Al transmon qubits on silicon-on-insulator for quantum device integration
Andrew J. Keller
, Paul B. Dieterle
, Michael Fang
, Brett Berger
, Johannes M. Fink
, and
Oskar Painter
Citation:
Appl. Phys. Lett.
111
, 042603 (2017); doi: 10.1063/1.4994661
View online:
http://dx.doi.org/10.1063/1.4994661
View Table of Contents:
http://aip.scitation.org/toc/apl/111/4
Published by the
American Institute of Physics
Al transmon qubits on silicon-on-insulator for quantum device integration
Andrew J.
Keller,
1,2
Paul B.
Dieterle,
1,2
Michael
Fang,
1,2
Brett
Berger,
1,2
Johannes M.
Fink,
1,2,3
and Oskar
Painter
1,2,
a)
1
Kavli Nanoscience Institute and Thomas J. Watson Laboratory of Applied Physics, California Institute
of Technology, Pasadena, California 91125, USA
2
Institute for Quantum Information and Matter, California Institute of Technology, Pasadena,
California 91125, USA
3
Institute for Science and Technology Austria, 3400 Klosterneuburg, Austria
(Received 3 April 2017; accepted 5 July 2017; published online 25 July 2017)
We present the fabrication and characterization of a
n aluminum transmon qubit on a silicon-on-insulator
substrate. Key to the qubit fabrication is the use of an anhydrous hydrofluoric vapor process which
selectively removes the lossy silicon oxide buried underneath the silicon device layer. For a 5.6 GHz
qubit measured dispersively by a 7.1 GHz resonator, we find
T
1
¼
3.5
l
s and
T
2
¼
2.2
l
s. This
process in principle permits the co-fabrication of silicon photonic and mechanical elements, providing
a route towards chip-scale integration of electro-opto-mechanical transducers for quantum network-
ing of superconducting microwave quantum circuits. The additional processing steps are compatible
with established fabrication techniques for aluminum transmon qubits on silicon.
Published by AIP
Publishing.
[
http://dx.doi.org/10.1063/1.4994661
]
In recent years, significant developments in experimen-
tal quantum information science
1
,
2
have been realized using
microwave superconducting qubits. These devices, consist-
ing of Josephson junctions (JJs) and linear circuit elements,
are typically coupled to high-
Q
superconducting microwave
cavities, which realizes the microwave analog of cavity
QED—so-called circuit QED.
3
–
5
The advent of the transmon
qubit
6
–
8
has provided a robust and scalable circuit QED
building block. The large vacuum coupling rate attainable
between qubit and cavity in the circuit QED architecture has
enabled, among other things, realization of the strong disper-
sive coupling regime,
5
,
9
creation of quantum gates with
state-of-the-art gate fidelities,
10
and most recently, circuits
capable of quantum error detection and correction.
11
,
12
Interfacing the circuit QED toolbox with other systems of
physical or technological intere
st—cavity optomechanical sys-
tems, for example, Refs.
13
and
14
—requires scalable fabrica-
tion techniques on compatible m
aterials systems. Many works
within the circuit QED community have focused on developing
fabrication methods that realize long qubit lifetimes and small
dephasing rates.
15
–
17
Two primary approaches have emerged:
the so-called planar approach wherein qubits are coupled to on-
chip resonators
18
and the 3D cavity approach wherein qubits
are coupled to 3D box cavities.
19
Whereas the former affords
higher device densities and more integration, the latter yields
longer coherence times. In recent years, silicon has become
favored as a substrate for its low dielectric loss and for the
diversity of available fabrication techniques,
20
–
22
resulting in
transmon qubits that have coherence times and gate fidelities
similar to or exceeding their sapphire counterparts.
10
,
18
,
23
–
25
Here, we present the fabrication and characterization of
a planar transmon qubit on silicon-on-insulator (SOI) with a
coherence time which is a factor of 20 improvement over
prior work in this material system.
26
These SOI qubit fabri-
cation methods not only realize viable qubits but are also
compatible with the integration of other photonic, electronic,
and MEMS components on the same SOI substrate.
27
Moreover, the processing steps for SOI may be simply added
to those required for a silicon qubit process, enabling either
Si or SOI qubits to be fabricated without a complete process
change.
Our qubit design [pictured in Fig.
1(b)
and shown sche-
matically in Fig.
