# Flip-chip-based microwave spectroscopy of Andreev bound states in a planar
Josephson junction

M. Hinderling  IBM Research Europe - Zurich, Säumerstrasse 4, 8803 Rüschlikon,
Switzerland  D. Sabonis  IBM Research Europe - Zurich, Säumerstrasse 4, 8803
Rüschlikon, Switzerland  S. Paredes  IBM Research Europe - Zurich,
Säumerstrasse 4, 8803 Rüschlikon, Switzerland  D. Z. Haxell  IBM Research
Europe - Zurich, Säumerstrasse 4, 8803 Rüschlikon, Switzerland  M. Coraiola
IBM Research Europe - Zurich, Säumerstrasse 4, 8803 Rüschlikon, Switzerland
S. C. ten Kate  IBM Research Europe - Zurich, Säumerstrasse 4, 8803
Rüschlikon, Switzerland  E. Cheah  Solid State Physics Laboratory, ETH Zurich,
Otto-Stern-Weg 1, 8093 Zürich, Switzerland  F. Krizek  IBM Research Europe -
Zurich, Säumerstrasse 4, 8803 Rüschlikon, Switzerland  Solid State Physics
Laboratory, ETH Zurich, Otto-Stern-Weg 1, 8093 Zürich, Switzerland  R. Schott
Solid State Physics Laboratory, ETH Zurich, Otto-Stern-Weg 1, 8093 Zürich,
Switzerland  W. Wegscheider  Solid State Physics Laboratory, ETH Zurich, Otto-
Stern-Weg 1, 8093 Zürich, Switzerland  F. Nichele<EMAIL_ADDRESS>IBM
Research Europe - Zurich, Säumerstrasse 4, 8803 Rüschlikon, Switzerland

###### Abstract

We demonstrate a flip-chip-based approach to microwave measurements of Andreev
bound states in a gate-tunable planar Josephson junction using inductively-
coupled superconducting low-loss resonators. By means of electrostatic gating,
we present control of both the density and transmission of Andreev bound
states. Phase biasing of the device shifted the resonator frequency,
consistent with the modulation of supercurrent in the junction. Two-tone
spectroscopy measurements revealed an isolated Andreev bound state consistent
with an average induced superconducting gap of 184 $\mu$eV and a gate-tunable
transmission approaching $0.98$. Our results represent the feasibility of
using the flip-chip technique to address and study Andreev bound states in
planar Josephson junctions, and they give a promising path towards microwave
applications with superconductor-semiconductor two-dimensional materials.

##  I INTRODUCTION

A class of gate-tunable superconducting-semiconducting materials have recently
been investigated as potential building blocks for hybrid amplifiers [1],
tunable quantum buses [2], non-reciprocal devices [3] and facilitation of
integration with cryogenic control systems [4]. Moreover, hybrid
superconducting devices based on Andreev bound states (ABSs) [5] hold promise
for field-compatible, gate- and flux-tunable qubits [6, 7, 8, 9, 10, 11, 12,
13, 14], potentially enabling long coherence times [15, 16]. Such ABS-based
qubits [17, 18] have been realized in hybrid Josephson junctions (JJs) using
nanowires [7, 8, 19], graphene [20] and two-dimensional hybrid systems [21].

Two-dimensional superconductor-semiconductor heterostructures grown on III-V
substrates, such as InP [22], offer several advantages in terms of
reproducibility and scalability, but pose difficulties in realizing high-
quality microwave components, e.g. coplanar waveguide (CPW) resonators. This
is due to, for example, the large dielectric losses in the host structure [23,
24, 25, 26] as well as to the piezoelectric properties of III-V materials
[27]. Furthermore, techniques used in the fabrication of microwave components
degrade semiconductor quality. Attempts to overcome these limitations included
removal of the buffer layers in the area surrounding the device [21], or by by
selective area growth of superconductor-semiconductor hybrid structures on
silicon [28]. Coupling quantum devices on III-V substrates to high-quality
microwave electronics via a flip-chip approach [29, 30, 31, 32, 33, 34] has
not yet been demonstrated. Owing to the promise of scalability presented by
planar systems, investigation of this platform is of particular interest for
future hybrid quantum systems.

Here, we demonstrate microwave spectroscopy of gate- and flux-tunable ABSs in
a planar JJ defined in an epitaxial InAs/Al heterostructure hosting a two-
dimensional electron gas (2DEG) using a flip-chip approach. Planar JJs and
coplanar microwave resonators were defined on different chips and inductively
coupled via a vacuum gap. We performed single- and two-tone spectroscopy
measurements of the ABS spectrum and found that the flip-chip approach,
yielding low-loss resonators, enabled readout of the JJ and control of the
individual ABSs. Moreover, when the electrostatically-defined transport
channel in the junction hosted an isolated ABS, its transmission could be
tuned up to $0.98$ via electrostatic gating.

Figure 1: Overview of the flip-chip approach, device design and resonator
characterization. (a) Optical image of the measured system. The Resonator Chip
provided microwave components on a low-loss Si substrate. The Device Chip
hosted a radio-frequency superconducting quantum interference device (rf-
SQUID) on the down-facing surface of an InP chip, underneath the green box.
(b) Simplified schematic cross-section of the flip-chip layout. The Resonator
Chip provided coplanar waveguide (CPW) resonators capacitively coupled to a
common transmission line (blue), drive lines (turquoise) and DC control lines
(light gray). The two chips were galvanically coupled via indium bump bonds so
that the ground of the rf-SQUID and the gates were connected to the DC control
lines. (c) False-colored scanning electron micrograph of the grounded rf-SQUID
(red), which was inductively coupled to a $\lambda$/4 CPW resonator and to a
drive line using the flip-chip technique. (d) Gated planar Josephson junction
(JJ) defined in a two-dimensional InAs/Al heterostructure, which was embedded
in the rf-SQUID in (c), marked with the orange box. The Al (red) was
selectively removed to form a $L\approx\mathrm{110}$ nm long and
$W\approx\mathrm{940}$ nm wide JJ. A split-gate (light gray) was evaporated on
top of the JJ for controllable depletion of the weak link in the exposed InAs
two-dimensional electron gas (dark gray). (e) Schematic representation of the
measured system, where the colors are defined in (b). (f) Magnitude (blue) and
phase (orange) response of the resonator when the JJ was insulating
($V_{\mathrm{G}}=-2.3$ V).

##  II DEVICE DESIGN, FLIP-CHIP TECHNIQUE AND RESONATOR CHARACTERIZATION

The investigated system was composed of two chips. The Device Chip was
fabricated from an InAs/Al heterostructure grown on an InP substrate [22]. The
Resonator Chip was realized with a high-resistivity Si substrate. After the
two chips were independently fabricated, they were coupled via flip-chip
bonding. An optical image of the measured system, where the Device Chip is
shown face-down on top of the Resonator Chip, is shown in Fig. 1(a) and a
simplified cross-section of the flip-chip layout is shown in Fig. 1(b).

Scanning electron micrographs of the Device Chip are shown in Figs. 1(c) and
(d). The active part of the Device Chip consists of a planar InAs JJ with
epitaxial Al leads (red), embedded in a radio-frequency superconducting
quantum interference device (rf-SQUID). The JJ had a length
$L\approx\mathrm{110}~{}\mathrm{nm}$ and a width
$W\approx\mathrm{940}~{}\mathrm{nm}$. Both the JJ and the rf-SQUID were
obtained by wet etching of the heterostructure and selective removal of the
Al. Two Ti/Au gates (light gray) with a minimal separation of
$90~{}\mathrm{nm}$ were evaporated on top of an insulating $\mathrm{HfO_{2}}$
layer, and used to electrostatically define a tunable constriction in the
InAs. Furthermore, the gates were arranged to fully cover InAs areas between
the mesa edges and the junction, suppressing parallel conductance paths. The
phase difference $\varphi$ across the JJ was tuned by applying a magnetic flux
$\it{\Phi}$ through the superconducting loop.

