Phononic bandgap nano-acoustic cavity with ultralong phonon lifetime
Gregory S. MacCabe,
1, 2,
∗
Hengjiang Ren,
1, 2,
∗
Jie Luo,
1, 2
Justin D. Cohen,
1, 2
Hengyun Zhou,
1, 2,
†
Alp Sipahigil,
1, 2
Mohammad Mirhosseini,
1, 2
and Oskar Painter
1, 2
1
Kavli Nanoscience Institute, California Institute of Technology, Pasadena, California 91125, USA
2
Institute for Quantum Information and Matter and Thomas J. Watson, Sr.,
Laboratory of Applied Physics, California Institute of Technology, Pasadena, California 91125, USA
(Dated: January 15, 2019)
We present measurements at millikelvin temperatures of the microwave-frequency acoustic prop-
erties of a crystalline silicon nanobeam cavity incorporating a phononic bandgap clamping structure
for acoustic confinement. Utilizing pulsed laser light to excite a co-localized optical mode of the
nanobeam cavity, we measure the dynamics of cavity acoustic modes with single-phonon sensitivity.
Energy ringdown measurements for the fundamental 5 GHz acoustic mode of the cavity shows an
exponential increase in phonon lifetime versus number of periods in the phononic bandgap shield,
increasing up to
τ
ph
,
0
≈
1
.
5 seconds. This ultralong lifetime, corresponding to an effective phonon
propagation length of several kilometers, is found to be consistent with damping from non-resonant
two-level system defects on the surface of the silicon device. Potential applications of these ultra-
coherent nanoscale mechanical resonators range from tests of various collapse models of quantum
mechanics to miniature quantum memory elements in hybrid superconducting quantum circuits.
In optics, geometric structuring at the nanoscale has
become a powerful method for modifying the electromag-
netic properties of a bulk material, leading to metamate-
rials [1] capable of manipulating light in unprecedented
ways [2]. In the most extreme case, photonic bandgaps
can emerge in which light is forbidden from propagating,
dramatically altering the emission of light from within
such materials [3–5]. More recently, a similar phononics
revolution [6] in the engineering of acoustic waves has
led to a variety of new devices, including acoustic cloaks
that can shield objects from observation [7], thermal crys-
tals for controlling the flow of heat [8, 9], optomechanical
crystals that couple photons and phonons via radiation
pressure [10], and phononic topological insulators whose
protected edge states can transport acoustic waves with
minimal scattering [11, 12].
Phononic bandgap structures, similar to their elec-
tromagnetic counterparts, can be used to modify the
emission or scattering of phonons. These ideas have
recently been explored in quantum optomechanics [13–
17] and electromechanics [18] experiments to greatly re-
duce the mechanical coupling to the thermal environment
through acoustic radiation. At ultrasonic frequencies
and below, one can combine phononic bandgap clamp-
ing with a form of ‘dissipation dilution’ in high stress
films [19] to realize quality (
Q
) factors in excess of 10
8
in two-dimensional nanomembranes [16] and approaching
10
9
in one-dimensional strain-engineered nanobeams [17].
At higher, microwave frequencies the benefit of stress-
loading of the film fades as local strain energy domi-
nates [17] and one is left once again to deal with intrinsic
material absorption.
∗
These authors contributed equally to this work.
†
Current Address: Department of Physics, Harvard University,
Cambridge, Massachusetts 02138, USA
To date, far less attention has been paid to the impact
of geometry and phononic bandgaps on acoustic mate-
rial absorption [20, 21]. Fundamental limits to sound
absorption in solids are known to result from the anhar-
monicity of the host crystal lattice [22–24]. At low tem-
peratures
T
, in the Landau-Rumer regime (
ωτ
th
1)
where the thermal phonon relaxation rate (
τ
−
1
th
) is much
smaller than the acoustic frequency (
ω
), a quantum
model of three-phonon scattering can be used to describe
phonon-phonon mixing that results in damping and ther-
malization of acoustic modes [22, 23]. Landau-Rumer
damping scales approximately as
T
α
, where
α
≈
4 de-
pends upon the phonon dispersion and density of states
(DOS) [23]. At the very lowest lattice temperatures
(
.
10 K), where Landau-Rumer damping has dropped
off, a residual damping emerges due to material de-
fects. These two-level system (TLS) defects [25], typi-
cally found in amorphous materials, correspond to a pair
of nearly degenerate local arrangements of atoms in the
solid which can have both an electric and an acoustic
transition dipole, and couple to both electric and strain
fields. Recent theoretical analysis shows that TLS inter-
actions with acoustic waves can be dramatically altered
in a structured material [20].
