of 47
Supplementary Materials for
Phonon engineering of atomic-scale defects in superconducting
quantum circuits
Mo Chen
et al.
Corresponding author: Oskar Painter, opainter@caltech.edu
Sci. Adv.
10
, eado6240 (2024)
DOI: 10.1126/sciadv.ado6240
This PDF file includes:
Supplementary Text
Tables S1 to S6
Figs. S1 to S24
References
I. METHODS
In this section, we describe considerations that underlie the design of the hybrid transmon qubit device with
Josephson junctions (JJs) embedded in acoustic bandgap structure. Our overarching goal is to strike a balance
between the simplicity of the transmon qubit device and its e
ectiveness in demonstrating the phonon engineering
of tunneling two-level systems (TLS) defects. This guiding principle is reflected in our decision of excluding Purcell
filters, as well as the inclusion of a shunt capacitor for the transmon qubit. Further discussions on the device design
will follow shortly. Along with the transmon qubit device design, we consider the acoustic metamaterials, as well
as their integration into the transmon device. We will also discuss the device fabrication process, wherein a single
resist layer Manhattan-style Josephson junction process plays a key role in the realization of our device. To conclude
the Methods section, we provide brief descriptions of our experimental measurement setup, the calculation of phonon
density of states using COMSOL, and a technique we used to generate new sets of TLS, known as thermal cycling.
A. Device design
The device serves two purposes: a) to identify individual TLS influenced by the engineered acoustic environment,
and b) to characterize their relaxation behavior. To achieve this, we direct our attention towards TLS that are
physically located inside the Josephson junction (JJ) tunnel barriers. This choice has three advantages. Firstly, their
strong couplings to the transmon qubit, due to the intense electric field within the JJ, set them apart from TLS at
circuit interfaces. Secondly, their physical confinement within a small area (the JJ) makes it convenient for phonon
engineering. Lastly, individual addressing and characterization of TLS inside the JJ are well-established
37,38,41
.
1. Transmon qubit
In the design of the transmon qubit, we decide to make our JJ an order-of-magnitude larger than typical JJs, with a
size of approximately
0
.
83
μ
m
2
, in order to increase occurrences of TLS inside the JJ. Such large JJ contributes to
a substantial junction capacitance. A transmon qubit, characterized by its qubit capacitance consisting mainly of the
junction capacitance, has been recently demonstrated in refs.
35,36
. In our design, the two JJs that form a symmetric
SQUID (superconducting quantum interference device) loop collectively contribute a
60 fF junction capacitance
to the transmon qubit. It is noteworthy that the junction capacitance is large enough that the JJs alone can make
up a transmon qubit, a configuration termed the ‘merged-element transmon’
35,36
. In this work, however, instead of
implementing a full merged-element transmon, we introduce a shunt capacitor. The shunt capacitor conveniently
facilitates coupling to control lines and readout resonators. The shunt capacitor accounts for
40 fF, resulting in a
total transmon capacitance of
100 fF. It is important to emphasize that the shunt capacitor is not protected by the
acoustic metamaterials, and its interaction with nearby resonant TLS is considered the major
T
1
relaxation channel
for the transmon qubit. Consequently, due to the influence of the shunt capacitor, we do not expect substantial e
ects
of phonon engineering on the transmon qubit in the current design.
To engineer the acoustic environment that the JJ and the TLS inside the JJ see, we position the JJ of the transmon
qubit on top of a rectangular platfrom consisting of an unpatterened Si suspended membrane, as shown in Fig.
S1.The
rectangular platform is tethered to the rest of the Si microchip through an acoustic metamaterial, which is designed
to exhibit a microwave-frequency acoustic bandgap centered around 5
.
1 GHz. Inside the bandgap, the acoustic
metamaterial shields the TLS from spontaneous phonon emission into the phonon modes of the bulk materials and
extends the lifetime of TLS.
The readout resonators are designed to situate
700 MHz above the transmon’s upper sweet spot frequency, with
a coupling strength of
70 MHz and a linewidth of
2 MHz. No Purcell filters are used, yielding a Purcell limit of
10
μ
s, a timescale that is on the same order of transmon’s
T
1
. We believe the Purcell limit serves as the secondary
contribution to the relaxation process of transmon, with the major contribution being resonant coupling to TLS at
the shunt capacitor, as mentioned earlier.
For the comprehensive characterization of phonon engineering of TLS and the acoustic bandgap, two distinct
microchips are designed, labelled Chip-A and Chip-B. Each chip accommodates four transmon qubits. On Chip-A,
the designed upper sweet spot frequencies of the four transmon qubits span the range of 6
6
.
5 GHz, strategically
chosen to resolve the upper edge of the acoustic bandgap. Conversely, the four transmon qubits on Chip-B are
designed to cover the frequency range 5
.