Figure 2: Single-tone spectroscopy measurements. (a) Magnitude
$\absolutevalue{S_{21}}$ of the resonator transmission as a function of offset
frequency $f_{\mathrm{r}}-f_{\mathrm{0}}$ and gate voltage $V_{\mathrm{G}}$ at
$\it{\Phi}=\mathrm{0}$. The strongest modulation of the resonance frequency
$f_{\mathrm{res.}}\equiv f_{\mathrm{r}}(\absolutevalue{S_{21}}=\mathrm{0})$
was for gate voltages between $-1.2$ V and $-1.4$ V, while for more positive
and more negative gate voltages the resonance frequency was largely
unaffected. The inset shows the internal quality factor $Q_{\mathrm{i}}$ of
the resonator as a function of $V_{\mathrm{G}}$. (b-e) Flux-dependence of
$\absolutevalue{S_{21}}$ at selected gate voltage values, marked with colored
bars in (a). The amplitude of the modulation decreased with more negative gate
voltages, until the Josephson junction was pinched-off at
$V_{\mathrm{G}}=-2.3$ V in panel (e).

The Resonator Chip was fabricated by sputtering of $200~{}\mathrm{nm}$ Nb and
dry etching. Key elements on the Resonator Chip are represented in Fig. 1(e)
and included three $\lambda/4$ CPW resonators capacitively coupled to a common
transmission line (blue), drive lines (turquoise), a ground plane (purple) and
DC control lines (light gray). Device Chip and Resonator Chip were connected
via several indium bumps, evaporated on the Resonator Chip. The bumps provided
connection to the DC control lines, a ground to the rf-SQUID via under-bump
metallization (UBM), and ensured mechanical stability of the structure.
Particular care was taken to align rf-SQUIDs with the shorted end of the
$\lambda/4$ resonators and with the drive lines to ensure the couplings with
inductances $M_{\mathrm{r}}$ and $M_{\mathrm{d}}$, respectively. The green
rectangle in Fig. 1(b) indicates the position of the rf-SQUID measured in this
work. Our flip-chip procedure allowed an alignment accuracy below
$1~{}\mathrm{\mu m}$. The separation between the two chips was
$5~{}\mathrm{\mu m}$, and was controlled by the size of the indium bumps and
the force used during the flip-chip bonding process. To minimize dielectric
losses, the Device Chip was considerably smaller than the Resonator Chip and
was placed to minimize its overlap with the resonators.

The resonators were measured with a vector network analyzer by sending a
signal $\it{P}_{\mathrm{in}}$ into the left port of the transmission line and
reading out the response $\it{P}_{\mathrm{out}}$ at the right port. Here, the
leftmost drive line in Fig. 1(b) was used to drive ABS pair transitions with a
continuous microwave signal $\it{P}_{\mathrm{d}}$. The parasitic coupling
between resonator and drive line is also reported [see Fig. S.2 in the
Supplemental Material (SM)]. The two gates were controlled by voltages $V_{1}$
and $V_{2}$, which could be set individually (see Fig. S.5 in SM). However, a
gate voltage $V_{\mathrm{G}}\equiv V_{1}=V_{2}$ was applied in the remainder
of this article. The magnetic flux in the rf-SQUID was produced via a home-
made superconducting coil mounted on top of the printed circuit board that
hosted the device.

##  III OPERATION AND CONTROLLABILITY

Figure 1(f) displays the magnitude $\absolutevalue{S_{21}}$ (blue) and phase
$\angle S_{21}$ (orange) of the readout resonator while the JJ was pinched-off
($V_{\mathrm{G}}=-2.3$ V). In this configuration, a bare resonator frequency
$f_{\mathrm{0}}\approx 5.9162~{}\mathrm{GHz}$ was detected. The asymmetry of
the resonance dip in $\absolutevalue{S_{21}}$ is explained by an impedance
mismatch in the circuit [35]. The internal quality factor was extracted using
the circle fit technique with diameter correction [36], yielding
$Q_{\mathrm{i}}\approx 5\cdot 10^{4}$. Similar values for the internal quality
factor of all three resonators on the resonator chip were found (see Fig. S.3
in SM), which highlights the prospects of the flip-chip approach for future
microwave experiments with two-dimensional heterostructures.

Figure 2 shows the magnitude of the complex scattering parameter
$\absolutevalue{S_{21}}$ in the vicinity of $f_{\mathrm{0}}$ as a function of
gate voltage $V_{\mathrm{G}}$ and magnetic flux $\it{\Phi}$ treading the rf-
SQUID. The effect on $\absolutevalue{S_{21}}$ is mediated by the variable
Josephson inductance of the JJ [37]. Figure 2(a) displays
$\absolutevalue{S_{21}}$ as a function of $V_{\mathrm{G}}$ measured at
$\it{\Phi}=\mathrm{0}$. A clear deviation of the resonance frequency
$f_{\mathrm{res.}}\equiv f_{\mathrm{r}}(\absolutevalue{S_{21}}=\mathrm{0})$
from $f_{\mathrm{0}}$ appeared for less negative values of $V_{\mathrm{G}}$.
The inset shows the internal quality factor $Q_{\mathrm{i}}$ of the resonator,
extracted from the data in Fig. 2(a). The flux dependence of
$\absolutevalue{S_{21}}$ for four selected values of $V_{\mathrm{G}}$,
indicated by the colored bars in Fig. 2(a), are shown in Fig. 2(b-e). Periodic
modulations of $f_{\mathrm{res.}}$ as a function of $\it{\Phi}$ were observed,
with maxima and minima of $f_{\mathrm{res.}}$ assigned to integer and half-
integer multiples of the flux quantum $\it{\Phi}_{\mathrm{0}}$, respectively.
While $f_{\mathrm{res.}}$ was modulated continuously over $\it{\Phi}$ at
$V_{\mathrm{G}}=-0.8$ V [Fig. 2(b)], amplitude variations were concentrated
around half-integers flux quanta as $V_{\mathrm{G}}$ lowered [Fig. 2(d)].

Measurements presented in Fig. 2 are consistent with gradual depletion of
lateral ABSs in the JJ [38]. For $V_{\mathrm{G}}>-1.4~{}\mathrm{V}$, a high
density of ABSs was present within the full width of the JJ. In such a
situation, a quasi-continuous distribution of ABSs populates the JJ [39],
including several ABSs with low and high transmission probabilities [40].
Therefore, the readout signal with frequency $f_{\mathrm{r}}$ could excite
transitions between ABSs at $\it{\Phi}=\mathrm{0}$, resulting in a shift of
$f_{\mathrm{res.}}$ and a decrease in $Q_{\mathrm{i}}$. When setting
$V_{\mathrm{G}}$ more negative, a narrow constriction is defined in the InAs,
allowing the presence of only a few ABSs orthogonal to the leads. In this
scenario, transitions between ABSs enabled by microwave driving are expected
only in the vicinity of half-integer multiples of $\it{\Phi}_{\mathrm{0}}$
[41, 42]. A more quantitative description of the situation and analysis of the
modulation of $f_{\mathrm{res.}}$, which follows Ref. 37, are presented in SM
and further support the interpretation above.

##  IV ANDREEV BOUND STATE SPECTROSCOPY

A direct visualization of the ABSs was obtained via two-tone spectroscopy
measurements as a function of $\it{\Phi}$, as presented in Fig. 3(a). For
these measurements, the JJ was tuned to the few-modes regime via
$V_{\mathrm{G}}$ and a continuous drive tone of variable frequency
$f_{\mathrm{d}}$ was applied to the drive line while monitoring magnitude and
phase of $S_{21}$ at $f_{\mathrm{r}}$ 111The frequency $f_{\mathrm{r}}$ had to
be determined for each value of $\it{\Phi}$. In this situation, the drive tone
excited transitions between ABS pairs through the superconducting gap, as
schematically shown in the inset of Fig. 3(a). For a short, single-channel
weak link the energy of the ABSs is given by:

$E_{\mathrm{A}}=\pm\Delta\sqrt{1-\tau\sin^{2}(\mathrm{\pi}\it{\Phi}/\it{\Phi}_{\mathrm{0}})},$
(1)

where $E_{\mathrm{A}}$ is the ABS energy, $\Delta$ is the induced
superconducting gap and $\tau$ is the transmission of the ABS [41, 42]. The
energy needed for an ABS pair transition is $\mathrm{2}E_{\mathrm{A}}$. Red
shading indicates the experimentally accessible energy range, which intersects
the ABS spectrum close to $\it{\Phi}=\it{\Phi}_{\mathrm{0}}/\mathrm{2}$. As
expected, ABSs pair transitions were observed only in the vicinity of
$\it{\Phi}=\it{\Phi}_{\mathrm{0}}/\mathrm{2}$. Accordingly, this two-tone
spectroscopy measurement supports the interpretation that the planar JJ was in
the ballistic short junction regime.

