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. Schwab
1
,*
1
Applied Physics, California Institute of Technology, Pasadena, California 91125, USA
2
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)
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
dB
of 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. The
interplay 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 and
elementary quantum states of an oscillator is a squeezed
state
[2]
, which is a minimum uncertainty state that has a
quadrature which is smaller than the zero-point level. Such
states have long been discussed in the context of gravita-
tional wave detection to improve the measurement sensi-
tivity
[3
–
5]
. It is well known that a coherent parametric
drive can be used to squeeze mechanical fluctuations
[6,7]
,
which is essentially equivalent to the technique first used to
squeeze ground-state optical fields
[8]
.However,the
maximum 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 steady
state 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 con-
tinuous quantum measurement and feedback
[9
–
12]
,butit
would substantially increase the experimental complexity.
Another method to generate a robust quantum state is
quantum reservoir engineering
[13]
, which has been used to
generate quantum squeezed states and entanglement with
trapped ions
[14,15]
and superconducting qubits
[16]
.It
can also be applied to an optomechanical system to
generate strong steady-state squeezing without quantum-
limited measurement and feedback
[17]
. By modulating the
optomechanical coupling with two imbalanced classical
drive tones, the driven cavity acts effectively as a squeezed
reservoir. When the engineered dissipation from the cavity
dominates the dissipation from the environment, the
mechanical resonator relaxes to a steady squeezed state.
This technique has been applied recently to generate
quantum squeezed states of macroscopic mechanical res-
onators
[18
–
20]
.
In addition to being a tool for state preparation, opto-
mechanics also provides a means to probe the quantum
behavior of macroscopic objects
[21
–
23]
. In particular, a
backaction-evading (BAE) measurement
[10,20,24
–
27]
of
a single motional quadrature can be implemented in an
optomechanical system. If the drive tones that modulate the
coupling are balanced, a continuous quantum nondemoli-
tion (QND) measurement of the mechanical quadrature can
be made. This technique can be used to fully reconstruct the
quantum state of the mechanical motion.
In this work, we combine reservoir engineering and
backaction-evading measurement with a microwave opto-
mechanical system to perform continuous QND measure-
ment of a quantum squeezed state. Among the previous
three squeezing experiments
[18
–
20]
, only Ref.
[20]
dem-
onstrated direct detection performed using a two-cavity
optomechanical system; here we implement both reservoir
engineering and BAE measurement simultaneously within
a simple single-cavity setup. In addition to the optome-
chanical interaction, a mechanical parametric effect is
observed. Contrary to previous works, where the mechani-
cal parametric effect produced parametric instability that
limited the precision of the BAE measurement
[26,28,29]
,
the interplay between the parametric drive and the engi-
neered dissipation enhances the mechanical squeezing. By
directly measuring the mechanical quadrature variances
with the BAE measurement, we demonstrate motional
quantum squeezing with squeezed quadrature variance
h
Δ
X
2
1
i¼
0
.
34
0
.
07
x
2
ZP
,
4
.
7
0
.
9
dB below the zero-
point level. This exceeds what is possible using only
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117,
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=
16
=
117(10)
=
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100801-1
© 2016 American Physical Society
parametric driving, even if one starts in the quantum ground
state. This is the first experiment to demonstrate more than
3 dB quantum squeezing in a macroscopic mechanical
system.
The mechanical oscillator in this work is a 100 nm thick,
40
×
40
μ
m
2
aluminum membrane, with fundamental reso-
nance frequency
ω
m
¼
2
π
×
5
.
8
MHz and mechanical line-
width
Γ
m
¼
2
π
×
8
Hz at 10 mK. It is capacitively coupled to
a lumped-element superconducting microwave resonator
with resonance frequency
ω
c
¼
2
π
×
6
.
083
GHz and damp-
ing rate
κ
¼
2
π
×
330
kHz [Fig.
1(a)
]. The mechanical
motion couples to the resonance frequency of the microwave
resonator through the modulation of the capacitance, with
an optomechanical coupling rate
g
0
¼ð
d
ω
c
=dx
Þ
x
ZP
¼
2
π
×
130
Hz, where
x
ZP
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ℏ
=
2
m
ω
m
p
¼
1
.
