of 12
Quantum nondemolition measurement of mechanical 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
K. C. Schwab
1
1
Applied Physics, California Institute of Technology, Pasadena, CA 91125, USA
2
Korea Research Institute of Standards and Science, Daejeon 305-340, Republic of Korea
3
Friedrich-Alexander-Universit ̈at Erlangen-N ̈urnberg, Staudtstr. 7, D-91058 Erlangen, Germany
4
Max Planck Institute for the Science of Light G ̈unther-Scharowsky-Straße 1/Bau 24, D-91058 Erlangen, Germany and
5
Department of Physics, McGill University, Montreal, Quebec, H3A 2T8 Canada
(Dated: May 30, 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 stan-
dard 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.
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]: a minimum uncertainty state has a quadrature
which is smaller than the zero-point level. Such states
have long been discussed in the context of gravitational
waves detection to improve the measurement sensitiv-
ity [3, 4]. It is well known that a coherent paramet-
ric drive can be used to squeeze mechanical fluctuations
[5, 6], which is essentially equivalent to the technique
first used to squeeze ground-state optical fields [7]. How-
ever, the maximum steady-state squeezing achieved by
this method is limited to 3 dB due to the onset of para-
metric instability. Therefore, it is in principle impossi-
ble 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 over-
come by combining continuous quantum measurement
and feedback [8–11], but it would substantially increase
the experimental complexity.
Another method to generate robust quantum state is
quantum reservoir engineering [12], which has been used
to generate quantum squeezed states and entanglement
with trapped ions [13, 14] and superconducting qubits
[15]. It can also applied to optomechanical system to
generate strong steady-state squeezing without quantum-
limited measurement and feedback [16]. By modulating
the optomechanical coupling with two imbalanced clas-
sical drive tones, the driven cavity acts effectively as
a squeezed reservoir. When the engineered dissipation
from the cavity dominates the dissipation from the en-
vironment, the mechanical resonator relaxes to a steady
squeezed state. This technique has been applied recently
to generate quantum squeezed states of macroscopic me-
chanical resonators [17–19].
In addition to being a tool for state preparation, op-
tomechanics also provides a means to probe the quantum
behavior of macroscopic objects [20–22]. In particular, a
backaction evading (BAE) measurement [9, 19, 23–26] 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 nonde-
molition (QND) measurement of the mechanical quadra-
ture 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 op-
tomechanical system to perform continuous QND mea-
surement of a quantum squeezed state.
Among the
previous three squeezing experiments [17–19], only [19]
demonstrated direct detection, performed using a two-
cavity optomechanical system; here we implement both
reservoir engineering and BAE measurement simultane-
ously within a simple single-cavity setup. In addition to
the optomechanical interaction, a mechanical paramet-
ric effect is observed. Contrary to previous works, where
the mechanical parametric effect produced parametric in-
stability that limited the precision of the BAE measure-
ment [25, 27, 28], the interplay between the parametric
drive and the engineered dissipation enhances the me-
chanical squeezing. By directly measuring the mechani-
cal quadrature variances with the BAE measurement, we
demonstrate motional quantum squeezing with squeezed
quadrature variance
X
2
1
= 0
.
34
±
0
.
07
x
2
zp
, 4
.
7
±
0
.
9
dB below the zero-point level. This exceeds what is pos-
sible using only 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 fundamen-
tal resonance frequency
ω
m
= 2
π
×
5
.
8 MHz and mechan-
ical linewidth
γ
m
= 2
π
×
8 Hz at 10 mK. It is capacitively
coupled to a lumped-element superconducting microwave
arXiv:1605.08148v2 [quant-ph] 27 May 2016
2
resonator with resonance frequency
ω
c
= 2
π
×
6
.
083 GHz
and damping rate
κ
= 2
π
×
330 kHz (Fig. 1a). 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
=
c
dx
x
zp
= 2
π
×
130 Hz, where
x
zp
=
~
2
m
= 1
.
8
fm is the amplitude of the zero-point fluctuation of the
mechanical oscillator with mass
m
= 432 pg. The system
is described by the Hamiltonian
ˆ
H
=
~
ω
c
ˆ
a
ˆ
a
+
~
ω
m
ˆ
b
ˆ
b
~
g
0
ˆ
a
ˆ
a
(
ˆ
b
+
ˆ
b
)
+
i
~
κ
in
(
α
(
t
a
α
(
t
a
) +
ˆ
H
diss
,
(1)
where ˆ
a
(
ˆ
a
)
is the annihilation (creation) operator of
the intra-cavity field,
ˆ
b
(
ˆ
b
)
is the mechanical phonon
annihilation (creation) operator,
κ
in
is the coupling rate
of the input coupler, and
α
(
t
) 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 [16]
̄
α
sqz
(
t
) = ( ̄
α
e
m
t
+ ̄
α
+
e
m
t
)
e
c
t
,
(2)
which is represented by the red and blue arrows in
Fig. 1b. Linearizing the cavity dynamics in the stan-
dard way, the pumps couple the microwave resonator to
the Bogoliubov mode of the mechanical motion with the
Hamiltonian
ˆ
H
sqz
=
~
G
(
ˆ
d
ˆ
β
+
ˆ
d
ˆ
β
)
,
(3)
where
ˆ
d
is the fluctuating part of the cavity field ˆ
a
,
and
ˆ
β
=
ˆ
b
cosh
r
+
ˆ
b
sinh
r
is the Bogoliubov-mode an-
nihilation operator whose ground state is a squeezed
state with squeezing parameter
r
= tanh
1
(
G
+
/G
).
