ARTICLE
Two-dimensional optomechanical crystal cavity
with high quantum cooperativity
Hengjiang Ren
1,2,3
, Matthew H. Matheny
1,2,3
, Gregory S. MacCabe
1,2,3
, Jie Luo
1,2,3
, Hannes Pfeifer
4,5
,
Mohammad Mirhosseini
1,2,3
& Oskar Painter
1,2,3
✉
Optomechanical systems offer new opportunities in quantum information processing and
quantum sensing. Many solid-state quantum devices operate at millikelvin temperatures
—
however, it has proven challenging to operate nanoscale optomechanical devices at these
ultralow temperatures due to their limited thermal conductance and parasitic optical
absorption. Here, we present a two-dimensional optomechanical crystal resonator capable of
achieving large cooperativity
C
and small effective bath occupancy
n
b
, resulting in a quantum
cooperativity
C
eff
≡
C
/
n
b
> 1 under continuous-wave optical driving. This is realized using a
two-dimensional phononic bandgap structure to host the optomechanical cavity, simulta-
neously isolating the acoustic mode of interest in the bandgap while allowing heat to be
removed by phonon modes outside of the bandgap. This achievement paves the way for a
variety of applications requiring quantum-coherent optomechanical interactions, such as
transducers capable of bi-directional conversion of quantum states between microwave
frequency superconducting quantum circuits and optical photons in a
fi
ber optic network.
https://doi.org/10.1038/s41467-020-17182-9
OPEN
1
Thomas J. Watson, Sr., Laboratory of Applied Physics, California Institute of Technology, Pasadena, CA 91125, USA.
2
Kavli Nanoscience Institute, California
Institute of Technology, Pasadena, CA 91125, USA.
3
Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA 91125,
USA.
4
Max Planck Institute for the Science of Light, Staudtstrasse 2, 91058 Erlangen, Germany.
5
Present address: Institut für Angewandte Physik, Universität
Bonn, Wegelerstraße 8, 53115 Bonn, Germany.
✉
email:
opainter@caltech.edu
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1
1234567890():,;
R
ecent advances in optomechanical systems, in which
mechanical resonators are coupled to electromagnetic
waveguides and cavities
1
,
2
, have led to a series of scienti
fi
c
and technical advances in areas such as precision sensing
3
,
4
,
nonlinear optics
5
,
6
, nonreciprocal devices
7
–
9
, and topological
wave phenomena
10
,
11
. In addition, such systems have been used
to explore macroscopic quantum phenomena, from initial
demonstrations of laser cooling of mechanical resonators into
their quantum ground state
12
–
16
to heralded preparation and
entanglement of mechanical quantum states
17
–
20
, generation of
squeezed light
6
,
21
, and coherent transduction between photons
with different energies
5
,
22
–
26
.
Optomechanical crystals (OMCs)
27
, where electromagnetic
and elastic waves overlap within a lattice, are patterned structures
that can be engineered to yield large radiation
–
pressure coupling
between cavity photons and phonons. Previous work has realized
one-dimensional (1D) silicon (Si) OMC cavities with extremely
large vacuum optomechanical coupling rates (
g
0
≈
1 MHz)
28
,
29
,
enabling a variety of applications in quantum optomechanics
including the aforementioned ground-state cooling
13
and remote
quantum entanglement of mechanical oscillators via an optical
channel
19
. An application area of growing interest for OMCs is in
hybrid quantum systems involving microwave-frequency super-
conducting quantum circuits
30
,
31
. Owing to the large ratio (×10
5
)
of the speed of light to the speed of sound in most materials,
OMCs operating at telecom-band optical frequency naturally
couple strongly to similar wavelength microwave-frequency
acoustic modes. Recent experimental demonstrations of
microwave-frequency phononic crystal cavities with ultralow
dissipation
32
and strong dispersive coupling to superconducting
qubits
33
indicate that there are potentially signi
fi
cant technical
advantages in forming an integrated quantum electrodynamic
and acoustodynamic circuit architecture for quantum informa-
tion processing
34
,
35
. In such an architecture, OMCs could provide
a quantum interface between microwave-frequency logic circuits
and optical quantum communication channels.
A signi
fi
cant roadblock to further application of OMC cavities
for quantum applications is the very weak, yet non-negligible
parasitic optical absorption in current devices
17
–
20
,
36
. Optical
absorption, thought to occur due to surface defect states
37
,
38
,
together with inef
fi
cient thermalization due to the 1D nature of Si
nanobeam OMC cavities currently in use, can yield signi
fi
cant
heating of the microwave-frequency acoustic mode of the device.
At ultralow temperatures (
≲
0.1 K), where microwave-frequency
systems can be reliably operated as quantum devices, optical
absorption leads to rapid (sub-microsecond) heating of the
acoustic cavity mode
36
. This has limited quantum optomecha-
nical experiments to schemes with high optical power and short
pulses
17
–
20
,
32
,
36
or very low continuous optical power
39
,
40
.
