of 7
ARTICLE
Topological phonon transport in an optomechanical
system
Hengjiang Ren
1,2,6,7,9
, Tirth Shah
3,4,9
, Hannes Pfeifer
3,8
, Christian Brendel
3
, Vittorio Peano
3
,
Florian Marquardt
3,4
& Oskar Painter
1,2,5
Light is a powerful tool for controlling mechanical motion, as shown by numerous applica-
tions in the
fi
eld of cavity optomechanics. Recently, small scale optomechanical circuits,
connecting a few optical and mechanical modes, have been demonstrated in an ongoing push
towards multi-mode on-chip optomechanical systems. An ambitious goal driving this trend is
to produce topologically protected phonon transport. Once realized, this will unlock the full
toolbox of optomechanics for investigations of topological phononics. Here, we report the
realization of topological phonon transport in an optomechanical device. Our experiment is
based on an innovative multiscale optomechanical crystal design and allows for site-resolved
measurements in an array of more than 800 cavities. The sensitivity inherent in our opto-
mechanical read-out allowed us to detect thermal
fl
uctuations traveling along topological
edge channels. This represents a major step forward in an ongoing effort to downscale
mechanical topological systems.
https://doi.org/10.1038/s41467-022-30941-0
OPEN
1
Thomas J. Watson, Sr., Laboratory of Applied Physics and Kavli Nanoscience Institute, California Institute of Technology, Pasadena, CA 91125, USA.
2
Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA 91125, USA.
3
Max Planck Institute for the Science of Light,
Staudtstr. 2, 91058 Erlangen, Germany.
4
Department of Physics, Friedrich-Alexander Universität Erlangen-Nürnberg, Staudtstr. 7, 91058 Erlangen, Germany.
5
AWS Center for Quantum Computing, Pasadena, CA 91125, USA.
6
Present address: Institute of High Performance Computing, Agency for Science,
Technology and Research (A*STAR), Singapore 138632, Singapore.
7
Present address: Anyon Computing Inc, Dover, DE 19901, USA.
8
Present address:
Institut für Angewandte Physik, Universität Bonn, Wegelerstr. 8, 53115 Bonn, Germany.
9
These authors contributed equally: Hengjiang Ren, Tirth Shah.
email:
opainter@caltech.edu
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1
1234567890():,;
R
ecent advances in cavity optomechanics
1
have now made it
possible to use light not just only as a passive measuring
device of mechanical motion
2
but also to manipulate the
motion of mechanical objects down to the level of individual
quanta of vibrations (phonons). At the same time, micro-
fabrication techniques have enabled small-scale optomechanical
circuits capable of on-chip manipulation of mechanical and
optical signals
3
12
. Building on these developments, theoretical
proposals have shown that larger-scale optomechanical arrays can
be used to modify the propagation of phonons, realizing a form of
topologically protected phonon transport
12
16
. This optomecha-
nical approach is part of a broader endeavor to realize topological
mechanical devices
17
in a variety of platforms ranging from
pendula
18
,
19
, to sound waves in
fl
uids
20
,
21
, and vibrations in
solids
22
26
, exploiting the general concepts of topologically robust
wave propagation
27
. It is motivated by the quest to reduce the
footprint of mechanical topological devices towards systems at
the nanoscale, where hypersonic frequency (
GHz) acoustic wave
circuits consisting of robust delay lines
28
and non-reciprocal
elements
29
32
may be implemented. Owing to the broadband
character of the topological channels, the control of the
fl
ow of
heat-carrying phonons may also be envisioned.
Here, we report the observation of topological phonon transport in
an optomechanical device comprising over 800 cavity-optomechanical
elements at room temperature fabricated on the surface of a silicon
microchip. Using sensitive, spatially resolved optical read-out
33
,
34
we
detect thermal phonons traveling along a topological edge channel.
We observe a substantial reduction of backscattering in a 0.325
0.34 GHz band demonstrating unprecedented carrier frequency and
bandwidth compared to the only existing nanoscale on-chip topolo-
gical mechanics implementations
25
,
26
. A key innovation of our work is
to introduce a design paradigm for optomechanical devices based on
multiscale optomechanical crysta
ls (OMCs). Standard single-scale
OMCs
35
38
are patterned free-standing s
tructures that can be engi-
neered to yield large radiation-pr
essure coupling between cavity
photons and phonons with similar wavelengths. In contrast, our
multiscale device consists of a superlattice structure, superimposing
two patterns with very different bu
t commensurate lattice spacings.
This multiscale approach adds an extra degree of
fl
exibility, decou-
pling the engineering of photonic and phononic modes. In our design,
the larger scale de
fi
nes a phononic crystal. Embedded within each unit
cell of the phononic crystal is a sma
ller scale photonic crystal, which
hosts a high-
Q
optical nanocavity for optical site-resolved read-out of
phonons. Local changes within the OMC lattice of the phononic
crystal unit cell are used to create
topologically distinct mechanical
domains, whose interface hosts phononic helical edge states based on
theValleyHalleffect
39
,
40
.
