# Strong Charge-Photon Coupling in Planar Germanium
Enabled by Granular Aluminium Superinductors

Marián Janík<EMAIL_ADDRESS>ISTA, Institute of Science and
Technology Austria, Am Campus 1, 3400 Klosterneuburg, Austria  Kevin Roux
ISTA, Institute of Science and Technology Austria, Am Campus 1, 3400
Klosterneuburg, Austria  Carla Borja Espinosa  ISTA, Institute of Science and
Technology Austria, Am Campus 1, 3400 Klosterneuburg, Austria  Oliver Sagi
ISTA, Institute of Science and Technology Austria, Am Campus 1, 3400
Klosterneuburg, Austria  Abdulhamid Baghdadi  ISTA, Institute of Science and
Technology Austria, Am Campus 1, 3400 Klosterneuburg, Austria  Thomas
Adletzberger  ISTA, Institute of Science and Technology Austria, Am Campus 1,
3400 Klosterneuburg, Austria  Stefano Calcaterra  L-NESS, Physics Department,
Politecnico di Milano, via Anzani 42, 22100, Como, Italy  Marc Botifoll
Catalan Institute of Nanoscience and Nanotechnology (ICN2), CSIC and BIST,
Campus UAB, Bellaterra, 08193 Barcelona, Catalonia, Spain  Alba Garzón Manjón
Catalan Institute of Nanoscience and Nanotechnology (ICN2), CSIC and BIST,
Campus UAB, Bellaterra, 08193 Barcelona, Catalonia, Spain  Jordi Arbiol
Catalan Institute of Nanoscience and Nanotechnology (ICN2), CSIC and BIST,
Campus UAB, Bellaterra, 08193 Barcelona, Catalonia, Spain  ICREA, Passeig de
Lluís Companys 23, 08010 Barcelona, Catalonia, Spain  Daniel Chrastina
L-NESS, Physics Department, Politecnico di Milano, via Anzani 42, 22100, Como,
Italy  Giovanni Isella  L-NESS, Physics Department, Politecnico di Milano, via
Anzani 42, 22100, Como, Italy  Ioan M. Pop  IQMT, Karlsruhe Institute of
Technology, 76131 Karslruhe, Germany  PHI, Karlsruhe Institute of Technology,
76131 Karlsruhe, Germany  Physics Institute 1, Stuttgart University, 70569
Stuttgart, Germany  Georgios Katsaros  ISTA, Institute of Science and
Technology Austria, Am Campus 1, 3400 Klosterneuburg, Austria

(August 28, 2024)

###### Abstract

High kinetic inductance superconductors are gaining increasing interest for
the realisation of qubits, amplifiers and detectors. Moreover, thanks to their
high impedance, quantum buses made of such materials enable large zero-point
fluctuations of the voltage, boosting the coupling rates to spin and charge
qubits. However, fully exploiting the potential of disordered or granular
superconductors is challenging, as their inductance and, therefore, impedance
at high values are difficult to control. Here we have integrated a granular
aluminium resonator, having a characteristic impedance exceeding the
resistance quantum, with a germanium double quantum dot and demonstrate strong
charge-photon coupling with a rate of $g_{\text{c}}/2\pi=$566\pm
2\text{\,}\mathrm{MHz}$$. This was achieved due to the realisation of a
wireless ohmmeter, which allows _in situ_ measurements during film deposition
and, therefore, control of the kinetic inductance of granular aluminium films.
Reproducible fabrication of circuits with impedances (inductances) exceeding
$13\text{\,}\mathrm{k\SIUnitSymbolOhm}$ ($1\text{\,}\mathrm{nH}$ per square)
is now possible. This broadly applicable method opens the path for novel
qubits and high-fidelity, long-distance two-qubit gates.

##  I Introduction

Superconducting circuits made from disordered superconductors with large
kinetic inductance are attracting attention for the development of qubits,
such as fluxonium [1, 2, 3, 4, 5], parametric amplifiers [6, 7, 8] and kinetic
inductance detectors [9, 10, 11, 12, 13, 14]. High characteristic impedance
devices, enabled by large kinetic inductors, are also paramount to
semiconductor quantum dot qubit applications, whereby several studies have
demonstrated strong charge-photon [15, 16, 17, 18] and spin-photon coupling
[19, 20, 21, 22]. Furthermore, the first long-distance coherent coupling
experiments between spin qubits have been reported [23, 24], leading to the
first photon-mediated long-distance two-qubit gate for electron spins in Si
[25]. Currently, one of the main challenges for achieving high-fidelity long-
distance two-qubit gates lies in increasing the spin-photon coupling strength
[25]. This can be achieved by enhancing the charge-photon coupling and the
spin-orbit interaction.

The charge-photon coupling strength $g_{\text{c}}$ is enhanced with high-
impedance $Z$ resonators, as $g_{\text{c}}\propto\sqrt{Z}$ [26, 3, 27, 28, 29,
30]. To date, quantum dot circuit quantum electrodynamics (cQED)
implementations utilising disordered nitride [31, 25, 21, 22] or Josephson
junction array [17, 18] resonators have not exceeded the characteristic
impedance of $\sim$ $3.8\text{\,}\mathrm{k\SIUnitSymbolOhm}$ (Extented Fig.
LABEL:fig:fige6). Outside the field of hybrid devices, much higher impedance
has been achieved, either relying on the combination of a large kinetic
inductance with a low stray capacitance of a suspended Josephson junction
array with an impedance of > $200\text{\,}\mathrm{k\SIUnitSymbolOhm}$ [32], or
a large mutual geometric inductance with a low stray capacitance of a
suspended planar coil with $\sim$ $31\text{\,}\mathrm{k\SIUnitSymbolOhm}$
[33]. These approaches cannot be readily implemented with standard
semiconductor spin qubits devices because of incompatibility with magnetic
fields and complex nanofabrication requirements. Other studies have focused on
the combination of a kinetic and mutual geometric inductance of meandered
structures of thin-film TiN, with a kinetic inductance of
$234\text{\,}\mathrm{pH}$ per square (${\mathrm{\Box}}^{-1}$) [34], or
granular aluminium (grAl) with
$220\text{\,}\mathrm{pH}\text{\,}{\mathrm{\Box}}^{-1}$ [3], both with
$Z\approx$ $28\text{\,}\mathrm{k\SIUnitSymbolOhm}$. While grAl offers magnetic
field resilience [35], lift-off compatibility [1], low losses [36], and sheet
kinetic inductance reaching
$2\text{\,}\mathrm{nH}\text{\,}{\mathrm{\Box}}^{-1}$ [37, 38, 39], it was not
yet exploited for quantum dot devices. This is presumably because of the poor
reproducibility of high-impedance films [40], as the film resistance, which
determines the kinetic inductance, is sensitive to evaporation parameters.

In this work, we developed a high-vacuum-compatible wireless ohmmeter with an
independently controlled rotary shutter, allowing for _in situ_ measurements
of the sheet resistance of the deposited film. This method allows us to
reliably realise superconducting coplanar waveguide (CPW) grAl resonators with
impedance exceeding $13\text{\,}\mathrm{k\SIUnitSymbolOhm}$, even reaching
$Z=$22.3\pm 0.3\text{\,}\mathrm{k\SIUnitSymbolOhm}$$, thanks to the large
sheet kinetic inductance up to $L_{\text{k}}=$2.7\pm
0.1\text{\,}\mathrm{nH}\text{\,}{\mathrm{\Box}}^{-1}$$. These resonators offer
magnetic field resilience of $B_{\perp}^{\text{max}}=$281\pm
1\text{\,}\mathrm{mT}$$ and $B_{\parallel}^{\text{max}}=$3.50\pm
0.05\text{\,}\mathrm{T}$$.

