of 21
Two-Dimensional Optomechanical Crystal Cavity with High Quantum Cooperativity
Hengjiang Ren,
1, 2
Matthew H. Matheny,
1, 2
Gregory S. MacCabe,
1, 2
Jie
Luo,
1, 2
Hannes Pfeifer,
3,
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
3
Max Planck Institute for the Science of Light, Staudtstrasse 2, 91058 Erlangen, Germany
(Dated: October 8, 2019)
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 demonstrate 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
.
3
>
1
under continuous-wave optical driving. This is realized using a two-dimensional phononic bandgap
structure to host the optomechanical cavity, simultaneously 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 fiber optic network.
Recent advances in optomechanical systems, in which
mechanical resonators are coupled to electromagnetic
waveguides and cavities [1, 2], have led to a series of sci-
entific and technical advances in areas such as precision
sensing [3, 4], nonlinear optics [5, 6], nonreciprocal de-
vices [7–9], and topological wave phenomena [10, 11]. In
addition, such systems have been used to explore macro-
scopic quantum phenomena, from initial demonstrations
of laser cooling of mechanical resonators into their quan-
tum ground state [12–16], to heralded preparation and
entanglement of mechanical quantum states [17–20], gen-
eration of squeezed light [6, 21], and coherent transduc-
tion between photons with different energies [5, 22–26].
Optomechanical crystals (OMCs) [27], where electro-
magnetic 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 vac-
uum optomechanical coupling rates (
g
0
1 MHz) [28,
29], enabling a variety of applications in quantum op-
tomechanics including the aforementioned ground-state
cooling [13] and remote quantum entanglement of me-
chanical oscillators via an optical channel [19]. An ap-
plication 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 experi-
Current Address: Institut f ̈ur Angewandte Physik, Universit ̈at
Bonn, Wegelerstrae 8, 53115 Bonn, Germany
opainter@caltech.edu; http://copilot.caltech.edu
mental demonstrations of microwave-frequency phononic
crystal cavities with ultralow dissipation [32] and strong-
dispersive coupling to superconducting qubits [33] indi-
cate that there are potentially significant technical ad-
vantages in forming an integrated quantum electrody-
namic and acoustodynamic circuit architecture for quan-
tum information processing [34, 35]. In such an architec-
ture, OMCs could provide a quantum interface between
microwave-frequency logic circuits and optical quantum
communication channels.
A significant roadblock to further application of OMC
cavities for quantum applications is the very weak, yet
non-negligible parasitic optical absorption in current de-
vices [17–20, 36]. Optical absorption, thought to oc-
cur due to surface defect states [37, 38], together with
inefficient thermalization due to the 1D nature of Si
nanobeam OMC cavities currently in use, can yield signif-
icant 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 optomechanical 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 figure-of-merit for quantum optome-
chanical applications is the effective quantum cooperativ-
ity (
C
eff
C/n
b
), corresponding to the standard photon-
phonon cooperativity (
C
) divided by the Bose factor of
the effective thermal bath (
n
b
) coupled to the acous-
tic mode of the cavity [5, 23, 36]. In previous experi-
ments with nanobeam OMC cavities at millikelvin tem-
peratures, the quantum cooperativity was substantially
degraded below unity (the relevant threshold for coher-
ent photon-phonon interactions) due to the heating and
damping caused by the optical-absorption-induced hot
arXiv:1910.02873v1 [quant-ph] 7 Oct 2019
2
e
c
d
150
170
190
210
230

o
(THz)
8.0
9.0
10.0
11.0

m
(GHz)
Γ
Χ
b
500nm
x
y
a
k
x
a
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 snowflake crystal.
Guided modes of the waveguide propagate along the
x
-axis. Insets: (left) ‘C’ shape parameters, (right) snowflake 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 band gaps of interest; green tick mark indicates the cavity
mode frequencies; gray regions denote the continua of propagating modes. In the photonic bandstructure only modes 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 profile (
E
y
component of the electric field) of the fundamental optical cavity resonance at
ω
c
/
2
π
= 194 THz, with red (blue) corresponding
to positive (negative) field amplitude.
e
, Simulated displacement profile 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).
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 due to the larger contact
area of the 2D structure with the bath, while the acous-
tic mode of interest is kept isolated from the environment
through the phononic bandgap of the structure. By keep-
ing the intrinsic damping of the acoustic mode low, this
method is a promising route to realizing
C
eff
>
1. Initial
work in this direction, performed at room temperature,
utilized snowflake-shaped holes in a Si membrane to cre-
ate a quasi-2D OMC with substantially higher optical
power handling capability, although with a relatively 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 (
×
42) compared
to 1D structures at millikelvin temperatures. Most im-
portantly, we demonstrate a
Q
-factor of 1
.