Results
Design of the multiscale OMC for topological phononics
.
Images of a fabricated multiscale OMC structure are shown in
Fig.
1
a, b (see Supplementary Note 2 for more details on device
fabrication). In our design, a triangular lattice of snow
fl
ake-
shaped holes with lattice spacing
a
m
=
16.02
μ
m is superimposed
onto another triangular lattice of cylindrical holes with a much
smaller spacing
a
o
=
450 nm. This hole pattern has been etched
into the thin (220 nm thickness) silicon device layer of a silicon-
on-insulator (SOI) microchip. After releasing the underlying
buried oxide layer, this produces an array of connected triangular
silicon membranes forming the phononic crystal, each hosting a
photonic crystal de
fi
ned by the smaller holes (see Fig.
1
b and
inset). The snow
fl
ake pattern is adopted from a well-known
single-scale OMC design
37
,
38
and has also been proposed theo-
retically as a platform for topological phononics
14
,
15
. In this work
we have increased the snow
fl
ake lattice spacing by a factor of ~30,
enabling every triangular membrane to harbor an optical nano-
cavity consisting of a localized defect in the triangular photonic-
crystal hole pattern. The purpose of using a cavity is to boost the
optomechanical interaction (see Supplementary Note 3 and
Supplementary Fig. 1). In Fig.
1
a, b, such a cavity is present only
in the downward-pointing triangular membranes, with the
upward-pointing triangular membranes having an unperturbed
photonic-crystal pattern. Although the two lattices (phononic and
photonic) are at vastly different scales, the patterning of the
photonic crystal within each triangular membrane does (weakly)
in
fl
uence the phononic properties, providing an extra knob to
trim the mechanical properties.
We employ these tuning knobs of the multiscale design to
realize a structure supporting robust helical edge states based on
the Valley Hall effect
39
. The Valley Hall effect is relevant for a
wide range of systems that support Dirac cones, including
electronic
39
,
40
, photonic
30
, and mechanical systems
20
,
23
. In this
context, valley refers to the quasi-momentum region around a
Dirac cone. In a time-reversal-symmetric system, the Dirac cones,
and thus the corresponding valleys, come in pairs mapped onto
each other by the operation of time reversal. Thus, the valley can
be viewed as a binary degree of freedom akin to the spin. In the
Valley Hall effect, valley-polarized edge excitations propagate in
opposite directions, analogous to spin-polarized edge states in the
Spin Hall effect.
As we are pursuing an optomechanical approach to the detection
of mechanical edge excitations, we focus here on the vibrational
modes that couple to light, the in-plane modes which are even
under the mirror operator
M
z
(
z
z
). For these modes, the
snow
fl
ake phononic crystal supports a pair of Dirac cones well-
isolated from the remaining bands
14
,
15
. In our experiment, the
Dirac cones have a center frequency of ~0.3 GHz, with linear Dirac-
like dispersion across a bandwidth of 70 MHz (see Fig.
1
c). These
cones are protected by a symmetry under
M
y
(see Supplementary
Note 4). We open the bulk bandgap that will host the helical edge
states by breaking this symmetry. Decreasing the size of the
photonic-crystal holes in the upward-pointing triangles by a factor
of 0.78 produces a bandgap of width 18 MHz (see Fig
1
d). The
underlying vibrational Bloch waves, calculated using
fi
nite-element
method (FEM) simulations (see Supplementary Note 1), are shown
in Fig.
1
e, f. A comparatively large unit-cell vacuum optomecha-
nical coupling (
g
0
=
2
π
× 33.7 kHz) is produced for the higher-
frequency mode in Fig.
1
f because it displays breathing motion
around the optical cavity. A detailed discussion of the optomecha-
nical coupling is provided in Supplementary Notes 3, 9.
In the Valley Hall effect, the topological transport takes place
through counter-propagating valley-polarized edge states which
exist at the domain walls separating two topologically distinct
domains of opposite so-called valley Chern number. By applying
the mirror operation
M
y
, we construct from the deliberately
mirror-symmetry-broken design described above a second
domain with opposite valley Chern numbers (see Fig.
1
g, h).
The key feature leading to robust transport is that edge
excitations can navigate a path with arbitrarily sharp angles
while still remaining con
fi
ned within the same valley region of
quasi-momentum space. On the other hand, backscattering
would require a large quasi-momentum transfer to reach a
different valley, and is thus strongly suppressed. Our
fi
t to the
Dirac Hamiltonian describing our anisotropic structure (see
Methods) shows both a dependence of the bandstructure on the
domain wall orientation and some deviations from the idealized
theoretical limiting case. For a horizontal domain wall, this leads
to in-gap edge states that extend only through part of the full
bandgap (see Fig.
1
i). Below we show that the transmission
around sharp corners remains robust nevertheless, with this
imperfection only reducing the relevant bandwidth.