The most relevant
fi
gure of merit for quantum optomechanical
applications, when there is thermal population in the system, is
the effective quantum cooperativity (
C
eff
≡
C
/
n
b
), corresponding
to the standard photon
–
phonon cooperativity (
C
≡
γ
OM
/
γ
b
)
divided by the Bose factor of the effective thermal bath (
n
b
)
coupled to the acoustic mode of the cavity
5
,
23
,
36
. Here
γ
b
repre-
sents the total coupling rate of the mechanical system to its
various thermal baths. In previous experiments with nanobeam
OMC cavities at millikelvin temperatures, the quantum coop-
erativity was substantially degraded owing to the heating and
damping caused by the optical-absorption-induced hot bath. The
heating of the acoustic cavity mode by the optically generated hot
bath can be mitigated through several different methods. The
simplest approach in a low temperature environment is to couple
the cavity more strongly to the surrounding cold bath of the chip
or through addition of another cold bath as in experiments in a
3
He buffer gas environment
41
,
42
. This method can be quite
effective in decreasing the acoustic mode thermal occupancy in
the presence of optical absorption; however, the effectiveness of
the method relies on increasing the coupling to baths other than
the optical channel, which necessarily decreases the overall
photon
–
phonon quantum cooperativity.
Here we employ a strategy that makes use of the frequency-
dependent density of phonon states within a phononic bandgap
structure to overcome this limitation. Using a two-dimensional
(2D) OMC cavity
43
–
45
, the thermal conductance between the hot
bath and the cold environment is greatly increased owing to the
larger contact area of the 2D structure with the bath, while the
acoustic mode of interest is kept isolated from the environment
through the phononic bandgap of the structure. By keeping the
intrinsic damping of acoustic mode low, this method is a pro-
mising route to realizing
C
eff
> 1. Initial work in this direction,
performed at room temperature, utilized snow
fl
ake-shaped holes
in a Si membrane to create a quasi-2D OMC with substantially
higher optical power handling capability, although with a rela-
tively low optomechanical coupling of
g
0
/2
π
=
220 kHz
45
. In this
work, we realize a Si quasi-2D OMC with over 50-fold
improvement in optomechanical back-action per photon and a
much higher thermal conductance (×68) compared to 1D struc-
tures at millikelvin temperatures. Most importantly, we demon-
strate a
Q
-factor of 1
:
2
þ
0
:
12
0
:
15
́
10
9
for the 10-GHz
optomechanically coupled acoustic mode of the cavity and a
C
eff
greater than unity under continuous-wave optical pumping, sui-
table for realizing applications such as signal transduction of
itinerant quantum signals
22
–
25
.
Results
Design of the quasi-2D OMC cavity
. The quasi-2D OMC cavity
in this work is designed around the silicon-on-insulator (SOI)
materials platform, which naturally provides for a thin Si device
layer of a few hundred nanometers in which both microwave-
frequency acoustic modes and near-infrared optical modes can be
guided in the vertical direction
46
. Patterning of the Si slab
through plasma etching is used to form a nanoscale lattice sup-
porting Bloch waves for both optical and acoustic modes. We
focus on the fundamental guided optical of even vector parity
about the center of the Si slab (
σ
z
=+
1). This choice is motivated
by the fact that for a connected lattice of low air-
fi
lling fraction
the fundamental
σ
z
=+
1 optical modes are the most strongly
guided in the Si slab, greatly reducing their sensitivity to scat-
tering loss. It is common to refer to these modes as transverse-
electric (TE-like), as their electric
fi
eld polarization lies pre-
dominantly in the plane of the slab. For a symmetric Si slab, only
the acoustic modes of
σ
z
=+
1 are coupled via radiation pressure
to the optical modes of the slab. The OMC cavity design consists
of three major steps.
First, we start with a periodically patterned quasi-2D slab
structure with both phononic and photonic bandgaps in which to
host the optomechanical cavity. Here we use the
“
snow
fl
ake
”
crystal with a hexagonal lattice
46
as shown in Fig.
1
a. The
snow
fl
ake crystal provides a pseudo-bandgap for TE-like optical-
guided waves and a full bandgap for all acoustic mode
polarizations. Finite-element-method (FEM) simulations of the
optical and acoustic modes of the snow
fl
ake crystal were
performed using the COMSOL software package
47
, with nominal
snow
fl
ake parameters corresponding to a Si slab thickness of
t
=
220 nm and (
a
,
r
,
w
)
=
(500, 205, 75) nm, resulting in a TE-like
guided-mode photonic bandgap extending over optical frequen-
cies of 180
–
240 THz (vacuum wavelength 1250
–
1667 nm) and an
acoustic bandgap covering 8.85
–
11.05 GHz.
Second, we create an in-plane waveguide in the snow
fl
ake
lattice. This is done by replacing one row of snow
fl
ake unit cells
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with a customized unit cell. Waveguiding to this line-defect
occurs for photon and phonon modes that lie within the
corresponding bandgaps of the surrounding snow
fl
ake lattice.
Here we chose to form the line-defect by replacing one row of
snow
fl
akes with a set of
“
C
”
-shaped holes. This design is inspired
by the 1D nanobeam OMCs reported in ref.