8
fm is the
amplitude of the zero-point fluctuation of the mechanical
oscillator with mass
m
¼
432
pg. The system is described by
the Hamiltonian
ˆ
H
¼
ℏ
ω
c
ˆ
a
†
ˆ
a
þ
ℏ
ω
m
ˆ
b
†
ˆ
b
−
ℏ
g
0
ˆ
a
†
ˆ
a
ð
ˆ
b
þ
ˆ
b
†
Þ
þ
i
ℏ
ffiffiffiffiffiffi
κ
in
p
ð
α
ð
t
Þ
ˆ
a
−
α
ð
t
Þ
ˆ
a
†
Þþ
ˆ
H
diss
;
ð
1
Þ
where
ˆ
a
ð
ˆ
a
†
Þ
is the annihilation (creation) operator of the
intracavity field,
ˆ
b
ð
ˆ
b
†
Þ
is the mechanical phonon annihilation
(creation) operator,
κ
in
is the coupling rate of the input
coupler, and
α
ð
t
Þ
is the external driving field. The term
ˆ
H
diss
accounts for dissipation.
To squeeze the mechanical motion, we drive the cavity
with a pair of pump tones at
ω
c
∓
ω
m
with intracavity
field
[17]
̄
α
sqz
ð
t
Þ¼ð
̄
α
−
e
i
ω
m
t
þ
̄
α
þ
e
−
i
ω
m
t
Þ
e
−
i
ω
c
t
;
ð
2
Þ
which is represented by the red and blue arrows in Fig.
1(b)
.
Linearizing the cavity dynamics in the standard way, the
pumps couple the microwave resonator to the Bogoliubov
mode of the mechanical motion with the Hamiltonian
ˆ
H
sqz
¼
−
ℏ
G
ð
ˆ
d
†
ˆ
β
þ
ˆ
d
ˆ
β
†
Þ
;
ð
3
Þ
where
ˆ
d
is the fluctuating part of the cavity field
ˆ
a
,
and
ˆ
β
¼
ˆ
b
cosh
r
þ
ˆ
b
†
sinh
r
is the Bogoliubov-mode anni-
hilation operator whose ground state is a squeezed state
with squeezing parameter
r
¼
tanh
−
1
ð
G
þ
=G
−
Þ
.
G
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
G
2
−
−
G
2
þ
p
is the coupling rate between the Bogoliubov
mode and the cavity.
G
∓
¼
g
0
ffiffiffiffiffiffi
n
∓
p
p
are the enhanced
optomechanical coupling rates, and
n
∓
p
¼j
̄
α
∓
j
2
are the
intracavity pump photon numbers corresponding to the
(a)
Δ
Δ
40
μ
m
X
1,2
2
x
zp
(b)
(c)
2
Γ
m
n
m
th
(rad/s)
n
c
th
(g)
(d)
(e)
(f)
n
p
+
/n
p
-
f-f
c
(MHz)
-1.5
-1
-0.5
0
0.5
1
1.5
0
2
4
6
S/P (10
-12
/Hz)
f-f
c
(MHz)
-1.5
-1
-0.5
0
0.5
1
1.5
0
1
2
S/P (10
-10
/Hz)
f-f
c
(kHz)
-80
-40
0
40
80
0
2.5
5
f-f
c
(kHz)
-5
-2.5
0
2.5
5
0.5
1
1.5
n
p
+
/n
p
-
0
0.2 0.4 0.6 0.8
1
-5
0
5
10
1
10
×
10
3
0
0.2
0.4
0.6
0.8
1
0.1
1
10
10
2
10
3
FIG. 1. (a) Optical micrograph of the device. The gray region is aluminum; the blue region is silicon. The square at the center is a
parallel 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
the squeezed states achieved at
n
tot
p
¼
1
.
35
×
10
4
(
n
tot
p
¼
1
.
85
×
10
5
). 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
n
þ
p
=n
−
p
¼
0
. The orange curves are
the predictions from Eq.
(4)
including cavity heating effect extracted from the experiment. (d) The cavity occupation
n
th
c
extracted from
the output spectra; the orange line is a linear fit of the pump ratio dependent heating. (e) The phonon bath heating rate
Γ
m
n
th
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 the
normalized 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 in
Fig.
1(c)
.
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squeezing pumps. The beam-splitter Hamiltonian in Eq.