G
=
G
2
G
2
+
is the coupling rate between the Bo-
goliubov mode and the cavity.
G
=
g
0
n
p
are the en-
hanced optomechanical coupling rates, and
n
p
=
|
̄
α
|
2
are the intracavity pump photon numbers corresponding
to the squeezing pumps.
The beam-splitter Hamiltonian in Eq. (3) enables us to
cool the Bogoliubov-mode into its ground state, produc-
ing a stationary mechanical squeezed state with quadra-
ture variances
ˆ
X
2
1
,
2
=
x
2
zp
{
Γ
m
Γ
eff
(
2
n
th
m
+ 1
)
+
Γ
opt
Γ
eff
(
2
n
th
c
+ 1
)
}
,
(4)
where Γ
eff
= Γ
m
+ 4
G
2
is the effective mechanical
linewidth and Γ
opt
= 4(
G
G
+
)
2
parameterizes the
phase-dependent driving of the mechanics by cavity fluc-
tuations. The quadrature variances depend on the intra-
cavity pump photon numbers
n
p
, as well as the cavity oc-
cupation
n
th
c
and the phonon bath occupation
n
th
m
, which
can be extracted from the output spectra (Fig. 1f,g). To-
gether with Eq. (4), the corresponding quadrature vari-
ances can be calculated [16, 29].
Fig. 1c shows the quadrature variances with various in-
tracavity pump photon ratio
n
+
p
/n
p
. We start by squeez-
ing the mechanical motion with total intracavity pump
photon number
n
tot
p
=
n
p
+
n
+
p
= 1
.
35
×
10
4
and pump
photon ratio
n
+
p
/n
p
= 0
.
5. This pump configuration
generates a mechanical squeezed state with the squeezed
quadrature variance
ˆ
X
2
1
= 1
.
54
±
0
.
59
x
2
zp
and the anti-
squeezed quadrature variance
ˆ
X
2
2
= 13
.
81
±
1
.
41
x
2
zp
,
indicated by the solid red circle and square in Fig. 1c.
The corresponding normalized output spectra and the fits
from the two-tone optomechanical model [16] are shown
in Fig. 1f. To further squeeze the mechanical motion,
we can increase the total pump photon number. The
blue circles (squares) in Fig. 1c are the squeezed (anti-
squeezed) quadrature variances at total intracavity pump
photon number
n
tot
p
= 1
.
85
×
10
5
. The solid (dashed)
blue curves are the predictions from Eq. (4) with con-
stant cavity and mechanical occupations extracted from
the output spectrum at
n
+
p
= 0; they agree with the
data at low pump photon ratio. At large pump photon
ratio, the cavity bath starts to heat up (Fig. 1d), which
increases the mechanical quadrature variances. The or-
ange curves in Fig. 1c are the predictions from Eq. (4)
including the cavity heating effect extracted from the
experiment (orange line in Fig. 1d). With the heating
effect, the minimum quadrature variance is achieved at
n
+
p
/n
p
= 0
.
43 with
ˆ
X
2
1
= 0
.
56
±
0
.
02
x
2
zp
(the solid
blue circle in Fig. 1d), 2
.
5
±
0
.
2 dB below the zero-point
level. The corresponding normalized output spectra and
fits are shown in Fig. 1g.
While inferring the level of squeezing from the cav-
ity output spectrum is convenient, it would be preferable
to have a more direct method that does not rely on as-
sumptions 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 backaction evading (BAE) probes (the pur-
ple arrows in Fig. 1b) at
ω
c
ω
m
∆ with intracavity
field [17, 26]
̄
α
BAE
(
t
) = 2 ̄
α
cos(
ω
m
t
+
φ
)
e
i
(
ω
c
∆)
t
,
(5)
where
φ
is the relative phase between the BAE probes
and the squeezing pumps. For a sideband-resolved sys-
tem (
ω
m

κ
), the modulation of the BAE probes
3
exclusively couples the mechanical quadrature
ˆ
X
φ
=
cos
φ
ˆ
X
1
sin
φ
ˆ
X
2
to the microwave resonance with the
interaction
ˆ
H
I
=
~
G
(
ˆ
d
e
i
t
+
ˆ
de
i
t
)
ˆ
X
φ
x
zp
,
(6)
where
G
=
g
0
n
p
is the enhanced optomechanical cou-
pling rate and
n
p
=
|
̄
α
|
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 me-
chanical quadrature. By sweeping the probe phase
φ
,
we can perform tomography of the mechanical quantum
state (Fig. 2a). In order to ensure no interference be-
tween the sidebands of the squeezing pumps and the BAE
probes, we detune the BAE sidebands from the cavity
resonance by ∆ = 2
π
×
160kHz

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 measured, from which we can extract the me-
chanical quadrature variance and linewidth. In the fol-
lowing, we will perform a BAE measurement to directly
characterize the weakly squeezed state corresponding to
the spectrum Fig. 1f and the strong squeezed state cor-
responding to the spectrum Fig. 1g.