We integrate a grAl CPW resonator with a double quantum dot (DQD) device
fabricated on a planar Ge/SiGe heterostructure confining holes, whose large
spin-orbit interaction enables fast and all-electric control of spin qubits
[41]. By using a resonator with a characteristic impedance of
$7.9\text{\,}\mathrm{k\SIUnitSymbolOhm}$ and sheet kinetic inductance of
$800\text{\,}\mathrm{pH}\text{\,}{\mathrm{\Box}}^{-1}$, we demonstrate a
strong hole-photon coupling with a rate of $g_{\text{c}}/2\pi=$566\pm
2\text{\,}\mathrm{MHz}$$ and cooperativity of $C=$251\pm 8$$.

##  II Granular Aluminium

Figure 1: grAl microstructure, evaporation characteristics and developed
holder. a Scanning transmission electron microscopy (STEM) electron energy
loss spectroscopy (EELS) core-loss qualitative compositional maps of Al (red),
O (green), and Si (blue) of a
$2.5\text{\,}\mathrm{k\SIUnitSymbolOhm}\text{\,}{\mathrm{\Box}}^{-1}$,
$25\text{\,}\mathrm{nm}$-thin grAl film deposited on a Si substrate, oriented
along the [011] zone axis. The oxygen map shows an aluminium oxide matrix with
embedded crystalline Al particles. The top $5\text{\,}\mathrm{nm}$ of the film
is fully oxidised. b Oxygen flow dependence of the sheet resistance
$R_{\textrm{s}}$ of a grAl film evaporated with a deposition rate of
$1\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-1}$ on a test glass piece with
10 squares as measured with a multimeter after the evaporation. The line is an
exponential fit. The red colour denotes the area above the SIT. c Sheet
resistance $R_{\textrm{s}}$ aiming at
$2.5\text{\,}\mathrm{k\SIUnitSymbolOhm}\text{\,}{\mathrm{\Box}}^{-1}$ for the
respective quadrants of the developed sample holder and without (w/o, blue)
the developed holder. The violin plots denote the distribution, points the
means and bars the standard deviation. Although the median value without the
holder is
$2.08\text{\,}\mathrm{k\SIUnitSymbolOhm}\text{\,}{\mathrm{\Box}}^{-1}$, the
sheet resistances spread from
$60\text{\,}\mathrm{\SIUnitSymbolOhm}\text{\,}{\mathrm{\Box}}^{-1}$ to
$16\text{\,}\mathrm{k\SIUnitSymbolOhm}\text{\,}{\mathrm{\Box}}^{-1}$, with the
mean value of
$2.64\text{\,}\mathrm{k\SIUnitSymbolOhm}\text{\,}{\mathrm{\Box}}^{-1}$ and a
standard deviation of
$2.3\text{\,}\mathrm{k\SIUnitSymbolOhm}\text{\,}{\mathrm{\Box}}^{-1}$. With
the holder, the resulting sheet resistance dispersion is significantly
decreased at the expense of the defined film thickness, as shown in the inset.
The resistances in the fourth quadrant are spread between
$1.68\text{\,}\mathrm{k\SIUnitSymbolOhm}\text{\,}{\mathrm{\Box}}^{-1}$ to
$4.42\text{\,}\mathrm{k\SIUnitSymbolOhm}\text{\,}{\mathrm{\Box}}^{-1}$, with
median of
$2.61\text{\,}\mathrm{k\SIUnitSymbolOhm}\text{\,}{\mathrm{\Box}}^{-1}$, mean
of $2.79\text{\,}\mathrm{k\SIUnitSymbolOhm}\text{\,}{\mathrm{\Box}}^{-1}$ and
standard deviation of
$0.8\text{\,}\mathrm{k\SIUnitSymbolOhm}\text{\,}{\mathrm{\Box}}^{-1}$. The
median of the thicknesses $25.4\text{\,}\mathrm{nm}$ matches the desired value
of $25\text{\,}\mathrm{nm}$, while they spread around the mean of
$27.1\text{\,}\mathrm{nm}$ with a standard deviation of
$8.8\text{\,}\mathrm{nm}$. d Schematics of the sample holder allowing _in
situ_ resistance measurements through a wireless connection. An independent
rotary shutter divides the holder into four quadrants. The rectangular test
pieces (green, red, purple and orange) have gold-plated ends connected via
cables (brown) with the two-probe measurement circuit inside the holder
through feedthroughs (gold). A square sample is shown in blue.

Granular aluminium has been studied for almost 60 years [42, 43] but has
recently attracted increased attention as a material of interest for qubits,
amplifiers and detectors [38, 1]. It consists of small crystalline grains of
$\approx$4\text{\,}\mathrm{nm}$$ in diameter (Fig. 1a, red and Extented Fig.
LABEL:fig:fige1) embedded in an amorphous aluminium oxide matrix (Fig. 1a,
green). The aluminium grains separated by an insulating barrier can be
modelled as a network of Josephson junctions, resulting in high kinetic
inductance [38]. The kinetic inductance of superconductors arises from the
inertia of moving Cooper pairs and increases with normal state resistance.
However, it cannot be increased infinitely; it is upper-bound by a
superconducting-to-insulator phase transition (SIT). For grAl films
$\sim$20\text{\,}\mathrm{nm}$$ thin, the SIT has been measured around
$5\text{\,}\mathrm{k\SIUnitSymbolOhm}\text{\,}{\mathrm{\Box}}^{-1}$ [44].

Figure 2: Characteristic impedance, quality factors and magnetic field
resilience of grAl CPW resonators. a Optical and b, c scanning electron
microscope images of six $200\text{\,}\mathrm{nm}$-wide grAl resonators side-
coupled to a feedline. Due to large sheet kinetic inductance $L_{\textrm{k}}$,
their length does not exceed $200\text{\,}\mathrm{\SIUnitSymbolMicro m}$. The
Nb ground plane around the grAl resonator is patterned with a vortex-trapping
hexagonal array of circular holes with $\sim$ $150\text{\,}\mathrm{nm}$
diamater [45]. d Magnitude $|S_{21}|$ of six bare resonators side-coupled to a
common feedline. The impedance of these resonators is extracted to be
$Z=$9.4\pm 0.2\text{\,}\mathrm{k\SIUnitSymbolOhm}$$ and the sheet kinetic
inductance to be $L_{\text{k}}=$916\pm
40\text{\,}\mathrm{pH}\text{\,}{\mathrm{\Box}}^{-1}$$. e Average photon number
dependence of the internal quality factor for $100\text{\,}\mathrm{nm}$\- and
$200\text{\,}\mathrm{nm}$-wide resonators. The impedance is $13.2\pm
0.5\text{\,}\mathrm{k\SIUnitSymbolOhm}$ [$L_{\text{k}}=$817\pm
62\text{\,}\mathrm{pH}\text{\,}{\mathrm{\Box}}^{-1}$$] and $8.9\pm
0.2\text{\,}\mathrm{k\SIUnitSymbolOhm}$ [$L_{\text{k}}=$816\pm
28\text{\,}\mathrm{pH}\text{\,}{\mathrm{\Box}}^{-1}$$] respectively. f Center
conductor width dependence of the characteristic impedance for different sheet
kinetic inductances. The circles correspond to resonators fabricated on Si,
while triangles to Ge/SiGe heterostructures with the Ge quantum well etched
away. The empty circles indicate samples where most resonators did not
resonate. The dashed lines represent the impedance calculated from Eq. 1 for
$25\text{\,}\mathrm{\SIUnitSymbolMicro m}$ separation of the center conductor
and the ground plane and relative permittivity of 11.8 (Methods). g Sheet
kinetic inductance dependence of the characteristic impedance for different
resonator widths. h (i) Applied out-of-plane (in-plane) magnetic field
dependence of the relative change in resonance frequency for different
resonator widths. The inset in h shows the center conductor width dependence
of the maximum out-of-plane field before the internal quality factor
significantly drops following $1.65\Phi_{0}/w^{2}$ (black line). Corresponding
quality factor dependencies are shown in Extented Fig. LABEL:fig:fige4.