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Site-resolved optomechanical read-out of topological vibra-
tions
. We have fabricated several devices where an internal
domain of type 2 is surrounded by an external domain of type 1.
The ensuing closed domain wall produces a topological
mechanical cavity. In a topological cavity, counter-propagating
running waves remain decoupled in spite of sharp turns and/or
disorder. This gives rise to a characteristic spectrum formed by a
series of doublets. These doublets are degenerate, with any slight
lifting of the degeneracy due to residual inter-valley scattering.
The
fi
rst topological cavity structure that we study is shown in
Fig.
2
a, b, consisting of an equilateral triangle of 28 snow
fl
ake unit
cells along each side. A schematic of our optical setup used to
measure the phononic properties of the topological cavity
structure is shown in Fig.
2
c. A tunable external cavity diode
laser coupled to an optical
fi
ber taper is used to optically excite
individual optical nanocavities within the multiscale OMC array.
The out-coupled laser light, which contains the local mechanical
motion of the structure imprinted as intensity modulations, is
detected on a photodiode and analyzed on an electronic spectrum
analyzer. Owing to the thermal nature of the measured
mechanical motion in this work, the measured electronic
spectrum analyzer signal represents a local mechanical noise
power spectral density (NPSD). By moving the taper position we
are able to address any unit cell of the larger-scale phononic
lattice, obtaining a site-resolved spectrum of the thermally
populated phonon modes (see Methods for further details). As
an example, we show in the top plot of Fig.
2
d the resulting
optically-transduced local mechanical spectrum for an optical
fi
ber taper position at site (d) in Fig.
2
a, which is in the bulk
region of domain 1. The measured spectrum is seen to be in close
agreement with our theoretical predictions based on FEM
simulations (bottom plot of Fig.
2
d), both of which show a bulk
bandgap that covers an interval from 316 to 338 MHz. We note
that compared to recent nanomechanical implementations
25
,
26
,
combining piezoelectric actuation and optical interferometric
read-out, the displacement sensitivity in our cavity-based
measurements is boosted by the cavity
fi
nesse. This has allowed
us to detect tiny thermal vibrations with amplitudes on the order
of 10 fm (Supplementary Note 7 and Supplementary Fig. 4).
We now focus on the domain wall region. Exploiting our
single-site resolution capability, we have measured the mechan-
ical NPSD as a function of read-out position, as shown in Fig.
2
e.
This reveals two dramatically different transport regimes. For the
mechanical cavity modes at lower frequencies (321
327 MHz), we
observe a strong modulation versus site position in each of the
mechanical mode peaks. These fringe-like features indicate that
thermal phonon excitations are re
fl
ected and form standing
waves. This is due to the absence of topological edge modes inside
the horizontal domain wall at these frequencies, resulting in
standing waves inside the slanted domain wall portions of the
mechanical cavity path. By contrast, we observe no such fringes in
the higher-frequency regime (327
337 MHz). This indicates
backscattering-immune running waves, providing a direct visual
signature of the formation of a topological mechanical cavity.
Fig. 1 Design of the multiscale optomechanical crystal for topological phononics. a
Optical microscope image showing the snow
fl
ake triangular lattice
(unit-cell dashed) with parameters (
d
,
r
,
w
,
a
m
)
=
(0.22, 5.77, 2.34, 16.02)
μ
m. The axes are aligned with the silicon crystal.
b
Focused Ion Beam (FIB) image
of unit-cell geometry with the simulated photonic-crystal cavity mode pro
fi
le (
E
[100]
component of the electric
fi
eld; red/blue indicates sign).
c
,
d
Simulated
phononic band structures with
M
y
mirror-symmetry intact and broken (design in
b
), respectively. Inset: Sketches of the Dirac cones.
e
,
f
Snapshots of the
mechanical mode deformation (colours indicate the local volume change,
u
; red corresponding to expansion and blue compression). The arrows in the
associated pictograms indicate the dynamics of the motion.
g
,
h
Optical microscope images and simulated mechanical mode pro
fi
les for two strip
con
fi
gurations, each comprising two topologically distinct domains (domain 1 as in
a
,
b
). The domain wall (dashed) has slope 0° (horizontal,
g
) or 240°
(slanted,
h
) relative to the [100] axis.
i
1-D band structures calculated for the horizontal and slanted con
fi
gurations. The red lines indicate the topological
edge state dispersion, the gray lines are the additional edge state modes localized at the top and bottom boundaries of the geometry (away from the
domain wall), the blue parts are bulk modes. The color shading inside the bulk bandgap identi
fi
es different transport regimes in systems where the two
types of domain walls are connected.
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3
Below, we refer to this frequency range as the topological
bandwidth. In between these regimes, there is a crossover region
(light gray in Fig.