2(c)
] is based on the Xmon qubit.
18
A long
rectangular capacitor is capacitively coupled to both a read-
out resonator and an XY-control line; the capacitor is con-
nected to ground through a SQUID loop [Fig.
1(c)
] that is
inductively coupled to a DC control line, which allows for
frequency tuning of the qubit.
18
Our readout resonator, con-
sisting of a
k
=
4 coplanar waveguide resonator, is inductively
coupled to a transmission line, which allows for dispersive
readout of the qubit.
18
We realize the as-measured parame-
ters of:
f
q
¼
x
q
=
2
p
¼
5
:
652 GHz,
g
=
2
p
¼
300 MHz,
x
r
=
2
p
¼
7
:
143 GHz, and
v
=
2
p
¼
3
:
5 MHz, where
x
q
¼
x
10
is
the qubit transition frequency,
g
¼ð
x
21
x
10
Þ
is the anhar-
monicity,
x
r
is the readout resonator frequency, and 2
v
¼
x
r
;
j
0
i
x
r
;
j
1
i
is the dispersive shift. These measured
values imply a Josephson energy
E
J
=
h
¼
14
:
8 GHz in the
transmon limit (
E
J
E
C
), where
h
x
q
ffiffiffiffiffiffiffiffiffiffiffiffiffi
8
E
J
E
C
p
E
C
and the charging energy
E
C
h
g
,aswellasavacuum
qubit-resonator coupling rate
g
=
2
p
¼
177 MHz where
g
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
D
v
ð
1
þ
D
=
g
Þ
p
and
D
¼
x
q
x
r
. The readout resona-
tor has intrinsic and extrinsic coupling
Q
sof
Q
i
¼
45
:
8
10
3
and
Q
e
¼
6
:
1
10
3
, measured at single-digit intra-
cavity photon numbers. These values are close to the
designed and expected values.
Our fabrication process is a multi-layer process pictured
in Fig.
1(a)
. We begin with a 10 mm
10 mm chip of SOI
[Si device layer: float zone grown, 220 nm,
3k
X
cm; bur-
ied oxide (BOX) layer: 3
l
m, silicon dioxide; Si handle:
Czochralski grown, 750
l
m,
5k
X
cm]. We then perform
the following main fabrication steps [from left to right in
Fig.
1(a)
]: (i) C
4
F
8
/SF
6
inductively coupled plasma reactive
a)
Electronic mail: opainter@caltech.edu
0003-6951/2017/111(4)/042603/4/$30.00
Published by AIP Publishing.
111
, 042603-1
APPLIED PHYSICS LETTERS
111
, 042603 (2017)
ion etch (ICP-RIE) of 50 nm radius holes through the device
layer to allow for release in step (v) below; (ii) electron
beam evaporation of 120 nm Al at 1 nm/s to define a ground
plane, the qubit capacitor, and the readout resonator; (iii)
double-angle electron beam evaporation of 60 nm and
120 nm of Al at 1 nm/s with an intervening 20 min oxidation
at 5 mbar and subsequent 2 min oxidation at 10 mbar to
forms the JJs; (iv) 5 min argon ion mill and 140 nm Al evap-
oration to form a “bandage” layer that electrically contacts
the Al layers defined in steps (ii) and (iii); and (v) HF vapor
etch of the underlying BOX layer. In the
supplementary
material
, we omit steps (i) and (v) to fabricate a qubit on a
high-resistivity Si substrate, with characterization confirming
that steps (ii)–(iv) alone yield a viable qubit on Si (as
opposed to SOI).
After steps (ii)–(iv), a liftoff process was performed in
n-methyl-2-pyrrolidone at 80
C for 2 h. In (i)–(iv), we use
electron beam lithography to pattern our resist. The above
process is similar to that described elsewhere
28
,
29
and, for
SOI samples, yields a device layer that is partially suspended
above the handle wafer. As highlighted by the yellow bound-
ary line in the scanning electron microscope image of Fig.
1(b)
, we etch 100
l
m into the BOX layer such that the circuit
is far from the lossy Si/SiO
2
interface.