Figure 3: Two-tone spectroscopy of Andreev bound states (ABSs) as a function
of magnetic flux. (a) Magnitude $\absolutevalue{S_{21}}$ of the resonator
transmission in the presence of a drive tone
$\it{P}_{\mathrm{d}}=\mathrm{-33}$ dBm with frequency $f_{\mathrm{d}}$ at
$V_{\mathrm{G}}=-1.5$ V as a function of applied flux $\it{\Phi}$. An ABS pair
transition was observed around $\it{\Phi}_{\mathrm{0}}/\mathrm{2}$, consistent
with the short junction limit. The inset shows a schematic representation of
isolated ABSs in the short junction limit. The red arrow indicates an ABS pair
transition and the red shaded area visualizes the accessible energy range with
our experimental set-up, which was limited to 26 GHz. (b) Andreev bound state
spectrum at $V_{\mathrm{G}}=-1.65$ V using $\it{P}_{\mathrm{d}}=\mathrm{-20}$
dBm. Aside from the ABS pair transition, weaker transitions were observed as
indicated by arrows. The white dashed line indicates the second harmonic of
the resonator. (c) A fit of the expression in (a) to the data extracted from
the ABS pair transition in (b) yielded the transmission $\tau=0.976\pm 0.001$
and the induced superconducting gap $\Delta=186\pm 2$ $\mu$eV of the observed
ABS.

At $V_{\mathrm{G}}=-1.65$ V, an ABS pair transition with a minimum frequency
of 14 GHz was probed [Fig. 3(b)]. In addition to the pair transition, weaker
features at lower and higher $f_{\mathrm{d}}$ than the main ABS transition
were observed, as indicated by arrows. These replicas, which are spaced by
$f_{\mathrm{res.}}$ from the ABS transition, are attributed to second-order
transitions involving absorption or emission of a resonator photon, satisfying
the energy relation $2E_{\mathrm{A}}\pm hf_{\mathrm{res.}}=hf_{\mathrm{d}}$,
where $\it{h}$ is Planck’s constant [32]. Several flux-independent lines
visible in the two-tone spectroscopy measurements are attributed to resonances
in the drive line and connected measurement circuit, resulting in a frequency-
dependent driving strength [44]. In particular, the strongest resonance,
marked with a white dashed line, corresponds to the second harmonic of the
resonator. All of these lines became more pronounced at higher driving powers
(see Fig. S.11 in SM).

The measured ABS pair transition was extracted from Fig. 3(b) and fitted with
$2E_{\mathrm{A}}$, with $E_{\mathrm{A}}$ from Eq. 1 to determine the induced
superconducting gap $\Delta$ and the transmission $\tau$ of the ABS. From the
fit, shown in Fig. 3(c), $\Delta=186\pm 2~{}\mu$eV and $\tau=0.976\pm 0.001$
were obtained. The value of $\Delta$ is consistent with previous studies of
hybrid superconductor-semiconductor heterostructures [45, 46].

##  V ELECTROSTATIC TUNING OF ISOLATED ANDREEV BOUND STATES

To further investigate the ABS spectrum in the planar JJ, the applied flux was
fixed to $\it{\Phi}=\it{\Phi}_{\mathrm{0}}/\mathrm{2}$ and two-tone
spectroscopy measurements were performed as a function of the gate voltage
$V_{\mathrm{G}}$, as shown in Fig. 4(a). No isolated ABS pair transitions
could be resolved for $V_{\mathrm{G}}>-1.4$ V, indicating a dense spectrum of
ABSs in the JJ. This is consistent with the observations from single-tone
measurements. At more negative $V_{\mathrm{G}}$, two isolated ABS pair
transitions were observed. Their transition frequencies increased with
decreasing $V_{\mathrm{G}}$ until, below $V_{\mathrm{G}}=-1.75$ V, no ABS
transitions were observed in the experimentally accessible frequency range.

Figure 4(b) depicts a zoom-in of the region marked with a box in Fig. 4(a),
measured at higher drive power to increase the visibility of ABS transitions
for $f_{\mathrm{d}}<14$ GHz. The frequency of the isolated ABS, indicated with
an arrow in panel (a), continuously decreased down to
$f_{\mathrm{d}}~{}=~{}10$ GHz towards $V_{\mathrm{G}}=-1.6$ V. The pair
transition signal disappeared in the vicinity of $V_{\mathrm{G}}=-1.6$ V (blue
marker) before its frequency increased again. This frequency modulation of the
ABS pair transition is explained by the formation of an accidental, incoherent
resonance in the junction. Such resonances were previously shown to enhance
the transmission $\tau$ of the JJ [47, 48, 23]. The single-tone measurement of
the resonator corresponding to the gate range in Fig. 4(a) is depicted in Fig.
4(c). Here, the incoherent resonance manifested itself as a local increase in
the resonator frequency at $V_{\mathrm{G}}=-1.6$ V (blue marker). Additional
flux-dependent single- and two-tone spectroscopy measurements in the vicinity
of the incoherent resonance are depicted in Fig. S.12 in SM.

Figure 4: Two-tone spectroscopy of Andreev bound states (ABSs) as a function
of gate voltage. (a) Magnitude $\absolutevalue{S_{21}}$ of the resonator
transmission in the presence of a drive tone
$\it{P}_{\mathrm{d}}=\mathrm{-24}$ dBm with frequency $f_{\mathrm{d}}$ at
$\it{\Phi}=\it{\Phi}_{\mathrm{0}}/\mathrm{2}$ as a function of gate voltage
$V_{\mathrm{G}}$. Isolated ABS pair transitions were observed from
$V_{\mathrm{G}}=-1.4$ V to $V_{\mathrm{G}}=-1.75$ V. At more positive gate
voltages, ABS transitions could no longer be resolved. (b) Zoomed-in
measurement of the region marked with a box in (a) using higher drive power,
to enhance the visibility of the ABS transition at lower frequencies. An
accidental resonance in the junction decreased the ABS frequency in the
vicinity of $V_{\mathrm{G}}=-1.6$ V (blue marker). (c) Single-tone
spectroscopy measurement corresponding to the gate voltage range in (a). (d)
Transmission $\tau$ and induced superconducting gap $\Delta$ extracted from
the ABS pair transition, indicated by the arrow in (a) as a function of gate
voltage $V_{\mathrm{G}}$.

Flux-dependent two-tone spectroscopy measurements of the isolated ABS,
indicated with an arrow in Fig. 4(a), were performed at multiple gate voltages
(see Fig. S.13 in SM). For each data set, $\Delta$ and $\tau$ were extracted
as in Fig. 3(c). The results are summarized in Fig. 4(d). The induced
superconducting gap was nearly constant with an average value of 184 $\mu$eV,
while the transmission was found to increase continuously with gate voltage,
approaching $\tau=0.98$ at $V_{\mathrm{G}}=-1.65$ V. The limited spectroscopic
window and the absence of reliable flux-dependent two-tone spectroscopy
measurements above $V_{\mathrm{G}}=-1.65$ V prevented the probing of a larger
gate voltage range. Nevertheless, the access and control of an isolated ABS
was successfully demonstrated.

##  VI CONCLUSION

We presented a framework for microwave spectroscopy measurements in two-
dimensional heterostructures using a flip-chip approach. We fabricated a
planar JJ where the conductive channel was defined by electrostatic gating,
allowing the probing of an isolated ABS with transmission up to $0.98$.
Efficient and reliable coupling between superconductor-semiconductor hybrid
devices and microwave electronics fulfills a crucial requirement towards
scalable architectures for Andreev qubits, Andreev molecules and topological
phenomena.

###### Acknowledgements.

We thank the Cleanroom Operations Team of the Binnig and Rohrer Nanotechnology
Center (BRNC) for their help and support. F. N. acknowledges support from the
European Research Council (grant number 804273) and the Swiss National Science
Foundation (grant number 200021_201082).

## Data availability

Data presented in this work will be available on Zenodo. The data that support
the finding of this study are available upon reasonable request from the
corresponding author.