1
5
.
5 GHz to resolve the lower edge of the acoustic bandgap. On each chip,
the four transmon qubits have identical JJs. Adjustments to the shunt capacitance of each transmon qubit tunes
its sweet spot frequency to the desired value. Between Chip-A and Chip-B, the geometry remains identical. The
di
erent transmon frequency ranges between the two chips are achieved by varying the oxidation condition during
the JJ fabrication process.
The fabricated Chip-A (Chip-B) covers upper sweet spot frequency ranges 6–6
.
5 GHz (5
.
4–5
.
8 GHz). Typical
parameters for all transmon qubits on both chips are presented in Table
S1. It’s worth noting that these parameters
are subject to minor changes during di
erent cool downs. On Chip-B, Q
4
remains operational, indicated by the
Lamb shift of the readout resonator and its susceptibility to flux tuning via the crosstalk from the Z lines of Q
1–3
.
However, Q
4
does not show frequency tuning through its own Z line. We suspect that potential defects in the Z line or
associated wirebonds lead to an open connection. We therefore exclude Q
4
from Chip-B in this study. Parameters of
the other seven transmon qubits align well with our design and simulations. Their
T
1
times span the range 1
.
5–6
.
0
μ
s,
corresponding to
Q
values between 0
.
5–2
10
5
, which are on par with the best SOI transmon qubits reported in
literature
33
. Fig.
S2 displays a representative Rabi chevron pattern measured on Q
4
of Chip-A.
2. Acoustic bandgap metamaterials
The acoustic metamaterial is formed from a periodic cross-shield pattern etched into the Si membrane layer. A
scanning electron microscope (SEM) image showcasing the JJ, the rectangular platform, and the surrounding acoustic
metamaterials is shown in Fig.
S1a. The platform region replaces the central 2
3 unit cells of the cross-shield
metamaterial, and is surrounded by nine periods of shielding. As shown in the SEM of the device (Fig.
S1a) and the
schematic ( Fig.
S1b), the JJ leads pass through several unit cells of the acoustic metamaterial, introducing additional
mass and perturbations to the band structures. Consequently, central to our design is the establishment of a large
acoustic bandgap in the presence of the JJ leads.
To account for these perturbations due to the JJ leads, we consider three di
erent unit cell types in the COMSOL
simulation of the acoustic band structure. These unit cell types are designed in accordance with the geometry of
our transmon device, which include
a
silicon only (enclosed by the blue dashed box in Fig.
S1b),
b
silicon with
30 nm thick Al leads from the JJ (enclosed by the red dashed box in Fig.
S1b), and
c
silicon with 50 nm thick
Al leads from the JJ (enclosed by the green dashed box in Fig.
S1b). Throughout all three cases, the Si geometry
stays the same. The dimensions used by the COMSOL simulations are derived from the SEM image of a sister chip
that is nominally identical to Chip-B. The results from the COMSOL simulations for the acoustic band structures in
these three scenarios, along special paths connecting highly symmetric points in the
k
-space, are shown in Fig.
S3.
Additionally, the corresponding bandedge frequencies are listed in Table
S2. The overlap of these three simulated
bandgaps yields an overall bandgap spanning 1
.
372 GHz, ranging from 4
.
442 GHz to 5
.
814 GHz, in the presence of
the JJ leads. The overall bandgap is 0
.
219 GHz narrower than the Si only unit cell (4
.
442–6
.
033 GHz), due to the
perturbations of Al leads.
Additionally, we conducted COMSOL simulations to explore the e
ects of the widths and thicknesses of the Al
leads that run through the Si cross-shield pattern. Of the two factors, thickness has the larger impact on the band
structures. As the thickness of the Al lead increases, the size of the bandgap decreases from 1
.
591 GHz until it vanishes
at approximately 100 nm Al thickness. Preserving a large acoustic bandgap therefore necessitates careful design of
the JJ geometry and the process of double-angle evaporation of Al. It is critical to ensure that any cross-shield
metamaterial unit cell undergoes no more than one metalization, or equivalently, avoiding the formation of parasitic
junctions on the cross-shield pattern. This condition ensures that the three scenarios simulated in Fig.
S3 faithfully
capture all the acoustic band structures encountered in our device. The elimination of parasitic junctions in the
cross-shield region is achieved by a geometric argument, which will be discussed in detail in the subsequent section
on device fabrication (Sec.
IB).
3. Localized acoustic phonon modes in the bandgap
In the center of the acoustic bandgap metamaterial, 2
3 unit cells are replaced by a suspended unpatterned Si
rectangular platform, on top of which the JJ sits. The platform region has a size comparable to the acoustic wavelength,
therefore supporting a few localized acoustic modes inside the acoustic bandgap frequency range. Through design,
we minimized the platform area needed for the junction angled evaporation, as detailed in Sec.