We prepare grAl by room temperature electron-beam evaporation of aluminium in
an oxygen atmosphere at $\sim$5\text{\times}{10}^{-5}\text{\,}\mathrm{mbar}$$
pressure at a rate of $1\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-1}$. The
resulting resistivity of the evaporated film depends on the oxygen flow and
the evaporation rate. To determine the sheet resistance, we initially used a
glass sample of 10 squares and measured its resistance using a multimeter
after the evaporation. The outcome of these evaporations is summarized in Fig.
1b. It is observed that the sheet resistance exponentially depends on the
oxygen flow for a given evaporation rate and is not reproducible. The poor
reproducibility is quantified in the blue points plot in Fig. 1c, which shows
individual evaporations of resonators and cQED samples targeting
$2.5\text{\,}\mathrm{k\SIUnitSymbolOhm}\text{\,}{\mathrm{\Box}}^{-1}$. Given
the poor reproducibility even for nominally identical consecutive
evaporations, it is challenging to systematically fabricate high-impedance CPW
resonators which operate within the designed range.

To overcome this experimental difficulty, a hermetically sealed high-vacuum
compatible wireless ohmmeter has been developed (Methods and Extented Fig.
LABEL:fig:fige2). It allows the _in situ_ two-probe electrical measurement of
the growing film resistance and wireless transmission of the data. This
enables the termination of the evaporation when the desired value is reached.
Such a procedure yields a reproducible resistance at the expense of the
uncertainty in the final film thickness. This can be decreased by equipping
the ohmmeter with an independent rotary shutter, as shown in Fig. 1d. It masks
three-quarters of the sample holder and is wirelessly controlled. In this
manner, three test samples are evaporated before the sample of interest
without having to break vacuum, thus allowing the operator to fine-tune the
oxygen flow and reach the desired film properties. As shown in Fig. 1c, this
procedure significantly reduces the uncertainty of both resistance and
thickness, making it possible to target the desired resonance frequency range.
These results demonstrate a successful implementation of our technical
solution, which is also transferable to other disordered superconductors and,
in general, thin-film systems.

##  III Granular aluminium resonators

We utilize the developed method to realize high-impedance grAl CPW $\lambda/2$
resonators. We first test the resonators on bare $\langle 100\rangle$ silicon
($\rho=$15\text{\,}\mathrm{\SIUnitSymbolOhm}\text{\,}\mathrm{cm}$$)
substrates. We intentionally choose low resistivity substrates as those
resonators are developed for use with semiconductor heterostructures with much
higher losses than highly resistive silicon or sapphire wafers [46, 27, 47].
The ground plane and feedline are comprised on Nb, with negligable kinetic
inductance (Methods) and employ different CPW geometries. The hanger geometry
(Fig. 2a) is preferred since it allows testing multiple resonators per chip,
as well as the precise determination of internal losses and coupling rates.
Figure 2b shows a scanning electron microscope image of a single grAl
resonator side-coupled to a common Nb feedline. Two strategies are employed in
order to maximise the impedance: first, increasing the kinetic inductance via
the sheet resistance and, second, decreasing the width of the center conductor
from $10\text{\,}\mathrm{\SIUnitSymbolMicro m}$ to $100\text{\,}\mathrm{nm}$.
Additionally, such narrow resonators exhibit an increased out-of-plane
magnetic field resilience [26], beneficial for materials with low in-plane
$g$-factors [41].

In Figs. 2f, g we summarise the characteristic impedances of the fabricated
resonators. Leveraging the developed wireless ohmmeter holder, we reproducibly
target sheet resistances of
$2.5\text{\,}\mathrm{k\SIUnitSymbolOhm}\text{\,}{\mathrm{\Box}}^{-1}$ and
obtain films with a kinetic inductance reaching $2.7\pm
0.1\text{\,}\mathrm{nH}\text{\,}{\mathrm{\Box}}^{-1}$. The impedance scales
with the kinetic inductance $L_{\text{k}}$ and the center conductor width $w$
according to

$Z\left(L_{\text{k}},w\right)=\sqrt{\frac{L_{\ell}}{C_{\ell}}}=\sqrt{\frac{L_{\ell}^{\text{k}}+L_{\ell}^{\text{g}}}{C_{\ell}}}=\sqrt{\frac{\frac{L_{\text{k}}}{w}+L_{\ell}^{\text{g}}\left(w\right)}{C_{\ell}^{\text{g}}\left(w\right)}},$
(1)

with $L_{\ell}$ ($C_{\ell}$) the inductance (capacitance) per length. For a
superconducting resonator, the inductance per length
$L_{\ell}=L_{\ell}^{\text{k}}+L_{\ell}^{\text{g}}$ is composed of a geometric
$L_{\ell}^{\text{g}}$ and a kinetic $L_{\ell}^{\text{k}}=L_{\text{k}}/w$
contribution (Methods). The highest impedance is reached for widths below
$200\text{\,}\mathrm{nm}$. However, we observe a limitation with three
$100\text{\,}\mathrm{nm}$-wide samples, with most devices failing to respond
to microwave excitation. These samples are highlighted with empty circles in
Figs. 2f, g. The highest characteristic impedance observed for such a sample
is $22.3\pm 0.3\text{\,}\mathrm{k\SIUnitSymbolOhm}$. Nevertheless, other
$100\text{\,}\mathrm{nm}$-wide samples demonstrate reliable performance.
Specifically, five samples comprising 20 resonators with impedance ranging
from $13\text{\,}\mathrm{k\SIUnitSymbolOhm}$ to
$17.3\text{\,}\mathrm{k\SIUnitSymbolOhm}$ show a $100\text{\,}\mathrm{\char
37\relax}$ yield. These results demonstrate that the impedance of the grAl CPW
resonators can reproducibly exceed $13\text{\,}\mathrm{k\SIUnitSymbolOhm}$,
strongly enhancing quantum fluctuations of the voltage.

We next measure the complex frequency response with respect to the driving
power and evaluate the quality factors as a function of the photon number
(Methods). Fig. 2e displays a power sweep of two resonators with widths of
$100\text{\,}\mathrm{nm}$ and $200\text{\,}\mathrm{nm}$. The resonators retain
a high-quality factor above 104 in the single photon regime, even with
impedance exceeding $13\text{\,}\mathrm{k\SIUnitSymbolOhm}$. We conclude that
high impedance grAl resonators exhibit losses that would not limit potential
cQED experiments, with loss rates in the order of $\mathrm{MHz}$, which is
comparable with the state-of-the-art charge/spin qubits decoherence rates
[15].