28
. Optomechanical
coupling in this sort of design is a result of both bulk
(photoelastic)
48
and surface (moving boundary)
49
effects. The
“
C
”
shape allows for large overlap of the acoustic mode stress
fi
eld
with the optical mode intensity in the bulk of the Si device layer,
while also focusing the optical mode at the air
–
Si boundary to
increase the moving boundary contribution to the optomecha-
nical coupling. The width of the line-defect and exact shape and
dimension of the
“
C
”
shape were optimized considering several
factors: (i) large guided-mode vacuum coupling rate
g
Δ
46
, (ii)
avoidance of leaky optical resonances of the slab, and (iii)
creation of guided acoustic bands with dispersion. Leaky optical
resonances are resonant with the Si slab yet lie above the light
cone; imperfections in the fabricated structure can result in large
coupling between the guided optical mode of interest and leaky
resonances at the same frequency, resulting in large scattering
loss. Acoustic bands of limited dispersion (
fl
at bands) are also
susceptible to fabrication imperfections as these acoustic modes
tends to localize around small defects resulting in poor overlap
with the more extended optical modes (this was a primary
dif
fi
culty in prior 2D snow
fl
ake OMC work
6
). Photonic and
phononic bandstructure diagrams of the optimized waveguide
unit cell are shown in Fig.
1
b, c, respectively. Shaded in light blue
are the optical guided-mode bandgap extending from 190 to 210
THz (vacuum wavelength 1430
–
1580 nm) and the acoustic
guided-mode bandgap extending from 10 to 10.6 GHz. Abutting
these bandgaps and plotted as solid blue curves are the optical
and acoustic waveguide bands of interest.
The
fi
nal step in the cavity design involves introducing a
tapering of the line-defect waveguide properties along the
waveguide propagation direction (
x
-axis). Here we utilize a
modulation of the
“
C
”
-shaped parameters that increases quad-
ratically in amplitude with distance along the
x
-axis of the line-
defect waveguide from a designated center position of the cavity.
This introduces an approximate quadratic shift of the frequency
of the waveguide modes with distance from the cavity center.
For waveguide modes near a band edge, this results in localization
of the modes as they are pushed into a bandgap away from the
cavity center. As detailed in Supplementary Note 1, a
Nelder
–
Mead simplex search algorithm was used to obtain a
tapered cavity structure with simultaneously high optical
Q
-factor
and large optomechanical coupling between co-localized optical
and acoustic modes. Figure
1
d, e displays the resulting simulated
fi
eld pro
fi
les of the fundamental optical resonance (
ω
c
/2
π
=
194
THz,
λ
c
=
1550 nm) and coupled acoustic resonance (
ω
m
/2
π
=
10.27 GHz) of the optimized 2D OMC cavity, respectively. The
co-localized modes have a theoretical vacuum optomechanical
coupling rate of
g
0
/2
π
=
1.4 MHz, and the optical mode has a
theoretical scattering-limited quality factor of
Q
c,scat
=
2.1 × 10
7
.
Test devices based on this new design were fabricated from a
SOI microchip with a 220-nm-thick Si device layer and an
underlying 3
μ
m buried oxide layer. A scanning electron
micrograph (SEM) image of a fabricated 2D OMC cavity and
optical waveguide for coupling light into the structure are shown
in Fig.
2
a, b. Several iterations of fabrication were performed in
order to improve the
fi
delity of the fabrication with respect to the
design structure. Between fabrication iterations, SEM images of
devices were analyzed to determine the fabricated geometrical
parameters of the cavity structure; this information was fed back
into the next fabrication iteration in order to realize devices with
a geometry as close as possible to the simulation-optimized
design parameters. An example of
fi
tted
“
C
”
-shaped and
snow
fl
ake holes are shown as red solid lines in the SEM image
of Fig.
2
b, with corresponding
fi
tted cavity parameters plotted in
Fig.
2
c.
Optomechanical coupling and mechanical damping
. Fabricated
devices were characterized both at room temperature (300 K) and
at cryogenic temperatures inside a fridge (
T
f
=
10 mK). A sim-
pli
fi
ed schematic of the optical measurement set-up is shown in
Fig.
2
d. Room temperature testing was performed using a dim-
pled optical
fi
ber taper to evanescently couple light into and out
of a chip-based Si coupling waveguide
50
; each Si coupling
waveguide is butt-coupled to a corresponding OMC cavity as
shown in Fig.
2
a (also see Supplementary Note 3). A typical
optical spectrum from one of the quasi-2D OMC cavities is
e
c
d
150
170
190
210
230
v
o
(THz)
8.0
X
9.0
10.0
11.0
v
m
(GHz)
b
500 nm
x
y
a
k
x
w
a
r
w
o
h
o
d
w
i
h
i
Fig. 1 Quasi-2D OMC cavity design. a
Unit cell schematic of a linear waveguide formed in the snow
fl
ake crystal. Guided modes of the waveguide
propagate along the
x
-axis. Insets: (left)
“
C
”
-shaped parameters; (right) snow
fl
ake parameters.
b
Photonic and
c
phononic bandstructure of the linear
waveguide. The solid blue curves are waveguide bands of interest; dashed lines are other guided modes; shaded light blue regions are bandgaps of inter
est;
green tick mark indicates the cavity mode frequencies; gray regions denote the continua of propagating modes. In the photonic bandstructure, only mo
des
of even vector parity about the center of the Si slab (
σ
z
=+
1) are shown. In the acoustic bandstructure, green dashed curves are for
σ
z
=+
1 and
σ
y
=
−
1
parity modes, and yellow dashed curves denote
σ
z
=
−
1 modes.
d
FEM-simulated mode pro
fi
le (
E
y
component of the electric
fi
eld) of the fundamental
optical cavity resonance at
ω
c
/2
π
=
194 THz, with red (blue) corresponding to positive (negative)
fi
eld amplitude.
e
Simulated displacement pro
fi
le of the
fundamental acoustic cavity resonance at
ω
c
/2
π
=
10.27 GHz. The magnitude of displacement is represented by color (large displacement in red, zero
displacement in blue).