Here we explore the limits of acoustic damping and
coherence of a microwave acoustic nanocavity with a
phononic crystal shield that possesses a wide bandgap
for all polarizations of acoustic waves. Our nanocav-
ity, formed from an optomechanical crystal (OMC)
nanobeam resonator [10, 14], supports an acoustic
breathing mode at
ω
m
/
2
π
≈
5 GHz and a co-localized op-
tical resonant mode at
ω
c
/
2
π
≈
195 THz (
λ
c
≈
1550 nm)
which allows us to excite and readout mechanical mo-
tion using radiation pressure from a pulsed laser source.
This minimally invasive pulsed measurement technique
avoids a slew of parasitic damping effects
−
typically as-
sociated with electrode materials and mechanical con-
tact [26], or probe fields for continuous readout
−
and
arXiv:1901.04129v1 [cond-mat.mes-hall] 14 Jan 2019
2
b
2 μm
coupling waveguide
acoustic shielding
nanobeam OMC
r
1
r
2
w
c
h
c
h
h
w
h
coupling waveguide
0
1
2
3
4
5
6
7
8
9
10
acoustic frequency (GHz)
e
Γ
X
M
Γ
z-even
z-odd
c
a
d
f
FIG. 1.
Nanobeam optomechanical crystal and phononic shield design. a
, Scanning electron microscope (SEM) image
of a full nanobeam optomechanical crystal (OMC) device fabricated on SOI with
N
= 7 periods of acoustic shielding. A central
coupling waveguide allows for fibre-to-chip optical coupling as well as side-coupling to individual nanobeam OMC cavities.
b
, SEM image of an individual nanobeam OMC and the coupling waveguide, with enlarged illustration of an individual unit
cell in the end-mirror portion of the nanobeam.
c
, FEM simulations of the mechanical (top; total displacement) and optical
(bottom; transverse electric field) modes of interest in the nanobeam. Distortion of the mechanical displacement profile is
exaggerated for clarity.
d
, SEM image showing the nanobeam clamping geometry.
e
, SEM image of an individual unit cell of
the cross-crystal acoustic shield. The dashed lines show fitted geometric parameters used in simulation, including cross height
(
h
c
= 474 nm), cross width (
w
c
= 164 nm), inner fillet radius (
r
1
), and outer fillet radius (
r
2
).
f
, Simulated acoustic band
structure of the realized cross-crystal shield unit cell, with the full acoustic bandgap highlighted in pink. Solid (dotted) lines
correspond to modes of even (odd) symmetry in the direction normal to the plane of the unit cell. The dashed red line indicates
the mechanical breathing-mode frequency at
ω
m
/
2
π
= 5
.
0 GHz.
allows for the sensitive measurement of motion at the
single phonon level [27]. The results of acoustic ring-
down measurements at millikelvin temperatures show
that damping due to radiation is effectively suppressed by
the phononic shield, with breathing mode quality factors
reaching
Q
= 4
.
9
×
10
10
, corresponding to an unprece-
dented frequency-
Q
product of
f
-
Q
= 2
.
6
×
10
20
. The
temperature and amplitude dependence of the residual
acoustic damping is consistent with relaxation damping
of non-resonant TLS, modeling of which indicates that
not only does the phononic bandgap directly eliminate
the acoustic radiation of the breathing mode but it also
reduces the phonon damping of TLS in the host material.
The devices studied in this work are fabricated from
the 220 nm device layer of a silicon-on-insulator (SOI) mi-
crochip. Details of the fabrication process are provided
in App. A. In Figs. 1(a-b) we show scanning electron
microscope images of a single fabricated device, which
consists of a coupling optical waveguide, the nanobeam
OMC cavities that support both the microwave acous-
tic and optical resonant modes, and the acoustic shield
that connects the cavity to the surrounding chip sub-
strate. Fig. 1(c) shows finite-element method (FEM)
simulations of the microwave acoustic breathing mode
and fundamental optical mode of the nanobeam cavity.