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## Supplemental Material: Flip-chip-based microwave spectroscopy of Andreev
bound states in a planar Josephson junction

##  I Material and fabrication

Resonator and Device Chips were made separately and subsequently flip-chip
bonded using indium bumps. The Device Chip was fabricated from a
heterostructure grown on an InP (001) substrate using molecular beam epitaxy
techniques. The heterostructure consisted of an $\mathrm{In_{x}Al_{1-x}As}$
buffer and a 8 nm thick InAs quantum well, confined by
$\mathrm{In_{0.75}Ga_{0.23}As}$ barriers 13 nm below the surface. A 10 nm
thick Al layer was deposited on top of the heterostructure, in the same
chamber as the III-V growth without breaking the vacuum. From gated Hall bar
measurements, after removal of the Al, the wafer had a peak mobility of $\mu=$
18000 $\mathrm{cm^{2}V^{-1}s^{-1}}$ and an electron density of $n=8\cdot
10^{11}~{}\mathrm{cm^{-2}}$, resulting in an elastic mean free path
$l_{\mathrm{e}}=\mu\hbar\sqrt{2\mathrm{\pi}n}/\mathrm{e}\approx$ 270 nm. The
superconducting coherence length was estimated as $\xi=\hbar
v_{\mathrm{F}}/(\pi\Delta)\approx$ 1.3 $\mu\mathrm{m}$ for a ballistic
junction and $\xi=\sqrt{\hbar D/\Delta}\approx$ 740 nm for a diffusive
junction, with diffusion coefficient $D=v_{\mathrm{F}}l_{\mathrm{e}}/2$, Fermi
velocity $v_{\mathrm{F}}=\hbar\sqrt{2\pi n}/m^{*}$ and induced superconducting
gap $\Delta=184~{}\mu\mathrm{eV}$ determined from two-tone spectroscopy
measurements. The sample was defined by first isolating the mesa structures by
selectively removing the top Al layer with Transene D etchant before etching
$\approx$ 290 nm into the III-V heterostructure using a chemical wet etch
solution of $\mathrm{H_{2}O:C_{6}H_{8}O_{7}:H_{3}PO_{4}:H_{2}O_{2}}$ (220 : 55
: 3 : 3). The under bump metalization (UBM) was deposited on top of the Al
contacts using lift-off techniques. The $L\approx 110$ nm long and $W\approx
940$ nm wide Josephson junction (JJ) was patterned on top of the mesa, by
selective removal of the Al with Transene D etchant. Then a 15 nm layer of
$\mathrm{HfO_{2}}$ was deposited using atomic layer deposition technique,
before evaporating metallic gate electrodes. In the region where the
resonators are close to the Device Chip, the InAlAs buffer was etched away. As
a last step, the $\mathrm{HfO_{2}}$ on top of the UBM was removed to have a
galvanic connection between the rf-SQUID on the Device Chip and the DC control
lines on the Resonator Chip.

The Resonator Chip was fabricated from a high resistivity
($\rho~{}\mathrm{>~{}10000~{}\Omega\cdot}$cm) intrinsic Si $\mathrm{\langle
100\rangle}$ wafer. After removing the native oxide using hydrofluoric acid,
200 nm of Nb was sputtered on the wafer. In a single lithographic step,
readout resonators, device drive and DC control lines, as well as holes in the
ground plane, were patterned into the Nb with a $\mathrm{Cl_{2}/Ar}$ dry
etching procedure. Afterwards, the In was evaporated directly on top of the Nb
using lift-off techniques. Hydrofluoric acid was used again to remove the
surface oxide of the Nb and In before flip-chip bonding.

The Device Chip was bonded to the Resonator Chip with a flip-chip bonder (Karl
Suss FC 150). To facilitate the alignment of both chips, alignment markers
were defined during the mesa etch and resonator etch step. After gluing the
sample into the printed circuit board (PCB) holder, the microwave and DC lines
from the PCB were connected to the corresponding bonding pads on the Resonator
Chip with Al bond wires. Moreover, Al air bridges across the transmission
line, resonators and drive lines connecting the different ground planes
ensured a homogeneous ground potential all over the Resonator Chip.

##  II Experimental setup

The measurements were carried out in a BlueFors BF-LD cryogen-free dilution
refrigerator with a mixing chamber base temperature of 9 mK. A schematic of
the wiring setup is depicted in Fig. S.1. The resonator readout and device
drive tones $P_{\mathrm{in}}$ and $P_{\mathrm{d}}$ were continuously applied
using a Keysight PNA-X vector network analyser (VNA). Signals were attenuated
in the lines by 66 dB and 39 dB, respectively for readout and drive tones.
Readout was performed in transmission mode, where the signal from the
transmission line on the Resonator Chip, after interacting with the device,
passed through a DC block, a circulator, a dual isolator and an RF low-pass
filter and was subsequently amplified with a cryogenic high electron mobility
(HEMT) amplifier (bandwidth 0.3-14 GHz, average gain 37 dB) followed by a
room-temperature amplifier (bandwidth 2-8 GHz, average gain 35 dB) before
reaching the VNA again ($P_{\mathrm{out}}$). The voltages on gates $V_{1}$ and
$V_{2}$ were controlled with DC voltages sourced with a QDevil QDAC. The
voltages applied via the DC loom were filtered with QDevil RF and RC filters
(cutoff above 80 MHz and 65 kHz, respectively) before reaching the Device
Chip. The chips were mounted on a QDevil high frequency PCB with no additional
shielding present. A vector magnet was present in the dilution refrigerator
but not used in this study and hence left grounded. Instead, the flux was
supplied by sourcing a current through a coil, mounted in the near vicinity of
the chips.

Figure S.1: Schematic of the measurement setup. The resonator readout and
device drive tones with frequencies $f_{\mathrm{r}}$ and $f_{\mathrm{d}}$ were
applied using a vector network analyzer (VNA), corresponding to
$P_{\mathrm{in}}$ and $P_{\mathrm{d}}$, respectively. After 66 dB and 39 dB
attenuation of $P_{\mathrm{in}}$ and $P_{\mathrm{d}}$ at different temperature
stages of the dilution refrigerator, the signals reached the Resonator Chip.
The resonator transmission signal passed through a DC block followed by a
circulator, a dual isolator and an RF low-pass filter. The signal was
amplified with a cryogenic HEMT amplifier at 4 K and a room-temperature
amplifier before being detected by the VNA ($P_{\mathrm{out}}$). An extra RF
line, which could be used for reflectometry measurements, stayed grounded
during the experiments. The gates were controlled with DAC DC voltage sources.
The DC lines were filtered using a homemade low-pass filter at room
temperature, QDevil RF and RC filters at base temperature and an RC filter on
the printed circuit board. The external flux $\it{\Phi}$ was supplied by
sourcing a DC current through a coil, mounted on top of the sample space.

##  III Resonator Chip characterization

The unintentional coupling between the drive line and the resonator was
characterized by performing a series of single-tone spectroscopy measurements
in the absence of Device Chip, as depicted in Fig. S.2. The frequency
$f_{\mathrm{r}}$ of the signal was swept across the resonance frequency
($f_{\mathrm{res.}}=6.363$ GHz) of the rightmost resonator in Fig. 1(a) of the
Main Text. In panel (a), a background noise of $-80$ dB was measured by
applying a signal with frequency $f_{\mathrm{r}}$ to an RF line, which was not
connected to the PCB, and monitoring the magnitude of transmission
$\absolutevalue{S_{\mathrm{21}}}$ by detecting the output power
$P_{\mathrm{out}}$ using a VNA. The background noise was independent of
$f_{\mathrm{r}}$ and consistent with the noise floor of the VNA. In (b), the
magnitude of transmission $\absolutevalue{S_{\mathrm{21}}}$ is shown when a
signal with power $P_{\mathrm{in}}$ was applied to the left port of the
transmission line and measured at its right port. The readout frequency
$f_{\mathrm{r}}$ was swept across $f_{\mathrm{res.}}$ and a background level
of $-25$ dB was observed. Panel (c) shows $\absolutevalue{S_{\mathrm{21}}}$
when the signal with power $P_{\mathrm{d}}$ was applied to the drive line and
$P_{\mathrm{out}}$ was measured at the right port of the transmission line. In
this configuration, the background was at $-50$ dB, which is higher than the
background noise of $-80$ dB, indicating an unintentional coupling between the
drive line and the resonator. The $-25$ dB attenuation between device drive
line and resonator was a trade-off between sufficiently strong coupling
between device drive line and rf-SQUID and minimization of the contribution of
losses to the resonator. However, this drive line did not limit the quality
factor of the resonator, instead, the Device Chip was the dominant component
of the loss tangent.

