of 5
RAPID COMMUNICATIONS
PHYSICAL REVIEW A
90
, 011803(R) (2014)
Silicon optomechanical crystal resonator at millikelvin temperatures
Se
́
an M. Meenehan,
1
Justin D. Cohen,
1
Simon Gr
̈
oblacher,
1
,
2
Jeff T. Hill,
1
Amir H. Safavi-Naeini,
1
Markus Aspelmeyer,
2
and Oskar Painter
1
,
*
1
Institute for Quantum Information and Matter and Thomas J. Watson, Sr., Laboratory of Applied Physics,
California Institute of Technology, Pasadena, California 91125, USA
2
Vienna Center for Quantum Science and Technology (VCQ), Faculty of Physics, University of Vienna, A-1090 Wien, Austria
(Received 24 March 2014; published 17 July 2014)
Optical measurements of a nanoscale silicon optomechanical crystal cavity with a mechanical resonance
frequency of 3
.
6 GHz are performed at subkelvin temperatures. We infer optical-absorption-induced heating
and damping of the mechanical resonator from measurements of phonon occupancy and motional sideband
asymmetry. At the lowest probe power and lowest fridge temperature (
T
f
=
10 mK), the localized mechanical
resonance is found to couple at a rate of
γ
i
/
2
π
=
400 Hz (
Q
m
=
9
×
10
6
) to a thermal bath of temperature
T
b
270 mK. These measurements indicate that silicon optomechanical crystals cooled to millikelvin
temperatures should be suitable for a variety of experiments involving coherent coupling between photons
and phonons at the single quanta level.
DOI:
10.1103/PhysRevA.90.011803
PACS number(s): 42
.
50
.
Wk
,
42
.
65
.
k
,
62
.
25
.
g
The coupling of a mechanical object’s motion to the
electromagnetic field of a high finesse cavity forms the basis
of various precision measurements [
1
], from large-scale grav-
itational wave detection [
2
] to microscale accelerometers [
3
].
Recent work utilizing both optical and microwave cavities
coupled to mesoscopic mechanical resonators has shown the
capability to prepare and detect such resonators close to their
quantum ground state of motion using radiation pressure back-
action [
4
7
]. Optomechanical crystals (OMCs), in which band
gaps for both optical and mechanical waves can be introduced
through patterning of a material, provide a means for strongly
interacting nanomechanical resonators with near-infrared
light [
8
]. Beyond the usual paradigm of cavity optomechanics
involving isolated single mechanical elements [
9
,
10
], OMCs
can be fashioned into planar circuits for photons and phonons,
and arrays of optomechanical elements can be interconnected
via optical and acoustic waveguides [
11
]. Such coupled OMC
arrays have been proposed as a way to realize quantum
optomechanical memories [
12
], nanomechanical circuits for
continuous variable quantum information processing [
13
] and
phononic quantum networks [
14
], and as a platform for
engineering and studying quantum many-body physics of
optomechanical metamaterials [
15
17
].
The realization of optomechanical systems in the quantum
regime is predicated upon the ability to limit thermal noise
in the mechanics while simultaneously introducing large
coherent coupling between optical and mechanical degrees
of freedom. In this regard, laser back-action cooling has
recently been employed in simple OMC cavity systems [
6
,
18
]
consisting of a one-dimensional (1D) nanobeam resonator
surrounded by a two-dimensional (2D) phononic band gap.
In this work, we optically measure the properties of such an
OMC cavity system in a helium dilution refrigerator down to
base temperatures of
T
f
=
10 mK.
The device studied here is formed from the top silicon
(Si) device layer of a silicon-on-insulator (SOI) wafer [
19
].
Figure
1(a)
shows a scanning electron microscope (SEM)
*
opainter@caltech.edu
image of a suspended device after processing. This device
consists of two nanobeam OMC cavities, optically coupled
to a common central waveguide [Fig.
1(b)
]. As described
in Ref. [
6
], the nanobeam cavities are patterned in such
a way as to support an optical resonance in the 1550 nm
wavelength band and a “breathing” mechanical resonance at
3
.
6 GHz. The highly localized optical and acoustic resonances
couple strongly via radiation pressure, with a theoretical
vacuum coupling rate of
g
0
/
2
π
=
870 kHz, corresponding
physically to the optical resonance shift due to the mechanical
resonator zero-point amplitude. Surrounding the waveguide
and nanobeam structure on three sides is a 2D “cross”
pattern [
11
] which has a full phononic band gap for all acoustic
waves in the 3
4 GHz frequency range. The fourth side
is left open, so as to allow close approach of an optical
fiber. The SOI sample is mounted to the mixing chamber
plate of a dilution refrigerator, and a set of stages are used
to align an antireflection-coated tapered lensed fiber (beam
waist
=
2
.
5
μ
m; focal distance
=
14
μ
m) to the coupling
waveguide of a given device under test [see Fig.
1(a)
and
Ref. [
19
]]. In order to aid efficient optical coupling, the Si
waveguide is tapered down to a tip of width 225 nm, providing
mode matching between fiber and waveguide [
20
,
21
].
Optical spectroscopy of a fiber-coupled device at a fridge
temperature of 4 K is shown in Fig.
1(c)
. In this measurement,
the frequency of a narrow linewidth external-cavity diode laser
is swept across the fundamental optical resonance of one of the
OMC cavities centered at
λ
c
=
1545 nm. The input laser light
is reflected from a photonic crystal mirror at the end of the Si
waveguide, and when resonant, evanescently couples in to the
nanobeam cavities (the other nanobeam cavity coupled to this
waveguide has a resonance several THz to the red). Some of the
light entering the cavity decays through intrinsic loss channels
at rate
κ
i
, while the remainder couples back into the central
waveguide at coupling rate
κ
e
, and is collected in reflection by
the lensed optical fiber. For this particular cavity, we observe
a total optical energy decay rate of
κ/
2
π
=
529 MHz, an
extrinsic coupling rate
κ
e
/
2
π
=
153 MHz, and an intrinsic
decay rate
κ
i
/
2
π
=
376 MHz. From the normalized reflection
signal level (
R
), the fraction of optical power reflected by the
1050-2947/2014/90(1)/011803(5)
011803-1
©2014 American Physical Society
RAPID COMMUNICATIONS
SE
́
AN M. MEENEHAN
et al.
PHYSICAL REVIEW A
90
, 011803(R) (2014)
10
μ
m
(a)
(d)
-7
-3.5
0
3.5
7
0.1
0.1
0.2
0.2
0.3
0.3
R
(c)
lensed
acoustic
radiation
shield
0.4
0.4
0
0
0.5
0.5
Δ/
2
π
(GHz)
1
μ
m
(b)
κ
e
κ
i
VOA
φ
-m
-
VC
Filter
RSA
λ
-meter
LO
Signal
κ = κ
e
+
κ
i
ω
m
− ω
m
laser
FIG. 1. (Color online) (a) SEM image of the OMC device. (b)
Zoomed in SEM image showing details of the waveguide-cavity
coupling region. (c) Normalized optical cavity reflection spectrum
(
R
). Detunings from resonance of