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3
displayed in Fig.
3
a, showing a fundamental optical resonance at a
wavelength of
λ
c
=
1558.8 nm with a loaded (intrinsic) optical
Q
-
factor of
Q
c
=
3.9 × 10
5
(
Q
c,i
=
5.3 × 10
5
).
In order to measure the coupled acoustic resonance(s) of the
OMC cavity, we used a pump
–
probe
“
electromagnetically induced
transparency
”
measurement
51
,
52
(see Supplementary Note 5).
Coherent detection of the beating of the pump and probe laser
tones with heterodyne spectroscopy produces a spectrum of the
coupled acoustic modes. For the new quasi-2D OMC cavity
design, we found a single, dominantly coupled acoustic mode
around
ω
m
/2
π
≈
10.2 GHz. A plot of the measured acoustic mode
spectrum at several optical pump powers is shown in Fig.
3
b for
the device of Fig.
2
a. Optomechanical back-action from the pump
laser can be seen to broaden the acoustic resonance; a plot of the
fi
t resonance linewidth (
γ
=
γ
0
+
γ
φ
+
γ
p
+
γ
OM
) versus intra-
cavity photon number of the pump laser tone (
n
c
) is shown as an
inset to Fig.
3
b. Here
γ
0
is the intrinsic energy decay rate,
γ
p
is the
optical absorption bath-induced damping,
γ
φ
is due to any pure
dephasing effects (frequency jitter) of the acoustic resonance, and
γ
OM
¼
4
g
2
0
n
c
=
κ
is the optomechanical back-action rate
32
. From
the slope of the back-action-broadened linewidth versus
n
c
,we
extract a vacuum coupling rate of
g
0
=
2
π
¼
1
:
09
þ
0
:
13
0
:
09
MHz, close to
the simulated optimum value of 1.4 MHz.
Following initial room temperature measurements, the new
optimized quasi-2D OMC cavities were tested at millikelvin
temperatures. We measured the intrinsic mechanical damping
rate, back-action cooling, and heating dynamics of the OMC
cavities in a dilution refrigerator (DR) with a temperature of
T
f
≈
10 mK at the base plate connected to the mixing chamber. The 1-
cm square sample containing an array of devices was directly
mounted on a copper mount attached to the mixing chamber
plate, and a 3-axis stage was used to align a lensed optical
fi
ber to
the tapered on-chip coupling waveguide of a given device under
test (see Fig.
2
a and Supplementary Note 3).
We measured the intrinsic mechanical
Q
-factor of the quasi-
2D OMC devices at millikelvin temperatures using an pulsed
optical scheme in which 10-
μ
s-long optical pulses excite and
read-out the energy in the acoustic mode (for details, see the
“
Methods
”
section). By varying the delay between the optical
pulses, this technique allows for the evaluation of the acoustic
energy ringdown while the laser light
fi
eld is off
32
,
53
. In Fig.
3
c,
we show the measured ringdown curve for a 10-GHz acoustic
mode of a quasi-2D snow
fl
ake OMC cavity with an additional
acoustic shield. In order to increase the acoustic isolation of the
acoustic cavity mode, on this device we added a periphery
consisting of eight periods of a cross-structure phononic bandgap
shield (see Supplementary Note 4). Fitting of the initial mode
occupancy at the beginning of an optical pulse (
n
i
) versus inter-
pulse delay (
τ
off
) yields an intrinsic acoustic energy decay rate of
γ
0
=
2
π
¼
8
:
28
þ
1
:
25
0
:
43
Hz, corresponding to a mechanical
Q
-factor of
Q
m
¼
1
:
2
þ
0
:
12
0
:
15
́
10
9
. This large mechanical
Q
-factor is consistent
with other measurements of Si nanobeam OMC cavities at
millikelvin temperatures
32
and is thought to result from a
suppression of acoustic absorption from near-resonant two-level
system defects
54
.
Optical-absorption-induced hot bath
. While the acoustic mode
Q
-factor measured in ringdown measurements is promising for
certain quantum memory applications
34
,
35
, it is measured with
the laser pump off. The prospects for performing coherent
quantum operations between photons and phonons depends
critically on the ability to minimize unwanted heating and
damping of the acoustic mode due to parasitic effects resulting
from optical absorption in the presence of an applied laser
fi
eld. A
model for the heating and damping in Si OMC cavities,
fi
rst
proposed in ref.
39
, is illustrated in Fig.
