
optomechanical system. If the drive tones that modulate thecoupling are balanced, a continuous quantum nondemolition (QND) measurement of the mechanical quadrature canbe made. This technique can be used to fully reconstruct thequantum state of the mechanical motion.
In this work, we combine reservoir engineering andbackaction-evading measurement with a microwave optomechanical system to perform continuous QND measurement of a quantum squeezed state. Among the previous
three squeezing experiments [18–20], only Ref. [20] demonstrated direct detection performed using a two-cavity
optomechanical system; here we implement both reservoir
engineering and BAE measurement simultaneously withina simple single-cavity setup. In addition to the optomechanical interaction, a mechanical parametric effect isobserved. Contrary to previous works, where the mechanical parametric effect produced parametric instability that
limited the precision of the BAE measurement [26,28,29],
the interplay between the parametric drive and the engineered dissipation enhances the mechanical squeezing. By
directly measuring the mechanical quadrature varianceswith the BAE measurement, we demonstrate motional
quantum squeezing with squeezed quadrature variancehΔX 21 i ¼ 0.34  0.07x2ZP , 4.7  0.9 dB below the zeropoint level. This exceeds what is possible using only

100801-1© 2016 American Physical Society

PRL 117, 100801 (2016)(creation) operator, κin is the coupling rate of the input
coupler, and αðtÞ is the external driving field. The termĤdiss accounts for dissipation.
To squeeze the mechanical motion, we drive the cavity
with a pair of pump tones at ωc ∓ ωm with intracavity
field [17]parametric driving, even if one starts in the quantum groundstate. This is the first experiment to demonstrate more than
3 dB quantum squeezing in a macroscopic mechanicalsystem.
The mechanical oscillator in this work is a 100 nm thick,
40 × 40 μm2 aluminum membrane, with fundamental resonance frequency ωm ¼ 2π × 5.8 MHz and mechanical linewidth Γm ¼ 2π × 8 Hz at 10 mK. It is capacitively coupled toa lumped-element superconducting microwave resonatorwith resonance frequency ωc ¼ 2π × 6.083 GHz and damping rate κ ¼ 2π × 330 kHz [Fig. 1(a)]. The mechanicalmotion couples to the resonance frequency of the microwave
resonator through the modulation of the capacitance, with
an optomechanical couplingpﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
rate g0 ¼
ﬃ ðdωc =dxÞxZP ¼
2π × 130 Hz, where xZP ¼ ℏ=2mωm ¼ 1.8 fm is theamplitude of the zero-point fluctuation of the mechanicaloscillator with mass m ¼ 432 pg. The system is described by
the HamiltonianĤ ¼ ℏωc â† â þ ℏωm b̂† b̂ − ℏg0 â† âðb̂ þ b̂† Þ
pﬃﬃﬃﬃﬃﬃ
þ iℏ κin ðα ðtÞâ − αðtÞâ† Þ þ Ĥdiss ;ᾱsqz ðtÞ ¼ ðᾱ− eiωm t þ ᾱþ e−iωm t Þe−iωc t ;Ĥsqz ¼ −ℏGðd̂† β̂ þ d̂β̂† Þ;

ð1Þ

n cth

(d) 10102

× 103

Γm n th(rad/s)m

X 21,2

2
xzp

1

(e)

10

1

40 μ m
0

0.2

0.4

Δ

0.6

0.8

n +p /n -p(f)
S/P (10-10/Hz)

(b)

5
0
-5

1

0

0.2

0.4

0.6

0.8

1

n +p /n -p
5

1.5
1
0.5
-5

-2.5

0

2.5

5

f-f c (kHz)
2
1
0
-1.5

10

(g)

-1

-0.5

0

0.5

f-f c (MHz)1

1.5

S/P (10-12/Hz)

0.1

ð3Þ

where d̂ is the fluctuating part of the cavity field â,and β̂ ¼ b̂ cosh r þ b̂† sinh r is the Bogoliubov-mode annihilation operator whose ground state is a squeezed statewith squeezing parameter r ¼ tanh−1 ðGþ =G− Þ. G ¼
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
G2− − G2þ is the coupling rate between the Bogoliubov
pﬃﬃﬃﬃﬃﬃ
mode and the cavity. G∓ ¼ g0 n∓
p are the enhanced
2
optomechanical coupling rates, and n∓
p ¼ jᾱ∓ j are theintracavity pump photon numbers corresponding to the(c) 103(a)ð2Þ

which is represented by the red and blue arrows in Fig. 1(b).
Linearizing the cavity dynamics in the standard way, thepumps couple the microwave resonator to the Bogoliubov
mode of the mechanical motion with the Hamiltonian

where âðâ† Þ is the annihilation (creation) operator of theintracavity field, b̂ðb̂† Þ is the mechanical phonon annihilationΔweek ending
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PHYSICAL REVIEW LETTERS