28
We characterize the qubit in a
3
He/
4
He dry dilution refrig-
erator with a base temperature of
T
f
7 mK using frequency-
domain and time-domain spectroscopy. We begin with
frequency-domain characterization and measure transmission
(
S
21
) through a coplanar waveguide feedline using a two-port
vector network analyzer (VNA). The Z control line is used to
carry a small current which produces an external flux bias,
U
ext
, in the SQUID loop of the qubit, thereby tuning the qubit
transition frequency,
f
q
.Foragiven
U
ext
, we identify
f
q
and
transitions to higher levels (from which we extract
g
)by
sweeping a continuous-wave (CW) microwave tone applied to
the XY drive line and monitoring the resonator response.
30
Having identified device parameters, we switch over to
time-domain characterization, using the measurement setup
summarized in Fig.
2
. We characterize the qubit using dis-
persive readout
31
(Fig.
3
), with
U
ext
set so that the qubit is at
a first-order flux-insensitive point.
6
,
18
Excited state popula-
tion decay [Fig.
3(b)
] and Ramsey oscillations [Fig.
3(c)
]
yield
T
1
¼
3.5
l
s and
T
2
¼
2.2
l
s, respectively. Comparative
work, involving superconducting phase qubits on much
thicker (2
l
m device layer) SOI, has previously realized
T
1
¼
1
:
6
l
s and
T
2
¼
110 ns.
26
We can estimate the Purcell-limited
T
1
by the simplistic
single-mode estimate
ð
D
=
g
Þ
2
=
j
r
¼
8
:
5
l
s, where
j
r
¼
x
r
=
Q
and 1
=
Q
¼
1
=
Q
i
þ
1
=
Q
e
. This is not much higher than our
measured
T
1
, implying that incorporating an on-chip Purcell
filter may improve our qubit lifetime.
32
,
33
Regarding the
measured
T
2
values, since obtaining these measurements, we
have identified and resolved some grounding issues in our
measurement setup that likely contributed to excess flux
noise coming from 60 Hz currents on our flux bias line. We
anticipate that these improvements may even be important at
the first-order flux insensitive point.
To characterize our gate fidelities, we utilized Clifford
group randomized benchmarking,
10
,
34
,
35
shown schemati-
cally in Figs.
4(a)
and
4(b)
. We measured three gates
(
X
p
;
X
p
=
2
, and
Y
p
). We realize an average gate fidelity of
f
ð
C
Þ¼
0
:
9860
ð
2
Þ
as well as individual gate fidelities
of
>
0.992 for all measured gates. It should be noted that
these gates have not yet been optimized to avoid phase errors
or leakage outside the computational basis.
36
In terms of the impact of the SOI device layer properties
or various fabrication steps on the resulting qubit decoherence
times, further systematic studies are required. In particular, the
FIG. 1. Qubit fabrication process and SEM images of the SOI device. (a) Five step fabrication process as detailed in the text. (b) SEM image of an SOI qubi
t.
The light (dark) gray regions are Al (exposed Si). The yellow outline demarcates the etch front of the HF vapor release, which extends
100
l
m under the
ground plane so as to isolate the qubit from the lossy Si-SiO
2
interface. The red box denotes the SQUID loop region of the device. (c) Zoom-in image of the
SQUID loop, formed by a double angle evaporation process. The green box bounds one junction. “Bandage” regions described in the main text are visible a
s
darker squares on both the qubit capacitor and the ground plane. (d) Zoom-in of an individual Josephson junction. Each junction has an approximate are
aof
(200 nm)
2
, corresponding to a zero-bias Josephson inductance of
L
J
;
0
¼
22 nH per junction under the conditions described in the main text. The lattice of tiny
dark circles faintly visible here are the etched holes that allow for HF vapor release. An orange arrow points to one such hole.
042603-2 Keller
etal.
Appl. Phys. Lett.
111
, 042603 (2017)
importance of using the vapor HF etch to remove native oxides
and (temporarily) passivate the Si surface before every evapo-
ration step of aluminum on the Si layer (including right before
the double angle evaporation used to form the JJs) needs to be
clarified further. Also, any residual effects of the underlying
BOX layer needs to be ruled out through the systematic studies
of qubit coherence versus undercut extent, in conjunction with
3D numerical modeling to determine more optimized qubit
and membrane geometries. Even while the precise physical
and materials limitations of ou
r system are unclear, current
coherence times are sufficient
for many quantum simulation
and quantum optics experiments
. Meanwhile, our realization
of a highly coherent SOI qubit rep
resents an essential building
block for hybrid electro-opto-mechanical systems on SOI.