We use the on-chip coupling waveguide to direct laser
light to the nanobeam OMC cavities. A pair of cavi-
ties with slightly different optical mode frequencies are
evanescently coupled to each waveguide. An integrated
photonic crystal back mirror in the waveguide allows for
optical measurement in a reflection geometry. The design
of the OMC cavities, detailed in Ref. [14], uses a taper-
ing of the etched hole size and shape in the nanobeam
to provide strong localization and overlap of the breath-
ing mode and the fundamental optical mode, resulting
in a vacuum optomechanical coupling rate [28] between
photons and phonons of
g
0
/
2
π
≈
1 MHz.
In order to minimize mechanical clamping losses, the
nanobeam is anchored to the Si bulk with a periodic cross
structure which is designed to have a complete phononic
bandgap at the breathing mode frequency [14]. Through
tuning of the cross height
h
c
and width
w
c
(c.f., Figs. 1(d-
e)), bandgaps as wide as
∼
3 GHz can be achieved as
shown in Fig. 1. We analyze SEM images of realized
structures to provide accurate structure dimensions for
our FEM models, and in particular we include in our
modeling a filleting of the inner and outer corners (
r
1
and
r
2
in Fig. 1e) of the crosses arising from technical
limitations of the patterning of the structure. To investi-
gate the efficacy of the acoustic shielding we fabricate and
characterize arrays of devices with a scaling of the cross
period number from
N
shield
= 0 to 10, with all other de-
sign parameters held constant. FEM modeling indicates
(see App. B) that the addition of the cross shield pro-
vides significant protection against nanometer-scale dis-
order which is inherently introduced during device fabri-
3
10
-5
10
-7
a
d
0
1
2
3
4
5
6
7
8
9
10
acoustic shield periods
10
5
10
6
10
7
10
8
10
9
10
10
10
11
Q
m
0
1
2
3
4
5
10
4
10
3
10
2
10
1
휏
(s)
e
measured
sim. (σ = 4 nm)
f
drive
read
drive
read
ω
c
ω
ω
c
ω
n
m
(phonon number)
1
10
0
0.2
0.4
0.6
0.8
1.0
휏
(s)
1.2
T
wait
T
wait
10
-6
10
-5
10
-4
휏
(s)
10
-7
b
t
(s)
10
-6
10
-5
10
-7
10
-6
10
-5
10
-7
c
...
...
time
exp
[-
γ
0
휏
]
heating
n
m
/n
m
f
n
m
i
n
m
f
n
m
f
n
m
i
n
m
(phonon number)
0
1
2
3
4
t
(
μs
)
0
1.0
2.0
3.0
n
m
/n
m
i
f
initial bin
T
pulse
T
pulse
휏
휏
휏
10
-6
(
n
m
=13.6)
i
n
m
= 4.2
f
FIG. 2.
Ringdown measurements of the acoustic breathing mode. a
, Illustration of the ringdown measurement
performed using a red-detuned (∆ = +
ω
m
) pulsed laser for excitation and readout.
b
, Normalized phonon occupancy measured
during (left) and after (right) the laser readout pulse (
n
c
= 569; optomechanical back-action rate
γ
OM
/
2
π
= 1
.
07 MHz) for a
6-shield device (device B). Squares are measured data points. Solid and dashed lines are a best fit to the dynamical model of
the hot bath (see App. D). The displayed pulse-on-state plot (left) corresponds to a delay of
τ
= 200
μ
s, with
n
i
m
= 4
.
2 and
n
f
m
= 13
.
6 phonons.
c
, Ringdown measurements of a 7-shield device (device C) for readout pulse amplitude of
n
c
= 320. The
series of inset panels show the measured (and fit; solid blue curve) anti-Stokes signal during the optical pulse at a series of pulse
delays.
d
, Plot of the measured breathing mode
Q
-factor versus number of acoustic shield periods
N
shield
. The solid green
line is a fit to the corresponding simulated radiation-limited
Q
-factor (see App. B) for devices with standard deviation (SD)
σ
= 4 nm disorder in hole position and size, similar to the value measured from device SEM image analysis. The shaded green
region corresponding to the range of simulated
Q
values (ensemble size 10) within one SD of the mean. The square purple data
points represents the measured
Q
in (
f
).
e
, Acoustic excitation is performed coherently by using either a blue-detuned pump
(upper diagram) to drive the breathing mode into self-oscillation, or using an RF-modulated red-detuned pump [29] (lower
diagram). See App. E for details of the coherent excitation and readout parameters.
f
, Ringdown measurements performed
on an eight-shield device (device D) at large phonon amplitude. For blue-detuned driving (red squares) the fit decay rate is
γ
0
/
2
π
= (0
.
122
±
0
.
020) Hz. For modulated-pump driving (purple circles) the fit decay rate is
γ
0
/
2
π
= (0
.
108
±
0
.
006) Hz. The
error bars are 90% confidence intervals of the measured values of
n
i
m
. The shaded regions are the 90% confidence intervals for
the exponential fit curves.
cation.
Optical measurements of the acoustic properties of the
OMC cavity are performed at millikelvin temperatures
in a dilution refrigerator. The sample containing an ar-
ray of different OMC devices is mounted directly on a
copper mount attached to the mixing chamber stage of
the fridge, and a single lensed optical-fiber is positioned
with a 3-axis stage to couple light into and out of each
device [27]. In a first set of measurements of acoustic
energy damping, we employ a single pulsed laser scheme
to perform both excitation and readout of the breathing
mode. In this scenario, depicted in Fig. 2(a), the laser
frequency (
ω
l
) is tuned to the red motional sideband of
the OMC cavity optical resonance, ∆
≡
ω
c
−
ω
l
≈
+
ω
m
,
and is pulsed on for a duration
T
pulse
and then off for a
variable time
τ
. This produces a periodic train of pho-
ton pulses due to anti-Stokes scattering of the probe laser
which are on-resonance with the optical cavity. The anti-
Stokes scattered photons are filtered from the probe laser
and sent to a single photon detector producing a photon
count rate proportional to the number of phonons in the
acoustic resonator (see Apps. C for details of the mea-
surement set-up and phonon number calibration meth-
ods).