ω
m
/
2
π
3
.
6 GHz are
denoted by the red and blue arrows, respectively. (d) Schematic
of the fiber-based heterodyne receiver used to perform optical and
mechanical spectroscopy of the OMC cavity.
λ
meter: wave meter,
φ
-m: electro-optic phase modulator, VOA: variable optical attenuator,
VC: variable coupler, RSA: real-time spectrum analyzer.
OMC cavity, collected by the lensed fiber, and detected on a
photodetector is estimated to be
η
cpl
=
34
.
7% [
19
].
The optical heterodyne detection scheme illustrated in
Fig.
1(d)
is used to measure the motion of the localized breath-
ing mode at frequency
ω
m
/
2
π
=
3
.
6 GHz. A high-power (
0
.
7 mW) local oscillator (LO) sets the gain of the heterodyne
receiver, and a low-power (1 nW–16
μ
W) optical signal beam
is used to probe the OMC cavity. In order to selectively detect
either the upper or lower motional sidebands generated on the
optical signal beam, the LO frequency (
ω
LO
) is shifted relative
to that of the signal beam (
ω
s
). For the measurements presented
here,
ω
LO
is adjusted such that the mechanical modulation beat
frequency
=
ω
LO
(
ω
s
±
ω
m
)
2
π
×
50 MHz, placing a
single motional sideband within the bandwidth (100 MHz) of
the photodetectors. In the case of a signal beam red-detuned
from cavity resonance (frequency
ω
c
)by

ω
c
ω
s
=
ω
m
, the resultant noise power spectral density (NPSD) as
transduced on a spectrum analyzer yields a resonant mechan-
ical signal component proportional to
S
bb
(
ω
)
=
n