Figure 3: Integration of a $7.9\text{\,}\mathrm{k\SIUnitSymbolOhm}$ grAl CPW
resonator with germanium quantum dots. a Optical microscope image of a hybrid
device implementing a grAl reflection resonator with a triple quantum dot
electrostatically defined by nine gates. In this experiment, it was operated
as a DQD, as indicated in the schematic cross-section of the device in the
inset. b Scanning electron microscope image showing detail of the region
highlighted with blue in a. The resonator grAl centre conductor (dark blue),
which is designed to be $\approx$111\text{\,}\mathrm{\SIUnitSymbolMicro m}$$
long and $200\text{\,}\mathrm{nm}$-wide, was evaporated with
$25\text{\,}\mathrm{nm}$ thickness. The red region highlights the mesa. c
Scanning electron microscope image showing detail of the region highlighted
with red in b. d Average photon number dependence of the internal quality
factor for the resonator implemented in a hybrid device with metallic gates
and a Ge hole gas in the vicinity of the tip of the resonator. The impedance
of the resonators is $\approx$7.9\text{\,}\mathrm{k\SIUnitSymbolOhm}$$. e
Stability diagram of the right DQD sensed by the reflected amplitude at the
drive frequency $\approx$2\text{\,}\mathrm{MHz}$$ below the resonance
frequency as a function of the two plunger gates defining the right DQD. The
number of confined holes is unknown. f A single interdot transition as seen by
the reflected amplitude. The black dashed lines highlight the range of
$V_{\text{G}}^{\text{L}}$, which is used for further analysis. g Line cut
taken along $V_{\text{G}}^{\text{R}}$ from the indicated range, centered
around zero DQD detuning representing the average of $\sim$ 20 line cuts,
showing the detuning dependence of the reflection amplitude.

To evaluate the potential of the developed CPW resonators for spin-photon
coupling experiments, we investigate their magnetic field behaviour. Since the
$g$-factor of holes is highly anisotropic, with rather small in-plane and
large out-of-plane $g$-factors [48], we probe their behaviour for both
magnetic field directions. In Fig. 2h, i, the relative frequency shift and the
internal quality factors as a function of the applied magnetic fields for
resonators with hundreds of photons are plotted. The superconducting depairing
parabolically lowers the resonance frequency due to increased kinetic
inductance, stemming from the decreasing superconducting gap. Furthemore, the
magnetic field induces losses by introducing Abrikosov vortices in the film,
whose dynamics contribute to microwave losses and resonator instability. The
creation of vortices is suppressed with narrow center conductors. Thus, the
maximal out-of-plane field the resonators withstand before the deterioration
$B_{\perp}^{\text{max}}$ increases with decreasing widths, and it
approximately follows
$B_{\perp}^{\text{max}}=1.65\Phi_{0}/d^{2}\leavevmode\nobreak\ $[49], where
$\Phi_{0}=h/\left(2e\right)$ is the superconducting flux quantum and $d$ is
the dimension perpendicular to the applied field (inset Fig. 2g). We note that
for $100\text{\,}\mathrm{nm}$-wide resonators, the magnetic field resilience
reaches $B_{\perp}^{\text{max}}=$281\pm 1\text{\,}\mathrm{mT}$$ out-of-plane.
Moreover, since the creation of vortices is also suppressed with thin films in
the in-plane field, resonators with thickness below $25\text{\,}\mathrm{nm}$
withstand $3\text{\,}\mathrm{T}$, the thinnest reaching
$B_{\parallel}^{\text{max}}=$3.50\pm 0.05\text{\,}\mathrm{T}$$.

In the context of spin qubits, the demonstrated combination of high impedance
and magnetic field resilience makes grAl a strong candidate for a long-range
coherent coupler. For silicon-based spin qubits ($g$-factor of 2), the spin
transition can be easily brought to the $48\text{\,}\mathrm{GHz}$ range where
microwave resonators are routinely operated. Moreover, such magnetic field
resilience allows one to readily implement grAl resonators with spin qubits
hosted in materials with highly anisotropic $g$-factors, such as Ge. In this
case, the spin qubit could be operated in an arbitrary magnetic field
orientation, an important feature of operating the qubit in the parameter
range where its coherence is maximized [50].

##  IV Strong charge-photon coupling

The strong coupling condition requires that the coherent coupling strength
$g_{c}$ exceeds both the charge ($\gamma$) and the resonator ($\kappa$) decay
rates. To coherently couple a charge with a photon, we integrate a grAl
resonator with a DQD formed in a Ge/SiGe heterostructure [51] (Figs. 3a to c).
We opted for a reflection geometry, which improves the signal-to-noise by a
factor of two relative to the hanger geometry, where the signal can leak back
to the input port [52]. The resonance frequency is measured to be
$7.262\text{\,}\mathrm{GHz}$, yielding a characteristic impedance of
$7.9\text{\,}\mathrm{k\SIUnitSymbolOhm}$ and sheet kinetic inductance of
$800\text{\,}\mathrm{pH}\text{\,}{\mathrm{\Box}}^{-1}$. The resonator is
designed to be over-coupled, so the coupling quality factor $Q_{\text{c}}$ is
$534.8\pm 4.7\text{\,}\mathrm{,}$ while the internal quality factor
$Q_{\text{i}}$ is $2000.4\pm 84.7\text{\,}\mathrm{w}$ith $\approx 1.9$ average
photons in the resonator, leading to the resonator field decay rate of
$\kappa/2\pi=$17.2\pm 0.3\text{\,}\mathrm{MHz}$$. The quality factors are
significantly smaller compared to a bare resonator. As described in Ref. [47],
low-impedance resonators fabricated on Ge-rich substrates featuring etched
quantum wells exhibit reduced internal quality factors of $\sim$ 104 in the
single photon regime [47]. Moreover, the metallic gates for defining the
quantum dots will cause significant microwave leakage, even more pronounced
for high characteristic impedance [27].

The triple quantum dot array is operated as a DQD formed on the right side,
with the two left-most gates grounded. The DQD is first tuned by measuring a
DC current. As observed in the stability diagram in Fig. 3e, the resonator is
dispersively shifted at the interdot transitions, _i.e._ when the DQD dipole
moment becomes sizeable, allowing for direct microwave readout of individual
interdot transitions (Fig. 3f). Line cuts along the right gate voltage
translated to detuning through lever arms (Extented Fig. LABEL:fig:fige3 and
the Methods) exhibit a sharp response in the complex resonator signal due to
the interaction with the DQD. The clear dispersive interaction between the DQD
and the resonator already hints towards a significant charge-photon
interaction strength. However, this type of measurement does not provide
sufficient information to extract the system parameters reliably [53].

In contrast, sweeping the drive frequency while stepping the DQD detuning
reveals the spectrum of the hybridized states (Fig. 4a), allowing for accurate
characterization of the system. Thus, for a different interdot transition, we
probe the system by reflection spectroscopy. A parabolic spectrum is expected
for a DQD occupied with an odd number of holes. A clear avoided crossing is
observed when the parabolic DQD charge transition frequency equals the
frequency of the resonator ($2t_{\text{c}}/h\approx\omega_{\text{r}}/2\pi$),
as seen in Fig. 4a. Since the strong coupling condition
$g_{\text{c}}>(\kappa,\gamma)$ is met, two well-resolved dressed states
separated by $2g_{\text{c}}/2\pi$ emerge. The resonant vacuum Rabi splitting
is observed (Fig. 4b), which constitutes the experimental evidence of the
strong coupling regime. From Fig. 4b we extract $g_{\text{c}}/2\pi$ of $566\pm
2\text{\,}\mathrm{MHz}$, with $\gamma/2\pi$ of $297\pm
9\text{\,}\mathrm{MHz}$, demonstrating strong charge-photon coupling with a
cooperativity $C=4g_{\text{c}}^{2}/\left(\gamma\kappa\right)$ of $251\pm 8$
[54]. The coupling strength agrees with the expected value
$g_{\text{c}}^{\text{e}}/2\pi=e\beta\left(\omega_{\text{r}}/2\pi\right)\sqrt{Z/\pi\hbar}/2\approx$570\text{\,}\mathrm{MHz}$$
with $e$ being the elementary charge, given a differential lever arm
$\beta=0.2$, $\omega_{\text{r}}/2\pi=$7.27\text{\,}\mathrm{GHz}$$ and
$Z=$8\text{\,}\mathrm{k\SIUnitSymbolOhm}$$.