We display in Fig. 2(b) a typical readout signal, show-
ing the normalized phonon occupancy during and imme-
diately after the application of a 4
μ
s pulse. The ini-
tial optomechanical back-action cooling of the acoustic
breathing mode is followed by a slower turn-on of heat-
ing of the mode during the pulse. After the pulse, with
4
the back-action cooling turned off, a transient heating
of the acoustic mode occurs over several microseconds.
The parasitic heating is attributable to very weak opti-
cal absorption of the probe pulse in the Si cavity which
produces a hot bath coupled to the breathing mode [27].
Here we use the transient heating of the acoustic mode
to perform ringdown measurements of the stored phonon
number. A phenomenological model of the dynamics of
the induced damping (
γ
p
) and effective occupancy (
n
p
) of
the hot bath (see App. D) allows us to fit the anti-Stokes
decay signal. Plotting the initial mode occupancy at the
beginning of the fit readout pulse (
n
i
m
) versus delay time
τ
between pulses (c.f., Fig. 2(a)), we plot the ringdown
of the stored phonon number in the the breathing mode
as displayed in Fig. 2(c) for a device with
N
shield
= 7.
Performing a series of ringdown measurements over a
range of devices with varying
N
shield
, and fitting an ex-
ponential decay curve to each ringdown we produce the
Q
-factor plot in Fig. 2(d). We observe an initial trend
in
Q
-factor versus shield number which rises on aver-
age exponentially with each additional shield period, and
then saturates for
N
shield
≥
5 to
Q
m
&
10
10
. As indi-
cated in Fig. 2(c) these
Q
values correspond to ringdown
of small, near-single-phonon level amplitudes. We also
perform ringdown measurements at high phonon ampli-
tude using two additional methods displayed schemat-
ically in Fig. 2(e) and described in detail in App. E.
These methods use two laser tones to selectively excite
the acoustic breathing mode using a
×
1000 weaker ex-
citation and readout optical pulse amplitude (
n
c
.
0
.
3).
The measured ringdown curves, displayed in Fig. 2(f),
show the decay from initial phonon occupancies of 10
3
-
10
4
of an 8-shield device (device D; square purple data
point in Fig. 2(d)).
The two methods yield similar
breathing mode energy decay rates of
γ
0
/
2
π
= 0
.
108 Hz
and 0
.
122 Hz, the smaller of which corresponds to a
Q
-
value of
Q
m
= 4
.
92
+0
.
39
−
0
.
26
×
10
10
and a phonon lifetime of
τ
ph
,
0
= 1
.
47
+0
.
09
−
0
.
08
s. Comparing all three excitation meth-
ods with widely varying optical-absorption-heating and
phonon amplitude, we consistently measure
Q
m
&
10
10
for devices with
N
shield
≥
5.
In order to understand the origin of the residual damp-
ing for large
N
shield
we also measured the temperature
dependence of the energy damping rate, breathing mode
frequency, and full width at half maximum (FWHM)
linewidth of the breathing mode for the highest
Q
8-
shield device (device D). In Fig. 3(a) we plot the en-
ergy damping rate which shows an approximately lin-
ear rise in temperature up to
T
f
≈
100 mK, and then
a much faster
∼
(
T
f
)
2
.
4
rise in the damping. Using
the two-tone coherent excitation method [29], we plot
in Fig. 3(b) the measured breathing mode acoustic spec-
trum at
T
f
= 7 mK and pump power
n
c
= 0
.
1. The
top plot shows rapid spectral scans (40 ms per scan) in
which the probe frequency is swept across the acoustic
resonance. These rapid scans show a jittering acoustic
line with a roughly ∆
ω
m
/
2
π
≈
1 kHz linewidth, consis-
tent with the predicted magnitude of optical back-action
T
f
(K)
10
-1
γ
0
/2π (Hz)
10
2
10
-1
10
0
10
1
10
-2
a
b
f
-
f
0
(kHz)
0
-4
-8
4
8
1.0
1.5
2.0
1.0
1.2
1.4
Count rate (×10
5
c.p.s.)
2.5
δf
m
n
c
(photons)
Δ
1/2
,
δf
(kHz)
4
0
0.2
0.4
0.6
0.8
10
-1
10
0
10
1
3
2
1
c
d
Δ
1/2
δf
m
10
1
3
5
T
f
(K)
Δ
1/2
Δ
1/2
10
0
T
f
(K)
10
-1
10
-2
10
2
10
-1
γ
0
/2π (Hz)
FIG. 3.
