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Chen
et al.
,
Sci. Adv.
10
, eado6240 (2024) 13 September 2024
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ien
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|
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PHYSICS
Phonon engineering of atomic-
scale def
ects in
superconducting quantum circuits
Mo Chen
1,2,3
, John Clai Owens
1,2,3
†, Harald Putterman
4
, Max Schäfer
1,2,3
‡, Oskar Painter
1,2,3,4
*
Noise within solid-
sta
te systems at low temperatures can typically be traced back to material defects. In amor
-
phous materials, these defects are broadly described by the tunneling two-
le
vel systems (TLSs) model. TLS have
recently taken on further relevance in quantum computing because they dominate the coherence limit of super
-
conducting quantum circuits. Efforts to mitigate TLS impacts have thus far focused on circuit design, material se
-
lection, and surface treatments. Our work takes an approach that directly modifies TLS properties. This is achieved
by creating an acoustic bandgap that suppresses all microwave-
fr
equency phonons around the operating fre
-
quency of a transmon qubit. For embedded TLS strongly coupled to the transmon qubit, we measure a pro
-
nounced increase in relaxation time by two orders of magnitude, with the longest
T
1
time exceeding 5 milliseconds.
Our work opens avenues for studying the physics of highly coherent TLS and methods for mitigating noise within
solid-
sta
te quantum devices.
INTRODUCTION
Glassy materials exhibit abnormal thermal transport behaviors at low
temperatures,
T
<
1 K. These anomalies include specific heat and ther
-
mal conductivity that deviate from predictions of the Debye model.
This is counter-
in
tuitive because the wavelengths of relevant phonons
at low temperatures are too long to distinguish between structurally
amorphous and crystalline solids. To address this mystery, Phillips (
1
)
and Anderson
et al.
(
2
) independently proposed the ubiquitous exis-
tence of microscopic two-
l
evel systems (TLSs), which are defect states
tunneling between two nearly equivalent local potential wells. These
TLS defects are known to distribute nearly uniformly over a broad fre-
quency range, and they have both elastic and electric dipoles that al-
low them to couple to strain and electric fields (
3
). The TLS model
successfully explains the aforementioned thermal anomalies of glassy
materials, as well as their acoustic and dielectric behaviors at low tem-
peratures. In addition, because of the omnipresence of TLS in amor
-
phous materials, their wide frequency distribution, and their ability to
couple through both phonons and photons, TLS have been associated
with noise in various solid-
s
tate quantum systems, including super
-
conducting (SC) quantum circuits (
4
8
), nanomechanical resonators
(
9
11
), and optomechanical cavities (
12
,
13
).
In the context of SC quantum circuits, TLS have been identified as a
primary limitation to the energy lifetime, coherence, and overall stability
of the physical qubits being explored for scalable quantum computing
architectures (
3
,
5
8
,
14
19
). TLS are thought to reside primarily at the
amorphous material interfaces that make up the physical qubit device
and cause dielectric loss through the interaction between their electric
dipoles and the electric field of the qubit. The interaction leads to a two-
s
tep energy dissipation process, where energy first transfers from the SC
qubit to resonant TLS and subsequently dissipates into the local environ-
ment of the TLS (
20
23
). Despite awareness of the two-
s
tep dissipation
process for SC qubit decay, past research efforts have focused on investi-
gating and mitigating the first step, namely, the energy decay from the
SC qubit to TLS. This choice is in part because of the challenge in access-
ing and controlling atomic-
s
cale TLS defects (
24
,
25
) and therefore in
one’s ability to modify the second step of the dissipation process. As a
result, TLSs have long been viewed as an intrinsic material defect to be
avoided (
14
,
19
,
26
,
27
). In the pursuit of better SC qubits, material inves-
tigations have focused on finding superconductors with a surface oxide
layer that has low TLS density (
28
30
). Similarly, circuit designs of SC
qubits aim to minimize the electric-
fi
eld strength of the electromagnetic
field produced by the qubit at material interfaces to reduce the interac-
tion between the SC qubit and TLS (
14
,
26
,
27
). These efforts have led to
microwave-
f
requency SC qubits with energy relaxation times that ex-
tend over hundreds of microseconds (
18
).
In the hopes of further understanding TLS and improving SC
qubits, in this work, we take direct aim at modifying the second step
of the dissipation process of SC qubits, namely, the interaction of
TLS with the reservoir of phonons of the material host as illustrated
in Fig. 1A. The phonon bath is targeted due to the roughly five-
o
rders-
o
f-
m
agnitude difference in the speed of sound and the speed
of light in materials, and the correspondingly much larger density of
states (DOS) at microwave frequencies for phonons versus photons.