γ/
[(
ω
ω
m
)
2
+
(
γ/
2)
2
][
22
]. Here

n

is the phonon occupancy of the
mechanical mode, and the total mechanical damping rate is
given by
γ
=
γ
i
+
γ
OM
, where
γ
i
is the intrinsic damping rate
of the mechanical resonator and
γ
OM
(

=
ω
m
)
=
4
g
2
0
n
c
is the optomechanically induced damping rate produced
by an intracavity photon number
n
c
. For a blue-detuned
probe (

=−
ω
m
), the resonant component of the NPSD is
proportional to

n
+
1 and
γ
=
γ
i
γ
OM
(

=
ω
m
).
Mechanical spectroscopy is first performed at a fridge
temperature of
T
f
=
4 K in order to calibrate the optome-
chanical transduction. The coupling rate
g
0
is determined
by observing the dependence of the mechanical linewidth on
n
c
for both red (

=
ω
m
) and blue (

=−
ω
m
) laser-cavity
detunings, as shown in Fig.
2(a)
. Above a threshold value,
n
c
>n
thr
1
.
5, optical amplification and self-oscillation of
10
1
10
2
10
0
10
1
<n>
(b)
(a)
10
-1
10
0
10
1
10
2
10
-34
10
-33
10
-35
10
-32
(c)
S
xx
(m
2
/Hz)
(
ω
-
ω
m
)/2
π
(MHz
)
γ
/2
π
(kHz)
01
-1
10
-2
10
-1
10
0
10
-2
10
-1
10
0
10
1
g
0
/2
π
= 840 kHz
γ
OM
/2
π
(kHz)
1.9
0.61
6.1
n
c
=0.06
19
61
153
n
c
n
c
n
f
n
thr
FIG. 2. (Color online) (a) Measured mechanical linewidth
γ
for

=
ω
m
(red) and
ω
m
(blue) at a fridge temperature of
T
f
=
4K.
The vertical blue dashed line indicates the threshold
n
c
beyond which
the mechanical resonance self-oscillates for

=−
ω
m
, resulting in a
40 dB increase in the mechanical signal level. Black circles indicate
the values of
γ
i
obtained by taking the average of the detuned data. The
inset shows
γ
OM
determined by subtracting
γ
i
from the red-detuned
γ
(circles), and from cooperativity
C
using the calibrated

n

(squares).
A linear fit (red line) yields
g
0
/
2
π
=
840 kHz. (b) Calibrated
mechanical mode occupancy

n

versus intracavity photon number
n
c
. Blue and red circles are measured with probe laser detunings

ω
m
, respectively. The mode occupancy
n
f
corresponding to
T
f
=
4 K is indicated by a black solid line. (c) Series of red-detuned
NPSD for range of
n
c
. Here the NPSD is plotted as
S
xx
=
x
2
zpf
S
bb
,
where
x
zpf
=
4
.
1 fm is the zero-point amplitude of the breathing
mode.
the mechanical resonator occurs for blue detuning. Below this
value, the optomechanical damping
γ
OM
can be found from
the difference between the red- and blue-detuned linewidths.
A linear fit of the derived
γ
OM
versus
n
c
yields a coupling rate
of
g
0
/
2
π
=
840 kHz. Using this value of
g
0
along with the
optical detection efficiency, the mechanical mode occupancy
versus
n
c
is calibrated from the area under the resonant part
of the measured NPSD [see Figs.
2(b)
and
2(c)
]. At high
n
c
the mechanical mode is seen to be cooled, whereas at low
n
c
the calibrated occupancy saturates to a constant value in good
agreement with the expected 4 K fridge occupancy (solid black
line).
As the fridge temperature is lowered into the subkelvin
range, a very different dependence of measured linewidth
and mode occupancy on optical probe power is observed.
In particular, the measured mechanical linewidth versus
n
c
,
shown in Fig.
3(a)
for
T
f
=
185 mK, increases with decreasing
probe power below an apparent minimum at
n
c
1. The
measured linewidth at low power is also too large to explain
the observed threshold of self-oscillation,
n
thr
0
.
1. These
inconsistencies indicate that the linewidth associated with the
true energy decay rate of the mechanics is likely obscured
in the time-averaged spectrum of Fig.
3(a)
due to frequency
011803-2
RAPID COMMUNICATIONS
SILICON OPTOMECHANICAL CRYSTAL RESONATOR AT . . .
PHYSICAL REVIEW A
90
, 011803(R) (2014)
10
2
10
1
signal power (
W
)
10
-11
10
-10
10
-12
23456
γ
/2
π
(kHz)
Δ
/2
π
(GHz)
n
c
10
-1
10
0
10
1
10
2
n
thr
γ
/2
π
(kHz)
22
18
14
10
8
23456
Δ
/2
π
(GHz)
(a)
(b)
(c)
FIG. 3. (Color online) (a) Mechanical linewidth versus
n
c
for
red- (red circles,