Figure S.2: Characterization of the unintentional coupling between device
drive line and readout resonator. (a) A background noise of $-80$ dB was
measured by applying a signal with frequency $f_{\mathrm{r}}$ to an RF line,
which was not connected to the printed circuit board, and monitoring the
magnitude of transmission $\absolutevalue{S_{\mathrm{21}}}$ by detecting the
output power $P_{\mathrm{out}}$ emerging from the transmission line using a
vector network analyzer (VNA). The frequency $f_{\mathrm{r}}$ was swept across
the resonance frequency ($f_{\mathrm{res.}}=6.363$ GHz) of the rightmost
resonator on the Resonator Chip. The background noise was independent of
$f_{\mathrm{r}}$ and consistent with the noise floor of the VNA. (b) The
signal with power $P_{\mathrm{in}}$ was applied to the left port and
$P_{\mathrm{out}}$ was measured at the right port of the transmission line.
The magnitude of transmission $\absolutevalue{S_{\mathrm{21}}}$ was monitored
as a function of frequency $f_{\mathrm{r}}$. In this case, the background was
at $-25$ dB with a pronounced resonance feature at $f_{\mathrm{r}}=6.363$ GHz
consistent with $f_{\mathrm{res.}}$. (c) A signal with power $P_{\mathrm{d}}$
was applied to the device drive line and $P_{\mathrm{out}}$ was measured at
the right port of the transmission line. Here, the background was at $-50$ dB
indicating an unintentional coupling with $-25$ dB attenuation between device
drive line and readout resonator.

The three resonators on the Resonator Chip in Main Text Fig. 1(a) were
characterized. The results are summarized in Fig. S.3. The measurements were
performed while the corresponding JJ on the Device Chip was fully depleted.
The circle fit technique with diameter correction from Ref. [36] was used to
extract the internal quality factor $Q_{\mathrm{i}}$ of the resonators. Values
of $Q_{\mathrm{i}}$ between $5\cdot 10^{4}$ and $1\cdot 10^{5}$ were obtained.
In comparison to previous work [23, 24, 25, 26], these consistently high
values of $Q_{\mathrm{i}}$ demonstrate the potential of employing flip-chip
techniques for microwave experiments with devices fabricated on III-V
substrates. Improving materials as well as fabrication, deep etching into the
silicon substrate [49] or reducing the amount of III-V materials in the
vicinity of the microwave components [21, 28] could further enhance the
quality factors of the microwave resonators.

Figure S.3: Resonator characterization. (a-c) For all three resonators on the
Resonator Chip in Main Text Fig. 1(a), real $\mathrm{Re}(S_{\mathrm{21}})$ and
imaginary $\mathrm{Im}(S_{\mathrm{21}})$ part of the transmission
$S_{\mathrm{21}}$ (blue) is plotted together with a fit (red) to determine the
internal quality factor $Q_{\mathrm{i}}$ of the resonators. The corresponding
magnitude response $\absolutevalue{S_{\mathrm{21}}}$ as a function of readout
frequency $f_{\mathrm{r}}$ is displayed in (d-f). Panels (a) and (d)
correspond to the resonator used in the measurements of the Main Text, while
panels (b) and (e) [(c) and (f)] correspond to the center [rightmost]
resonator in Fig. 1(a) of the Main Text. The measurements were performed while
the corresponding Josephson junction was pinched-off. The circle fit technique
with diameter correction from Ref. [36] was used to extract the internal
quality factors $Q_{\mathrm{i}}$ of the resonators, yielding values between
$5\cdot 10^{4}$ and $1\cdot 10^{5}$ as shown in (d-f).

##  IV Measurement procedure and data processing

In single-tone spectroscopy measurements, a signal $P_{\mathrm{in}}$ with
frequency $f_{\mathrm{r}}$, swept around the readout resonator frequency, was
sent to the transmission line. Ater the interaction with the readout
resonator, the magnitude $\absolutevalue{S_{21}}$ and phase $\angle S_{21}$ of
the transmitted part of the input signal ($P_{\mathrm{out}}$) was measured
using a VNA. To acquire flux- and gate-dependent measurements, this procedure
was repeated at multiple values of $\it{\Phi}$ or $V_{\mathrm{G}}\equiv
V_{1}=V_{2}$, respectively. We assigned $\it{\Phi}=$ 0 to the central DC
current value, where the resonator frequency was maximum and independent of
small deviations of DC current. Since the resonator resonance frequency
modulates with a period of one flux quantum $\it{\Phi}_{\mathrm{0}}$,
$\it{\Phi}=$ 0 was defined within the flux quantum when no current was applied
to the coil.

For two-tone spectroscopy measurements, the readout frequency $f_{\mathrm{r}}$
was first swept around the resonance frequency of the readout resonator to
determine the minimum in the magnitude $\absolutevalue{S_{21}}$ corresponding
to $f_{\mathrm{res.}}$. The readout frequency was then offset by 80 kHz,
$f_{\mathrm{r}}=f_{\mathrm{res.}}+80$ kHz, to position the readout tone on the
slope of the resonator response to increase the sensitivity of shifts in
$f_{\mathrm{res.}}$. The spectroscopy measurement was then performed by
monitoring the resonator response at the aforementioned fixed
$f_{\mathrm{r}}$, while simultaneously, a continuous drive tone
$P_{\mathrm{d}}$ at frequency $f_{\mathrm{d}}$ was applied to the device drive
line. The drive tone with frequency $f_{\mathrm{d}}$ was swept while measuring
the magnitude and phase of the resonator signal $P_{\mathrm{out}}$. Finally,
the background of the resonator response was removed by subtracting the median
of the $f_{\mathrm{d}}$ sweep. For flux- and gate-dependent two-tone
spectroscopy measurements this procedure was repeated at each value of
$\it{\Phi}$ or $V_{\mathrm{G}}$, respectively.

In all single- and two-tone spectroscopy plots, the colorbar of the magnitude
$\absolutevalue{S_{21}}$ was rescaled to [0,1]. This was done by finding the
minimum value of $\absolutevalue{S_{21}}$ for each dataset. Then, this minimum
magnitude was added to all values of $\absolutevalue{S_{21}}$ in the dataset,
such that the new dataset consisted of only positive values. Afterwards, the
maximum value of $\absolutevalue{S_{21}}$ was found and all
$\absolutevalue{S_{21}}$ were normalized to this value.

The transmission $\tau$ and induced superconducting gap $\Delta$ of the JJ
were obtained by fitting two-tone spectroscopy data representing a
$\it{\Phi}$-dependent Andreev bound state (ABS) pair transition using Eq. S.1.
In the short junction limit, an ABS pair transition frequency
$f_{\mathrm{ABS}}$ is described by:

$f_{\mathrm{ABS}}=2E_{\mathrm{A}}=2\Delta\sqrt{1-\tau\sin^{2}(\mathrm{\pi}\it{\Phi}/\it{\Phi}_{\mathrm{0}})},$
(S.1)

where $E_{\mathrm{A}}$ is the energy of a single-channel ABS [41, 42]. First,
resonances caused by the device drive line and measurement circuit were
filtered out by taking the gradient along $\it{\Phi}$ for each
$f_{\mathrm{d}}$. Then, a Sobel filter was applied to extract all edges of the
color plot. To remove the weaker higher-order transitions, the maximum value
of the response along the extracted edges was taken at every flux $\it{\Phi}$.
The remaining data points were fitted to Eq. S.1. Due to the spurious
resonances detected in the measurement circuit, it was not possible to fit the
two-tone spectrum at every flux $\it{\Phi}$ with a Lorentzian function to
extract the data points corresponding to the energy of the ABS pair
transition. To overcome this problem, the window function could be used, but
this required assumptions about the flux-dependence of the ABS energy.