2
f
g), where the horizontal edge already
supports edge states but backscattering is still possible because
small quasi-momentum transfers are suf
fi
cient to
fl
ip right-
moving into left-moving horizontal edge states due to their
proximity to the Brillouin zone boundary of the horizontal edge
structure (see bandstructure plot in Fig.
1
i). We note that
signi
fi
cant backscattering for edge states based on the Valley Hall
effect should be expected whenever the wavefunction is not well
localized within one valley. For our experiment and other setups
featuring sharp domain walls, this sets a limit on the achievable
topological bandwidth which could be bypassed by introducing
smooth domain walls
41
.
We further substantiate the absence of backscattering in the
topological bandwidth by comparing the frequency dependence
of the measured NPSD with theory predictions that assume
perfect transmission at the corners. They are based on scattering
matrix calculations that take FEM simulations as input (see
Supplementary Note 7 and Supplementary Figs. 3, 5). The
theoretical spectra are in good agreement with measurement
results both on the slanted and the horizontal edges, as shown in
Fig.
2
f, g, respectively. Even the behavior of the peak heights,
distinctly different for both types of edges, is captured very well
by including both the group velocity dispersion and the
frequency-dependent vacuum optomechanical coupling
g
ð
1D
Þ
0
(see Fig.
2
h) in our analysis.
Robustness against backscattering
. While the triangle geometry
is the simplest closed-loop geometry, already producing a topo-
logical mechanical cavity, we also sought to test the robustness
and immunity to waveguide imperfections in more complex
cavity structures where we could independently vary the length of
waveguide segments between sharp corners. The effects of such
variations should be most pronounced in geometry with appre-
ciable backscattering at the corners, eventually producing separate
standing wave patterns in the segments whose free spectral range
would depend on the segment length. By contrast, the ideal case
of robust topological transport should only be sensitive to the
overall length of the domain wall circumference. Producing
samples with different local geometrical details, but the same
circumference, allows us to test these ideas by comparing their
spectra.
To this end, we designed and fabricated two tree-shaped cavity
structures. Each of these has a total domain wall circumference of
Domain 1
Domain 2
(f)
(g)
(d)
28
a
m
Laser
EDF
A
-mete
r
Spectrum Analyzer
FPC
PD2
PD1
SW1
Ta p e r
310
320
330
340
Frequency (MHz)
4
0
c
a
Arb. units
2
Frequency (MHz)
4
0
d
f
10
8
334
326
14
20
18
16
12
0
n
G
n
S
mech,n
/
t
[10
-3
]
326
330
334
322
g
2
26
Frequency (MHz)
1
0
Site #
e
b
n
G
n
S
mech,n
/
t
[10
-3
]
n
G
n
S
mech,n
/
t
[10
-3
]
g
0
/2
[kHz]
1D
h
g
0
/2
[kHz]
1D
Arb. units
Arb. units
Fig. 2 Characterization of topological edge states using optomechanical read-out. a
Optical microscope image of triangular topological mechanical cavity
(Domain wall: dashed line. Read-out cavities for the measurements in
d
,
f
, and
g
: yellow dots).
b
, Zoom-in of the topological cavity corner (green box in
a
).
c
Experimental setup. Mechanical side-bands are imprinted on a laser beam transmitted through an optical cavity, detecting the NPSD of the mechanical
waves. Acronyms: optical wave meter (
λ
meter), variable optical attenuator (VOA),
fi
ber polarization controller (FPC), optical switch (SW), erbium-doped
fi
ber ampli
fi
er (EDFA), photodetector (PD).
d
,
f
,
g
Measured (top) and numerically estimated (bottom) NPSD, respectively, in the bulk of domain 1, on a
slanted edge, and on a horizontal edge. Insets in
f
, and
g
: Sketches showing read-out positions and the expected local density of states.
e
Measured NPSD
as a function of frequency and read-out position on a slanted edge (highlighted in black in the sketch). Red dashed line corresponds to the spectrum in
f
.
The low-frequency region (dark gray in
f
and
g
) harbors modes only inside the slanted edges (cf Fig.
1
i). Data calibration is required to compare
measurements from different read-out cavities (see Supplementary Note 8 and Supplementary Fig. 6).
h
Optomechanical coupling for edge states in
slanted (top) and horizontal (bottom) domain walls (see Supplementary Note 7).
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96 unit cells and includes seven 60° corners, but individual
segment lengths differ. Figure
3
c shows the mechanical spectra
measured near the horizontal edge of both tree geometries,
superimposed onto each other. The most important observation
is that, outside of the gray region, the two spectra agree almost
perfectly, despite the different geometries. This is a clear and
direct experimental signature of the near-perfect absence of
backscattering, as predicted for the topological edge states. The
gray region is close to the bandgap for the horizontal edge, where
no suppression of backscattering is expected (see above).
In order to estimate the sensitivity of the spectra to
backscattering, we performed calculations assuming varying
levels of backscattering for both tree geometries (Fig.