For typical Debye-
l
evel electric dipole moments and eV-
l
evel defor
-
mation potentials of TLS, this makes the dominant bath that of pho-
nons. We design and fabricate a frequency-
t
unable transmon qubit
(
31
,
32
) with its Josephson junctions (JJs) embedded in an engi-
neered acoustic structure that features a GHz-
w
ide acoustic band-
gap, as shown in Fig. 1 (B to F). TLS within the junctions and with
transition frequency within the acoustic bandgap range, experience
a suppressed two-
dim
ensional (2D) phonon DOS, which effectively
shields them from resonant decay into the phonon bath. Using the
transmon qubit as a quantum sensor, we are able to individually ad-
dress and characterize the coherence properties of strongly coupled
TLS within the JJs of the electric qubit. Our experimental results
show that the
T
1
times for TLS with resonant frequency lying inside
the acoustic bandgap are, on average, extended by two orders of
magnitude compared to when the TLS frequency lies outside the
acoustic bandgap. The coherence and the temperature dependence
of the
T
1
relaxation process of the acoustically shielded TLS are also
1
t
homas J
. Watson, Sr.,
l
aboratory of Applied Physics, California
i
nstitut
e of
t
ech
-
nology, Pasadena, CA 91125, USA.
2
i
nstitut
e for Quantum
i
nf
ormation and Matter,
California
i
nstitut
e of
t
echnology
, Pasadena, CA 91125, USA.
3
Kavli
n
anoscience
i
nstitut
e, California
i
nstitut
e of
t
echnology
, Pasadena, CA 91125, USA.
4
AWS Center
for Quantum Computing, Pasadena, CA 91125, USA.
*Corresponding author.
e
mail: opainter@
calt
ech.
edu
†P
resent address: AWS Center for Quantum Computing, Pasadena, CA 91125, USA.
‡Present address:
d
epar
tment of Applied Physics and Physics, Yale University,
n
ew
h
aven, C
t
06520, USA.
Copyright © 2024
t
he
A
uthors, some rights
reserved; exclusive
licensee American
Association for the
Advancement of
Science. no claim to
original U.S.
Government Works.
distributed under a
Creative Commons
Attribution license 4.0
(CC BY).
Downloaded from https://www.science.org at California Institute of Technology on October 31, 2024
Chen
et al.
,
Sci. Adv.
10
, eado6240 (2024) 13 September 2024
SCienCe Adv
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studied, indicating that coherence is limited to the microsecond
level due to low-
frequency noise, and thermally activated relaxation
channels open up above 75 mK.
RESULTS
Transmon qubit with acoustically shielded junctions: Design
and fabrication
We fabricate our transmon qubit device on a silicon-
on-
insulator
(SOI) substrate (
33
), which allows for nanoscale fabrication of high-
quality acoustic bandgap structures in the microwave frequency
range (see the “Methods” section in the Supplementary Materials
for design and fabrication details). Figure 1 (B, E, and F) shows
scanning electron micrographs (SEM) of various parts of the device.
The transmon qubit has a shunt capacitor, which is used to couple to
a coplanar-
waveguide resonator for readout of the qubit state. It
also has two aluminum-
aluminum oxide-
aluminum (Al-
AlO
x
- Al)
JJs forming a magnetic flux sensitive SQUID loop for tuning of the
electric qubit state via a current-
carrying
Z
control line. An
XY
con
-
trol line is added for direct charge excitation of the qubit. The trans-
mon qubit is unremarkable in its design, except for the fact that each
JJ in the SQUID loop is located on top of a suspended Si platform
formed by the release of the 220-
nm-
thick Si device layer from the
SOI substrate and tethered to the SOI substrate by nine periods of
an acoustic bandgap structure (Fig. 1, E and F). The acoustic band-
gap structure of this work is based on a cross-
shield design (
13
,
34
),
which has a 1.372-
GHz-
wide acoustic bandgap centered at 5.128 GHz
(see the “Acoustic bandgap metamaterials” section in the Supple
-
mentary Materials). In theory, this effectively isolates the JJs of
the transmon qubit, and any TLS defects that may be within the
amorphous oxide layer of each junction, from acoustic modes of the
SOI substrate (see the “Localized acoustic phonon modes in the
bandgap” section in the Supplementary Materials for a discussion
on localized acoustic phonon modes).
The TLS within the acoustically isolated JJs are distinguished
from other TLS in different regions of the circuit by their signature
strong coupling to the transmon qubit due to the strong electric field
of the qubit mode in the atomically thin AlO
x
barrier layer of the JJs.
To increase the occurrence of these TLS of interest, the JJs of our
device are chosen to have a relatively large area of 0.83
μ
m
2
each.