2
×
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, suitable for realizing applications
such as signal transduction of itinerant quantum sig-
nals [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 acous-
tic 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 lat-
tice supporting Bloch waves for both optical and acous-
tic modes. We focus on the fundamental guided optical
modes 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-filling fraction the funda-
mental
σ
z
= +1 optical modes are the most strongly
guided in the Si slab, greatly reducing their sensitiv-
ity to scattering loss. It is common to refer to these
modes as transverse-electric-like (TE-like), as their elec-
tric field polarization lies predominantly in the plane of
3
the slab. For a symmetric Si slab only the acoustic modes
of
σ
z
= +1 are coupled via radiation pressure to the op-
tical 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 ‘snowflake’ crystal with a hexagonal
lattice [46] as shown in Fig. 1a. The snowflake crys-
tal provides a pseudo-bandgap for TE-like optical guided
waves and a full bandgap for all acoustic mode polar-
izations. Finite-element-method (FEM) simulations of
the optical and acoustic modes of the snowflake crys-
tal were performed using the COMSOL software pack-
age [47], with nominal snowflake parameters (
a,r,w
) =
(500
,
205
,
75) nm and Si slab thickness
t
= 220 nm, re-
sulting in a TE-like guided mode photonic band gap
extending over optical frequencies 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
snowflake lattice. This is done by replacing one row of
snowflake unit cells with a customized unit cell. Waveg-
uiding to this line-defect occurs for photon and phonon
modes that lie within the corresponding bandgaps of the
surrounding snowflake lattice. Here, we chose to form the
line-defect by replacing one row of snowflakes with a set
of ‘C’-shaped holes. This design is inspired by the one-
dimensional nanobeam OMCs reported in Ref. [28]. Op-
tomechanical coupling in this sort of design is a result of
both bulk (photoelastic) [48] and surface (moving bound-
ary) [49] effects. The ‘C’-shape allows for large overlap
of the acoustic mode stress field 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 in-
crease the moving boundary contribution to the optome-
chanical coupling. The width of the line-defect, and ex-
act shape and dimension of the ‘C’-shape were optimized
considering several factors: (i) large guided-mode vac-
uum coupling rate
g
[46], (ii) avoidance of leaky optical
resonances of the slab, and (iii) creation of guided acous-
tic bands with dispersion. Leaky optical resonances are
resonant with the Si slab yet lie above the light cone; im-
perfections 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 (flat
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 difficulty in prior 2D
snowflake OMC work [6]). Photonic and phononic band-
structure diagrams of the optimized waveguide unit cell
are shown in Figs. 1b and 1c, respectively. Shaded in
light blue are the optical guided-mode bandgap extending
from 190 THz to 210 THz (vacuum wavelength 1430 to
1580 nm) and the acoustic guided-mode bandgap extend-
ing from 10 GHz to 10
.
6 GHz. Abutting these bandgaps
and plotted as solid blue curves are the optical and acous-
tic waveguide bands of interest.
The final step in the cavity design involves intro-
ducing a tapering of the line-defect waveguide proper-
ties along the waveguide propagation direction (
x
-axis).
Here we utilize a modulation of the ‘C’-shape parame-
ters that increases quadratically in amplitude with dis-
tance along the
x
-axis of the line-defect waveguide from
a designated center position of the cavity. This intro-
duces 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 re-
sults in localization of the modes as they are pushed into
a bandgap away from the cavity center. As detailed in
App. A, a Nelder-Mead simplex search algorithm was
used to obtain a tapered cavity structure with simulta-
neously high optical
Q
-factor and large optomechanical
coupling between co-localized optical and acoustic cavity
modes. Figures 1d and 1e display the resulting simu-
lated field profiles of the fundamental optical resonance
(
ω
c
/
2
π
= 194 THz,
λ
c
= 1550 nm) and coupled acous-
tic 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 theoreti-
cal 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 scan-
ning electron microscope (SEM) image of a fabricated
2D OMC cavity and optical waveguide for coupling light
into the structure are shown in Figs. 2a and 2b. Several
iterations of fabrication were performed in order to im-
prove the fidelity of the fabrication with respect to the
design structure. Between fabrication iterations SEM im-
ages of devices were analyzed to determine the fabricated
geometrical parameters of the cavity structure; this infor-
mation 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 fitted ‘C’-shape and snowflake holes are
shown as red solid lines in the SEM image of Fig. 2b, with
corresponding fitted cavity parameters plotted in Fig. 2c.