3
d), where
t
2
(1
t
2
) is the transmission (re
fl
ection) probability at each
corner. These results show that even a small re
fl
ection probability
of the order of 5% is enough to produce clearly visible differences
between the spectra, including the splitting of the peaks. Since no
such deviations are visible in the direct measurements, we
conclude that the transmission surpasses 95% in our experiment,
which can be seen as a
fi
gure-of-merit. This con
fi
rms that the
phononic topological edge states robustly transmit through sharp
corners. In addition, we have theoretically investigated the
backscattering in the presence of fabrication disorder, see
Supplementary Note 11 and Supplementary Fig. 8. Introducing
a standard deviation of 10 nm in selected geometrical parameters
of both the snow
fl
ake and the cylindrical holes, we have estimated
Frequency (MHz)
(c,d)
(c,d)
Domain 1
Domain 2
Domain 1
Domain 2
14
a
m
6
a
m
9
a
m
18
a
m
6
a
m
7
a
m
ab
c
d
e
4
0
4
0
4
0
322 323
324
325
326
327
328
329
Frequency (MHz)
321
330
339
338
337
336
335
334
333
332
331
(a) (b)
(a) (b)
(a) (b)
(a) (b)
n
G
n
S
mech,n
/
t
[10
-3
]
8
(a)
(
(
Arb. units
Arb. units
Fig. 3 Robustness against backscattering. a
,
b
Optical microscope image of two different tree-shaped topological mechanical cavities.
c
Comparison of
measurement results for the two trees.
d
Theoretical prediction for three backscattering strengths. Darker/lighter spectra correspond to the tree in
a
/
b
.In
c
, the crossover region where backscattering can occur without requiring large quasi-momentum transfer is highlighted in light gray.
e
Measurements for a
trivial waveguide mechanical cavity (see Supplementary Note 10). Transduced NPSD measured at a slanted (dark blue) and a horizontal (light blue) edg
e,
revealing strong backscattering.
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5
a round trip re
fl
ection probability
r
rt
2
for the triangular
mechanical cavity (Fig.
2
) of around 1%. We expect the actual
fabrication disorder using our electron-beam lithography to be in
the range of 2
4nm
42
.
For further comparison, we also designed and fabricated a
trivial cavity. It is created by pulling a band into the Dirac cone
gap of the surrounding bulk along a line defect embedded into an
otherwise uniform domain 1. In this case, the mechanical spectra
measured at two different locations (on a slanted and a horizontal
edge) show signatures of backscattering from the sharp corners
(see Fig.
3
e), with irregular peak spacing and different peak
locations for the two spectra. As described in more detail in
Supplementary Note 10 and Supplementary Fig. 7, the funda-
mental reason for larger re
fl
ections occurring in the trivial cavity
is that the trivial waveguide supports both forward and backward
moving modes within the same valley, greatly reducing the
required quasi-momentum transfer for backscattering.
Discussion
In conclusion, we have demonstrated a multiscale optomecha-
nical crystal and observed topological transport of thermal pho-
nons in the 0.3 GHz band over a bandwidth of 15 MHz. This
design opens the door to implementing on-chip phononic
circuits
13
,
15
,
16
,
43
with robust topological waveguides that have
access to the full toolbox of optomechanics. Beyond cooling,
mechanical lasing, sensitive read-out, and optical generation of
nonclassical quantum states, this would also include the active
optical control of topological circuits via local manipulation of
mechanical modes (e.g., switching links between edge states).
Another very promising avenue for applications consists in
pushing towards even higher frequencies in the hypersonic
regime
up to 100 GHz should be possible with advanced
lithographic methods
inverting the scale hierarchy between
photonics and phononics. This would allow one to manipulate
thermal phonons in myriad of new ways, including broadband
cooling of entire microscale objects, not just individual mechan-
ical modes. Unidirectional edge channels like those found in a
Chern insulator would allow one to implement thermal diodes,
and, when supplemented by an energy pump, topologically pro-
tected phonon ampli
fi
cation and lasing
29
32
. An exciting long-
term perspective is to use topological phononic circuits as the
basis of a new platform to explore quantum acoustodynamics for
quantum information processing and storage, with coupling to
dopants or superconducting qubits.
Methods
Valley Hall effect: theoretical model with anisotropy
. In the Valley Hall effect,
the relevant topological invariant is the so-called valley Chern number
C
v
39
,
44
. The
valley Chern number is de
fi
ned within one valley in the framework of an effective
two-band description and assumes two possible half-integer values,
C
v
=
±1/2.
Interfaces between regions with opposite valley Chern numbers support in-gap
valley-polarized edge states. Since the two valleys are mapped into each other by
time reversal, their edge states are counter-propagating.