4
a. In this model, the
acoustic mode of the OMC cavity is weakly coupled to the sur-
rounding DR environment at a rate
γ
0
, while simultaneously
GHz
~
Laser
Spectrum
analyzer
EDFA
EOM
-
BPD
a
c
b
d
Dilution refrigerator (
T
f
~ 10 mk)
SPD
l
ω
c
ω
l
ω
c
-
Δ
ω
AOM
Circulator
Sideband filter
VC
80
1
3
57
120
160
200
“C” shape holes
Fitted parameters (nm)
1
μ
m
OMC cavity region
Cross-shield
Coupling waveguide
Lensed fiber
10
μ
m
Sideband filter
Fig. 2 Device fabrication and measurement set-up. a
SEM image of a full quasi-2D snow
fl
ake OMC device fabricated on SOI. This device is an
“
8-shield
device
”
in which an additional eight periods of cross-structure phononic bandgap shielding is applied at the periphery (see Supplementary Note 4). A
lensed optical
fi
ber is used to couple light into a tapered on-chip waveguide, which is butt-coupling to the quasi-2D snow
fl
ake OMC cavity.
b
SEM image of
the center of the cavity region. Fitting to the geometries for
“
C
”
-shaped holes and snow
fl
ake holes in the cavity region are shown as red solid lines.
c
Measured
“
C
”
parameters
fi
t from SEM images of a fabricated cavity (dots) along with their design values (dashed curves). Shown over one half of the
cavity (
fi
rst 8
“
C
”
shapes on the right side of the cavity) are the parameters (from top to bottom):
h
o
(cyan circles),
w
o
/2 (green squares),
h
i
(blue crosses),
and
w
i
/2 (red triangles). Error bars in geometrical parameters correspond to standard deviation of ten sets of measured SEM images.
d
Experimental set-
up for characterization of the quasi-2D snow
fl
ake OMC cavity. Multiple optical switches are used to switch between continuous wave and pulsed optical
excitation and between heterodyne spectroscopy and single photon detection. AOM acousto-optic modulator, EOM electro-optic modulator, EDFA
erbium-doped
fi
ber ampli
fi
er, VC variable coupler, BDP balanced photodetector, SPD single photon detector. A more detailed schematic and description of
the measurement set-up is provided in Supplementary Note 2.
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being coupled to an optically generated hot bath at a rate
γ
p
. The
source of the hot bath in Si OMC devices is thought to be due to
linear optical absorption via electronic defect states at the surface
of the etched Si
37
,
38
, which through phonon-mediated relaxation
processes produces a bath of hot phonons that pile up above the
acoustic bandgap of the OMC structure due to phonon bottleneck
effects
32
. Assuming a phonon density of states corresponding to
that of a 2D plate for the hot bath above the OMC bandgap, and
weak coupling to the localized acoustic cavity mode via 3-phonon
scattering, such a model predicts heating of the localized acoustic
mode that scales as
n
1
=
3
c
and damping that scales as
n
2
=
3
c
.
Here we explore the optically induced parasitic heating and
damping for the quasi-2D OMC cavity. As for the previous 1D
nanobeam measurements, optical measurements were performed
using a continuous-wave pump laser tuned to the optical cavity
resonance (
Δ
=
0) to avoid optomechanical back-action cooling
and damping (see
“
Methods
”
). Results of the optically induced
heating and damping of the 10 GHz breathing-like mode of the
2D OMC cavity are displayed in Fig.
4
b, c, respectively. In Fig.
4
b,
the inferred hot bath occupancy
n
p
(left vertical axis) and the
corresponding bath temperature
T
p
(right vertical axis) are
plotted versus intra-cavity photon number
n
c
(lower horizontal
axis) and the corresponding input power (upper horizontal axis).
We
fi
nd that the hot bath occupancy is accurately
fi
t by the power
law,
n
p
¼ð
1
:
1
Þ
́
n
0
:
3
c
. This is the same power-law scaling as
found for 1D nanobeam OMCs in ref.
32
(red dashed line in
Fig.
4
b); however, the overall magnitude of the hot bath
occupancy has substantially dropped for the quasi-2D cavity by
a factor of 7.2. Using a modi
fi
ed thermal conductance model
that assumes ballistic phonon transport and a power-law
dependence on temperature consistent with our measurements,
C
th
¼
ε
ð
T
p
Þ
α
¼
2
:
3
, FEM numerical simulations of the 1D and 2D
cavity structures (see Supplementary Note 6) predict a greatly
enhanced thermal conductance coef
fi
cient for the quasi-2D cavity
ε
2D
/
ε
1D
=
42. This yields a hot bath occupancy ratio between
nanobeam and quasi-2D cavities of 6.2, in good correspondence
to the measured value. In alignment with our design strategy, the
reduction in mode heating according to this model is primarily
the result of a geometric effect due to the fact that the quasi-2D
cavity is connected to the surrounding chip structure over a larger
in-plane solid angle than that of the 1D nanobeam and thus has a
larger number of phonon modes to carry heat away (i.e., a larger
thermal conductance).
In Fig.
4
c, we plot the spectral linewidth of the breathing mode
of the quasi-2D cavity, determined in this case by measuring the
acoustic mode thermal noise spectrum imprinted on the re
fl
ected
0
0.04
0.08
0.12
off
(s)
10
–2
10
–1
10
0
‹
n
› (phonons)
10
–7
10
–6
10
–5
n
f
n
i
T
~ 10 mK
Q
m
= 1.2 × 10
9
0
200
3
4
5
6
7
γ
/2
π
(MHz)
n
c
T
~ 300 K
Q
m
~ 3500
10.21
10.23
10.19
1
2
3
4
5
Modulation frequency (GHz)
–20
0
20
0
Reflection (normalized)
Reflection (normalized)
1
Δ
(pm)
0
b
a
c
~ 4 pm
Q
c
~ 390,000
Fig. 3 Optomechanical characterization. a
Optical spectrum of a quasi-2D snow
fl
ake OMC cavity measured using a swept laser scan.