##  V Additional measurements

###  V.1 Additional single-tone spectroscopy data

The influence of readout power on resonator response is shown in Fig. S.4. For
$V_{\mathrm{G}}=0$ V and $V_{\mathrm{G}}=-1.66$ V, the readout power
$P_{\mathrm{in}}$ was swept at fixed flux $\it{\Phi}=\mathrm{0}$. The green
markers in Fig. S.4(a) and (c) indicate the value of $P_{\mathrm{in}}$ at
which the measurements in the Main Text were taken ($-30$ dBm). The change in
resonance frequency at high readout power in Fig. S.4(a) is commonly observed
in coupled resonator-rf-SQUID systems under strong driving, and could signify
non-linear effects such as over-driving of the resonator or saturation of the
available ABSs [50, 37]. Moreover, the internal quality factor
$Q_{\mathrm{i}}$ was also reduced as the readout power was increased beyond
$-30$ dBm as shown in panel (c). Yet, when the gate voltage was set to
$V_{\mathrm{G}}=-1.66$ V, these effects were no longer present: the resonance
frequency and $Q_{\mathrm{i}}$ remained constant as the readout power was
increased as depicted in panels (b) and (d). Based on these results, we
conclude that the change in resonance frequency and $Q_{\mathrm{i}}$ were
related to the density of ABSs. At $V_{\mathrm{G}}=-1.66$ V, no ABSs were
available for microwave-induced transitions at $\it{\Phi}=\mathrm{0}$, because
in this configuration the JJ was in the ballistic short junction regime
($L<l_{\mathrm{e}}\ll\xi$). At $V_{\mathrm{G}}=0$ V lateral ABSs, which were
longer than $l_{\mathrm{e}}$ and comparable to $\xi$, were present in the JJ,
yielding a minigap with many ABSs with low and high transmission probabilities
[39, 40]. Therefore, at high readout power the number of transitions between
ABSs was increased, resulting in a larger number of loss channels and thus a
decrease in $Q_{\mathrm{i}}$ in Fig. S.4(c) [51]. Moreover, the transitions to
higher ABSs with lower mobility reduced the supercurrent in the rf-SQUID,
yielding an increase of the inductance of the JJ for high readout powers and
thus a decrease in the resonance frequency.

Figure S.4: Resonator response as a function of readout power. (a) Magnitude
$\absolutevalue{S_{21}}$ of the resonator transmission as a function of
readout tone power $P_{\mathrm{in}}$ as well as frequency $f_{\mathrm{r}}$ at
$\it{\Phi}=\mathrm{0}$ and $V_{\mathrm{G}}=0$ V. The Josephson junction (JJ)
hosted a high density of Andreev bound states (ABSs), thereby shifting the
resonance frequency downwards at large $P_{\mathrm{in}}$. (b) Same as (a) for
$V_{\mathrm{G}}=-1.66$ V. Only few ABSs were present in the JJ. In this
situation, the resonance frequency remained constant with increasing
$P_{\mathrm{in}}$. Internal quality factors $Q_{\mathrm{i}}$ as a function of
$P_{\mathrm{in}}$ were extracted from the resonance curves in (a) and (b) and
are displayed in panels (c) and (d), respectively. The green markers at
$P_{\mathrm{in}}=-30$ dBm. indicate the power of the resonator probe tone
$P_{\mathrm{in}}$ used for the measurements in the Main Text Figure S.5:
Resonator response at $\it{\Phi}=\it{\Phi}_{\mathrm{0}}/\mathrm{2}$ when
tuning both gates of the Josephson junction (JJ) individually (see Fig. 1(c)
of the Main Text). (a) Magnitude $\absolutevalue{S_{21}}$ was monitored around
the resonance frequency of the readout resonator as a function of gate voltage
$V_{\mathrm{1}}$, while $V_{\mathrm{2}}=0$ V. (b) Same as in (a), but
$\absolutevalue{S_{21}}$ as a function of $V_{2}$ with $V_{1}=0$ V. For both
gates, a decrease in gate voltage led the resonance frequency of the resonator
to increase (after an initial plateau) before it saturated below
$V_{\mathrm{1,2}}\approx-1.4$ V. Moreover, the JJ could be pinched-off
($f_{\mathrm{r}}=f_{\mathrm{0}}$) only if both gates were energized. Figure
S.6: Characterization of gate hysteresis. Magnitude $\absolutevalue{S_{21}}$
was measured around the resonance frequency of the readout resonator while
sweeping the gate voltage $V_{\mathrm{G}}$ (a) down and (b) up at
$\it{\Phi}=\it{\Phi}_{\mathrm{0}}/\mathrm{2}$. The small gate hysteresis did
not affect the two-tone spectroscopy experiments.

Sweeping the gate voltage $V_{\mathrm{G}}$ from -2.3 V (JJ pinched-off) to 0 V
at three values of flux $\it{\Phi}$ led to different responses of the
resonator frequency as depicted in Fig. S.7. For
$V_{\mathrm{G}}\leq\mathrm{-2.3}$ V the JJ was pinched-off, and the resonance
frequency approached $f_{\mathrm{0}}$, independent of $\it{\Phi}$. With
increasing $V_{\mathrm{G}}$ the resonator resonance frequency
$f_{\mathrm{res.}}$ deviated from the bare resonance frequency
$f_{\mathrm{0}}$ by [37]:

$\delta f_{\mathrm{0}}\equiv
f_{\mathrm{res.}}-f_{\mathrm{0}}\approx\frac{8}{\mathrm{\pi}^{2}}\frac{M^{2}}{L_{\mathrm{R}}(L_{\mathrm{J}}+L_{\mathrm{Loop}})},$
(S.2)

where $M\approx 14.7$ pH is the calculated mutual inductance between the
resonator and the rf-SQUID using the model of a rectangular loop next to a
straight current-carrying wire,
$L_{\mathrm{R}}=L_{\mathrm{G}}^{\mathrm{R}}+L_{\mathrm{K}}^{\mathrm{R}}\approx
2.33$ nH is the calculated total (geometric and kinetic) inductance of the
resonator using conformal mapping techniques [52, 53], $L_{\mathrm{J}}$ is the
inductance of the JJ and
$L_{\mathrm{Loop}}=L_{\mathrm{G}}^{\mathrm{Loop}}+L_{\mathrm{K}}^{\mathrm{Loop}}\approx
201$ pH is the calculated total inductance of the superconducting loop. The
inverse Josephson inductance $L_{\mathrm{J}}^{\mathrm{-1}}$ is related to a
change in the supercurrent $I_{\mathrm{s}}$ of the rf-SQUID by:

$L_{\mathrm{J}}^{\mathrm{-1}}=\frac{2\pi}{\it{\Phi}_{\mathrm{0}}}\frac{\partial
I_{\mathrm{s}}(\varphi)}{\partial\varphi},$ (S.3)

where
$\varphi=2\pi(\it{\Phi}-L_{\mathrm{Loop}}I_{\mathrm{s}}(\varphi))/\it{\Phi}_{\mathrm{0}}$
is the phase difference across the JJ [54, 55]. The Josephson inductance
$L_{\mathrm{J}}$ can be extracted from the current-phase relation (CPR) [see
Fig. S.9(a)]. Therefore, $L_{\mathrm{J}}$ can be positive or negative, leading
to positive or negative frequency shifts $\delta f_{\mathrm{0}}$ of the bare
resonator frequency $f_{\mathrm{0}}$.