We note that due to both our elongated cavity design and the anisotropic silicon
crystal (see Supplementary Note 5 and Supplementary Fig. 2), our system is not
invariant under
C
3
-rotations. This is a notable difference compared to previous
larger-scale implementations of the Valley Hall effect
20
,
23
,
30
,
40
,
45
,
46
. Taking into
account the residual bulk symmetry
T
M
x
,we
fi
nd that our system is approximated
by the effective two-band Dirac Hamiltonian (see Supplementary Note 4)
^
H
D
¼

Ω
þð
v
0
þ
v
x
^
σ
x
Þ
^
p
x
þ
v
y
^
σ
y
^
p
y
þ
Θ
ð
^
r
Þ
;
m
þ
m
0
^
p
x


^
σ
z
:
ð
1
Þ
Here, we set
=
1,
^
σ
x
;
y
;
z
are the Pauli matrices, {, } denotes the anti-commutator,
and
Θ
(
r
)
=
1/2 (
Θ
(
r
)
=
1/2) inside domain 1 (domain 2). Moreover,
p
=
(
p
x
,
p
y
)
is the quasi-momentum counted from a point on the
k
x
-axis where the Bloch waves
are mapped into each other via
M
y
, see Fig.
1
e, f. The most obvious difference
between
C
3
-symmetric systems is that the speed of the edge state now depends on
the domain wall orientation. The solutions for slanted and horizontal domain walls
and other surprising features are discussed in Supplementary Note 4.
We now focus on the valley close to the
K
point. Fixing the gauge by choosing
σ
z
=
1 for the Bloch wave shown in Fig.
1
f, a
fi
t yields
m
=
2
π
× 10.8 MHz,
m
0
=
a
m
¼
2
π
́
5
:
4MHz,
v
x
/
a
m
=
2
π
× 12.5MHz, and
v
y
/
a
m
=
2
π
× 14.9 MHz.
The valley Chern number for the lowest band is
C
v
=
sign(
Θ
(
r
)
mv
x
v
y
)/2, see
Supplementary Note 4. Thus, we
fi
nd
C
v
=
1/2 (
C
v
=
1/2) for domain 1 (domain
2). According to the bulk-boundary correspondence, the edge state will be a right-
mover if one crosses the domain wall from domain 1 to domain 2
47
. This is
consistent with our strip FEM simulations, see Fig.
1
i. The expansion leading to Eq.
(
1
) is valid if
m
=
a
m

v
y
;
ð
v
2
x
þ
m
0
2
Þ
1
=
2
(see Supplementary Note 4). This
condition is not strictly ful
fi
lled in our experiment, which leads to the deviations
from the ideal case remarked upon in the main text.
Measuring the mechanical thermal
fl
uctuations
. The thermal-mechanical
motion of phonons within the multiscale OMCs of this work is measured by
driving the system with the laser locked to a blue detuning of 340 MHz from the
optical nanocavity resonance. This frequency offset is chosen to align with the
center frequency of the mechanical Dirac cones, increasing the sensitivity of the
optical read-out for phonons propagating in the topological edge states. An optical
fi
ber taper with a localized dimple region couples light evanescently into and out of
an individual optical cavity with high ef
fi
ciency. By moving the taper, we can
address any unit cell of the larger-scale phononic lattice. Mechanical motion is
imprinted on the phase of the laser light inside the optical nanocavity, which when
extracted via the optical
fi
ber taper maps the mechanical motion into intensity
modulations in the transmitted laser light. The transmitted laser signal in the
optical
fi
ber is sent through an erbium-doped
fi
ber ampli
fi
er (EDFA) to amplify the
optical intensity modulations, and then onto a high-speed photoreceiver. The RF
voltage from the photoreceiver is sent into a spectrum analyzer to determine the
noise power spectral density (NPSD). The NPSD of the photocurrent contains a
component proportional to the sum of the mechanical NPSD
S
mech,
n
of the
mechanical normal modes of the structure, weighted by the square of the local
optomechanical coupling
G
n
(
j
), where
n
labels the mechanical mode and
j
labels
the (unit cell of the) read-out cavity (see Supplementary Note 6). Since only the
vibrations within a single unit cell contribute to the optomechanical coupling
G
n
(
j
),
the transduced mechanical NPSD can be viewed as a (coarse-grained) mechanical
local density of states.
Data availability
The data supporting the results presented in this article are available at Zenodo open-
access repository under [
https://doi.org/10.5281/zenodo.6414313
]
48
. Additional data that
support the
fi
ndings of this study are available from the corresponding author (O.P.)
upon reasonable request.
Code availability
The code supporting the results presented in this article are available at Zenodo open-
access repository under [
https://doi.org/10.5281/zenodo.6414313
]
48
.
Received: 17 June 2021; Accepted: 24 May 2022;
References
1. Aspelmeyer, M., Kippenberg, T. J. & Marquardt, F. Cavity optomechanics.
Rev. Mod. Phys.
86
, 1391
1452 (2014).
2. de Groot, P. J. A review of selected topics in interferometric optical metrology.
Rep. Prog. Phys.
82
, 056101 (2019).
3. Massel, F. et al. Multimode circuit optomechanics near the quantum limit.
Nat. Commun.