b
Pump
–
probe
measurement of the mechanical mode spectrum of interest centered around
ω
m
/2
π
=
10.21 GHz for different optical pump powers (from lowest to highest
peak re
fl
ection:
n
c
=
33,
n
c
=
104, and
n
c
=
330). Inset: measured mechanical mode linewidth versus
n
c
. The device measured in
a
,
b
is a cavity with zero
additional acoustic shield periods (zero-shield device).
c
Pulsed ringdown measurement of a quasi-2D snow
fl
ake OMC cavity with 8 periods of additional
cross-structure shielding (8-shield device). The measured optomechanical parameters of this device are (
κ
,
κ
e
,
g
0
,
ω
m
,
γ
0
)
=
2
π
(1.19 GHz, 180 MHz, 1.18
MHz, 10.02 GHz, 8
:
28
þ
1
:
25
0
:
43
Hz). The phonon occupancy at the beginning of each 10-
μ
s optical pulse,
n
i
(orange squares), is measured using a peak power
corresponding to
n
c
=
60. Inset: mode heating within the optical pulse, with blue circles corresponding to data and the solid line a
fi
t to the heating curve.
n
f
is the mode occupancy at the end of the optical pulse, used here to check for consistent excitation across the different inter-pulse
τ
off
measurements.
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5
laser pump
fi
eld using a balanced heterodyne receiver (see Fig.
2
d
and
“
Methods
”
). In such a measurement, with resonant pump
laser
fi
eld (
Δ
=
0), the acoustic mode linewidth is expected to
contain contributions from the intrinsic energy decay rate
γ
0
, the
optical absorption bath-induced damping
γ
p
, and any pure
dephasing effects (frequency jitter) of the acoustic resonance
γ
φ
.
Referring to Fig.
4
c, we see that the linewidth dependence on
n
c
can be separated into three different regimes: (i) at the lowest
powers (
n
c
≲
10) the linewidth begins to saturate to a constant
value (in our
fi
t, this is given by
γ
φ
(
γ
0
is entirely negligible on this
scale), (ii) a low-power regime (100 <
n
c
<
n
th
=
1000) with a
relatively strong dependence of linewidth on optical power,
and (iii) a high-power regime (
n
c
>
n
th
=
1000) with a second,
weaker dependence of linewidth on optical power, where
n
th
is
the threshold for
n
c
used between (ii) and (iii) in the
fi
tting
(see Supplementary Note 7). Fitting the low-power regime
with a power-law dependence on
n
c
,we
fi
nd
γ
=
2
π
¼
γ
φ
=
2
π
þð
1
:
1 kHz
Þ
́
n
0
:
61
c
, with
γ
φ
/2
π
=
14.54 kHz. At these
lower powers, we
fi
nd a power-law scaling and overall magnitude
of damping of the breathing mode of the quasi-2D OMC cavity
which is close to that for the 1D nanobeam cavities of ref.
32
(see
the dashed red curve in Fig.
4
c). In the high-power regime, we
fi
nd a
fi
t given by
γ
=
2
π
¼
23
:
91 kHz
þð
9
:
01 kHz
Þ
́
n
0
:
29
c
, with a
power-law exponent that is approximately half that in the low-
power regime. Determining the exact mechanism of the
γ
p
slow
down versus
n
c
(or indirectly
T
p
) in the high-power regime is
outside the scope of this article; however, possibilities include a
change in the phonon scattering rate with increasing phonon
frequency in the nanostructured Si
fi
lm
55
,
56
or a transition from
Landau
–
Rumer scattering to Akhiezer-type damping as the
effective bath temperature rises
57
.
Before moving on to measurements of back-action cooling in
the quasi-2D OMC cavity, we note one important distinction
between the geometry of the optical coupling in the new 2D
devices in comparison to previously studied 1D nanobeam
devices. Whereas in the 1D nanobeam devices the coupling
waveguide is evanescently coupled to the OMC cavity
—
and is
thus not in direct mechanical contact with the nanobeam cavity
—
in the quasi-2D devices the optical coupling waveguide is
physically connected to one end of the OMC cavity region (see
Fig.
2
a). Optical absorption in the coupling waveguide of the
quasi-2D OMC devices may thus also lead to heating of the
acoustic cavity mode. This effect is further corroborated by FEM
simulations, detailed in Supplementary Note 8, that show that a
weak cavity is formed between the end of the coupling waveguide
and the quasi-2D OMC cavity. Optical absorption of the input
power (
P
in
) in the coupling waveguide can be modeled as an
effective waveguide photon number,
n
wg
, where
n
wg
=
β
P
in
for
some
fi
xed constant
β
independent of cavity detuning. Assuming
that the dependence of
n
p
on
n
wg
is the same as that for
n
c
, we can
10
0
10
0
10
1
10
–3
10
–2
10
–1
10
o
10
1
10
–1
10
0
10
1
10
2
10
4
‹
n
› (phonons)
γ
/2
π
(kHz)
n
c
10
1
10
1
10
0
10
1
10
3
10
2
10
2
10
4
n
c
T
P
(K)
bc
1D nanobeam
P
in
(
μ
W)
P
in
(
μ
W)
(
n
c
)
0.3
(
n
c
)
0.61
(
n
c
)
0.29
1D nanobeam
<n>
γ
p
(
n
c
,
n
wg
)
n
f
(
T
f
)
10 mK
γ
OM
n
c
n
p
,
T
p
,
γ
p
(
n
c
,
n
wg
)
γ
0
a
Acoustic
shielding
“Hot” bath
Optical cavity
Lensed fiber
Coupling waveguide
γ
φ
n
wg
DR bath
Fig. 4 Optical-absorption-induced hot bath. a
Diagram illustrating the proposed model of heating of the mechanics due to optical absorption and the
various baths coupled to the localized mechanical mode.