2.5
0
-80

-40

0

40

80

f-f c (kHz)
6
4
2
0
-1.5

-1

-0.5

0

0.5

1

1.5

f-f c (MHz)FIG. 1. (a) Optical micrograph of the device. The gray region is aluminum; the blue region is silicon. The square at the center is aparallel plate capacitor which is coupled to a spiral inductor to form a microwave resonator. The top gate of the capacitor is a compliant
membrane whose fundamental motion is being studied. (b) Schematic of the pumps (red and blue arrows) and probes (purple arrows)
relative to the cavity resonance. The inset shows the schematic of the BAE probe sideband spectrum. (c) The squeezed quadrature
variances (circles) and antisqueezed quadrature variances (squares) inferred from the output spectra. The red (blue) symbols represent
4
tot
5
the squeezed states achieved at ntot
p ¼ 1.35 × 10 (np ¼ 1.85 × 10 ). The blue shaded region indicates sub-zero-point squeezing. The−
blue curves are the predictions from Eq. (4) with constant cavity and mechanical occupations at nþ
p =np ¼ 0. The orange curves are
the predictions from Eq. (4) including cavity heating effect extracted from the experiment. (d) The cavity occupation nth
c extracted fromthe output spectra; the orange line is a linear fit of the pump ratio dependent heating. (e) The phonon bath heating rate Γm nth
m extracted
from the output spectra. (f),(g) The output spectra normalized by the transmitted power of the red-detuned pump. The upper panel is thenormalized noise spectrum of the mechanical sideband, the lower panel is the noise spectrum of the cavity resonance. (f) The normalized
output spectra correspond to the solid red circle in Fig. 1(c). (g) The normalized output spectra correspond to the solid blue circle inFig. 1(c).

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squeezing pumps. The beam-splitter Hamiltonian in Eq. (3)enables us to cool the Bogoliubov mode into its groundstate, producing a stationary mechanical squeezed statewith quadrature variances
hΔX̂21;2 i ¼ x2zp


Γ∓
Γm
opt
th
th
ð2nm þ 1Þ þ
ð2nc þ 1Þ ;
Γeff
Γeff

ð4Þ

where Γeff ¼ Γm þ 4G2 =κ is the effective mechanical line2
width, and Γ∓
opt ¼ 4ðG− ∓ Gþ Þ =κ parametrizes the phasedependent driving of the mechanics by cavity fluctuations.
To extract the mechanical quadrature variance, wemeasure the normalized noise spectra of the cavity resonance [the lower panels of Figs. 1(f) and 1(g)] and themechanical sideband (the upper panels of Figs. 1(f) and
1(g)]. By fitting the normalized spectra with the optomechanical model [30], we can extract the cavity occupationthnth
c and the phonon bath occupation nm . Together with thecalibrations of the enhanced optomechanical coupling rateG , we can calculate the quadrature variances with
Eq. (4) [17,18].
Figure 1(c) shows the quadrature variances with various−
intracavity pump photon ratio nþ
p =np . We start by squeezing the mechanical motion with total intracavity pump−þ4photon number ntot
p ¼ np þ np ¼ 1.35 × 10 and pump−
photon ratio nþ
p =np ¼ 0.5. This pump configuration generates a mechanical squeezed state with the squeezed
quadrature variance hΔX̂21 i ¼ 1.54  0.59x2ZP and the antisqueezed quadrature variance hΔX̂22 i ¼ 13.8  1.4x2ZP indicated by the solid red circle and square in Fig. 1(c). The
corresponding normalized output spectra and the fits fromthe two-tone optomechanical model [17] are shown in
Fig. 1(f). To further squeeze the mechanical motion, we canincrease the total pump photon number. The blue circles(squares) in Fig. 1(c) are the squeezed (antisqueezed)
quadrature variances at total intracavity pump photon
5
number ntot
p ¼ 1.85 × 10 . The solid (dashed) blue curves
are the predictions from Eq. (4) with constant cavity andmechanical occupations extracted from the output spectrum
at nþ
p ¼ 0; they agree with the data at low pump photon
ratio. At large pump photon ratio, the cavity bath starts toheat up [Fig. 1(d)], which increases the mechanical quadrature variances. The orange curves in Fig. 1(c) are thepredictions from Eq. (4) including the cavity heating effectextracted from the experiment [orange line in Fig. 1(d)].
With the heating effect, the minimum quadrature variance2−2
is achieved at nþ
p =np ¼ 0.43 with hΔX̂ 1 i ¼ 0.56  0.02xZP[the solid blue circle in Fig. 1(d)], 2.5  0.2 dB below thezero-point level. The corresponding normalized output
spectra and fits are shown in Fig. 1(g).
While inferring the level of squeezing from the cavity
output spectrum is convenient, it would be preferable tohave a more direct method that does not rely on assumptions about the mechanical dynamics. This can be achieved