Already, electromechanical and optomechanical coherent
transduction bandwidths exceed the bandwidth of our qubit by
a factor of two,
14
,
28
,
37
a prerequisite for high-fidelity, bi-direc-
tional microwave-to-optical quantum state transduction—an
interesting and challenging research program in its own right,
with many potential realizations.
Overall, our fabrication and measurements of Al qubits
on SOI represent a modest but important technical stepping
stone on the path to a variety of potential quantum informa-
tion and quantum science goals. Taken together with com-
plementary advances in the fields of cavity opto- and electro-
mechanics,
14
,
38
and in the context of competing systems,
18
,
25
we are optimistic about the potential of hybrid quantum sys-
tems and circuit QED on silicon-on-insulator.
FIG. 2. (a) Time-domain measurement scheme. Near the top, Gaussian fil-
ters are indicated by enclosed Gaussian lineshapes (lines
i
and
i
are filtered
individually). CW microwave sources with
Z
¼
50
X
are indicated by the ac
voltage symbols. The microwave source used for readout is followed by a
power divider (we use just two ports and terminate others with 50
X
).
Attenuators are indicated by rectangles with labeled power attenuation.
Capacitor symbols show inner/outer DC blocks. All low pass filters are
reflective except for the 64 kHz filter, which is a dissipative RCR filter
(R
¼
499
X
,C
¼
10 nF). (b) Photograph of 1 cm
2
chip wire-bonded to
printed circuit board. (c) Schematic of the device, including the layout of
qubit, readout resonator, control lines, and cavity feedline.
FIG. 3. Qubit characterization. (a) Excited state population (normalized to
the unit interval) as a function of XY drive frequency and pulse duration
s
exhibits a chevron pattern typical of a qubit undergoing Rabi oscillations.
(b) Natural log of the excited state population, normalized to the unit inter-
val, shows exponential decay as a function of waiting time
s
with lifetime
T
1
¼
3.5
l
s (points are data, blue trace is fit). (c) By applying two off-
resonance
p
=
2 pulses with a variable intervening delay
s
, the excited state
population shows Ramsey oscillations (points are data; blue trace is fit). The
decay of the envelope yields coherence time and
T
2
;
S
OI
¼
2.2
l
s. In (a)–(c),
we use a rectangle-windowed 500 ns readout pulse and 30 ns X
p
and X
p
/2
pulses.
FIG. 4. Randomized benchmarking. (a) A schematic of Clifford group ran-
domized benchmarking, described in the
supplementary material
. (b)
Excited state probability as a function of
N
reveals the gate fidelity through
the slope of the resultant line on a semilog plot and the relations described in
the
supplementary material
. The limit of perfect fidelity is shown as a dashed
line. (c) Gate fidelity (with an arbitrary offset given by the readout fidelity)
as a function of
N
using 30 ns pulses. Error bars are 1 standard error in the
measurements averaged over 50 random Clifford sequences. Uncertainties
in the gate fidelities represent 1 standard deviation of
f
due to the statistical
uncertainty of the parameter
p
in the exponential fit.
042603-3 Keller
etal.
Appl. Phys. Lett.
111
, 042603 (2017)
See
supplementary material
for details concerning the
measurement setup and the randomized benchmarking, and
for characterization of an Al-on-Si transmon fabricated by
omitting the first and last steps of our SOI fabrication
process.
We gratefully acknowledge the Martinis Group (UCSB/
Google) for their amplifier and filter designs, Dan Vestyck
for his support of our uEtch HF vapor tool, and Mark
Rosamond for discussions. This work was supported by the
AFOSR MURI Quantum Photonic Matter (Grant No.
16RT0696), the AFOSR MURI Wiring Quantum Networks
with Mechanical Transducers (Grant No. FA9550-15-1-
0015), the Institute for Quantum Information and Matter, an
NSF Physics Frontiers Center (Grant No. PHY-1125565)
with the support of the Gordon and Betty Moore Foundation,
and the Kavli Nanoscience Institute at Caltech. A.J.K.
acknowledges the IQIM Postdoctoral Fellowship.
1
M. Devoret and R. J. Schoelkopf,
Science
339
, 1169 (2013).
2
A. Houck, H. Tureci, and J. Koch,
Nat. Phys.