Figure 4: Strong charge-photon coupling. a Resonator microwave spectroscopy as
a function of drive frequency $\omega_{\text{d}}$ and DQD detuning
$\varepsilon$. Two well-isolated dressed states are observed. At
$\varepsilon=0$ and with the charge transition set resonant with the
resonator, the vacuum Rabi splitting is observed, demonstrating the strong
coupling regime. The dotted lines show the bare resonator frequency and
energies of the bare charge transition, while the dashed lines show the
dressed transitions. The inset shows a theoretical reproduction of the
experimental data (Methods). b Magnitude line cuts along the frequency axes,
taken at the detunings indicated by the dashed blue and red labels. A fit to
the line-cut at zero detuning in the right panels gives
$(g_{\text{c}}/2\pi,\gamma/2\pi,t_{\text{c}}/h)=(566,297,3626)\pm(2,9,2)$ MHz,
yielding a cooperativity of $251\pm 8$. The red curve is offset by 0.15 for
clarity.

##  V Discussion and outlook

We have developed a wireless ohmmeter which enables us to _in situ_ define the
sheet resistance of grAl CPWs, allowing for reproducible fabrication of grAl
resonators with characteristic impedance exceeding
$13\text{\,}\mathrm{k\SIUnitSymbolOhm}$, even reaching
$22\text{\,}\mathrm{k\SIUnitSymbolOhm}$. The reported values are more than
four times higher than the state-of-the-art for semiconductor circuit quantum
electrodynamics experiments [17].

Furthermore, we have shown that grAl resonators remain low-loss and are
resilient to magnetic field up to $3.5\text{\,}\mathrm{T}$ in-plane and
$281\text{\,}\mathrm{mT}$ out-of-plane. By integrating a
$7.9\text{\,}\mathrm{k\SIUnitSymbolOhm}$ grAl resonator with a DQD formed in
planar Ge we demonstrate strong hole-photon coupling with a rate of
$g_{\text{c}}/2\pi=$566\pm 2\text{\,}\mathrm{MHz}$$. The demonstrated charge-
photon coupling strength and magnetic field resilience of the grAl resonators
combined with the strong spin-orbit coupling opens the path towards strong
spin-photon coupling in planar Ge [41]. Even for modest spin-orbit
interaction, similar to that created by a gradient magnetic field of a
micromagnet employed in Si [23, 24], spin-photon coupling rates could exceed
$100\text{\,}\mathrm{MHz}$. Such high rates suggest that two-qubit operations
between distant hole spins with high fidelities are within reach [55, 56, 25].

In conclusion, we have proposed controllably deposited grAl as an alternative
to Josephson junction arrays and atomically disordered superconductors for
semiconductor-superconductor circuit quantum electrodynamics experiments. Its
key advantages lie in being high impedance, magnetic field resilient, lift-off
compatible and low-loss. By further optimising the design to reach
$\beta\approx 0.25$ and using an impedance of
$16\text{\,}\mathrm{k\SIUnitSymbolOhm}$, charge-photon coupling strengths
above $1\text{\,}\mathrm{GHz}$ are within reach. Moreover, the impedance can
be increased beyond these values by meandering the grAl resonator and
utilizing the mutual geometric inductance [34, 3]. Given the state-of-the-art
quantum dot technologies and using a grAl resonator _ceteris paribus_ , spin-
photon coupling could enter the ultra-strong coupling regime, following in the
footsteps of the superconducting qubits and opening an avenue towards
unexplored physics [57] and advanced quantum information processing
applications.

## Methods

### Vacuum-compatible wireless ohmmeter

The wireless ohmmeter was designed to fit the MEB550S Plassys HV electron-beam
evaporator, equipped with a motorized evaporation stage. The stage can tilt
between the loading and evaporation positions and rotate along the evaporation
axis to ensure a uniform distribution of the deposited thin film. The ohmmeter
was designed to communicate wirelessly with the handheld device outside the
vacuum chamber so as not to restrict the movement of the rotary stage. The
designed device uses the surface of the ohmmeter enclosure lid as a mounting
surface for the samples and measuring probes. The ohmmeter circuit is placed
inside the hermetically sealed enclosure, and the measuring electrodes are
connected to the ohmmeter through hermetically sealed electrical cable
feedthroughs. The wireless _in situ_ resistance measurement enables the user
to monitor the sheet resistance of the deposited film and to interrupt the
process when the desired resistance is reached.

An independently controlled rotary shutter is magnetically coupled to a servo
motor enclosed inside the holder, as magnetic coupling is a straightforward
method for transferring motion from a vacuum-non-compatible motor to the high
vacuum side.

The battery-powered wireless ohmmeter includes components to precisely measure
the growing thin film resistance and transmit the measured values to the
recording device outside the vacuum chamber. The design is based on the
ATmega2560 microcontroller to communicate with the rest of the components,
which include a 2.4GHz wireless transceiver, a barometer to record the
temperature and pressure inside the hermetic enclosure for detecting potential
pressure leaks, a servo motor to move the magnetically coupled shutter, and a
precision analog-to-digital converter to measure resistance using the constant
voltage method. The highest sensitivity is achieved by setting the range
resistor closest to the measured value. Since the resistance of the deposited
grAl spans several orders of magnitude, we use a set of four different
resistors selected by switching mechanical relays to cover a large measurement
range.

### Sample fabrication

The bare resonators are fabricated on $\langle 100\rangle$ silicon substrates
($\rho\approx$15\text{\,}\mathrm{\SIUnitSymbolOhm}\text{\,}\mathrm{cm}$$).
First, 5/$60\text{\,}\mathrm{nm}$ Cr/Au alignment markers are patterned using
$100\text{\,}\mathrm{keV}$ electron beam lithography (EBL), high-vacuum (HV)
electron beam (e-beam) evaporation and lift-off process. Next, a
$20\text{\,}\mathrm{nm}$ Nb feedline and ground plane are defined using EBL,
ultra-high vacuum (UHV) e-beam evaporation and lift-off. The grAl is not
employed in this layer since achieving $50\text{\,}\mathrm{\SIUnitSymbolOhm}$
matching with a kinetic inductance of
$2\text{\,}\mathrm{nH}\text{\,}{\mathrm{\Box}}^{-1}$ would require a
$1\text{\,}\mathrm{mm}$-wide feedline separated from the ground plane by
$100\text{\,}\mathrm{nm}$. Finally, the $\sim$25\text{\,}\mathrm{nm}$$ grAl
center conductors are patterned with EBL, HV e-beam evaporation in an oxygen
atmosphere and lift-off. After the evaporation, the film is subject to a
$5\text{\,}\mathrm{min}$ static post-oxidation.