=
ω
m
) and blue-detuned (blue circles,

=−
ω
m
)
probes at
T
f
=
185 mK. The blue dashed line indicates the self-
oscillation threshold. (b) Measured resonant signal power (circles)
versus optical detuning,

,for
n
c
1
.
5. The red curve shows a best
fit to the data, yielding
C
=
γ
OM
i
=
3
.
9at

=
ω
m
.Herewehave
assumed, to good approximation when
ω
m

1and(

ω
m
)
2

(

+
ω
m
)
2
, a back-action damping
γ
OM
(

)
=
(4
g
2
0
n
c
)[1
+
4(

ω
m
)
2
2
]
1
, cooled mode occupancy

n
∝
[1
+
γ
OM
(

)
i
]
1
,and
heterodyne resonant signal power
∝
n

γ
OM
. The black dashed curve
shows the expected signal in the absence of back-action cooling
(
C
0), consistent with assuming
γ
i
is equal to the measured time-
averaged linewidth. (c) Corresponding measured (circles) mechanical
linewidth versus

. The red curve is a fit with
C
(

=
ω
m
) constrained
(
=
3
.
9), but assuming a Voigt line shape with additional Gaussian
frequency jitter term. The dashed black curve is the best fit for
C
constrained (
=
3
.
9) but with no additional frequency jitter term.
jitter [
23
,
24
]. Due to the long averaging times (minutes to
hours) required at low optical probe power, direct observation
of the mechanical frequency jitter is not possible. Indirect
confirmation of the frequency jitter is ascertained by studying
the detuning (

) dependence of the transduced signal power
and linewidth for a fixed
n
c
1
.
5, as shown in Fig.
3(b)
.
Such a measurement keeps constant any effects such as optical
heating or frequency jitter that might depend on the intracavity
photon number. The resulting fit to the data (red curves; see
caption for details), yields
γ
i
/
2
π
=
2
.
3 kHz for the intrinsic
mechanical energy decay rate,
γ
G
/
2
π
=
6
.
1kHzforthe
Gaussian frequency jitter, and
g
0
/
2
π
=
805 kHz, consistent
with the
T
f
=
4 K value. In what follows we use additional
on-resonance heating measurements to determine
γ
i
over the
entire range of
n
c
.
Heating of the mechanical mode by optical absorption
becomes significant at subkelvin temperatures due to the
sharp drop in thermal conductance with temperature [
25
]. The
source of optical absorption in our structures is most likely
due to electronic defect states at the surface of Si [
26
,
27
].
This heating mechanism is investigated at
T
f
=
10 mK, for
which
n
f
is negligible, by measuring

n

and
γ
using an
optically resonant probe (

=
0;
γ
OM
=
0). We observe in
Fig.
4(a)
that

n
∝
n
1
/
4
c
. This weak power-law dependence
is consistent with indirect coupling of the breathing mode to
an optically generated bath with thermal conductance scaling
as
G
th
T
3
. Phonons with wavelengths small relative to the
dimensions of the cavity structure are expected to have such a
conductance scaling [
28
], although their estimated escape time
from the nanobeam (
γ
1
THz
1–10 ns) suggests that they must
quickly come into thermal equilibrium. Given the slow rates of
most bulk relaxation processes [
29
], this fast thermalization is
likely due to a relatively large degree of diffusive and inelastic
scattering at the surfaces of the patterned nanobeam [
30
].
Based on these on-resonance observations, a proposed
microscopic model for the optical absorption heating and
damping is illustrated in Fig.
4(c)
. Here the long-lived
breathing mode is weakly coupled (
γ
0
) through the phononic
shield to the exterior fridge environment, and is locally
coupled via phonon-phonon scattering (
γ
p
)[
31
] to the optically
generated high frequency phonons within the acoustic cavity.
A phenomenological model based upon this microscopic
picture is shown schematically in Fig.
4(d)
. We parametrize the
coupling of the mechanical resonator to the separate thermal
baths by decomposing the mechanical damping rate into
γ
L
=
γ
0
+
γ
p
+
γ
OM
, where the fridge bath (occupancy
n
f
) couples
at rate
γ
0
, the optical-absorption-induced bath (temperature
T
p
,
occupancy
n
p
at
ω
m
) couples at rate
γ
p
, and the intracavity laser
field (effective zero-temperature bath) couples at rate
γ
OM
.The
resulting average mechanical mode occupation is then given
by