Temperature dependence of acoustic damp-
ing, frequency, and frequency jitter.
a, Plot of the
measured breathing mode energy damping rate,
γ
0
/
2
π
, as
a function of fridge temperature (
T
f
). Dashed green (ma-
genta) curve is a fit with temperature dependence
γ
0
∼
T
1
.
01
f
(
γ
0
∼
T
2
.
39
f
). Error bars are 90% confidence intervals of the
exponential fit to measured ring down curves. Inset: Plot
of measured damping data with estimated energy damping
from a TLS model (see App. G). The shaded blue region
corresponds to the standard deviation of log (
γ
0
/
2
π
) for 100
different random TLS distributions.
b
, Two-tone coherent
spectroscopy signal. Upper plot: three individual spectrum
of rapid frequency sweeps with a frequency step size of 500 Hz
and dwell time of 1 ms (RBW
≈
0
.
5 kHz). Lower plot:
average spectrum of rapid scan spectra taken over minutes,
showing broadened acoustic response with FWHM linewidth
of ∆
1
/
2
/
2
π
= 4
.
05 kHz. The large on-resonance response
corresponds to an estimated optomechanical cooperativity of
C
≡
γ
OM
/
(
γ
0
+
γ
p
)
&
1
.
1, consistent with the predicted mag-
nitude of back-action damping
γ
OM
/
2
π
≈
817 Hz and bath-
induced damping
γ
p
/
2
π
≈
120 Hz at the measurement pump
power level
n
c
= 0
.
1.
c
, Breathing mode resonance frequency
shift
δf
and ensemble average FWHM-linewidth ∆
1
/
2
as a
function of pump photon number
n
c
. Solid red curve is fit
to back-action limited linewidth, yielding
g
0
/
2
π
= 1
.
15 MHz.
d
, Measured
δf
(upper plot) and ∆
1
/
2
(lower plot) versus
T
f
.
Error bars are 90% confidence intervals from Voigt fit to mea-
sured spectra. Data presented in (
a
-
d
) are for device D. The
solid magenta curves in the inset of (
a
) and top panel of (
d
)
corresponds to simulations of a single random TLS distribu-
tion.
(
γ
OM
/
2
π
≈
820 Hz) and hot bath damping (
γ
p
/
2
π
≈
120 Hz) at the
n
c
= 0
.
1 measurement power. An ensem-
ble average of these scans, taken over several minutes,
yields a broadened and reduced contrast acoustic line of
FWHM ∆
1
/
2
/
2
π
= 4
.
05 kHz.
Note that in Fig. 3(b) we are measuring the acous-
5
tic line with the laser light on, as opposed to the ring-
down measurements of Fig. 2 in which the laser light is
off. Lowering the optical pump power to reduce back-
action and absorption-induced damping limits further
the already low signal-to-noise ratio, and scanning more
slowly begins to introduce frequency jitter into the mea-
sured line. As such, we can only bound the intrinsic
low temperature coherence time of the breathing mode
to
τ
coh
,
0
&
2
/
∆
ω
m
≈
0
.
3 ms. Further information can,
however, be gleaned by measuring the linewidth and cen-
ter frequency of the ensemble averaged spectrum as a
function of
n
c
(Fig. 3(c)) and
T
f
(Fig. 3(d)). The width
of the frequency jitter spectrum, averaged over minutes,
is roughly independent of optical pump power and tem-
perature down to the lowest measurable pump powers
(
n
c
= 0
.
02) and up to
T
f
= 800 mK. The center fre-
quency, on the other hand, shifts up in frequency with
both temperature and optical power. The frequency shift
versus
T
f
is consistent with the frequency shift versus
n
c
if the hot bath temperature (see App. D) is used as a
proxy for the fridge temperature.
Estimates of the magnitude of Landau-Rumer damp-
ing of the breathing mode (see App. F) indicate that
3-phonon scattering in Si is far too weak at
T
f
.
1 K
to explain the measured damping. Analysis of the inter-
actions of TLS with the localized acoustic modes of the
confined geometry of the OMC cavity structure, how-
ever, show that TLS interactions can explain all of the
observed breathing mode behavior. In this analysis, de-
tailed in App. G, FEM simulation is used to find the
frequencies and radiation-limited damping rates of the
acoustic quasi-normal modes of the OMC cavity struc-
ture. An estimate of the spectral density of TLS within
the breathing mode volume (
V
m
≈
0
.
11 (
μ
m)
3
) is ascer-
tained from estimated surface oxide (
∼
0
.