The hybrid cQED devices are fabricated on a Ge/SiGe heterostructure grown by
low-energy plasma-enhanced chemical vapour deposition with forward grading
[51]. The two-dimensional hole gas is self-accumulated in the Ge quantum well
$\approx$20\text{\,}\mathrm{nm}$$ below the surface.

The process is as follows: First, the ohmic contacts are patterned with EBL.
Before the deposition of $60\text{\,}\mathrm{nm}$ of Pt at an angle of
$5\text{\,}\mathrm{\SIUnitSymbolDegree}$, a few nanometres of native oxide and
SiGe spacer are removed by argon bombardment. Next, the hole gas is dry-etched
with SF6-O2-CHF3 reactive ion etching everywhere except a small
$\sim$60\text{\,}\mathrm{nm}$$ high mesa area extending to two bonding pads to
form a conductive channel. Subsequently, the native SiO2 is removed by a
$10\text{\,}\mathrm{s}$ dip in buffered HF before the gate oxide is deposited.
The oxide is a $\sim$10\text{\,}\mathrm{nm}$$ atomic-layer-deposited aluminium
oxide (Al2O3) film grown at $300\text{\,}\mathrm{\SIUnitSymbolCelsius}$. Next,
a $20\text{\,}\mathrm{nm}$ Nb feedline and ground plane are fabricated with
EBL, UHV evaporation and lift-off. The separation of the grAl center conductor
from the Nb ground plane is $25\text{\,}\mathrm{\SIUnitSymbolMicro m}$ for
resonators with a width less than $1\text{\,}\mathrm{\SIUnitSymbolMicro m}$.
Further retraction of the ground plane would not increase the impedance
appreciably. The single-layer top gates for defining the QDs are patterned in
three different steps of EBL, evaporation and lift-off. The barrier and
plunger gates are defined separately close to the edge of the mesa with Ti/Pd
3/$20\text{\,}\mathrm{nm}$. An additional Ti/Pd
3/$97\text{\,}\mathrm{nm}$-thick gate metal layer defines microwave filters
and the bonding pads, connects to the previously defined gates and overcomes
the slanted mesa edge. Finally, the grAl center conductors are defined as in
the bare resonator samples.

### Measurement setup

The reported low-temperature measurements were performed in a cryogen-free
dilution refrigerator with a base temperature of $10\text{\,}\mathrm{mK}$. The
sample was mounted on a custom-printed circuit board thermally anchored to the
mixing chamber of the cryostat. The electrical connections were made via wire
bonding. The schematic of the measurement setup is shown in Extented Fig.
LABEL:fig:fige5.

### Impedance evaluation

Since the characteristic impedance cannot be measured directly, we extract it
from the resonance frequency. $C_{\ell}=C_{\ell}^{\text{g}}\left(w\right)$ and
$L_{\ell}^{\text{g}}\left(w\right)$ in Eq. 1 are solely given by the design
parameters (the center conductor width $w$, the separation between the ground
plane and the center conductor $s$ and the relative permittivity of the
substrate $\varepsilon_{\text{r}}$) and can be analytically calculated. We
extract the characteristic impedance from the resonance frequency, which is
given as

$\omega_{\text{r}}=\frac{1}{\sqrt{L_{\lambda/2}\left(C_{\lambda/2}+C_{\text{c}}\right)}},$
(2)

where $L_{\lambda/2}=\left(2\ell/\pi^{2}\right)L_{\ell}$ and
$C_{\lambda/2}=\left(\ell/2\right)C_{\ell}$ are lumped-element equivalents of
the distributed parameters of the resonator. For longer, lower-impedance
resonators, the coupling capacitance $C_{\text{c}}$ is usually dominated by
$C_{\lambda/2}$ and can be neglected. In the case of our compact high-
impedance, they are of the same order of magnitude, and both have to be
considered. We simulate the coupling capacitance using COMSOL electrostatic
simulations. We measure the complex frequency response of a resonator and fit
the signal to evaluate the resonance frequency. We calculate the sheet kinetic
inductance $L_{\text{k}}$ by numerically solving Eq. 2 with the simulated
coupling capacitance.

### Fitting

The Heisenberg-Langevin equation of motion for a bare hanger resonator in the
rotating frame is

$\displaystyle\dot{\hat{a}}$
$\displaystyle=-\frac{i}{\hbar}[\hat{a},\hbar(\omega_{\text{r}}-\omega_{\text{d}})\hat{a}^{\dagger}\hat{a}]-\frac{\kappa}{2}\hat{a}-\sqrt{\frac{\kappa_{\text{c}}}{2}}\hat{b}_{\text{in}}$
$\displaystyle=-i(\omega_{\text{r}}-\omega_{\text{d}})\hat{a}-\frac{\kappa}{2}\hat{a}-\sqrt{\frac{\kappa_{\text{c}}}{2}}\hat{b}_{\text{in}},$
(3)

where $\hat{a}$ is the resonator field and $\hat{b}_{\text{in}}$ is the input
field. For the steady state ($\dot{\hat{a}}=0$), it becomes

$\hat{a}=\frac{-\sqrt{\frac{\kappa_{\text{c}}}{2}}}{i(\omega_{\text{r}}-\omega_{\text{d}})+\frac{\kappa}{2}}\hat{b}_{\text{in}}.$
(4)

Using the input-output relation
$\hat{b}_{\text{out}}=\hat{b}_{\text{in}}+\sqrt{\frac{\kappa_{\text{c}}}{2}}\hat{a}$,
where $\hat{b}_{\text{out}}$ is the output field, yields the complex
transmission of a bare hanger-type resonator

$S_{21}(\omega_{\text{d}})=\frac{\langle\hat{b}_{\text{out}}\rangle}{\langle\hat{b}_{\text{in}}\rangle}=1-\frac{\kappa_{\text{c}}e^{i\phi}}{\kappa+2i\left(\omega_{\text{r}}-\omega_{\text{d}}\right)},$
(5)

with an inserted term $e^{i\phi}$ accounting for the impedance mismatch.
Before fitting, the feedline transmission S21 is first normalized by a
background trace to remove the standing wave pattern obtained from a scan with
a shifted resonance frequency, either due to a magnetic field or a DQD-induced
dispersive shift. The average intraresonator photon number at resonance was
calculated as

$\langle
n\rangle=\frac{2}{\hbar\omega_{\text{r}}}\frac{\kappa_{\text{c}}}{\kappa^{2}}P_{\text{d}},$
(6)

where $P_{\text{d}}$ is the drive power after accounting for input
attenuation. In the case of a single port resonator, Eq. 6 gains an extra
factor of two.

For a DQD occupied with an odd number of holes, the cavity-DQD Hamiltonian in
the rotating frame and rotating wave approximation can be described by

$\hat{H}/\hbar=\left(\omega_{\text{r}}-\omega_{\text{d}}\right)\hat{a}^{\dagger}\hat{a}+\frac{\left(\omega_{\text{q}}-\omega_{\text{d}}\right)}{2}\hat{\sigma}_{z}+g_{\text{eff}}\left(\hat{a}^{\dagger}\hat{\sigma}_{-}+\hat{a}\hat{\sigma}_{+}\right),$
(7)

where $\omega_{\text{d}}$ is the drive frequency, $\omega_{\text{r}}$ is the
resonance frequency, $\omega_{\text{q}}\leavevmode\nobreak\
=\leavevmode\nobreak\ \sqrt{\varepsilon^{2}+4t_{\text{c}}^{2}}/h$ is the
charge transition frequency with $\varepsilon$ being the DQD detuning and
$t_{\text{c}}$ the tunnel coupling.
$g_{\text{eff}}=g_{\text{c}}\frac{2t_{\text{c}}}{\hbar\omega_{\text{q}}}$ is
the effective charge-photon coupling strength, $\hat{a}^{\dagger}$ ($\hat{a}$)
is the photon creation (annihilation) operator and $\hat{\sigma}_{z}$ is the
Pauli $z$ matrix and $\hat{\sigma}_{-}$ ($\hat{\sigma}_{+}$) is the qubit
lowering (raising) operator.