The AlO
x
barrier layer is also grown slightly thicker to keep the
junction energy
E
J
uninfluenced by the change in junction size (see
the “Device fabrication” section in the Supplementary Materials for
details). As a result, the JJs have a substantial junction capacitance
in addition to the nonlinear inductance, which is similar to the
merged-
element transmon qubit (
35
,
36
). In the current design, the
JJs account for approximately 60 fF or 60% of the total transmon
capacitance. The remaining qubit capacitance comes from an addi-
tional planar shunt capacitor, which is varied for each transmon qu-
bit. This variation in shunt capacitance produces a series of qubit
devices with upper sweet-
spot frequency covering a range of 5.44 to
6.48 GHz, facilitating the characterization of the entire acoustic
bandgap of the designed acoustic shield.
Transmon qubit and TLS characterization
Characterization of transmon qubit devices, and any coupled TLS, is
performed in a dilution refrigerator, where the chip-
scale sample
containing the devices is mounted to the mixing chamber plate of
the fridge. The fridge reaches a base temperature of 7 mK, which
cools down both the SC qubit and TLS close to their respective
C
A
BD
F
E
Fig. 1.
A hybrid platform for phonon engineering in superconducting quantum circuits.
(
A
) Schematic of the two-
step energy dissipation process in a SC qubit. en-
ergy first decays from the SC qubit to a bath of near-
resonant
tlS, with coupling rate
g
, and then dissipates into the local environment of
tlS.
this environment is pre
-
sumed to be dominated by a bath of phonon modes

b
, relaxing at a rate
κ
, and interacting with the elastic dipole
γ
of the
tlS. (
B
) SeM image (false colored) of the
fabricated hybrid transmon qubit device. each qubit couples to its dedicated
λ
/4 readout (RO) resonator (turquoise),
Z
control line (green) and
XY
control line (blue).
the
entire device, outlined in pink, is suspended on the 220-
nm-
thick Si device layer, which is released from the underlying 3-
μ
m-
thick oxide BOX layer of the SOi chip.
(
C
) Circuit diagram of the transmon qubit and readout resonator. Approximately 40% of the transmon capacitance comes from the shunt capacitor (orange), and 60%
from the JJs. (
D
) Simulated acoustic band structure of the Si cross-
shield unit cell, with the acoustic bandgap centered around 5.1 Gh
z shaded in pink. (
E
) Zoomed-
in view
of the SQUid loop of the transmon qubit device.
the SQUid loop is formed with two JJs in parallel between the shunt capacitor and ground. each JJ is fabricated on top
of a micron-
scale Si platform tethered to the SOi substrate by a cross-
shield acoustic bandgap structure. (
F
) d
etailed SeM image of one of the JJs, showing the cross-
shield
patterning of the acoustic bandgap structure.
the contacts from the shunt capacitor and ground to the top and bottom electrodes of the JJ, respectively, are visible as
narrow Al leads that run across the connected cross-
shield lattice.
Downloaded from https://www.science.org at California Institute of Technology on October 31, 2024
Chen
et al.
,
Sci. Adv.
10
, eado6240 (2024) 13 September 2024
SCienCe Adv
AnCeS
|
ReSeARCh AR
tiCle
3 of 9
ground states. The transmon qubit is first characterized in the time
domain with pulsed excitation and dispersive readout. The trans-
mon qubits of this work are measured to have excited-
to-
ground
state relaxation times of
T
1
3
μ
s, which is comparable to the best
reported SOI qubit (
33
). We attribute the qubit
T
1
relaxation to both
dielectric loss at the shunt capacitor and Purcell decay to the readout
resonator. Further details of the qubit characterization and the qubit
parameters for all seven qubit devices studied in this work can be
found in the “Device design” section in the Supplementary Materials.
Once characterized, the transmon qubit is used as a quantum
sensor to identify individual TLS in the JJs that experience a struc-
tured acoustic environment. Using the pulse sequence depicted in
Fig. 2A, we perform pulsed microwave spectroscopy to explore the
electrically-
active transitions of a given transmon qubit device. The
measured microwave spectrum of one such transmon qubit device
(Chip- A, Q
1
) is shown in Fig. 2C, with qubit frequency tuned be-
tween 5.5 and 6.3 GHz using a flux bias pulse via the
Z
control line.
Strong couplings of the transmon to five TLS (labeled TLS1 to TLS5)
manifest as avoided crossings in the spectrum, from which we ex-
tract the TLS frequencies and coupling strengths
g
to the transmon
qubit. In these experiments, the microwave power on the
XY
con
-
trol line is chosen to yield approximately 100-
ns transmon
π
- pulses,
which determines the power-
broadened qubit linewidth and allows
us to resolve TLS with a coupling strength of
g
5 MHz. The strong
couplings of TLS1 to TLS5 are a signature that these TLS are physi-
cally located inside the JJs of Q
1
on Chip-
A and, hence, inside the
acoustic bandgap structures.