8
, 292 (2012).
3
A. Blais, R.-S. Huang, A. Wallraff, S. Girvin, and R. Schoelkopf,
Phys.
Rev. A
69
, 062320 (2004).
4
A. Blais, J. Gambetta, A. Wallraff, D. Schuster, S. Girvin, M. Devoret,
and R. Schoelkopf,
Phys. Rev. A
75
, 032329 (2007).
5
M. Devoret, S. Girvin, and R. Schoelkopf,
Ann. Phys.
16
, 767 (2007).
6
J. Koch, T. Yu, J. Gambetta, A. Houck, D. Schuster, J. Majer, A. Blais, M.
Devoret, S. Girvin, and R. Schoelkopf,
Phys. Rev. A
76
, 042319 (2007).
7
J. Schreier, A. Houck, J. Koch, D. Schuster, B. Johnson, J. Chow, J.
Gambetta, J. Majer, L. Frunzio, M. Devoret, S. Girvin, and R. Schoelkopf,
Phys. Rev. B
77
, 180502 (2008).
8
A. A. Houck, J. Koch, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf,
Quantm Inf. Proccess.
8
, 105 (2009).
9
A. Wallraff, D. Schuster, A. Blais, L. Frunzio, R.-S. Huang, J. Majer, S.
Kumar, S. Girvin, and R. Schoelkopf,
Nature
431
, 162 (2004).
10
R. Barends, J. Kelly, A. Megrant, A. Veitia, D. Sank, J. Jeffrey, T. White,
J. Mutus, A. Fowler, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A.
Dunsworth, C. Neill, P. O’Malley, P. Roushan, A. Vainsencher, J.
Wenner, A. Korotkov, A. Cleland, and J. Martinis,
Nature
508
, 500
(2014).
11
M. Reed, L. DiCarlo, S. Nigg, L. Sun, L. Frunzio, S. Girvin, and R.
Schoelkopf,
Nature
482
, 382 (2012).
12
N. Ofek, A. Petrenko, R. Heeres, P. Reinhold, Z. Leghtas, B. Vlastakis, Y.
Liu, L. Frunzio, S. Girvin, L. Jiang, M. Mirrahimi, M. Devoret, and R.
Schoelkopf,
Nature
536
, 441 (2016).
13
A. Safavi-Naeini and O. Painter,
New J. Phys.
13
, 013017 (2011).
14
R. Andrews, R. Peterson, T. Purdy, K. Cicak, R. Simmonds, C. Regal, and
K. Lehnert,
Nat. Phys.
10
, 321 (2014).
15
C. Quintana, A. Megrant, Z. Chen, A. Dunsworth, B. Chiaro, R. Barends,
C. Campbell, Y. Chen, I.-C. Hoi, E. Jeffrey, J. Kelly, J. Mutus, P.
O’Malley, C. Neill, P. Roushan, D. Sank, A. Vainsencher, J. Wenner, T.
White, A. Cleland, and J. Martinis,
Appl. Phys. Lett.
105
, 062601 (2014).
16
C. Wang, C. Axline, Y. Gao, T. Brecht, Y. Chu, L. Frunzio, M. Devoret,
and R. Schoelkopf,
Appl. Phys. Lett.
107
, 162601 (2015).
17
R. McDermott,
IEEE Trans. Appl. Supercond.
19
, 2 (2009).
18
R. Barends, J. Kelly, A. Megrant, D. Sank, E. Jeffrey, Y. Chen, Y. Yin, B.
Chiaro, J. Mutus, C. Neill, P. O’Malley, P. Roushan, J. Wenner, T. White,
A. Cleland, and J. Martinis,
Phys. Rev. Lett.
111
, 080502 (2013).
19
C. Axline, M. Reagor, R. Heeres, P. Reinhold, C. Wang, K. Shain, W.
Pfaff, Y. Chu, L. Frunzio, and R. Schoelkopf,
Appl. Phys. Lett.
109
,
042601 (2016).
20
A. D. O’Connell, M. Ansmann, R. C. Bialczak, M. Hofheinz, N. Katz, E.
Lucero, M. N. C. McKenney, H. Wang, E. M. Weig, A. N. Cleland, and J.
M. Martinis,
Appl. Phys. Lett.
92
, 112903 (2008).