The Heisenberg-Langevin equations of motion for a system described by Eq. 7
are [58]

$\displaystyle\dot{\hat{a}}$
$\displaystyle=-\frac{i}{\hbar}[\hat{a},\hat{H}]-\kappa\hat{a}-\sqrt{\kappa_{\text{c}}}\hat{b}_{\text{in}}$
$\displaystyle=-i(\omega_{\text{r}}-\omega_{\text{d}})\hat{a}-\kappa\hat{a}-\sqrt{\kappa_{\text{c}}}\hat{b}_{\text{in}}-ig_{\text{eff}}\hat{\sigma}_{-},$
(8) $\displaystyle\dot{\hat{\sigma}}_{-}$
$\displaystyle=-\frac{i}{\hbar}[\hat{\sigma}_{-},\hat{H}]-\frac{\gamma}{2}\hat{\sigma}_{-}$
$\displaystyle=-i(\omega_{\text{q}}-\omega_{\text{d}})\hat{\sigma}_{-}-\frac{\gamma}{2}\hat{\sigma}_{-}-ig_{\text{eff}}\hat{a}.$
(9)

For the steady state ($\dot{\hat{\sigma}}_{-}=0$), Eq. 9 becomes

$\hat{\sigma}_{-}=\frac{g_{\text{eff}}}{-(\omega_{\text{q}}-\omega_{\text{d}})+i\gamma/2}\hat{a},$
(10)

which inserted in Eq. 8 with $\dot{\hat{a}}=0$ gives

$\hat{a}=\frac{-\sqrt{\kappa_{\text{c}}}}{i(\omega_{\text{r}}-\omega_{\text{d}})+\frac{\kappa}{2}+\frac{ig_{\text{eff}}^{2}}{-(\omega_{\text{q}}-\omega_{\text{d}})+i\gamma/2}}\hat{b}_{\text{in}}.$
(11)

Using
$\hat{b}_{\text{out}}=\hat{b}_{\text{in}}+\sqrt{\kappa_{\text{c}}}\hat{a}$
yields the complex reflection of a single-port resonator coupled to a DQD
charge transition

$S_{11}(\omega_{\text{d}})=\frac{\langle\hat{b}_{\text{out}}\rangle}{\langle\hat{b}_{\text{in}}\rangle}=1-\frac{2\kappa_{\text{c}}e^{i\phi}}{\kappa+2i\left(\omega_{\text{r}}-\omega_{\text{d}}\right)+\frac{2ig_{\text{eff}}^{2}}{-(\omega_{\text{q}}-\omega_{\text{d}})+i\gamma/2}}.$
(12)

The translation between the gate voltages and DQD detuning is given by [59]

$\varepsilon=\mu_{\text{L}}-\mu_{\text{R}}=\left(\alpha_{\text{LL}}-\alpha_{\text{RL}}\right)\Delta
V_{\text{G}}^{\text{L}}-\left(\alpha_{\text{RR}}-\alpha_{\text{LR}}\right)\Delta
V_{\text{G}}^{\text{R}},$ (13)

where the lever arms are extracted from bias triangles of the DQD, shown in
Extented Fig. LABEL:fig:fige3.

## Data Availability

All data included in this work will be available at the Institute of Science
and Technology Austria repository.

## Acknowledgements

We acknowledge Franco De Palma, Mahya Khorramshahi, Fabian Oppliger, Thomas
Reisinger, Pasquale Scarlino and Xiao Xue for helpful discussions. This
research was supported by the Scientific Service Units of ISTA through
resources provided by the MIBA Machine Shop and the Nanofabrication facility.
This research and related results were made possible with the support of the
NOMIS Foundation, the HORIZON-RIA 101069515 project, the FWF Projects with
DOI:10.55776/P32235, DOI:10.55776/I5060 and DOI:10.55776/P36507. IMP
acknowledges funding from the Deutsche Forschungsgemeinschaft (DFG – German
Research Foundation) under project number 450396347 (GeHoldeQED). ICN2
acknowledges funding from Generalitat de Catalunya 2021SGR00457. We
acknowledge support from CSIC Interdisciplinary Thematic Platform (PTI+) on
Quantum Technologies (PTI-QTEP+). This research work has been funded by the
European Commission – NextGenerationEU (Regulation EU 2020/2094), through
CSIC’s Quantum Technologies Platform (QTEP). ICN2 is supported by the Severo
Ochoa program from Spanish MCIN/AEI (Grant No.: CEX2021-001214-S) and is
funded by the CERCA Programme/Generalitat de Catalunya. Part of the present
work has been performed in the framework of Universitat Autònoma de Barcelona
Materials Science PhD program. AGM has received funding from Grant
RYC2021-033479-I funded by MCIN/AEI/10.13039/501100011033 and by European
Union NextGenerationEU/PRTR. M.B. acknowledges support from SUR Generalitat de
Catalunya and the EU Social Fund; project ref. 2020 FI 00103. The authors
acknowledge the use of instrumentation and the technical advice provided by
the Joint Electron Microscopy Center at ALBA (JEMCA). ICN2 acknowledges
funding from Grant IU16-014206 (METCAM-FIB) funded by the European Union
through the European Regional Development Fund (ERDF), with the support of the
Ministry of Research and Universities, Generalitat de Catalunya. ICN2 is a
founding member of e-DREAM [60].

## Author contribution

M.J. developed the fabrication recipes, designed the resonator and hybrid cQED
devices and analysed data. K.R. developed and characterized the grAl
evaporation process for the wireless ohmmeter with the help of M.J.. M.J.,
K.R. and C.B.E. fabricated samples and performed experiments. A.B. and T.A.
conceived, designed and built the wireless ohmmeter. M.J. and O.S. developed
the microwave technology for the Ge/SiGe heterostructures. S.C., D.C. and G.I.
designed and grew the Ge/SiGe heterostructure. M.B., A.G.M. and J.A. performed
the atomic resolution scanning transmission electron microscopy structural and
electron energy-loss spectroscopy compositional-related characterization.
M.J., K.R. and G.K. contributed to discussions and the preparation of the
manuscript with input from the rest of the authors. G.K. and I.P. initiated,
and G.K. supervised the project.

## Competing interests

The authors declare no competing interests.