n

(
n
c
)
=
[
γ
0
n
f
+
γ
p
(
T
p
)
n
p
(
T
p
)]
/
[
γ
0
+
γ
p
(
T
p
)
+
γ
OM
(
n
c
)],
where
T
p
(
n
c
).
The calibrated mechanical mode occupation for a red-
detuned (

=
ω
m
) probe is plotted against
n
c
in Fig.
4(e)
for
T
f
=
10 mK (purple) and 635 mK (green). Both curves exhibit
a series of heating and cooling trends, and in fact coincide
for
n
c

1. At the lowest optical probe powers (
n
c
=
0
.
016)
and lowest fridge temperature (
T
f
=
10 mK), the calibrated
phonon occupancy reaches a minimum

n
=
0
.
98
±
0
.
11,
corresponding to
T
270 mK. The complex behavior of these
two cooling curves can be understood by comparing to the
proposed phenomenological model. For this model
n
f
is taken
to correspond to the measured
T
f
,
n
p
(
n
c
) is ascertained by
extrapolating the on-resonance measurement of

n

[Fig.
4(a)
],
and
γ
OM
(
n
c
) is found from the fit value
g
0
/
2
π
=
840 kHz to
the high power region of the red-detuned mechanical linewidth
[red circles in Fig.
4(b)
]. Assuming a common
γ
p
(
n
c
) and
γ
0
,
the resulting
γ
i
(
n
c
) curve that best fits the measured

n

data
for both
T
f
=
10 and 635 mK fridge temperatures is plotted
in Fig.
4(b)
(blue circles). Also shown in Fig.
4(b)
are the
best-fit value of the coupling to the fridge bath
γ
0
/
2
π
=
306
±
28 Hz (dashed black horizontal curve) and a smooth spline
curve fit to the inferred values of
γ
p
(
n
c
) (red solid curve). A
plot of the best-fit model is shown alongside the measured

n

cooling curves in Fig.
4(e)
. In addition to the good agreement
of the model for both fridge temperatures, the inferred intrinsic
energy damping rate is also consistent with the measured
self-oscillation threshold [Fig.
3(a)
]. At the lowest probe
powers (
n
c
=
0
.
016), the energy damping mechanical
Q
factor
reaches an impressively high value of
Q
m
=
9
×
10
6
.
Alongside the calibrated mode occupancy

n

,wehave
also measured the sideband asymmetry
ξ
shown in the
inset to Fig.
4(e)
. The sideband asymmetry is defined
as
ξ
=
I
/I
+
1[
32
], where
I
±
is the area under the
Lorentzian part of the NPSD for an optical probe with
011803-3
RAPID COMMUNICATIONS
SE
́
AN M. MEENEHAN
et al.
PHYSICAL REVIEW A
90
, 011803(R) (2014)
<n>
10
2
10
1
(d)
(c)
linewidth (kHz)
10
1
10
0
10
-1
10
-1
10
0
10
1
10
2
n
c
0
(a)
(b)
c
T
p
(
n
c
)
T
f
,
n
f
THz
m
,
<n>
THz
GHz
MHz
XM
p
E
v
P
b 0,1
E
c
0
OM
n
=0
n
p
(
T
p
)
<n>
n
f
n
c
(e)
0.8
1.0
1.2
1.4
1.6
1.8
2.0
0.025
0.03
0.035
n
c
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
<n>
10
-1
10
0
10
1
10
2
n
c
p
(
T
p
)
T
p
(
n
c
)
T
p
n
c
1/4
10
1
10
0
T
p
(K)
0.02
FIG. 4. (Color online) (a) Measured phonon occupancy versus intracavity photon number for a resonant probe at
T
f
=
10 mK. The red line
shows a power-law fit to the occupancy. The right axis shows the equivalent bath temperature,
T
p
. (b) Measured mechanical linewidth versus
n
c
for resonant (gray circles) and red-detuned (red circles) probes. The blue circles show the best-fit values of
γ
i
=
γ
0
+
γ
p
to both
T
f
=
10
and 635 mK data sets. The best-fit value for
γ
0
and a smooth curve fit to the inferred
γ
p
are shown as a black dashed line and a solid red line,
respectively. (c) Diagram illustrating the proposed model of optical-absorption-driven heating of the mechanics and (d) schematic showing the
various baths coupled to the localized mechanical mode (see text for details). (e) Calibrated mode occupancy versus
n
c
for
T
f
=
10 mK and
635 mK. Best fits to both temperature data sets using the proposed heating model are shown as solid curves, with shaded regions representing
the variation in the fit for
γ
0
/
2
π
=
306
±
28 Hz. The inset shows the measured asymmetry
ξ
as a function of
n
c
, with the prediction of the
best-fit model shown as solid curves.
T
f
=
10 mK data (fits) are shown as purple circles (solid curves).
T
f
=
635 mK data (fits) are shown as
green circles (solid curves).
detuning