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in our system without needing to introduce an additional
cavity resonance: we continue to use the cavity density ofstates near resonances to generate mechanical squeezing
but now use the density of states away from resonances tomake an independent, backaction-evading measurement ofa single mechanical quadrature. In this way, our single
cavity effectively plays the role of two: it both generates
squeezing and permits an independent detection of thesqueezing.
To directly measure a single mechanical quadrature, in
addition to the squeezing pumps, we introduce another pairof weak BAE probes [the purple arrows in Fig. 1(b)] atωc ∓ ωm − Δ with intracavity field [18,27]
ᾱBAE ðtÞ ¼ 2ᾱ cosðωm t þ ϕÞe−iðωc −ΔÞt ;ð5Þ

where ϕ is the relative phase between the BAE probes andthe squeezing pumps. For a sideband-resolved system(ωm ≫ κ), the modulation of the BAE probes exclusively
couples the mechanical quadrature X̂ϕ ¼ cos ϕX̂1 −sin ϕX̂2 to the microwave resonance with the interactionĤI ¼ −ℏGðd̂† e−iΔt þ d̂eiΔt Þ

X̂ϕ
;
xZP

ð6Þ

pﬃﬃﬃﬃﬃ
where G ¼ g0 np is the enhanced optomechanical coupling rate, and np ¼ jᾱj2 is the intracavity pump photon
number corresponding to the BAE probes. Since X̂ϕ is aconstant of motion of the system, the interaction (6) enables
a continuous QND measurement of the the mechanicalquadrature. By sweeping the probe phase ϕ, we canperform tomography of the mechanical quantum state[Fig. 2(a) ]. In order to ensure no interference between
the sidebands of the squeezing pumps and the BAE probes,
we detune the BAE sidebands from the cavity resonance by
Δ ¼ 2π × 160 kHz ≫ Γeff . The power of the BAE probes
are set about 10 dB weaker than the power of the squeezing
pumps to avoid extra heating. In the experiment, themotional sideband spectrum of the BAE probes is measured, from which we can extract the mechanical quadrature variance and linewidth. In the following, we will
perform a BAE measurement to directly characterize theweakly squeezed state corresponding to the spectrum
Fig. 1(f) and the strong squeezed state corresponding to
the spectrum Fig. 1(g).
Figure 2(b) shows the mechanical quadrature variance asa function of the probe phase ϕ. The red circles are thequadrature variances of the weakly squeezed state measured with the BAE technique. The red curve is thequadrature variance inferred from the corresponding output
spectrum [Fig. 1(f)]. In this case, the results from the BAE
measurement are in good agreement with the results
inferred from the output spectra. Similarly, the blue circles
are the quadrature variances of the strong squeezed state

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PRL 117, 100801 (2016)(c) 15

(b)
X
X

x2zp

10
5
0
0.5
0.25
0
-40

10 -1
-160

Γ m/2 (kHz)

m

10

0

SX

X

4

X2

2
xzp

10 1

-20

0

20

-80

5

40

f-f m (kHz)
-120

10

-40

0

40

0
-160

-120

-80

-40

0

40

FIG. 2. (a) Schematic of dissipative mechanical squeezing. The gray circle represents the initial thermal state in phase space. The
engineered reservoir generates phase-dependent dissipation that relaxes the mechanics into a squeezed state, which is represented by
the blue ellipse. The gray dashed circle represents the zero-point level. (b) Mechanical quadrature variance as a function of probe phase.
The blue shaded region indicates sub-zero-point squeezing. The red (blue) circles are the quadrature variances of the weakly (strong)squeezed state as measured using the BAE technique. The solid curves are the quadrature variances inferred from the corresponding
output spectra assuming no mechanical parametric drive. The dashed curves are the predictions of an optomechanical model including
the mechanical parametric effect. The insets are the mechanical quadrature spectra of the strong squeezed state with phase ϕ at −70°(red), −50° (green), −20° (yellow), 0° (blue). The gray Lorentzian in the lower inset represents the spectrum with quadrature varianceequal to half of the zero-point fluctuation (the 3 dB limit). (c) Mechanical quadrature linewidth as a function of probe phase. The red(blue) circles are the measured mechanical quadrature linewidths of the weakly (strong) squeezed state. The solid lines are the theoreticalpredictions from the ideal optomechanical model. The dashed curves are the fits with the optomechanical model including themechanical parametric interaction.