(3)
enables us to cool the Bogoliubov mode into its ground
state, producing a stationary mechanical squeezed state
with quadrature variances
h
Δ
ˆ
X
2
1
;
2
i¼
x
2
zp
Γ
m
Γ
eff
ð
2
n
th
m
þ
1
Þþ
Γ
∓
opt
Γ
eff
ð
2
n
th
c
þ
1
Þ
;
ð
4
Þ
where
Γ
eff
¼
Γ
m
þ
4
G
2
=
κ
is the effective mechanical line-
width, and
Γ
∓
opt
¼
4
ð
G
−
∓
G
þ
Þ
2
=
κ
parametrizes the phase-
dependent driving of the mechanics by cavity fluctuations.
To extract the mechanical quadrature variance, we
measure the normalized noise spectra of the cavity reso-
nance [the lower panels of Figs.
1(f)
and
1(g)
] and the
mechanical sideband (the upper panels of Figs.
1(f)
and
1(g)
]. By fitting the normalized spectra with the optome-
chanical model
[30]
, we can extract the cavity occupation
n
th
c
and the phonon bath occupation
n
th
m
. Together with the
calibrations of the enhanced optomechanical coupling rate
G
, we can calculate the quadrature variances with
Eq.
(4) [17,18]
.
Figure
1(c)
shows the quadrature variances with various
intracavity pump photon ratio
n
þ
p
=n
−
p
. We start by squeez-
ing the mechanical motion with total intracavity pump
photon number
n
tot
p
¼
n
−
p
þ
n
þ
p
¼
1
.
35
×
10
4
and pump
photon ratio
n
þ
p
=n
−
p
¼
0
.
5
. This pump configuration gen-
erates a mechanical squeezed state with the squeezed
quadrature variance
h
Δ
ˆ
X
2
1
i¼
1
.
54
0
.
59
x
2
ZP
and the anti-
squeezed quadrature variance
h
Δ
ˆ
X
2
2
i¼
13
.
8
1
.
4
x
2
ZP
indi-
cated by the solid red circle and square in Fig.
1(c)
. The
corresponding normalized output spectra and the fits from
the two-tone optomechanical model
[17]
are shown in
Fig.
1(f)
. To further squeeze the mechanical motion, we can
increase the total pump photon number. The blue circles
(squares) in Fig.
1(c)
are the squeezed (antisqueezed)
quadrature variances at total intracavity pump photon
number
n
tot
p
¼
1
.
85
×
10
5
. The solid (dashed) blue curves
are the predictions from Eq.
(4)
with constant cavity and
mechanical occupations extracted from the output spectrum
at
n
þ
p
¼
0
; they agree with the data at low pump photon
ratio. At large pump photon ratio, the cavity bath starts to
heat up [Fig.
1(d)
], which increases the mechanical quad-
rature variances. The orange curves in Fig.
1(c)
are the
predictions from Eq.
(4)
including the cavity heating effect
extracted from the experiment [orange line in Fig.
1(d)
].
With the heating effect, the minimum quadrature variance
is achieved at
n
þ
p
=n
−
p
¼
0
.
43
with
h
Δ
ˆ
X
2
1
i¼
0
.
56
0
.
02
x
2
ZP
[the solid blue circle in Fig.
1(d)
],
2
.
5
0
.
2
dB below the
zero-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 to
have a more direct method that does not rely on assump-
tions about the mechanical dynamics. This can be achieved
in our system without needing to introduce an additional
cavity resonance: we continue to use the cavity density of
states near resonances to generate mechanical squeezing
but now use the density of states away from resonances to
make an independent, backaction-evading measurement of
a 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 the
squeezing.
To directly measure a single mechanical quadrature, in
addition to the squeezing pumps, we introduce another pair
of weak BAE probes [the purple arrows in Fig.