3
,1
6 (2012).
4. Zhang, M., Shah, S., Cardenas, J. & Lipson, M. Synchronization and phase
noise reduction in micromechanical oscillator arrays coupled through light.
Phys. Rev. Lett.
115
, 163902 (2015).
5. Xu, H., Mason, D., Jiang, L. & Harris, J. Topological energy transfer in an
optomechanical system with exceptional points.
Nature
537
,80
83 (2016).
6. Kharel, P. et al. High-frequency cavity optomechanics using bulk acoustic
phonons.
Sci. Adv
.
5
, eaav0582 (2019).
7. Ruesink, F., Miri, M.-A., Alu, A. & Verhagen, E. Nonreciprocity and magnetic-
free isolation based on optomechanical interactions.
Nat. Commun.
7
,1
8
(2016).
8. Peterson, G. A. et al. Demonstration of ef
fi
cient nonreciprocity in a microwave
optomechanical circuit.
Phys. Rev. X
7
, 031001 (2017).
9. Bernier, N. R. et al. Nonreciprocal recon
fi
gurable microwave optomechanical
circuit.
Nat. Commun.
8
, 604 (2017).
10. Fang, K. et al. Generalized non-reciprocity in an optomechanical circuit via
synthetic magnetism and reservoir engineering.
Nat. Phys.
13
, 465 (2017).
ARTICLE
NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-022-30941-0
6
NATURE COMMUNICATIONS
| (2022) 13:3476 | https://doi.org/10.1038/s41467-022-30941-0 | www.nature.com/naturecommunications
11. Xu, H., Jiang, L., Clerk, A. & Harris, J. Nonreciprocal control and cooling of
phonon modes in an optomechanical system.
Nature
568
,65
69 (2019).
12. Mathew, J. P., Pino, J. D. & Verhagen, E. Synthetic gauge
fi
elds for phonon
transport in a nano-optomechanical system.
Nat. Nanotechnol.
15
, 198
202
(2020).
13. Peano, V., Brendel, C., Schmidt, M. & Marquardt, F. Topological phases of
sound and light.
Phys. Rev. X
5
, 031011 (2015).
14. Brendel, C., Peano, V., Painter, O. J. & Marquardt, F. Pseudomagnetic
fi
elds
for sound at the nanoscale.
Proc. Natl Acad. Sci. USA
114
, E3390
E3395
(2017).
15. Brendel, C., Peano, V., Painter, O. & Marquardt, F. Snow
fl
ake phononic
topological insulator at the nanoscale.
Phys. Rev. B
97
, 020102 (2018).
16. Sanavio, C., Peano, V. & Xuereb, A. Nonreciprocal topological phononics in
optomechanical arrays.
Phys. Rev. B
101
, 085108 (2020).
17. Nassar, H. et al. Nonreciprocity in acoustic and elastic materials.
Nat. Rev.
Mater
.
5
,1
19 (2020).
18. Süsstrunk, R. & Huber, S. D. Observation of phononic helical edge states in a
mechanical topological insulator.
Science
349
,47
50 (2015).
19. Nash, L. M. et al. Topological mechanics of gyroscopic metamaterials.
Proc.
Natl Acad. Sci. USA
112
, 14495
14500 (2015).
20. Lu, J. et al. Observation of topological valley transport of sound in sonic
crystals.
Nat. Phys.
13
, 369
374 (2017).
21. Deng, W. et al. Acoustic spin-chern insulator induced by synthetic spin
orbit
coupling with spin conservation breaking.
Nat. Commun.
11
,1
7 (2020).
22. Mousavi, S. H., Khanikaev, A. B. & Wang, Z. Topologically protected elastic
waves in phononic metamaterials.
Nat. Commun.
6
, 8682 (2015).
23. Miniaci, M., Pal, R. K., Morvan, B. & Ruzzene, M. Experimental observation of
topologically protected helical edge modes in patterned elastic plates.
Phys.
Rev. X
8
, 031074 (2018).
24. Yu, S.-Y. et al. Elastic pseudospin transport for integratable topological
phononic circuits.
Nat. Commun.
9
, 3072 (2018).
25. Cha, J., Kim, K. W. & Daraio, C. Experimental realization of on-chip
topological nanoelectromechanical metamaterials.
Nature
564
, 229
233
(2018).
26. Ma, J., Xi, X., Li, Y. & Sun, X. Nanomechanical topological insulators with an
auxiliary orbital degree of freedom.
Nat. Nanotechnol.
16
, 576
583 (2021).
27. Hasan, M. Z. & Kane, C. L. Colloquium: topological insulators.
Rev. Mod.
Phys.
82
, 3045 (2010).
28. Hafezi, M., Demler, E. A., Lukin, M. D. & Taylor, J. M. Robust optical delay
lines with topological protection.
Nat. Phys.
7
, 907
912 (2011).
29. Bandres, M. A. et al. Topological insulator laser: experiments.
Science
359
,
eaar4005 (2018).
