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