1(b)
]at
ω
c
∓
ω
m
−
Δ
with intracavity field
[18,27]
̄
α
BAE
ð
t
Þ¼
2
̄
α
cos
ð
ω
m
t
þ
φ
Þ
e
−
i
ð
ω
c
−
Δ
Þ
t
;
ð
5
Þ
where
φ
is the relative phase between the BAE probes and
the squeezing pumps. For a sideband-resolved system
(
ω
m
≫
κ
), the modulation of the BAE probes exclusively
couples the mechanical quadrature
ˆ
X
φ
¼
cos
φ
ˆ
X
1
−
sin
φ
ˆ
X
2
to the microwave resonance with the interaction
ˆ
H
I
¼
−
ℏ
G
ð
ˆ
d
†
e
−
i
Δ
t
þ
ˆ
de
i
Δ
t
Þ
ˆ
X
φ
x
ZP
;
ð
6
Þ
where
G
¼
g
0
ffiffiffiffiffi
n
p
p
is the enhanced optomechanical cou-
pling rate, and
n
p
¼j
̄
α
j
2
is the intracavity pump photon
number corresponding to the BAE probes. Since
ˆ
X
φ
is a
constant of motion of the system, the interaction
(6)
enables
a continuous QND measurement of the the mechanical
quadrature. By sweeping the probe phase
φ
, we can
perform 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
2
π
×
160
kHz
≫
Γ
eff
. 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, the
motional sideband spectrum of the BAE probes is mea-
sured, from which we can extract the mechanical quad-
rature variance and linewidth. In the following, we will
perform a BAE measurement to directly characterize the
weakly 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 as
a function of the probe phase
φ
. The red circles are the
quadrature variances of the weakly squeezed state mea-
sured with the BAE technique. The red curve is the
quadrature 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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measured with the BAE technique. The blue curve is
the quadrature variance inferred from the corresponding
output spectra [Fig.
1(g)
]. The minimum quadrature vari-
ance is achieved at
φ
¼
0
° with
h
Δ
ˆ
X
2
φ
i¼
0
.
34
0
.
07
x
2
ZP
,
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 measure-
ment 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 our
observation of a phase dependence of the quadrature
linewidth in the BAE measurement [Fig.
2(c)
]. Similar
induced mechanical parametric driving has been observed
in 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 this
mechanical parametric driving, we phenomenologically
add the mechanical parametric interaction to our otherwise
ideal optomechanical model
[30]
:
ˆ
H
para
¼
−
ℏ
λ
ð
e
i
ψ
ˆ
b
2
þ
e
−
i
ψ
ˆ
b
†
2
Þ
;
ð
7
Þ
where
λ
is the amplitude of the parametric interaction, and
ψ
is the relative phase between the parametric drive and the
squeezing pumps.
We fit the observed phase-dependent quadrature line-
width to our model, thus, extracting the amplitude and
phase of the parametric drive
[30]
. By assuming the phase
of 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 phase
shift), the model captures the observed phase-dependence
behavior of the quadrature linewidth, as shown by the
dashed curves in Fig.
2(c)
. Surprisingly, if one instead
assumes that the parametric driving is a result of the main
squeezing tones (i.e., take
ψ
a constant independent of
φ
),
one cannot capture the observed phase dependence of the
quadrature linewidth
[30]
. These results suggest that the
parametric drive is induced by the BAE probes.
The dashed curves in Fig.
2(b)
are the predicted quad-
rature 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 parametric
drive. We stress that our treatment of the spurious mechani-
cal parametric drive is phenomenological; we do not know
the precise microscopic mechanism which causes this
driving. Nonetheless, it allows us to explain both surprising
features of the BAE measurements (the observed phase-
dependent mechanical quadrature linewidth and the
enhanced squeezing) and provide a direction to engineer
the parametric drive to increase the squeezing
[30]
.
In conclusion, we combine reservoir engineering
and backaction-evading measurement in a microwave
optomechanical system to demonstrate a continuous
QND measurement of mechanical squeezed states. A
(a)
(b)
(c)
X
X
X
-160
-120
-80
-40
0
40
10
-1
10
0
10
1
f-f
m
(kHz)
-40
-20
0
20
40
0
0.25
0.5
0
5
10
X
2
x
zp
2
-160
-120
-80
-40
0
40
Γ
m
/2
(kHz)
0
5
10
15
x
zp
2
S
X
4
m
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 variance
equal 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 theoretical
predictions from the ideal optomechanical model. The dashed curves are the fits with the optomechanical model including the
mechanical parametric interaction.
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100801-4
spurious mechanical parametric effect is observed and
provides additional squeezing. Together with the spurious
mechanical parametric drive, the reservoir engineering
technique produces more than 3 dB squeezing below the
zero-point level. The present scheme can be applied to
generate and characterize more complicated quantum states
by 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 informa-
tion processing
[34]
, as well as fundamental study of
quantum decoherence
[35,36]
.
This work is supported by funding provided by the
Institute for Quantum Information and Matter, a NSF
Physics Frontiers Center with support of the Gordon and
Betty 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. NSF-
EEC 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.
*
schwab@caltech.edu
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