30. Zeng, Y. et al. Electrically pumped topological laser with valley edge modes.
Nature
578
, 246
250 (2020).
31. Peano, V., Houde, M., Marquardt, F. & Clerk, A. A. Topological quantum
fl
uctuations and traveling wave ampli
fi
ers.
Phys. Rev. X
6
, 041026 (2016).
32. Mittal, S., Goldschmidt, E. A. & Hafezi, M. A topological source of quantum
light.
Nature
561
, 502
506 (2018).
33. Teufel, J. D., Donner, T., Castellanos-Beltran, M., Harlow, J. W. & Lehnert, K.
W. Nanomechanical motion measured with an imprecision below that at the
standard quantum limit.
Nat. Nanotechnol.
4
, 820
823 (2009).
34. Wilson, D. et al. Measurement-based control of a mechanical oscillator at its
thermal decoherence rate.
Nature
524
, 325
329 (2015).
35. Eichen
fi
eld, M., Chan, J., Camacho, R. M., Vahala, K. J. & Painter, O.
Optomechanical crystals.
Nature
462
,78
82 (2009).
36. Safavi-Naeini, A. H. & Painter, O. Design of optomechanical cavities and
waveguides on a simultaneous bandgap phononic-photonic crystal slab.
Opt.
Express
18
, 14926
14943 (2010).
37. Safavi-Naeini, A. H. et al. Two-dimensional phononic-photonic band gap
optomechanical crystal cavity.
Phys. Rev. Lett.
112
, 153603 (2014).
38. Ren, H. et al. Two-dimensional optomechanical crystal cavity with high
quantum cooperativity.
Nat. Commun.
11
,1
10 (2020).
39. Martin, I., Blanter, Y. M. & Morpurgo, A. F. Topological con
fi
nement in
bilayer graphene.
Phys. Rev. Lett.
100
, 036804 (2008).
40. Ju, L. et al. Topological valley transport at bilayer graphene domain walls.
Nature
520
, 650
655 (2015).
41. Shah, T., Marquardt, F. & Peano, V. Tunneling in the Brillouin zone: theory of
backscattering in valley Hall edge channels.
Phys. Rev. B
104
, 235431 (2021).
42. MacCabe, G. S. et al. Nano-acoustic resonator with ultralong phonon lifetime.
Science
370
, 840
843 (2020).
43. Habraken, S. J. M., Stannigel, K., Lukin, M. D., Zoller, P. & Rabl, P.
Continuous mode cooling and phonon routers for phononic quantum
networks.
N. J. Phys
.
14
, 115004 (2012).
44. Zhang, F., MacDonald, A. H. & Mele, E. J. Valley chern numbers and
boundary modes in gapped bilayer graphene.
Proc. Natl Acad. Sci. USA
110
,
10546
10551 (2013).
45. Gao, F. et al. Topologically protected refraction of robust kink states in valley
photonic crystals.
Nat. Phys.
14
, 140
144 (2018).
46. Schaibley, J. R. et al. Valleytronics in 2D materials.
Nat. Rev. Mater.
1
, 16055
(2016).
47. Asbóth, J. K., Oroszlány, L. & Pályi, A. A Short Course on Topological
Insulators: Band Structure and Edge States in One and Two Dimensions.
Lecture Notes in Physics
(Springer International Publishing, 2016).
48. Ren, H. et al. Topological phonon transport in an optomechanical system
[data set].
Zenodo
https://doi.org/10.5281/zenodo.6414313
(2022).
Acknowledgements
We would like to thank Sameer Sonar and Utku Hatipoglu for the help with nanofab-
rication and measurement. This work was supported by the Gordon and Betty Moore
Foundation (award #7435) and the Kavli Nanoscience Institute at Caltech. H.R. was
supported by the National Science Scholarship from A
*
STAR, Singapore. T.S. and F.M.
acknowledge support from the European Union?s Horizon 2020 research and innovation
programme under the Marie Sklodowska-Curie grant agreement No. 722923 (OMT).
V.P. acknowledges support by the Julian Schwinger Foundation (Grant No. JSF-16-03-
0000). F.M. acknowledges support from the European Union?s Horizon 2020 Research
and Innovation program under Grant No. 732894, Future and Emerging Technologies
(FET)-Proactive Hybrid Optomechanical Technologies (HOT).
Author contributions
H.R., T.S., C.B., F.M., V.P., and O.P. came up with the concept and planned the
experiment. H.R., T.S., H.P., and C.B. performed the device design and fabrication. H.R.
performed the measurements. H.R., T.S., F.M., V.P., and O.P. analyzed the data. All
authors contributed to the writing of the manuscript.
Competing interests
The authors declare no competing interests.
Additional information
Supplementary information
The online version contains supplementary material
available at
https://doi.org/10.1038/s41467-022-30941-0
.
Correspondence
and requests for materials should be addressed to Oskar Painter.
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