measured with the BAE technique. The blue curve isthe quadrature variance inferred from the corresponding
output spectra [Fig. 1(g)]. The minimum quadrature variance is achieved at ϕ ¼ 0° with hΔX̂2ϕ i ¼ 0.34  0.07x2ZP ,4.7  0.9 dB below the zero-point level. This is lower than
the quadrature variance inferred from the output spectrum,
implying that there is additional dynamics at play (beyond
the ideal optomechanical interaction).
The enhanced squeezing observed in the BAE measurement suggests an additional squeezing mechanism beyond
the dissipative mechanism discussed above; an obvious
candidate is direct parametric driving of the mechanics. The
presence of such driving is further corroborated by ourobservation of a phase dependence of the quadrature
linewidth in the BAE measurement [Fig. 2(c)]. Similar
induced mechanical parametric driving has been observedin other BAE measurements; they can arise via a number of
mechanisms including thermal effects as well as higher
nonlinearities [28,29]. To understand the effects of thismechanical parametric driving, we phenomenologically
add the mechanical parametric interaction to our otherwise
ideal optomechanical model [30]:Ĥpara ¼ −ℏλðeiψ b̂2 þ e−iψ b̂†2 Þ;

ð7Þ

where λ is the amplitude of the parametric interaction, andψ is the relative phase between the parametric drive and thesqueezing pumps.
We fit the observed phase-dependent quadrature linewidth to our model, thus, extracting the amplitude andphase of the parametric drive [30]. By assuming the phaseof the parametric drive ψ follows the phase of the BAE
probe (i.e., ψ ¼ ϕ þ ψ 0 , where ψ 0 is a constant phaseshift), the model captures the observed phase-dependence
behavior of the quadrature linewidth, as shown by thedashed curves in Fig. 2(c). Surprisingly, if one instead
assumes that the parametric driving is a result of the mainsqueezing tones (i.e., take ψ a constant independent of ϕ),one cannot capture the observed phase dependence of thequadrature linewidth [30]. These results suggest that theparametric drive is induced by the BAE probes.
The dashed curves in Fig. 2(b) are the predicted quadrature variances including the mechanical parametric effect.
The model suggests that the combination of the reservoir
engineering with the mechanical parametric drive provides
extra squeezing. However, the model does not fully capture
the observed quadrature variance in the BAE measurement.
The deviation may be due to the complicated heating
effects associated with the underlaying nonlinearities[28,29] that cause the spurious mechanical parametricdrive. We stress that our treatment of the spurious mechanical parametric drive is phenomenological; we do not know
the precise microscopic mechanism which causes thisdriving. Nonetheless, it allows us to explain both surprisingfeatures of the BAE measurements (the observed phasedependent mechanical quadrature linewidth and theenhanced squeezing) and provide a direction to engineer
the parametric drive to increase the squeezing [30].
In conclusion, we combine reservoir engineeringand backaction-evading measurement in a microwave
optomechanical system to demonstrate a continuous
QND measurement of mechanical squeezed states. A

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spurious mechanical parametric effect is observed andprovides additional squeezing. Together with the spurious
mechanical parametric drive, the reservoir engineering
technique produces more than 3 dB squeezing below thezero-point level. The present scheme can be applied togenerate and characterize more complicated quantum statesby carefully engineering the nonlinear interaction [32,33].
The ability to generate and measure a strong quantum
squeezed state of a macroscopic mechanical object would
be useful for ultrasensitive detection [3], quantum information processing [34], as well as fundamental study ofquantum decoherence [35,36].
This work is supported by funding provided by theInstitute for Quantum Information and Matter, a NSF
Physics Frontiers Center with support of the Gordon andBetty Moore Foundation (Grant No. NSF-IQIM 1125565),
by the Defense Advanced Research Projects Agency(Grant No. DARPA-QUANTUM HR0011-10-1-0066), by
the NSF (Grants No. NSF-DMR 1052647 and No. NSFEEC 0832819), and by the Semiconductor Research
Corporation and Defense Advanced Research Project
Agency (DARPA) through STARnet Center for Function
Accelerated nanoMaterial Engineering. J. S. was supported
by Basic Science Research Program through the
National Research Foundation of Korea funded by the
Ministry of Science, ICT Future Planning (Grants
No. 2016R1C1B2014713 and No. 2016R1A5A1008184).
A. C., F. M., and A. K. acknowledge support from the
DARPA ORCHID program through a grant from
AFOSR, F. M. and A. K. from ITN cQOM and the ERC
OPTOMECH, and A. C. from NSERC.

*

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