25 nm [30])
and etch-damage (
∼
15 nm [31]) layer thicknesses in
the Si device, and bulk TLS density found in amor-
phous materials [25, 32]. Using the resulting effective
spectral density of interacting TLS,
n
0
,m
≈
20 GHz
−
1
,
and average TLS transverse and longitudinal deforma-
tion potentials of
̄
M
≈
0
.
04 eV and
̄
D
≈
3
.
2 eV, re-
spectively, yields breathing mode damping and frequency
shifts which are in excellent agreement with the measured
data (see Fig. 3). The estimated level of frequency jitter
is also found in agreement with the measured value, as-
suming all TLS are being pumped via the same optical
absorption that drives the hot bath.
Several key observations can be drawn from the TLS
damping modeling. The first is that the typical
T
3
de-
pendence of TLS relaxation damping of acoustic waves
is dependent on the phonon bath DOS into which the
TLS decay [20, 21]. In the OMC cavity the phonon DOS
is strongly modified from a three-dimensional bulk ma-
terial. This directly results in the observed weak tem-
perature dependence of the acoustic damping for
T
f
.
100 mK, where the thermally acitvated TLS interact res-
onantly with an approximately one-dimensional phonon
DOS. A second point to note is that the TLS reso-
nant damping is strongly suppressed due to the phononic
bandgap surrounding the OMC cavity. Estimates of the
phonon-induced spontaneous decay rate of TLS in the
bandgap is on the order of Hz; combined with the discrete
number of TLS in the small mode volume of the breath-
ing mode, acoustic energy from the breathing mode can-
not escape via resonant coupling to TLS. The observed
lack of saturation of the breathing mode energy damp-
ing with either temperature or phonon amplitude is fur-
ther evidence that non-resonant relaxation damping
−
due to dispersive coupling to TLS
−
is dominant [25].
Finally, the small average number of estimated TLS in
V
m
which are thermally activated at the lowest temper-
atures (
∼
2), leads to significant variation in the simu-
lated TLS relaxation damping at
T
f
∼
10 mK (see shaded
blue region of the inset to Fig. 3(a)). This is consistent
with the observed fluctuations from device-to-device in
the low-temperature
Q
m
for devices with
N
shield
>
5 (see
Fig. 2(d)).
Utilizing the advanced methods of nanofabrication and
cavity optomechanics has provided a new toolkit to ex-
plore quantum acoustodynamics in solid-state materials.
Continued studies of the behavior of TLS in similar engi-
neered nanostructures to the OMC cavity of this work
may lead to, among other things, new approaches to
modifying the behavior of quasi-particles in supercon-
ductors [33], mitigating decoherence in superconduct-
ing [34, 35] and color center [36, 37] qubits, and even
new coherent TLS-based qubit states in strong coupling
with an acoustic cavity [38]. The extremely small mo-
tional mass (
m
eff
= 136 fg [14]) and narrow linewidth
of the OMC cavity also make it ideal for precision mass
sensing [39] and in exploring limits to alternative quan-
tum collapse models [40]. Perhaps most intriguing is
the possibility of creating a hybrid quantum architecture
consisting of acoustic and superconducting quantum cir-
cuits [41–48], where the small scale, reduced cross-talk,
and ultralong coherence time of quantum acoustic de-
vices may provide significant improvements in connec-
tivity and performance of current quantum hardware.
ACKNOWLEDGMENTS
This work was supported by the ARO Quantum Opto-
Mechanics with Atoms and Nanostructured Diamond
MURI program (grant N00014-15-1-2761), the ARO-LPS
Cross-Quantum Systems Science & Technology program
(grant W911NF-18-1-0103), the Institute for Quantum
Information and Matter, an NSF Physics Frontiers Cen-
ter (grant PHY-1733907) with support of the Gordon and
Betty Moore Foundation, and the Kavli Nanoscience In-
stitute at Caltech. H.R. gratefully acknowledges support
from the National Science Scholarship from A*STAR,
Singapore.
6
[1] D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-
Nasser, and S. Schultz, Phys. Rev. Lett.
84
, 4184 (2000).
[2] J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and
R. D. Meade,
Photonic Crystals: Molding the Flow of
Light
, 2nd ed. (Princeton University Press, 2008).
[3] E. Yablonovitch, T. Gmitter, J. P. Harbison,
and
R. Bhat, Appl. Phys. Lett.
51
, 2222 (1987).
[4] S. John and J. Wang, Physical Review B
43
, 12772
(1991).
[5] M. Fujita, S. Takahashi, Y. Tanaka, T. Asano, and
S. Noda, Science
308
, 1296 (2005).
[6] M. Maldovan, Nature
503
, 209 (2013).
[7] S. Zhang, C. Xia, and N. Fang, Phys. Rev. Lett.
106
,
024301 (2011).
[8] S. Narayana and Y. Sato, Phys. Rev. Lett.