Optomechanical coupling and mechanical damp-
ing.
Fabricated devices were characterized both at room
temperature (300 K) and at cryogenic temperatures in-
side a fridge (
T
f
= 10 mK). A simplified schematic of the
optical measurement set-up is shown in Fig. 2d. Room
temperature testing was performed using a dimpled op-
tical fiber taper to evanescently couple light into and out
of a chip-based Si coupling waveguide [50]; each Si cou-
pling waveguide is butt-coupled to a corresponding OMC
cavity as shown in Fig. 2a (also see App. C). A typical
optical spectrum from one of the quasi-2D OMC cavi-
ties is displayed in Fig. 3a, showing a fundamental op-
tical resonance at a wavelength of
λ
c
= 1558
.
8 nm with
a loaded (intrinsic) optical
Q
-factor of
Q
c
= 3
.
9
×
10
5
4
GHz
~
d
Dilution Refrigerator (
T
f
~ 10mK)
Laser
Spectrum
Analyzer
EDFA
sideband lter
SPD
EOM
-
BPD
ω
c
ω
l
ω
c
ω
l
ω
c
ω
l
sideband lter
10 μm
ω
c
ω
l
ω
c
ω
l
ω
c
-
ω
l
AOM
circulator
a
VC
c
1
3
5
7
80
120
160
200
`C’ shape holes
Fitted parameters (nm)
b
1μm
OMC cavity region
cross-shield
coupling waveguide
lensed ber
10
μ
m
FIG. 2.
Device fabrication and measurement setup. a
, SEM image of a full quasi-2D snowflake 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 App. D). A lensed optical fiber is used to couple light into a tapered on-chip waveguide which
is butt-coupling to the quasi-2D snowflake OMC cavity.
b
, SEM image of the center of the center of the cavity region. Fitting
to the geometries for ‘C’ shape holes and snowflake holes in the cavity region are shown as red solid lines.
c
, Measured ‘C’
parameters fit from SEM images of a fabricated cavity (dots) along with their design values (solid curves). Shown over one
half of the cavity (first 8 ‘C’ shapes on the right side of the cavity) are the parameters:
h
i
(blue),
h
o
(cyan),
w
i
/
2 (red), and
w
o
/
2 (green).
d
, Experimental setup for characterization of the quasi-2D snowflake OMC cavity. Multiple optical switches are
used to switch between continuous-wave and pulsed optical excitation, and between heterodyne spectroscopy and single photon
detection. Acronyms: Acousto-Optic Modulator (AOM), Electro-Optic Modulator (EOM), Erbium-Doped Fiber Amplifier
(EDFA), Variable Coupler (VC), Balanced Photodetector (BPD), Single Photon Detector (SPD). A more detailed schematic
and description of the measurement setup is provided in App. B.
(
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 scheme in-
volving a laser pump tone of frequency
ω
p
fixed near
the red motional sideband of the optical cavity (∆
ω
c
ω
p
+
ω
m
). In this scheme, electro-optic modu-
lation of the pump laser is used to create a weak laser
probe tone of frequency
ω
s
that we tune across the
OMC cavity resonance by sweeping the modulation fre-
quency [51]. The reflected pump and probe laser tones
from the OMC cavity contain the driven response of
those optomechanically-coupled acoustic modes that are
in two-photon resonance with the laser drive fields (i.e.,
ω
m
=
ω
s
ω
p
). Coherent detection of the beating of
the pump and probe laser tones on a high speed pho-
todetector 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. 3b for the device of Fig. 2a. Optomechanical back-
action from the pump laser can be seen to broaden the
acoustic resonance; a plot of the fit resonance linewidth
(
γ
) versus intra-cavity photon number of the pump laser
tone (
n
c
) is shown as an inset to Fig. 3b. From the slope
of the back-action-broadened linewidth versus
n
c
we ex-
tract a vacuum coupling rate of
g
0
/
2
π
= 1
.
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 mil-
likelvin temperatures. We measured the intrinsic me-
chanical damping rate, backaction 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 10 mm
by 5 mm sample containing an array of devices was di-
rectly mounted on a copper mount attached to the mix-
ing chamber plate, and a 3-axis stage was used to align
a lensed optical fiber to the tapered on-chip coupling
waveguide of a given device under test (see Fig. 2a and
App. C).
We measured the intrinsic mechanical
Q
-factor of the
quasi-2D OMC devices at millikelvin temperatures us-
ing a pulsed optical scheme in which 10-microsecond-
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 en-