IB3. This minimal
Si platform and the junction structure on top of it minimizes the total number of localized acoustic modes.
Accurate simulation of these modes are demanding, due to the large total area involved and small feature size
of the device geometry. We estimated the localized acoustic modes in COMSOL. The frequency and quality factor
of simulated eigenmodes are shown in Fig.
S4, considering a structure of the Si platform, surrounding Si acoustic
bandgap metamaterial, and aluminum JJ on top. There is a set of 10 discrete localized acoustic modes, represented
by open circles, with a free spectral range on the order of 100 MHz, and quality factors on the order of 10
7
in Fig.
S4.
To ensure that we are not just measuring these long-lived harmonic acoustic modes, we have checked the anhar-
monicity of the detected TLS states. As discussed in more detail in Sec.
IIF , the TLS we measured are e
ectively
two-level systems with very large anharmonicity, resolving the concern that these detected states are high-Q harmonic
acoustic modes. However, these local acoustic modes could still play a role when they hybridize with TLS. In this
case, the measured strong anharmonicity of TLS indicates that they would still be predominantly TLS-like hybrid
modes.
B. Device fabrication
Our fabrication process of the hybrid device stems from the fabrication recipe for transmon qubit on silicon-on-
insulator (SOI) substrate outlined in ref.
33
. Our modified process is illustrated in Fig.
S5. We start with an SOI wafer
(SEH) with the following specifications: silicon device layer, 220 nm in thickness, resistivity
5k
·
cm; buried
silicon dioxide layer, 3
μ
m in thickness; and a silicon handle, 750
μ
minthickness,
5k
·
cm. First, the wafer is
diced along the
h
100
i
direction into chips of dimensions 20 mm
10 mm. We then perform the following fabrication
steps, all using 100 keV electron-beam lithography (Raith EBPG5200) for patterning, and electron beam evaporation
(Plassys MEB 550S) for metalization: (i) Si device layer patterning using inductively coupled plasma reactive ion
etching (ICP-RIE) with C
4
F
8
/SF
6
(Oxford Plasmalab 100) to define the cross-shield acoustic metamaterials, as well
as the release holes for device suspension. (ii) 30
double-angle evaporation for the Manhattan-style JJ (30 nm/50 nm)
using a single layer photo resist (ZEP520A). The oxidation steps are performed at 130 mbar, for a duration of 84
0
and 102
0
for Chip-A and Chip-B, respectively. (iii) Al ground plane patterning by lifto
. (iv) Ar ion milling, bandage
deposition and lifto
. (v) Device release in anhydrous vapor-HF (SPTS uEtch).
Our fabrication process is fine-tuned to accurately realize our design and ensure the preservation of a large acoustic
bandgap. The key in our fabrication is to minimize any perturbations to the acoustic structure, in particular: a)
Preventing any metal deposition on the vertical sidewalls of the silicon acoustic structures, and b) eliminating the
formation of parasitic junctions on the silicon cross-shield structures during the double-angle evaporation process.
1. Preventing metal deposition on the Si sidewalls
To avoid any undesired metal deposition on the vertical sidewalls of the silicon acoustic structures, we make sure 1.
the JJ lead is small enough to completely locate on top of the Si structure, and 2. there is good alignment between
the JJ and acoustic metamaterial patterns. For the first part, we make the width of the JJ leads narrower than the
width of the Si tether, which is the narrowest part of the cross-shield acoustic metamaterial that the JJ lead runs
through, given by
w
lead
= 45 nm
<w
tether
= 72 nm, as illustrated in Fig.
S1b. For the second part, we employed
local markers during the e-beam lithography, which contributes to consistent alignment accuracy, resulting in small
misalignment of
.
10 nm between the acoustic metamaterial pattern and the JJ, as evidenced in Fig.
S1a. Even
considering a worst case scenario with a 10 nm misalignment, the metal deposition of the JJ leads remains confined
to the top silicon surface, avoiding any undesired metal deposition onto the sides of the silicon acoustic structures.
2. Eliminating the formation of parasitic junctions
To eliminate the formation of parasitic junctions on the cross-shield metamaterials, we implement a geometric
strategy in the Manhattan-style JJ, as illutrated in Fig.
S6. During the angled evaporation process, an unmetalized
‘shadow’ area of size
l
shadow
=
d
tan
is created, where
d
is the thickness of photoresist and
the evaporation angle
from normal incidence. In our process, the shadow size is approximately
l
shadow
150 nm. In the design of the
Manhattan-style JJ, we enforce the condition that
w
lead
l
shadow
,where
w
lead
= 45 nm is the width of our JJ
leads. This condition guarantees that only one layer of Al is metalized on the Si structure, avoiding the formation
of parasitic junctions in the Manhattan-style JJ configuration. Specifically, the fabrication of Manhattan-style JJ
involves two separate Al evaporations, whose in-plane evaporation angles are perpendicular to each other. When the
in-plane direction of the evaporation aligns with the length direction of the JJ lead (into the plane in Fig.