Next, to probe further the properties of the strongly coupled TLS,
we calibrate coherent SWAP operations between transmon qubit
and TLS states using SWAP spectroscopy (
37
,
38
), as illustrated in
Fig. 2B. Figure 2D shows a representative measurement of SWAP
spectroscopy performed on device Q
1
of Chip-
A, where five pro-
nounced vacuum Rabi oscillation patterns appear at the same
pulsed flux-
bias amplitudes as the previously measured anticross-
ings for TLS1 to TLS5. This confirms that these fringes arise from
the resonant exchange interactions between the transmon qubit and
individual strongly coupled TLS and signifies the coherent nature of
these TLS. On the basis of these vacuum Rabi oscillations, we iden-
tify optimal SWAP gates for each TLS.
Using the SWAP gate, we are able to selectively prepare any one
of the five TLS in their first excited state. Moreover, sequential ap-
plication of this technique allows us to look for the presence of sec-
ond (and higher-
order) excited states of the TLS. The absence of
observable higher-
order excited states (see the “Anharmonicity of
TLS” section in the Supplementary Materials) indicates that the
strongly coupled TLS to the transmon qubits measured here are
highly anharmonic. This eliminates any concerns that the TLS-
like
behavior observed in this study originates from high-
Q
harmonic
acoustic modes of the acoustic bandgap structure.
T1 lifetime of acoustically shielded TLS
To characterize the lifetime of TLS, we begin by preparing a TLS in
its excited state using the SWAP gate and then let the TLS relax for a
variable amount of time. Last, we map the TLS state back to the
transmon using a second SWAP gate and measure the final trans-
mon state using its dispersive readout circuit (Fig. 3A). During the
TLS relaxation time, the transmon qubit is typically tuned to its up-
permost frequency, far from resonance with the TLS to avoid Pur
-
cell decay of the TLS through the transmon qubit. The resulting
T
1
relaxation curves for the five characterized TLS of Q
1
on Chip-
A are
shown in Fig. 3B, with TLS1 to TLS3 having
T
1
2
μ
s, while TLS4
and TLS5 exhibit two to three orders-
of-
magnitude longer relax-
ation times of
T
1,TLS4
=
215
±
15
μ
s and
T
1,TLS5
=
1100
±
200
μ
s,
respectively (here the one SD uncertainty in the
T
1
is quoted). Nota-
bly, both TLS4 and TLS5 have transition frequencies that lie within
the expected acoustic bandgap of the crosss-
shield structure based
on numerical finite-
element simulations, whereas TLS1 to TLS3 all
have frequencies above the simulated bandgap.
TLS1
TLS3
TLS2
TLS5
TLS4
5.72
5.58
0.62
0.7
5.65
0.66
Population
0
1
AB
XY
Z
Readout
0.35
0.4
0.45
0.5
0.55
0.
60
.7
0.65
Z
flux bias (V)
250
200
150
100
50
6.2
6
5.8
5.6
XY
frequency (GHz)
Z
duration (ns)
C
D
Fig. 2.
Characterization of a hybrid transmon-
TLS system.
(
A
) Pulse sequence
used for microwave spectroscopy of the qubit and (
C
) corresponding measured
transmon qubit spectrum for
Q
1
of Chip-
A.
in this measurement protocol, a
Z
- pulse
flux biases the transmon qubit away from its flux-
insensitive sweet spot. An over
-
lapping
XY
pulse of 100 ns duration probes the excitation of the transmon qubit at
different flux biases. When the transmon is in resonance with a
tlS, their hybridiza-
tion results in avoided crossings. For
Q
1
of Chip-
A, we measure five distinct
tlS, la-
beled
tlS1 to
tlS5. notably, the avoided crossings of
tlS4 and
tlS5 are situated
within the simulated acoustic bandgap, with the upper frequency bandedge indi-
cated by the green dashed line.
the inset provides a magnified view of the avoided
crossing of
tlS5.
the red solid line in the inset is a fitting curve with
ω
tlS5
/2
π
=
5.6563 Gh
z and
g
/2
π
=
21.7 Mh
z. (
B
) Pulse sequence of the transmon-
tlS SWAP
gate spectroscopy and (
D
) corresponding measured transmon-
tlS SWAP spec
-
trum for device
Q
1
of Chip-
A. i
n this measurement protocol, the transmon is first
excited by an
XY
π
- pulse and then tuned by a
Z
pulse with varying amplitude and
duration, and lastly the transmon qubit population is dispersively read out upon
tuning back to its starting frequency.
the resulting chevron patterns correspond to
vacuum Rabi oscillations between the transmon qubit and the strongly coupled
tlS1 to tlS5.
Downloaded from https://www.science.org at California Institute of Technology on October 31, 2024