b
Plot of measured
n
p
versus
n
c
. The solid line is a
fi
t to the data giving
n
p
¼ð
1
:
1
Þ
́
n
0
:
3
c
. Dashed
line indicates
n
p
versus
n
c
for a 1D nanobeam device measured in ref.
32
for comparison.
c
Plot of measured linewidth
γ
(blue dots) versus
n
c
. The blue solid
line is a power-law
fi
t to the data, where in the low
n
c
regime
γ
=
2
π
γ
φ
=
2
π
þ
γ
p
=
2
π
¼
14
:
54 kHz
þð
1
:
1 kHz
Þ
́
n
0
:
61
c
and in the high
n
c
regime
γ
=
2
π
¼
23
:
91 kHz
þð
9
:
01 kHz
Þ
́
n
0
:
29
c
. The red solid curve is the resulting
fi
t for
γ
p
by itself. For comparison, the dashed red curve is a plot of
γ
p
versus
n
c
for a 1D nanobeam device from ref.
32
. Error bars correspond to 90% con
fi
dence interval in
fi
t to Lorentzian spectral linewidth (error bars smaller than the
symbol are not shown). In
b
,
c
the quasi-2D OMC cavity measurements are for the same 8-shield device as in Fig.
3
c.
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write
n
p
(
n
c
,
P
in
)
=
n
p
(
n
c
+
n
wg
). Similarly,
γ
p
(
n
c
,
P
in
)
=
γ
p
(
n
c
+
n
wg
). In the measurements above with resonant pumping at
Δ
=
0, very small input powers were required to build up large intra-
cavity photon numbers, and as such
n
c
≫
n
wg
. In what follows,
where we perform back-action cooling with
Δ
=
ω
m
≫
κ
, the
input power required to yield a given
n
c
is much larger and
n
wg
cannot be ignored.
Effective quantum cooperativity
. The ability of a cavity opto-
mechanical systems to perform coherent quantum operations
between the optical and mechanical degrees of freedom requires
both large cooperativity
C
≡
γ
OM
/
γ
b
and a mechanical mode
thermal occupancy
〈
n
〉
< 1 (the thermal noise in the optical mode
is assumed negligible)
5
,
23
. Here
γ
b
represents the total coupling
rate of the mechanical system to its various thermal baths, which
in the case of the Si OMC cavities, is given by
γ
b
=
γ
0
+
γ
p
. The
relevant
fi
gure of merit is then the effective quantum coopera-
tivity
C
eff
≡
C
/
n
b
2
, where
n
b
is the total effective bath occupancy
de
fi
ned by the relation
γ
b
n
b
≡
γ
0
(
n
0
+
1)
+
γ
p
(
n
p
+
1), where the
“
+
1
”
terms correspond to spontaneous decay and
n
0
≲
10
−
3
is
the bath occupancy in the surrounding chip region of the OMC
cavity (see Supplementary Note 9). As
γ
0
≪
γ
p
for the optical
powers used in this work and
n
0
≪
1, in what follows
γ
b
n
b
≈
γ
p
(
n
p
+
1).
A measurement of the quantum cooperativity
C
eff
can be made
by observing the cooled mechanical occupancy under optical
back-action cooling of the coupled mechanical mode,
h
n
i¼
γ
p
n
p
þ
γ
0
n
0
γ
0
þ
γ
OM
þ
γ
p
¼
n
b
1
C
þ
1
n
b
;
C
1
1
C
eff
;
ð
1
Þ
where we have implicitly assumed that the optical pump laser
responsible for producing back-action damping
γ
OM
has a zero
effective noise occupancy. Back-action cooling is most ef
fi
cient in
the resolved sideband limit (
κ
/2
ω
m
< 1) when the optical pump is
applied on the red-detuned motional sideband of the cavity,
Δ
=
ω
m
. In Fig.
5
a, we show the measured cooling curve of the quasi-
2D snow
fl
ake cavity under continuous-wave optical pumping at
Δ
=
ω
m
. We infer the measured 10.2 GHz acoustic cavity mode
occupancy (blue dots) from calibration of the photon counts of
the anti-Stokes sideband of the re
fl
ected optical pump laser (see
Fig.
2
d and
“
Methods
”
).
We can also predict the back-action cooling curve of the
acoustic mode based on our independent measurements of
γ
OM
(Fig.
3
b),
γ
p
(Fig.
4
b), and
n
p
(Fig.
4
c) versus
n
c
. Using Eq. (
1
), we
plot the theoretical acoustic mode cooling versus
n
c
(i.e.,
β
=
0) as
a solid red curve in Fig.