S6), Al is
deposited to the Si structure. When the in-plane direction of the evaporation is perpendicular to the length direction
of the JJ lead, as shown in Fig.
S6b, Al only deposits onto the resist, and is subsequently lifted o
. Consequently, no
parasitic junctions are formed during the double-angle evaporation process. This is important to the preservation of
a large acoustic bandgap, as excessive Al deposition on the cross-shield acoustic metamaterial can quickly diminish
the bandgap, as discussed previously.
We note that parasitic junctions still exist in our fabrication process, where we broaden up the JJ leads for the
bandage. However, these parasitic junctions are strategically positioned outside the acoustic metamaterial region. As
a result, the increased thickness and weight of Al in these areas do not a
ect the acoustic bandgap. These parasitic
junctions are shorted by a bandage at the final stage of the fabrication (step v). We remark that for a single JJ qubit
(fixed frequency qubit), it is indeed possible to completely eliminate the parasitic junctions, through purely geometric
considerations. This is important in future work when we embed the whole merged-element transmon qubit
35,36
into
the acoustic structure and remove the shunt capacitor from our design.
3. Single resist layer JJ process
In conjunction with the criterion
w
lead
l
shadow
, we make an additional e
ort to minimize the size of the shadow
area
l
shadow
. The shadow area is inherently part of the rectangular platform on which the JJ resides, as depicted
in Fig.
S1. As such, a larger shadow region requires a larger rectangular platform. A larger platform in turn
supports more localized acoustic phonon modes inside the acoustic bandgap, which might potentially influence TLS
performance.
In order to suppress the number of these localized acoustic phonon modes, and avoid their potential couplings to
TLS, we have developed a single resist layer JJ fabrication process, similar to that outlined in ref.
72
, using ZEP520A
instead of the more conventional PMMA-MMA double layer resist process. This process minimizes the size of the
shadow area to
l
shadow
150 nm, resulting in free spectral range of the localized acoustic phonon modes within the
acoustic bandgap on the order of
100 MHz according to COMSOL simulations shown in Sec.
IA2.
It is important to note, however, that we have noticed the formation of free-standing vertical Al sidewalls post the
lifto
process in some of our devices, as shown in Fig.
S7. This phenomenon is anticipated when single layer resist is
used in angled evaporation, without an undercut. In this scenario, the metal deposited on the sidewall of the resist,
as indicated in Fig.
S6a, might not be entirely removed through the lifto
process. However, despite the presence of
these residual vertical Al sidewalls, we have not observed impacts on the performance of the transmon qubits.
4. JJ oxidation
The oxidation condition for the merged-element-style JJs are determined based on measurements of previous cal-
ibration chips, as shown in Fig.
S8. An extended oxidation duration at high static oxygen pressure grows a thicker
AlO
x
barrier layer of the JJ, which decreases the Josephson energy
E
J
and the transmon frequency. Calibration data
show an empirical linear dependence between the transmon frequency and the oxidation duration, to which we fit and
inform our fabrication of Chip-A and Chip-B. The one outlier is a chip that aged for approximately one month prior
to measurement, which explains the atypical behavior. The frequencies of fabricated Q
1
’s of Chip-A and Chip-B,
represented by pentagrams agree well with the empirical linear fit.
C. Measurement setup
Fig.
S9 shows a schematic of the measurement setup inside the cryogen-free dilution refrigerator (Bluefors LD400),
which includes standard shielding and filtering for superconducting transmon qubit experiments
73,74
. The refrigerator
consists of multiple temperature stages, which in descending order of temperature are 300 K, 50 K, 4 K, still, cold
plate (CP), and mixing chamber (MXC) flanges. The experimental sample is mounted to the MXC plate. Under
standard operating conditions, the MXC plate achieves a base temperature of 7 mK, providing the low temperature
environment required for the experiments.
The frequency control of each transmon qubit is achieved by a bias current that generates a magnetic field threading
through the SQUID loop of the transmon qubit. The bias current consists of two parts: the static DC bias (slow Z)
and the dynamic RF pulse (fast Z). The static DC bias is generated by a stable DC voltage source (QDevil QDAC)
passing through a 2
.