Figure S.7: Gate-dependent resonator response at three flux values. Magnitude
$\absolutevalue{S_{21}}$ of the resonator response was measured around the
resonance frequency of the readout resonator while sweeping the gate voltage
$V_{\mathrm{G}}$ at (a) $\it{\Phi}=\mathrm{0}$, (b)
$\it{\Phi}=\it{\Phi}_{\mathrm{0}}/\mathrm{4}$ and (c)
$\it{\Phi}=\it{\Phi}_{\mathrm{0}}/\mathrm{2}$. When the Josephson junction
(JJ) was pinched-off ($V_{\mathrm{G}}=-2.3$ V) the resonance frequency
approached $f_{\mathrm{0}}$, independent of $\it{\Phi}$. Increasing
$V_{\mathrm{G}}$ resulted in a flux-dependent shift $\delta f_{\mathrm{0}}$ of
the bare resonance frequency $f_{\mathrm{0}}$. The magnitude and sign of
$\delta f_{\mathrm{0}}$ depended on the inductance of the JJ. Moreover, at
high $V_{\mathrm{G}}$ the linewidth of the resonator resonance was larger
towards $\it{\Phi}=\it{\Phi}_{\mathrm{0}}/\mathrm{2}$. In this regime, Andreev
bound states existed all over the superconducting gap, thus the resonant
resonator signal $P_{\mathrm{in}}$ was able to excite transitions between
these subgap states, which therefore increased the linewidth. Figure S.8:
Flux-dependent resonator response at selected gate voltages $V_{\mathrm{G}}$.
(a) Magnitude $\absolutevalue{S_{21}}$ was measured around the resonance
frequency of the readout resonator as a function of flux $\it{\Phi}$ over 20
periods at $V_{\mathrm{G}}=\mathrm{0}$ V. (b-m) Flux-dependent magnitude
$\absolutevalue{S_{21}}$ was monitored in the same frequency range as in (a).
The measurements at selected gate voltages $V_{\mathrm{G}}$ extend the data in
Fig. 2(b-e) of the Main Text. These data sets were used to map characteristic
quantities of the Josephson junction as a function of $V_{\mathrm{G}}$, i.e.
critical current $I_{\mathrm{c}}$, effective transmission
$\mathrm{\bar{\tau}}$ and Josephson inductance $L_{\mathrm{J}}$, summarized in
Fig. S.9. Figure S.9: Characteristic quantities of the Josephson junction (JJ)
as a function of gate voltages $V_{\mathrm{G}}$. (a) The current-phase
relation (CPR), which is the supercurrent $I_{\mathrm{s}}$ as a function of
phase $\varphi$ across the JJ, is plotted for the three gate voltages
$V_{\mathrm{G}}$. The CPR was extracted from flux-dependent single-tone
measurements in Fig. S.8 using the iterative fitting method described in Ref.
[37]. (b) Critical current $I_{\mathrm{c}}$ as a function of $V_{\mathrm{G}}$.
The $I_{\mathrm{c}}$ decreased with decreasing $V_{\mathrm{G}}$, consistent
with the reduction of the density of ABSs in the junction. (c) Josephson
inductance ($L_{\mathrm{J}}$) as a function of $V_{\mathrm{G}}$ at
$\it{\Phi}=\mathrm{0}$ was extracted from CPRs using Eq. S.3. (d) Effective
transmission $\bar{\tau}$, describing the transmission of an effective single
ABS replacing all physical ABSs, as a function of $V_{\mathrm{G}}$ was
extracted from CPRs using Eq. S.5. (e) Normalized readout resonator shift
$\delta f_{\mathrm{0}}$ and normalized linewidth of the resonator $\Delta
f_{\mathrm{FWHM}}$ as a function of $V_{\mathrm{G}}$ at
$\it{\Phi}=\mathrm{0}$.

By iteratively fitting the periodic shifts of the resonator frequency in Fig.
S.8 in a self-consistent way under consideration of self-screening effects
[37], the CPR was obtained for each gate voltage $V_{\mathrm{G}}$. The
resulting CPR for three gate voltages, which were used in Fig. 2(b-d) of the
Main Text, is shown in Fig. S.9(a). The CPR contains in its amplitude
information about the density of ABSs and in its deviation from sinusoidal
shape an indication of whether high transmission modes are present in the
junction [56].

The critical current $I_{\mathrm{c}}$, which is the maximal amplitude of the
CPR, as a function of $V_{\mathrm{G}}$ is depicted in Fig. S.9(b). For gate
voltages down to $V_{\mathrm{G}}=-1.2$ V, the junction had a high density of
ABSs, supported by the fact that each ABS carries a maximal supercurrent

$I_{\mathrm{s,i}}(\varphi)=-\frac{\mathrm{2e}}{h}\frac{\mathrm{\partial}E_{\mathrm{A,i}}}{\mathrm{\partial}\varphi}<\mathrm{7~{}nA},$
(S.4)

where $E_{\mathrm{A,i}}$ is the energy of the individual ABSs, which depends
on its individual transmission [57]. By further decreasing $V_{\mathrm{G}}$,
the critical current was rapidly reduced to $I_{\mathrm{c}}\approx\mathrm{13}$
nA. Therefore, in this gate voltage regime only few high-transmissive ABSs
could be present in the junction. For $V_{\mathrm{G}}<-1.4$ V,
$I_{\mathrm{c}}$ decreased slowly indicating a reduction in the transmission
of the individual ABSs rather than its density, which is consistent with
observations from two-tone spectroscopy measurements in Fig. 4 of the Main
Text. In addition, the Josephson inductance $L_{\mathrm{J}}$ was extracted
from CPR using Eq. S.3, which is shown in Fig. S.9(c) for
$\it{\Phi}=\mathrm{0}$.

Since it is not feasible to assign a transparency to each individual ABS, the
transmission of the junction was instead described macroscopically with an
effective transmission $\mathrm{\bar{\tau}}$, which is identical for all ABSs.
The effective transmission $\mathrm{\bar{\tau}}$ was determined by fitting the
CPR using:

$I_{\mathrm{s}}(\varphi)=I_{\mathrm{0}}\frac{\mathrm{\bar{\tau}sin(\it{\varphi})}}{\sqrt{\mathrm{1-\bar{\tau}sin^{2}(\it{\varphi}/\mathrm{2})}}},$
(S.5)

where $I_{\mathrm{0}}=(\mathrm{e}/\mathrm{2}\hbar)\bar{N}\Delta$, with
$\bar{N}$ is the effective number of ABSs in the junction [58]. Figure S.9(d)
summarizes the $\mathrm{\bar{\tau}}$ as a function of $V_{\mathrm{G}}$. Across
the full gate voltage range a high $\mathrm{\bar{\tau}}$ was observed, which
is consistent with the interpretation of the junction being in the diffusive
long regime for $V_{\mathrm{G}}>-1.4$ V and ballistic short regime for
$V_{\mathrm{G}}<-1.4$ V, respectively. In the diffusive regime, there are many
ABSs with low transmission as well as high transmission following Dorokhov’s
bimodal distribution [40, 59], resulting in a relatively high
$\mathrm{\bar{\tau}}$. For $V_{\mathrm{G}}<-1.4$ V, the junction hosted only
few ABSs as discussed in Fig. S.9(b). Two-tone spectroscopy measurements in
Fig. 4 of the Main Text revealed that some of the ABSs had high transmission,
but there were also some with low transmission, so that $\mathrm{\bar{\tau}}$
remained approximately constant.

The resonance frequency shift as a function of $V_{\mathrm{G}}$, together with
$Q_{\mathrm{i}}$, in Fig. 2(a) of the Main Text revealed more details about
different gate voltage regimes. Figure S.9(e) shows the percentage change of
both the frequency shift $\delta f_{\mathrm{0}}$ and the linewidth $\Delta
f_{\mathrm{FWHM}}$ of the resonator resonance as a function of
$V_{\mathrm{G}}$ at $\it{\Phi}=\mathrm{0}$. For gate voltages down to
$V_{\mathrm{G}}=-0.8$ V, the resonance frequency remained nearly unaffected
because ABSs with low transmission were depleted, which had a minor
contribution to the supercurrent. However, they likely acted as loss channels
and thus $\Delta f_{\mathrm{FWHM}}$ decreased. As $V_{\mathrm{G}}$ was tuned
to more negative values, both $\delta f_{\mathrm{0}}$ and $\Delta
f_{\mathrm{FWHM}}$ decreased, indicating a further reduction of the ABS
density in the junction. For $V_{\mathrm{G}}<-1.4$ V, the resonance frequency
still decreased while the linewidth of the resonator remained unchanged. This
is attributed to the absence of available ABSs for microwave-induced
transitions at $\it{\Phi}=\mathrm{0}$ and a reduction in the transmission of
individual ABSs. This interpretation is consistent with the two-tone
spectroscopy data in Figs. 3 and 4 of the Main Text.