Fig. 2b shows the mechanical quadrature variances
from the BAE measurement as a function of the probe
phase
φ
. The red circles are the quadrature variances of
the weakly squeezed state measured with the BAE tech-
nique. The red curve is the inferred quadrature variances
from the corresponding output spectrum (Fig. 1f). In
this case, the results from the BAE measurement are in
good agreement with the results inferred from the output
spectrum. Similarly, the blue circles are the quadrature
variances of the strong squeezed state measured with the
BAE technique. The blue curve is the inferred quadra-
ture variance from the corresponding output spectrum
(Fig. 1g). The minimum quadrature variance is achieved
at
φ
= 0
with
ˆ
X
2
φ
= 0
.
34
±
0
.
07
x
2
zp
, 4
.
7
±
0
.
9 dB be-
low the zero-point level. This is lower than the quadra-
ture variance inferred from the output spectrum, imply-
ing that there is additional dynamics at play (beyond the
ideal optomechanical interaction).
The enhanced squeezing observed in the BAE mea-
surement suggests an additional squeezing mechanism
beyond the dissipative mechanism discussed above; an
obvious candidate is direct parametric driving of the me-
chanics. The presence of such driving is further corrob-
orated by our observation of a phase dependence of the
quadrature linewidth in the BAE measurement (Fig. 2c).
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 [27, 28]. To understand
the effects of this mechanical parametric driving, we phe-
nomenologically add the mechanical parametric interac-
tion to our otherwise ideal optomechanical model [29]:
ˆ
H
para
=
~
λ
(
e
ˆ
b
2
+
e
ˆ
b
2
)
,
(7)
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
linewidth to our model, thus extracting the amplitude
and phase of the parametric drive. [29]. By assuming
the phase of the parametric drive
ψ
follows the phase of
the BAE probe (i.e.
ψ
=
φ
+
ψ
0
, where
ψ
0
is a constant
phase shift), the model captures the observed phase de-
pendence behavior of the quadrature linewidth, as shown
by the dashed curves in Fig. 2c. Surprisingly, if one in-
stead assumes that the parametric driving is a result of
the main squeezing tones (i.e. take
ψ
a constant inde-
pendent of
φ
), one cannot capture the observed phase
dependence of the quadrature linewidth [29]. These re-
sults suggest that the parametric drive is induced by the
BAE probes.
The predicted squeezed quadrature variance for the
strong-pumps configuration is
ˆ
X
2
1
= 0
.
50
±
0
.
05
x
2
zp
,
as shown by the blue star in Fig. 2b. We stress that our
treatment of the spurious mechanical parametric drive
is phenomenological; we do not know the precise micro-
scopic mechanism which causes this driving. Nonethe-
less, it allows us to explain both surprising features of the
BAE measurements (the observed phase-dependent me-
chanical quadrature linewidth, and the enhanced squeez-
ing).
In conclusion, we combine reservoir engineering and
backaction evading measurement in a microwave optome-
chanical system to demonstrate a continuous QND mea-
surement of a mechanical quantum squeezed state. From
the BAE measurement, 4
.
7
±
0
.
9 dB of squeezing below
the zero point level has been observed, surpassing the
3 dB limit of the standard parametric squeezing tech-
nique. In addition, a phase dependence of the quadra-
ture linewidth is observed and explained by including a
mechanical parametric interaction to the ideal optome-
chanical model. The interplay between the optomechan-
ical interaction and the mechanical parametric interac-
tion enhances the mechanical squeezing and provides a
qualitative explanation to the BAE measurement results.
The present scheme can be applied to generate and char-
acterize more complicated quantum states by carefully
engineering the nonlinear interaction [30, 31]. The abil-
ity to generate and measure a strong quantum squeezed
state of a macroscopic mechanical object would be use-
ful for ultra-sensitive detection [3], quantum information
processing [32], as well as fundamental study of quantum
decoherence [33, 34].
This work is supported by funding provided by the
Institute for Quantum Information and Matter, an NSF
4
Physics Frontiers Center with support of the Gordon and
Betty Moore Foundation (NSF-IQIM 1125565), by the
Defense Advanced Research Projects Agency (DARPA-
QUANTUM HR0011-10-1-0066), by the NSF (NSF-
DMR 1052647 and NSF-EEC 0832819), and by the
Semiconductor Research Corporation (SRC) and Defense
Advanced Research Project Agency (DARPA) through
STARnet Center for Function Accelerated nanoMaterial
Engineering (FAME). 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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