8 k
resistor at room temperature. The DC current is filtered by a RC low-pass filter (QDevil
QFilter) at 65 kHz placed at the 4 K stage. The DC bias provides a broad tuning range and high tuning precision for
the transmon qubit frequency. The static DC bias is combined with a dynamic RF pulse (fast Z) through a DC-coupled
bias tee (Mini-Circuits ZFBT-4R2GW+ with the capacitor shorted). The fast Z pulse is generated directly by an
arbitrary waveform generator (AWG, Keysight M3202A), introducing dynamic tuning capabilities for the transmon
qubit frequency. For the present experiment, we have not performed corrections for Z line distortions, as discussed
in ref.
75
. Consequently, a slight drift in the patterns of vacuum Rabi oscillations at short Z duration is observed, as
seen in Fig. 2 of the main text and Fig.
S19.
The resonant control of transmon qubit is achieved by the XY line, which couples capacitively to the transmon
qubit through a coupling capacitance of approximately
80 aF. We use a total of 50 dB attenuation (XMA cryogenic
attenuators) in the fridge XY lines, to accommodate the need of higher microwave driving power for the direct control
of TLS (to be discussed in Sec.
III1 ). The microwave signal is generated at room temperature. A pair of intermediate
frequency (IF) signals from the AWG (Keysight M3202A), in conjugation with a local oscillator (LO) signal from a
microwave signal generator (Rhode&Schwarz SMB100A), undergoes IQ mixing (Marki Microwave MMIQ-0218L) and
generates a single sideband microwave signal that achieves the XY control of the transmon qubit as well as TLS.
For the readout (RO) of the transmon qubits, a microwave RO input signal (generated and filtered similarly to the
XY signal) is passed down to the feedline of the sample. The RO output signal from the feedline is first amplified
by a JTWPA (Josephson traveling wave parametric amplifier) which is sandwiched by two sets of circulators (Low
Noise Factory LNF-CICI4
12), a HEMT (high electron mobility transistor, Low Noise Factory LNF-LNC4
16B or
LNF-LNC0.3
14A) amplifer, a low-noise room temperature amplifier (MITEQ LNA-30-0400800-07-10P), a high pass-
filter (Mini-Circuits VHF-4600+), a tunable attenuator (Vaunix Lab Brick LDA-133), and another MITEQ low-noise
amplifier (MITEQ LNA-30-0400800-07-10P). The RO output signal is then downconverted at room temperature by
an IQ mixer and the same LO used to generate the RO input signal. The resulting in-phase (I) and quadrature (Q)
signals are filtered (Mini-Circuits VLF-160+), amplified (Mini-Circuits ZFL-500HLNB+), and digitized (Keysight
M3102A) for qubit readout. In addition to the aforementioned filtering, low-pass filters (Mini-Circuits VLFX-400+,
K&L Microwave 6L250-12000/T26000) and infrared Eccosorb filters (custom made) are added at the MXC plate
where appropriate. All the microwave instruments are synchronized to an external 10 MHz reference clock from a
Rubidium frequency standard (Stanford Research Systems FS725). The AWG and digitizer are both triggered by a
delay generator (Stanford Research Systems DG645).
D. Phonon density of states
In this section we describe the process of finding the phonon density of states (DOS) based on the COMSOL
simulated band structures. To achieve this, we expand upon the simulations presented in Fig.
S3, which focus on
special paths connecting points of high degrees of symmetry. These simulations are e
cient in finding the bandgap
frequencies. However, they do not represent the entire band structures, and consequently, the phonon DOS. To extract
the phonon DOS, we leverage symmetries in our structure and uniformly sample one quarter of the first Brilloin zone
in the two-dimensional
k
-space, given by
k
x
,k
y
2
[0
,
/a
], using
N
steps for the
k
x
,k
y
values. Here,
a
denotes the
length of the square unit cell. We then count the total number of
k
states in the first Brilloin zone, accounting for
symmetries. The results are then grouped into frequency bins of 80 MHz interval based on the frequencies of the
eigenstates, and normalized by a factor of 1
/
(2
N
2)
2
to arrive at the phonon DOS. The resulting phonon DOS for
all three unit cell types is shown in Fig.
S10, and a zoom-in view is displayed in Fig. 3e in the main text.
E. Thermal cycling of the device
In our experiment, we employ a method known as thermal cycling to generate new distributions of TLS on the same
devices. This method has been shown to be e
ective in ref.
39
when the MXC plate temperature rises above
20 K.
In this study, we perform thermal cycling of the fridge to room temperature, then back down to the base temperature,
to ensure the absence of correlation between the two sets of TLS characterized during di
erent cool-down cycles.