108
, 214303
(2012).
[9] M. Maldovan, Phys. Rev. Lett.
110
, 025902 (2013).
[10] M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and
O. Painter, Nature
459
, 550 (2009).
[11] C. Brendel, V. Peano, O. Painter, and F. Marquardt,
Phys. Rev. B
97
, 020102(R) (2018).
[12] J. Cha, K. W. Kim, and C. Daraio, Nature
564
, 229
(2018).
[13] T. P. M. Alegre, A. Safavi-Naeini, M. Winger,
and
O. Painter, Opt. Express
19
, 5658 (2011).
[14] J. Chan, A. H. Safavi-Naeini, J. T. Hill, S. Meenehan,
and O. Painter, Appl. Phys. Lett.
101
, 081115 (2012).
[15] P.-L. Yu, K. Cicak, N. S. Kampel, Y. Tsaturyan, T. P.
Purdy, R. W. Simmonds, and C. A. Regal, App. Phys.
Lett.
104
, 023510 (2014).
[16] A. B. Y. Tsaturyan, E. S. Polzik, and A. Schliesser,
Nature Nanotech.
12
, 776 (2017).
[17] A. H. Ghadimi, S. A. Fedorov, N. J. Engelsen, M. J.
Bereyhi, R. Schilling, D. J. Wilson, and T. J. Kippen-
berg, Science
360
, 764 (2018).
[18] M. Kalaee, M. Mirhosseini, P. B. Dieterle, M. Peruzzo,
J. M. Fink, and O. Painter, arXiv:1808.04874 (2018).
[19] Q. P. Unterreithmeier, T. Faust, and J. P. Kotthaus,
Phys. Rev. Lett.
105
, 027205 (2010).
[20] R. O. Behunin, F. Intravaia, and P. T. Rakich, Phys.
Rev. B
93
, 224110 (2016).
[21] B. D. Hauer, P. H. Kim, C. D. F. Souris, and J. P. Davis,
Phys. Rev. B
98
, 214303 (2018).
[22] L. Landau and G. Rumer, Phys. Z. Sowjet.
11
(1937).
[23] G. P. Srivastava,
The Physics of Phonons
(Taylor &
Francis Group, 1990).
[24] T. O. Woodruff and H. Ehrenreich, Phys. Rev.
123
, 1553
(1961).
[25] W. A. Phillips, Rep. Prog. Phys.
50
, 1657 (1987).
[26] S. Galliou, M. Goryachev, R. Bourquin, P. Abbe, J. P.
Aubry, and M. E. Tobar, Scientific Reports
3
, 2132
(2013).
[27] S. M. Meenehan, J. D. Cohen, G. S. MacCabe, F. Marsili,
M. D. Shaw, and O. Painter, Phys. Rev. X
5
, 041002
(2015).
[28] M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt,
Cavity Optomechanics: Nano- and Micromechanical Res-
onators Interacting with Light
(Springer-Verlag, 2014).
[29] A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichen-
field, M. Winger, Q. Lin, J. T. Hill, D. Chang, and
O. Painter, Nature
472
, 69 (2011).
[30] N. Yabumoto, M. Oshima, O. Michikami, and S. Yoshii,
Jpn. J. Appl. Phys.
20
, 893 (1981).
[31] G. S. Oehrlein, Materials Science and Engineering: B
4
,
441 (1989).
[32] R. N. Kleiman, G. Agnolet, and D. J. Bishop, Phys. Rev.
Lett.
59
, 2079 (1987).
[33] K. Rostem, P. J. de Visser, and E. J. Wollack, Phys.
Rev. B
98
, 014522 (2018).
[34] J. Gao,
The Physics of Superconducting Microwave Res-
onators
, Ph.D. thesis, California Institute of Technology
(2008).
[35] J. M. Martinis and A. Megrant, arXiv:1410.5793 (2014).
[36] Y.-I. Sohn, S. Meesala, B. Pingault, H. A. Atikian,
J. Holzgrafe, M. G ̈undo ̆gan, C. Stavrakas, M. J. Stan-
ley, A. Sipahigil, J. Choi, M. Zhang, J. L. Pacheco,
J. Abraham, E. Bielejec, M. D. Lukin, M. Atat ̈ure, and
M. Lonˇcar, Nat. Commun.
9
, 2012 (2018).
[37] T. Astner, J. Gugler, A. Angerer, S. Wald, S. Putz,
N. J. Mauser, M. Trupke, H. Sumiya, S. Onoda, J. Isoya,
J. Schmiedmayer, P. Mohn, and J. Majer, Nat. Mat.
17
,
313 (2018).
[38] T. Ramos, V. Sudhir, K. Stannigel, P. Zoller, and T. J.
Kippenberg, Phys. Rev. Lett.
110
, 193602 (2013).