Figure S.10: Time stability of the resonator response at gate voltage
$V_{\mathrm{G}}$ and flux $\it{\Phi}$ sensitive points. Magnitude
$\absolutevalue{S_{21}}$ was monitored around the resonance frequency of the
readout resonator as a function of flux $\it{\Phi}$ at (a)
$V_{\mathrm{G}}=\mathrm{0}$ V and (b) $V_{\mathrm{G}}=\mathrm{-1.4}$ V to
illustrate the configuration at which the data in (c-f) were taken, indicated
by the colored bars. Evolution of the magnitude $\absolutevalue{S_{21}}$
around the resonance frequency for (c) $V_{\mathrm{G}}=\mathrm{0}$ V and (d)
$V_{\mathrm{G}}=\mathrm{-1.4}$ V within 10 minutes. (e, f) Same as (c, d) at a
flux $\it{\Phi}$, at which the resonance frequency shifted remarkably for
varying $\it{\Phi}$. Absence of fluctuations and drifts in the resonance
frequency at the flux insensitive point $\it{\Phi}=\mathrm{0}$ for both
$V_{\mathrm{G}}=\mathrm{0}$ V and $V_{\mathrm{G}}=\mathrm{-1.4}$ V suggested
gate voltage induced charge noise was negligible. The resonance frequency
stayed stable for periods over 10 minutes at the flux sensitive point in (e)
and (f), indicating that there were no magnetic field drifts observable during
that time. Small fluctuations in the resonator response on a shorter time
scale caused by magnetic field fluctuations were observed. However, they did
not limit the two-tone spectroscopy measurements in this work, since these
were performed around the flux insensitive point
$\it{\Phi}=\it{\Phi}_{\mathrm{0}}/\mathrm{2}$.

###  V.2 Additional two-tone spectroscopy measurements for two selected drive
powers

Figure S.11: Gate-dependent two-tone spectroscopy measurements for two
selected drive powers. (a) Magnitude $\absolutevalue{S_{21}}$ of the readout
resonator response as a function of drive tone frequency $f_{\mathrm{d}}$ at
$\it{\Phi}=\it{\Phi}_{\mathrm{0}}/\mathrm{2}$, as well as gate voltage
$V_{\mathrm{G}}$. The drive tone power was set to
$P_{\mathrm{d}}=\mathrm{-33}$ dBm. (b) Probing the region marked with a box in
(a) with higher resolution. (c) Same as (a) with $P_{\mathrm{d}}=\mathrm{-15}$
dBm. (d) Zoomed-in measurement of the region marked with a box in (c).
Horizontal lines in the two-tone spectroscopy measurements were present
predominantely above $f_{\mathrm{d}}=14$ GHz and only at gate voltage and flux
values at which an Andreev bound state (ABS) pair transition was observed (see
also Fig. 3 of the Main Text). Moreover, within this range the strength of the
horizontal lines remained constant as a function of $V_{\mathrm{G}}$ and
$\it{\Phi}$, but increased at stronger driving powers $P_{\mathrm{d}}$. We
attribute these horizontal lines to resonances in the drive line and connected
measurement circuit [44]. Stronger drive powers and the unintentional
resonances in the drive line were responsible for the increased linewidth of
the ABS pair transition spectrum [60]. Replicas of ABSs, marked with arrows in
panel (c), were observed at strong driving powers. Following Ref. [32], we
assign these replicas to second-order transitions involving absorption or
emission of a resonator photon.

###  V.3 Additional data for the study of electrostatic tuning of isolated
Andreev bound states

Figure S.12: Flux-dependent single- and two-tone spectroscopy data in the
vicinity of the incoherent resonance at $V_{\mathrm{G}}=\mathrm{-1.63}$ V as
discussed in Fig. 4 of the Main Text. (a) Frequency response of the readout
resonator as a function of flux $\it{\Phi}$ at $V_{\mathrm{G}}=\mathrm{-1.63}$
V. An avoided crossing, symmetric around
$\it{\Phi}=\it{\Phi}_{\mathrm{0}}/\mathrm{2}$, was observed. Following Ref.
[23], this avoided crossing is a sign of virtual resonator-Andreev bound state
(ABS) photon exchange, described by a Jaynes-Cummings interaction. (b) Two-
tone spectroscopy data corresponding to the single-tone data in (a). The ABS
pair transition at $\it{\Phi}=\it{\Phi}_{\mathrm{0}}/\mathrm{2}$ with
$f_{\mathrm{d}}=10$ GHz is consistent with observations in Fig. 4 of the Main
Text and Fig. S.11. In the vicinity of
$\it{\Phi}=\it{\Phi}_{\mathrm{0}}/\mathrm{2}$, the transition frequency was
reduced due to the incoherent resonance before it increased again. Figure
S.13: Flux-dependent two-tone spectroscopy measurements at selected gate
voltages $V_{\mathrm{G}}$. (a-m) Magnitude $\absolutevalue{S_{21}}$ of the
readout resonator response as a function of drive tone frequency
$f_{\mathrm{d}}$ and flux $\it{\Phi}$. These data sets were used to extract
the induced superconducting gaps $\Delta$ and transmissions $\tau$, which are
plotted in Fig. 4(d) of the Main Text. By comparing the data with the results
in Fig. S.11, replicas of the ABS pair transition were identified and removed
from the data set.

##  VI Estimate of characteristic time scales for qubits

Coherent manipulation of isolated ABSs is limited by uncontrolled interactions
between the ABSs and degrees of freedom in its environment. We estimated both
relaxation time $T_{1}$ and dephasing time $T_{2}$ of an Andreev level qubit
from our two-tone spectroscopy data. Relaxation of the qubit can be caused by
many dissipative channels, for example dielectric losses, charge noise, flux
noise, quasiparticle fluctuations and Purcell noise. Therefore, all
contributions add to $T_{1}$ as:

$\frac{1}{T_{1}}=\frac{1}{T^{D}}+\frac{1}{T^{N_{g}}}+\frac{1}{T^{\it{\Phi}}}+\frac{1}{T^{QP}}+\frac{1}{T^{P}}.$
(S.6)

From time stability measurements at the gate voltage and flux sensitive points
[Fig. S.10], we conclude that the charge noise and flux noise were not the
limiting mechanisms for $T_{1}$, since the Andreev level qubit would be
operated at a gate voltage and flux insensitive point, a so-called sweet spot.
At zero magnetic field, it is expected that quasiparticle fluctuations have
been very weak [10] and since the qubit frequency $f_{Q}=14$ GHz was much
larger than the resonator frequency $f_{\mathrm{r}}=5.9162$ GHz, Purcell loss
is expected to have been very small. The materials used and the absence of
surface treatments suggests that the dominant source of relaxation was due to
dielectric losses [61, 62]. In this case, the relaxation time $T_{1}$ is given
by Eq. S.7

$T_{1}\approx T^{D}=\frac{Q_{\mathrm{i}}}{2\pi f_{Q}},$ (S.7)

where $Q_{\mathrm{i}}$ is the internal quality factor of the resonator. We
estimated the dielctric losses to find an upper limit of $T_{1}\approx 600$ ns
for $Q_{\mathrm{i}}=5\cdot 10^{4}$ and $f_{Q}=14$ GHz, corresponding to the
resonator used for the measurements and the minimum frequency of the ABS pair
transition shown in Fig. 3(c). Possible attempts to improve the lifetime
$T_{1}$ could include using materials with lower loss tangent, providing
better shielding in the device environment and optimizing the fabrication of
the Resonator Chip.

The dephasing time $T_{2}$ describes the loss of coherence of a superposition
state and is given by:

$\frac{1}{T_{2}}=\frac{1}{2T_{1}}+\frac{1}{T_{2}^{*}},$ (S.8)

where $T_{2}^{*}$ is the pure dephasing time. Pure dephasing comprises the
noise which causes the qubit frequency to fluctuate. Therefore, $T_{2}^{*}$
can be estimated by the full width at half maximum (FWHM) $2\gamma$ of the ABS
transition signal

$T_{2}^{*}=\frac{1}{\pi 2\gamma}.$ (S.9)

The FWHM of the ABS transition was extracted from line cuts of Fig. S.11(a)
and (b) at $-1.72$ V and $-1.73$ V as well as $-1.55$ V and $-1.56$ V by
fitting the spectrum with a Lorentzian function, yielding $2\gamma=280-400$
MHz and thus $T_{2}^{*}=0.8-1.1$ ns. Pure dephasing could be reduced by better
filtering of input signals and shielding of the sample space from the
environment.