II. DISCUSSIONS
In this section we provide analysis and discussions regarding the phonon engineering of TLS. We start by presenting
a comprehensive list of parameters for all 56 TLS characterized and analyzing their distributions. Next, we derive the
frequencies of the average acoustic bandgap, shared across all devices, driven by the TLS data. This is followed by an
examination of individual devices, where we identify the distinctive acoustic bandgaps corresponding to each of the
seven fabricated devices. The results extend and complement the data presented in the main text. Additionally, this
analysis unveils disorders in the frequencies of individual device bandgaps, providing an explanation for some of the
outlier data points mentioned in the main text. We then provide an explanation for the significant variations observed
in the TLS
T
1
relaxation times, based on the confined geometry of the device, thereby addressing the remaining outlier
data points mentioned in the main text. Following this, we present experiments and data that corroborates the
anharmonicity of TLS. Our experimental findings suggest that TLS is highly anharmonic. Intriguingly, our data also
implies a three-mode coupling involving TLS, the transmon qubit, and an additional TLS. Furthermore, we present
data and analysis on the temperature-dependent relaxation of both the transmon qubit and TLS. This motivates a
detailed discussion on possible relaxation channels for the TLS, o
ering a comprehensive view on the temperature-
dependent TLS relaxation. Finally, we showcase direct XY control for TLS, which has been used to characterize both
the energy relaxation
T
1
and dephasing
T
2
of TLS. The result raises intriguing questions regarding the interactions
between quasiparticles and TLS.
A. TLS parameters
A complete list of TLS frequencies, their respective coupling strengths
g
to the transmon qubit, and
T
1
relaxation
times, measured on Chip-A and Chip-B, is provided in Table
S3 and Table
S4, respectively. This dataset that
includes 56 distinct TLS has been acquired across seven transmon devices. The TLS characterized span frequencies
from 3
.
7421 GHz to 6
.
3935 GHz, and their
T
1
values range from 0
.
25
±
0
.
02
μ
s to 5400
±
800
μ
s (Fig.
S11). The TLS
frequencies and coupling strengths
g
are extracted through fitting the avoided crossings in the microwave spectroscopy
of transmon qubits to the transmon-TLS interaction model
H
=
!
q
2
ˆ
z
q
+
!
TLS
2
ˆ
z
TLS
+
ˆ
H
int
,
H
int
=
g
+
q
ˆ
TLS
q
ˆ
+
TLS
)
,
(S1)
where
!
denote their frequencies, ˆ
z
,
ˆ
±
are the Pauli operators.
We remark that certain TLS
T
1
’s measurements are conducted using a strong microwave pulse that directly drives
the TLS to its excited-state. These particular TLS
T
1
values are distinctly marked by
in both Table
S3 and Table
S4).
We note that the
T
1
values obtained by this method appear comparatively shorter than those measured using SWAP
with the transmon qubit. Moreover, TLS relaxation curves measured by this method can sometimes deviate from a
simple exponential decay. When such deviations are evident, we report the relaxation values derived from fitting to
a double exponential model (Eq.
S36). We attribute both phenomenon to interactions with the quasiparticles (QP)
induced by the strong microwave pulse
76,77
, which will be discussed later in Sec.
III3 .
During the TLS
T
1
measurements, the transmon qubit also acts as an energy dissipation channel for the TLS,
inducing a Purcell decay on the TLS
T
1
lifetime. The Purcell-limit follows (
g
2
/
2
)
·
1
,q
,where
g
is the TLS-
transmon coupling strength,
their frequency detuning during the
T
1
relaxation, and
1
,q
=1
/T
1
,q
the relaxation
rate of the transmon qubit. When measuring TLS
T
1
relaxation, the transmon qubit is usually tuned to its upper
sweet spot, away from the TLS, to maximize the Purcell-limit. There are a few cases when the transmon at the
upper sweet spot might still impose a Purcell-limit. One such case is for TLS18–23 (Chip A Q
3
during CD1), where
qubit upper sweet spot (6
.
11 GHz) hybridizes with TLS18 (6
.
0877 GHz) with a coupling strength
g
= 30
.
2 MHz. To
increase the Purcell-limit, we tuned the transmon to 5
.
799 GHz during TLS
T
1
measurements. We then used a fast
flux to tune the transmon back to the upper sweet spot for readout. Another case is for TLS35 (Chip B Q
1
during
CD1), where we DC flux tuned the transmon qubit to 5
.
454 GHz.
In Fig.
S12 we show the ratio between the estimated Purcell-limit and the measured
T
1
lifetime for each TLS,
represented by blue crosses. TLS35 exhibits
T
1
= 261
±
21
μ
s, close to the Purcell limit from the transmon qubit (Q
1
of Chip-B). In light of this proximity to the Purcell limit, and the likelihood that the measured TLS35
T
1
value falls
short of its intrinsic
T
1
, TLS35 is excluded from all the median and mean
T
1
statistics. Among the remaining 55 TLS
characterized, their Purcell limits from the transmon are considerably higher than their measured relaxation times,
by at least a factor of 3
, indicated by the red dashed line as a guide for the eye. As such, the
T
1
’s of these 55 TLS
are likely not Purcell-limited, and they are all used in the calculation of the median and mean values reported in this
study.