ω
m
. The asymmetry is sensitive to both the
absolute mode occupancy and to the sum of
γ
0
and
γ
p
through
the cooperativity
C
=
γ
OM
/
(
γ
0
+
γ
p
). Good correspondence
can also be seen between the best-fit model (solid curves) and
the measured
ξ
(circles).
Although significant work remains to determine the exact
microscopic details of the optical absorption heating and
frequency jitter observed in the measurements of the quasi-
1D OMC cavity studied here, there are nonetheless several
interesting points to note. First, from the measured time-
averaged mechanical linewidth and optically induced bath
temperature (
T
p
)inFig.
4
we find that the frequency noise of
the mechanical resonator drops with increasing temperature
as
T
0
.
9
p
[
19
]. Such a frequency noise behavior is similar to
that found for two-level systems coupled to superconducting
microwave resonators [
24
], and may be due to native oxide
formation at the Si surfaces. Secondly, the phononic shield
provides excellent mechanical isolation of the breathing mode,
while at the same time providing good mechanical coupling to
the fridge bath for heat carrying phonons above the acoustic
band gap. Thirdly, although lower phonon occupancies could
have been measured using thinner phononic shields, effectively
increasing the coupling rate
γ
0
to the fridge bath at
T
f
,this
would come with a commensurate reduction in cooperativity
C
=
γ
OM
i
. Coherent quantum interactions between the
optical cavity field and the mechanical resonator require
C>
1
and