[39] M. S. Hanay, S. I. Kelber, C. D. OConnell, P. Mulvaney,
J. E. Sader, and M. L. Roukes, Nature Nanotech.
10
,
339 (2015).
[40] S. Nimmrichter, K. Hornberger,
and K. Hammerer,
Phys. Rev. Lett.
113
, 020405 (2014).
[41] M. H. Devoret and R. J. Schoelkopf, Science
339
, 1169
(2013).
[42] J.-M. Pirkkalainen, S. U. Cho, J. Li, G. S. Paraoanu,
P. J. Hakonen, and M. A. Sillanp ̈a ̈a, Nature
494
, 211
(2013).
[43] M. V. Gustafsson, T. Aref, A. F. Kockum, M. K. Ekstrm,
G. Johansson, and P. Delsing, Science
346
, 207 (2014).
[44] Y. Chu, P. Kharel, W. H. Renninger, L. D. Burkhart,
L. Frunzio, P. T. Rakich, and R. J. Schoelkopf, Science
358
, 199 (2017).
[45] R. Manenti, A. F. Kockum, A. Patterson, T. Behrle,
J. Rahamim, G. Tancredi, F. Nori, and P. J. Leek, Nat.
Commun.
8
, 975 (2017).
[46] K. J. Satzinger, Y. P. Zhong, H.-S. Chang, G. A. Peairs,
A. Bienfait, M.-H. Chou, A. Y. Cleland, C. R. Conner,
E. Dumur, J. Grebe, I. Gutierrez, B. H. November, R. G.
Povey, S. J. Whiteley, D. D. Awschalom, D. I. Schuster,
and A. N. Cleland, arXiv:1804.07308 (2018).
[47] B. A. Moores, L. R. Sletten, J. J. Viennot, and K. Lehn-
ert, Phys. Rev. Lett.
120
, 227701 (2018).
[48] P. Arrangoiz-Arriola, E. A. Wollack, M. Pechal, J. D.
Witmer, J. T. Hill, and A. H. Safavi-Naeini, Phys. Rev.
X
8
, 031007 (2018).
[49] J. D. Cohen, S. M. Meenehan, G. S. MacCabe,
S. Gr ̈oblacher, A. H. Safavi-Naeini, F. Marsili, M. D.
Shaw, and O. Painter, Nature
520
, 522 (2015).
[50] S. M. Meenehan, J. D. Cohen, S. Gr ̈oblacher, J. T. Hill,
A. H. Safavi-Naeini, M. Aspelmeyer, and O. Painter,
Phys. Rev. A
90
, 011803 (2014).
[51] P. S. Zyryanov and G. G. Taluts, Soviet Physics JETP-
USSR
22
, 1326 (1966).
[52] M. G. Holland, Phys. Rev.
132
, 2461 (1963).
[53] J. Callaway, Phys. Rev.
113
, 1046 (1959).
7
[54] D. ter Haar,
Collected Papers of L. D. Landau
, 2nd ed.
(Elsevier, 2013).
[55] A. Akhiezer, J. Phys. (Moscow)
1
, 277 (1939).
[56] R. Lifshitz and M. L. Roukes, Phys. Rev. B
61
, 5600
(2000).
[57] N. Moiseyev,
Non-Hermitian Qauntum Mechanics
(Cam-
bridge University Press, 2011).
[58] M. Field, C. G. Smith, M. Pepper, D. A. Ritchie, J. E. F.
Frost, G. A. C. Jones, and D. G. Hasko, Phys. Rev. Lett.
70
, 1311 (1993).
[59] H. Lefebvre-Brion and R. W. Field,
The Spectra and
Dynamics of Diatomic Molecules: Revised and Enlarged
Edition
(Elsevier Academic Press, 2004).
[60] W. B. Gauster, Phys. Rev. B
4
, 1288 (1971).
[61] J. J. Hall, Phys. Rev.
161
, 756 (1967).
[62] H. J. McSkimin, J. Appl. Phys.
24
, 988 (1953).
[63] G. S. Oehrlein and J. F. Rembetski, IBM J. Res. Develop.
36
, 140 (1992).
[64] Y. H. Lee, G. S. Oehrlein, and C. Ransom, Radiation
Effects and Defects in Solids: Incorporating Plasma Sci-
ence and Plasma Technology
111-112
, 221 (1989).
[65] M. Morita, T. Ohmi, E. Hasegawa, M. Kawakami, and
K. Suma, Appl. Phys. Lett.
55
, 562 (1989).
[66] M. Morita and T. Ohmi, Jpn. J. Appl. Phys.
33
, 370
(1994).
[67] W. A. Phillips, Phys. Rev. Lett.
61
, 2632 (1988).
[68] J. L. Black and B. I. Halperin, Phys. Rev. B
16
, 2879
(1977).