B. TLS distributions
In this section, we analyze three distributions of the TLS parameters: 1. the distribution of TLS
T
1
, 2. the
distribution of their coupling strengths
g
to the transmon qubit, and 3. the distribution of TLS
T
1
against
g
.
1. TLS
T
1
distribution
To begin, we look at the distribution of TLS
T
1
from all 55 TLS, in supplementary to Fig. 3c in the main text.
In Fig.
S13, we present the cumulative distribution of TLS
T
1
values for family A (blue squares) and family B
(red triangles), respectively. To characterize this distribution, we employ three commonly used models: the normal
distribution, the exponential distribution, and the log-normal distribution. These models are given by their cumulative
distribution functions (CDF),
CDF
norm
(
x
)=
1
2
[1 + erf(
x
μ
p
2
)]
,
CDF
exp
(
x
)=1
exp(
x
)
,
CDF
logn
(
x
)=
1
2
[1 + erf(
ln
x
μ
p
2
)]
.
(S2)
Based on the fittings using the three models in Fig.
S13, represented by the solid lines, we identify the log-normal
distribution as the best representation for our data. The resulting parameters yield distinct median
T
1
values of
4
.
1
±
0
.
2
μ
s for family A and 414
±
17
μ
s for family B. These fitted values are consistent with those outlined in
Table
S5 based on the frequencies of the acoustic bandgap, which will be discussed later.
2. TLS coupling strength distribution
Next, we show the distribution of TLS coupling strengths
g
to the transmon qubit in Fig.
S14. According to the
standard tunneling model (STM), this distribution is a reflection of the electric dipoles of TLS, which has a density
of
37
d
2
N/dEdg
=
A
p
1
g
2
/g
2
max
/g,
(S3)
where
E, A,
are the energy of TLS, the area of the JJ, and the TLS density, respectively. The measured TLS
distribution over coupling strength
g
overall aligns well with STM predictions. To determine the TLS density
,we
normalize the fitted parameter by the total size of JJ in the transmon qubit, which is approximately 1
.
66
μ
m
2
, and
the collective frequency span of the seven transmons in our search for these TLS, which amounts to 22 GHz. This
results in a TLS density of
=0
.
6 GHz
1
μ
m
2
, in agreement with literature
24,36,37
.
3. TLS
T
1
distribution over coupling strength
In addition, the STM ascribes TLS relaxation to spontaneous phonon emission
1,2
via the interaction between TLS’
elastic dipole and the acoustic environment. In this context, the TLS’ elastic dipole is proportional to its electric
dipole, governed by
/
0
/E
/
~
d
. Here
0
is the tunneling energy,
E
the eigenenergy, and
~
d
the electric dipole of
the TLS. As discussed above, the coupling strength
g
reflects the electric dipole of TLS. Consequently, a power-law
dependence of 1
/T
1
/
g
is expected, and has indeed been observed for TLS located inside the Josephson Junctions
of a phase qubit
39
.
We remark that this power-law dependence is not unique to the spontaneous phonon emission process. Based on
the STM, any relaxation process mediated through either the electric or elastic dipole of TLS would yield a power-law
dependence. Therefore, it could also apply to our device, where the spontaneous phonon emission has been suppressed.
In Fig.
S15 we show the TLS
T
1
distribution against their coupling strengths
g
to the transmon qubit. Here, the blue
and red filled circles respectively represent TLS located outside and inside the average acoustic bandgap. At first
glance, our data does not readily exhibit a clear power-law dependence for TLS either within or outside the average
acoustic bandgap, partly due to the wide spread of the
T
1
data points that obscures any underlying correlations.
To address this, we follow the method in ref.
39
, and group the data into bins based on their
g
values. We then
compute the mean and standard deviation in each bin, which are represented by the open markers and their errorbars,
respectively, with corresponding colors in Fig.
S15. These data points reveal a trend of negative correlation between
the mean
T
1
values and the coupling strength
g
, in alignment with expectations from the STM. However, this trend
does not convincingly conform to a power-law dependence. We attribute this deviation to the relatively limited size
of our available dataset. Additionally, the deviation could arise from the extreme ways in which the acoustic bandgap
metamaterial structures the acoustic environment. This influence can even extend to frequencies outside the acoustic
bandgap. In such cases, the substantial alteration in the acoustic DOS, rather than the susceptibility to the acoustic
environment, may prevail in determining the TLS
T
1
distribution.