n

<
1, and although the devices of this work closely
approach this limit, a move to quasi-2D Si OMC devices [
33
]
with orders of magnitude larger thermal conductance should
enable future work in the quantum regime as envisioned in
recent proposals [
13
,
15
17
].
The authors would like to thank Michael Roukes, Ron
Lifshitz, and Michael Cross for helpful discussions regarding
the proposed thermal model, as well as Jasper Chan, Witlef
Wiezcorek, and Jason Hoelscher-Obermaier for support in
the early stages of the experiment. This work was supported
by the DARPA ORCHID and MESO programs, the Insti-
tute for Quantum Information and Matter, an NSF Physics
Frontiers Center with support of the Gordon and Betty
Moore Foundation, and the Kavli Nanoscience Institute at
Caltech. A.S.N. acknowledges support from NSERC. S.G.
was supported by a Marie Curie International Outgoing Fel-
lowship within the 7th European Community Framework Pro-
gramme. S.M.M., J.D.C., and S.G. contributed equally to this
work.
[1] V. Braginsky and A. Manukin,
Measurement of Weak Forces
in Physics Experiments
(University of Chicago Press, Chicago,
1977).
[2] The LIGO Scientific Collaboration,
Nat. Phys.
7
,
962
(
2011
).
[3] A. G. Krause, M. Winger, T. D. Blasius, Q. Lin, and O. Painter,
Nat. Photon.
6
,
768
(
2012
).
[4] T. Rocheleau, T. Ndukum, C. Macklin, J. B. Hertzberg, A. A.
Clerk, and K. C. Schwab,
Nature (London)
463
,
72
(
2010
).
[5] J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman,
K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W.
Simmonds,
Nature (London)
475
,
359
(
2011
).
[6] J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill,
A. Krause, S. Gr
̈
oblacher, M. Aspelmeyer, and O. Painter,
Nature (London)
478
,
89
(
2011
).
[7] E. Verhagen, S. Del
́
eglise, S. Weis, A. Schliesser, and T. J.
Kippenberg,
Nature (London)
482
,
63
(
2012
).
011803-4
RAPID COMMUNICATIONS
SILICON OPTOMECHANICAL CRYSTAL RESONATOR AT . . .
PHYSICAL REVIEW A
90
, 011803(R) (2014)
[8] M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and
O. Painter,
Nature (London)
462
,
78
(
2009
).
[9] S. Gr
̈
oblacher, J. B. Hertzberg, M. R. Vanner, G. D. Cole,
S. Gigan, K. C. Schwab, and M. Aspelmeyer,
Nat. Phys.
5
,
485
(
2009
).
[10] J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Marquardt,
S. M. Girvin, and J. G. E. Harris,
Nature (London)
452
,
72
(
2008
).
[11] A. H. Safavi-Naeini and O. Painter,
Opt. Express
18
,
14926
(
2010
).
[12] D. Chang, A. H. Safavi-Naeini, M. Hafezi, and O. Painter,
New J. Phys.
13
,
023003
(
2011
).
[13] M. Schmidt, M. Ludwig, and F. Marquardt,
New J. Phys.
14
,
125005
(
2012
).
[14] S. J. M. Habraken, K. Stannigel, M. D. Lukin, P. Zoller, and
P. R a b l ,
New J. Phys.
14
,
115004
(
2012
).
[15] A. Tomadin, S. Diehl, M. D. Lukin, P. Rabl, and P. Zoller,
Phys.Rev.A
86
,
033821
(
2012
).
[16] M. Ludwig and F. Marquardt,
Phys. Rev. Lett.
111
,
073603
(
2013
).
[17] M. Schmidt, V. Peano, and F. Marquardt,
arXiv:1311.7095
.
[18] J. Chan, Ph.D. thesis, California Institute of Technology, 2012.
[19] See Supplemental Material at
http://link.aps.org/supplemental/
10.1103/PhysRevA.90.011803
for details of device design,
fabrication, and characterization.
[20] V. R. Almeida, R. R. Panepucci, and M. Lipson,
Opt. Lett.
28
,
1302
(
2003
).
[21] J. D. Cohen, S. M. Meenehan, and O. Painter,
Opt. Express
21
,
11227
(
2013
).
[22] A. H. Safavi-Naeini, J. Chan, J. T. Hill, S. Gr
̈
oblacher, H. Miao,
Y. Chen, M. Aspelmeyer, and O. Painter,
New J. Phys.
15
,
035007
(
2013
).
[23] Y. T. Yang, C. Callegari, X. L. Feng, and M. L. Roukes,
Nano Lett.
11
,
1753
(
2011
).
[24] J. Gao, Ph.D. thesis, California Institute of Technology, 2008.
[25] M. G. Holland,
Phys. Rev.
132
,
2461
(
1963
).
[26] A. Stesmans,
Appl. Phys. Lett.
68
,
2076
(
1996
).
[27] M. Borselli, T. J. Johnson, C. P. Michael, M. D. Henry, and
O. Painter,
Appl. Phys. Lett.
91
,
131117
(
2007
).
[28] R. Chen, A. I. Hochbaum, P. Murphy, J. Moore, P. Yang, and
A. Majumdar,
Phys. Rev. Lett.
101
,
105501
(
2008
).
[29] N. Mingo,
Phys. Rev. B
68
,
113308
(
2003
).
[30] J. B. Hertzberg, M. Aksit, O. O. Otelaja, D. A. Stewart, and
R. D. Robinson,
Nano Lett.
14
,
403
(
2014
).
[31] P. S. Zyryanov and G. G. Taluts, J. Exp. Theor. Phys.
22
, 1326
(1966).
[32] A. H. Safavi-Naeini, J. Chan, J. T. Hill, Thiago P. Mayer Alegre,
A. Krause, and O. Painter,
Phys. Rev. Lett.
108
,
033602
(
2012
).
[33] A. H. Safavi-Naeini, J. T. Hill, S. Meenehan, J. Chan, S.
Gr
̈
oblacher, and O. Painter,
Phys.Rev.Lett.
112
,
153603
(
2014
).
011803-5