21
S. J. Weber, K. W. Murch, D. H. Slichter, R. Vijay, and I. Siddiqi,
Appl.
Phys. Lett.
98
, 172510 (2011).
22
A. Bruno, G. de Lange, S. Asaad, K. L. van der Enden, N. K. Langford,
and L. DiCarlo,
Appl. Phys. Lett.
106
, 182601 (2015).
23
A. D. C
orcoles, E. Magesan, S. J. Srinivasan, A. W. Cross, M. Steffen, J.
M. Gambetta, and J. M. Chow,
Nat. Commun.
6
, 6979 (2015).
24
S. Asaad, C. Dickel, N. K. Langford, S. Poletto, A. Bruno, M. A. Rol, D.
Deurloo, and L. DiCarlo,
npj Quantum Inf.
2
, 16029 (2016).
25
Y. Chu, C. Axline, C. Wang, T. Brecht, Y. Gao, L. Frunzio, and R.
Schoelkopf,
Appl. Phys. Lett.
109
, 112601 (2016).
26
U. Patel, Y. Gao, D. Hover, G. J. Ribeill, S. Sendelbach, and R.
McDermott,
Appl. Phys. Lett.
102
, 012602 (2013).
27
S. Barzanjeh, M. Wulf, M. Peruzzo, M. Kalaee, P. B. Dieterle, O. Painter,
and J. M. Fink, e-print
arXiv:1706.00376
.
28
P. B. Dieterle, M. Kalaee, J. M. Fink, and O. Painter,
Phys. Rev. Appl.
6
,
014013 (2016).
29
A. Dunsworth, A. Megrant, C. Quintana, Z. Chen, R. Barends, B. Burkett,
B. Foxen, Y. Chen, B. Chiaro, A. Fowler, R. Graff, E. Jeffrey, J. Kelly, E.
Lucero, J. Y. Mutus, M. Neeley, C. Neill, P. Roushan, D. Sank, A.
Vainsencher, J. Wenner, T. C. White, and J. M. Martinis, e-print
arXiv:1706.00879
.
30
J. M. Fink, “Quantum nonlinearities in strong coupling circuit QED,”
Ph.D. thesis, ETH Zurich (2010).
31
D. I. Schuster, A. Wallraff, A. Blais, L. Frunzio, R.-S. Huang, J. Majer, S.
M. Girvin, and R. J. Schoelkopf,
Phys. Rev. Lett.
94
, 123602 (2005).
32
M. D. Reed, B. R. Johnson, A. A. Houck, L. DiCarlo, J. M. Chow, D. I.
Schuster, L. Frunzio, and R. J. Schoelkopf,
Appl. Phys. Lett.
96
, 203110
(2010).
33
E. Jeffrey, D. Sank, J. Y. Mutus, T. C. White, J. Kelly, R. Barends, Y.
Chen, Z. Chen, B. Chiaro, A. Dunsworth, A. Megrant, P. J. J. O’Malley,
C. Neill, P. Roushan, A. Vainsencher, J. Wenner, A. N. Cleland, and J. M.
Martinis,
Phys. Rev. Lett.
112
, 190504 (2014).
34
J. Chow, J. Gambetta, L. Tornberg, J. Koch, L. Bishop, A. Houck, B.
Johnson, L. Frunzio, S. Girvin, and R. Schoelkopf,
Phys. Rev. Lett.
102
,
090502 (2009).
35
E. Magesan, J. M. Gambetta, B. R. Johnson, C. A. Ryan, J. M. Chow, S. T.
Merkel, M. P. da Silva, G. A. Keefe, M. B. Rothwell, T. A. Ohki, M. B.
Ketchen, and M. Steffen,
Phys. Rev. Lett.
109
, 080505 (2012).
36
F. Motzoi, J. M. Gambetta, P. Rebentrost, and F. K. Wilhelm,
Phys. Rev.
Lett.
103
, 110501 (2009).
37
J. Witmer, J. Valery, P. Arrangoiz-Arriola, C. Sarabalis, J. Hill, and A.
Safavi-Naeini, e-print
arXiv:1612.02421
.
38
M. Aspelmeyer, T. Kippenberg, and F. Marquardt,
Rev. Mod. Phys.
86
,
1391 (2014).
042603-4 Keller
etal.
Appl. Phys. Lett.
111
, 042603 (2017)