C. Identification of the average acoustic bandgap
As described in the main text, we select a
T
1
cuto
between 35
μ
s and 85
μ
s to categorize all TLS into two groups:
family A, characterized by shorter TLS
T
1
, and family B, characterized by longer TLS
T
1
. Remarkably, we observe a
strong correlation between this categorization based solely on
T
1
values and the frequency distribution of TLS within
the two families. This correlation motivates us to identify an average acoustic bandgap across all seven transmon
devices, using the following cost function,
C
(
f
1
,f
2
) = log[1
F
A
(
f
1
,f
2
)
F
B
(
f
1
,f
2
)]
,
(S4)
where the frequency band is specified between
f
1
and
f
2
.
F
A
(
f
1
,f
2
) denotes the fraction of TLS in family A whose
frequencies lie outside this defined frequency band, while
F
B
(
f
1
,f
2
) represents the fraction of TLS in family B that
fall within this frequency band.
The landscape of the cost function
C
(
f
1
,f
2
) is provided in Fig.
S16 as a function of the lower bandedge frequency
f
1
and upper bandedge frequency
f
2
. The minimum in the landscape yields
C
min
=
1
.
98, which identifies the average
acoustic bandgap present across all seven transmon devices. This average bandgap is characterized by
f
1
,
avg
.
bg
2
[4
.
510
,
4
.
547] GHz
,
f
2
,
avg
.
bg
2
[5
.
690
,
5
.
735] GHz
.
(S5)
This average bandgap, in turn, yields a median TLS
T
1
of
M
out
,
2D
(
T
1
)=4
.
4
μ
s outside the bandgap and
M
in
,
2D
(
T
1
)=
505
μ
s inside the bandgap. Our preference for using median over mean is justified by the large skewness for the TLS
T
1
distributions.
It’s important to note that the average bandgap, shared across di
erent fabricated devices and chips, represents a
lower-bound estimate, due to fabrication disorder on individual devices, which will be discussed shortly in Sec.
IID .
Despite this, the average acoustic bandgap still boasts a width exceeding 1 GHz. Furthermore, it exhibits a remarkable
similarity to the COMSOL simulated bandgap, di
ering by merely
.
100 MHz, as shown in Table
S5. This high
degree of agreement underscores the reproducibility and robustness of the overall fabrication process for the acoustic
bandgap metamaterial.
Lastly, we emphasize that the determination of the frequencies of the average acoustic bandgap does not depend
on any a priori knowledge of the existence of an acoustic forbidden band. Instead, these bandgap frequencies arise
naturally from the TLS data itself.
D. Acoustically-shielded TLS on individual devices
Using the full set of TLS data collected from all seven fabricated transmon devices, we present additional details
complementing the information in Fig. 3 from the main text, and demonstrate the robust TLS
T
1
enhancement on
all devices from the acoustic bandgap. In Fig.
S17 and Fig.
S18, we present the TLS
T
1
relaxation data measured on
individual transmon devices, for Chip-A and Chip-B, respectively. These plots reveal the existence of two families of
TLS, based on their frequencies and
T
1
times, for each device. We use the same method for identifying the average
acoustic bandgap to analyze the bandgaps of these individual devices. For each transmon device, we search for the
frequency range of the acoustic bandgap [
f
1
,f
2
] that minimizes the cost function
C
(
f
1
,f
2
). This analysis yields the
frequencies of either one or both of the bandedges, depending on the frequency ranges and total number of TLS
characterized on the particular transmon device. The determined frequencies of the bandedges
f
1
and
f
2
are depicted
using gray shading in Fig.
S17 and Fig.
S18. For reference, we also plot the average bandgap frequencies
f
1
,
avg
.
bg
and
f
2
,
avg
.
bg
determined above in Sec.
IIC , using pink shading. The overlap between the individual device bandgaps and
the average bandgap highlights the robustness of the fabrication process of acoustic bandgap metamaterial. These
experimentally identified frequencies of the bandedges are listed in Table
S5, along with the frequency range of the
bandgap given by COMSOL simulations. The table also includes the median and mean TLS
T
1
values both inside
and outside the corresponding bandgaps.
Upon comparing the experimentally identified bandgaps across all seven devices, we observe disorder in the bandgap
frequencies, which is most pronounced in Chip-A Q
2
, as illustrated in Fig.
S17b. In this case, we identify a bandgap
that is up-shifted in frequency, which likely stems from fabrication disorder in the acoustic metamaterials. We remark
that the upward shift in the bandgap frequencies for Chip-A Q
2
results in the TLS in family A, circled out in black in
Fig. S17b, appearing as an outlier when using the average acoustic bandgap for analysis. However, when we apply the
acoustic bandgap specific to this individual device, the TLS falls outside the bandgap, aligning with our expectation
for family B. Similarly, we claim that in Fig.
S17a, the TLS circled out on the left side is also misclassified as an