5
a. Not only does the
β
=
0 theoretical
curve predict substantially more cooling of the acoustic mode
than measured but also the shape of the measured and theoretical
curves are quite different. As alluded to at the end of the previous
section, one signi
fi
cant difference between the back-action
cooling measurements and the measurements of
n
p
and
γ
p
is
that the cooling measurements are performed at
Δ
=
ω
m
,
requiring a 100-fold increase in the optical pump power to reach
a given intra-cavity photon number
n
c
. For reference, we have
plotted the corresponding input power
P
in
on the top horizontal
axes of Figs.
4
b, c and
5
a. Adding an additional waveguide photon
number
n
wg
=
β
P
in
to the intra-cavity photon number
n
c
in
determining
n
p
and
γ
p
,we
fi
nd a modi
fi
ed cooling curve (
β
=
15
μ
W
−
1
; solid blue curve) that
fi
ts both the magnitude and shape of
the measured cooling curve. In Fig.
5
b, we plot the corresponding
quantum cooperativity curve showing that
C
eff
reaches
above unity.
Although we have achieved
C
eff
> 1 under continuous-wave
optical driving in the newly designed quasi-2D OMC cavities,
looking forward, signi
fi
cant further increases can be achieved.
One clear method is to thermally decouple the input coupling
waveguide from the OMC cavity in order to eliminate the
parasitic heating from
n
wg
. This can be accomplished, for
instance, by using evanescent side-coupling instead of butt-
coupling of the coupling waveguide to the cavity. A second
approach to dramatically improving
C
eff
is through improve-
ments in optical quality factor, where similar quasi-2D planar
photonic crystal devices have already been demonstrated with
optical
Q
-factors approaching 10
7
58
. In Fig.
5
c, we estimate
achievable
〈
n
〉
and
C
eff
as a function of
n
c
and
Q
c
assuming that
n
wg
has been successfully eliminated. We
fi
nd that, for a
Q
-factor
of
Q
c
=
3.90 × 10
5
, equal to that of the zero-shield device of
Fig.
3
a, it should be possible to reach
〈
n
〉
≈
0.1 and
C
eff
≈
5 for
optical pump powers at the single photon level.
Discussion
In conclusion, we have presented the design, fabrication, and
characterization of a new quasi-2D OMC cavity with a breathing-
like acoustic mode of 10 GHz frequency and large vacuum
optomechanical coupling rate
g
0
/2
π
≳
1 MHz. By employing an
engineered 2D phononic and photonic bandgap material in
which to host the OMC cavity, the acoustic breathing-like mode
of interest is well protected from its environment, while phonon
modes above the acoustic bandgap serve as additional channels
for removing heat from the cavity region. Through this dual role
of the 2D bandgap structure, we demonstrate at millikelvin
temperatures a localized acoustic cavity mode with intrinsic
Q
-
factor of 1
:
2
þ
0
:
12
0
:
15
́
10
9
and a greatly increased (×68) thermal
conductance between the cavity and the cold bath reservoir of the
surrounding chip compared to previous 1D nanobeam OMC
devices. These properties of the quasi-2D OMC cavity allow us to
achieve a quantum cooperativity
C
eff
> 1 under continuous-wave
optical pumping. They also point the way for signi
fi
cant further
improvements in quantum cooperativity through modi
fi
cation in
the optical coupling geometry.
This result ushers forth a variety of quantum optomechanical
applications using chip-scale OMCs. In particular, in the case of
hybrid superconducting and acoustic microwave quantum
circuits
33
,
59
–
61
, optomechanical devices that can operate in con-
tinuous mode in the high
C
eff
regime at millikelvin temperatures
would enable bi-directional conversion of itinerant optical and
microwave quantum signals (see Supplementary Note 10),
forming the critical interface necessary to realize an optical
quantum network
62
of superconducting quantum circuit nodes.
These advances may also allow quantum optomechanical mea-
surements to be performed at even lower temperatures than
currently possible with OMC cavities, which, given their
demonstrated ultralow rates of intrinsic acoustic dissipation
32
,
will allow for further studies of theories related to gravitationally
induced decoherence
63
,
64
and nonlinearities and dephasing in
mechanical systems
65
.
Methods
Device fabrication
. The devices were fabricated from a SOI wafer (SEH, 220 nm Si
device layer, 3
μ
m buried-oxide layer) using electron beam lithography followed by
inductively coupled plasma reactive ion etching. The Si device layer is then masked
by photoresist to de
fi
ne a
“
trench
”
region of the chip to be etched and cleared to
which a lens
fi
ber can access the chip coupling waveguides. In the unprotected
trench region of the chip, the buried-oxide layer is removed with a highly aniso-
tropic plasma etch, and the handle Si layer is removed to a depth of 120
μ
m using
an isotropic plasma etch. The devices were then released in vapor hydro
fl
uoric acid
(HF) and cleaned in a piranha solution (3:1 H
2
SO
4
:H
2
O
2
) before a
fi
nal diluted HF
etch to remove any surface oxides.
Device characterization
. Fabricated devices are characterized using a
fi
ber-cou-
pled, wavelength-tunable external cavity diode laser. The laser light is sent through
a 50-MHz bandwidth-tunable
fi
ber Fabry
–
Perot
fi
lter (Micron Optics FFP-TF2) to
reject laser phase noise at the mechanical frequency. After this pre
fi
ltering, the light
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