## References

  * Grünhaupt  _et al._ [2019] L. Grünhaupt, M. Spiecker, D. Gusenkova, N. Maleeva, S. T. Skacel, I. Takmakov, F. Valenti, P. Winkel, H. Rotzinger, W. Wernsdorfer, _et al._ , Granular aluminium as a superconducting material for high-impedance quantum circuits, Nature materials 18, 816 (2019).
  * Hazard  _et al._ [2019] T. Hazard, A. Gyenis, A. Di Paolo, A. Asfaw, S. Lyon, A. Blais, and A. Houck, Nanowire superinductance fluxonium qubit, Physical review letters 122, 010504 (2019).
  * Kamenov  _et al._ [2020] P. Kamenov, W.-S. Lu, K. Kalashnikov, T. DiNapoli, M. T. Bell, and M. E. Gershenson, Granular aluminum meandered superinductors for quantum circuits, Physical Review Applied 13, 054051 (2020).
  * Pita-Vidal  _et al._ [2020] M. Pita-Vidal, A. Bargerbos, C.-K. Yang, D. J. Van Woerkom, W. Pfaff, N. Haider, P. Krogstrup, L. P. Kouwenhoven, G. De Lange, and A. Kou, Gate-tunable field-compatible fluxonium, Physical Review Applied 14, 064038 (2020).
  * Rieger  _et al._ [2023] D. Rieger, S. Günzler, M. Spiecker, P. Paluch, P. Winkel, L. Hahn, J. Hohmann, A. Bacher, W. Wernsdorfer, and I. Pop, Granular aluminium nanojunction fluxonium qubit, Nature Materials 22, 194 (2023).
  * Xu  _et al._ [2023] M. Xu, R. Cheng, Y. Wu, G. Liu, and H. X. Tang, Magnetic field-resilient quantum-limited parametric amplifier, PRX Quantum 4, 010322 (2023).
  * Frasca  _et al._ [2024] S. Frasca, C. Roy, G. Beaulieu, and P. Scarlino, Three-wave-mixing quantum-limited kinetic inductance parametric amplifier operating at 6 T near 1 K, Physical Review Applied 21, 024011 (2024).
  * Parker  _et al._ [2022] D. J. Parker, M. Savytskyi, W. Vine, A. Laucht, T. Duty, A. Morello, A. L. Grimsmo, and J. J. Pla, Degenerate parametric amplification via three-wave mixing using kinetic inductance, Physical Review Applied 17, 034064 (2022).
  * Gao  _et al._ [2012] J. Gao, M. Vissers, M. Sandberg, F. Da Silva, S. W. Nam, D. Pappas, D. Wisbey, E. Langman, S. Meeker, B. Mazin, _et al._ , A titanium-nitride near-infrared kinetic inductance photon-counting detector and its anomalous electrodynamics, Applied Physics Letters 101 (2012).
  * Swenson  _et al._ [2013] L. Swenson, P. Day, B. Eom, H. Leduc, N. Llombart, C. McKenney, O. Noroozian, and J. Zmuidzinas, Operation of a titanium nitride superconducting microresonator detector in the nonlinear regime, Journal of Applied Physics 113 (2013).
  * Bueno  _et al._ [2014] J. Bueno, P. Coumou, G. Zheng, P. De Visser, T. Klapwijk, E. Driessen, S. Doyle, and J. Baselmans, Anomalous response of superconducting titanium nitride resonators to terahertz radiation, Applied Physics Letters 105 (2014).
  * Szypryt  _et al._ [2016] P. Szypryt, B. A. Mazin, G. Ulbricht, B. Bumble, S. Meeker, C. Bockstiegel, and A. Walter, High quality factor platinum silicide microwave kinetic inductance detectors, Applied Physics Letters 109 (2016).
  * Dupré  _et al._ [2017] O. Dupré, A. Benoît, M. Calvo, A. Catalano, J. Goupy, C. Hoarau, T. Klein, K. Le Calvez, B. Sacépé, A. Monfardini, _et al._ , Tunable sub-gap radiation detection with superconducting resonators, Superconductor Science and Technology 30, 045007 (2017).
  * Valenti  _et al._ [2019] F. Valenti, F. Henriques, G. Catelani, N. Maleeva, L. Grünhaupt, U. von Lüpke, S. T. Skacel, P. Winkel, A. Bilmes, A. V. Ustinov, _et al._ , Interplay between kinetic inductance, nonlinearity, and quasiparticle dynamics in granular aluminum microwave kinetic inductance detectors, Physical review applied 11, 054087 (2019).
  * Mi  _et al._ [2017a] X. Mi, J. Cady, D. Zajac, P. Deelman, and J. R. Petta, Strong coupling of a single electron in silicon to a microwave photon, Science 355, 156 (2017a).
  * Stockklauser  _et al._ [2017] A. Stockklauser, P. Scarlino, J. V. Koski, S. Gasparinetti, C. K. Andersen, C. Reichl, W. Wegscheider, T. Ihn, K. Ensslin, and A. Wallraff, Strong coupling cavity QED with gate-defined double quantum dots enabled by a high impedance resonator, Physical Review X 7, 011030 (2017).
  * Scarlino  _et al._ [2022] P. Scarlino, J. H. Ungerer, D. J. van Woerkom, M. Mancini, P. Stano, C. Müller, A. J. Landig, J. V. Koski, C. Reichl, W. Wegscheider, _et al._ , In situ tuning of the electric-dipole strength of a double-dot charge qubit: Charge-noise protection and ultrastrong coupling, Physical Review X 12, 031004 (2022).
  * De Palma  _et al._ [2023] F. De Palma, F. Oppliger, W. Jang, S. Bosco, M. Janík, S. Calcaterra, G. Katsaros, G. Isella, D. Loss, and P. Scarlino, Strong hole-photon coupling in planar Ge: probing the charge degree and Wigner molecule states, arXiv preprint arXiv:2310.20661 (2023).
  * Mi  _et al._ [2018] X. Mi, M. Benito, S. Putz, D. M. Zajac, J. M. Taylor, G. Burkard, and J. R. Petta, A coherent spin–photon interface in silicon, Nature 555, 599 (2018).
  * Samkharadze  _et al._ [2018] N. Samkharadze, G. Zheng, N. Kalhor, D. Brousse, A. Sammak, U. Mendes, A. Blais, G. Scappucci, and L. Vandersypen, Strong spin-photon coupling in silicon, Science 359, 1123 (2018).
  * Yu  _et al._ [2023] C. X. Yu, S. Zihlmann, J. C. Abadillo-Uriel, V. P. Michal, N. Rambal, H. Niebojewski, T. Bedecarrats, M. Vinet, É. Dumur, M. Filippone, _et al._ , Strong coupling between a photon and a hole spin in silicon, Nature Nanotechnology , 1 (2023).
  * Ungerer  _et al._ [2024] J. H. Ungerer, A. Pally, A. Kononov, S. Lehmann, J. Ridderbos, P. P. Potts, C. Thelander, K. A. Dick, V. F. Maisi, P. Scarlino, _et al._ , Strong coupling between a microwave photon and a singlet-triplet qubit, Nature Communications 15, 1068 (2024).
  * Borjans  _et al._ [2020] F. Borjans, X. Croot, X. Mi, M. Gullans, and J. Petta, Resonant microwave-mediated interactions between distant electron spins, Nature 577, 195 (2020).
  * Harvey-Collard  _et al._ [2022] P. Harvey-Collard, J. Dijkema, G. Zheng, A. Sammak, G. Scappucci, and L. M. Vandersypen, Coherent spin-spin coupling mediated by virtual microwave photons, Physical Review X 12, 021026 (2022).
  * Dijkema  _et al._ [2023] J. Dijkema, X. Xue, P. Harvey-Collard, M. Rimbach-Russ, S. L. de Snoo, G. Zheng, A. Sammak, G. Scappucci, and L. M. Vandersypen, Two-qubit logic between distant spins in silicon, arXiv preprint arXiv:2310.16805 (2023).
  * Samkharadze  _et al._ [2016] N. Samkharadze, A. Bruno, P. Scarlino, G. Zheng, D. DiVincenzo, L. DiCarlo, and L. Vandersypen, High-kinetic-inductance superconducting nanowire resonators for circuit QED in a magnetic field, Physical Review Applied 5, 044004 (2016).
  * Harvey-Collard  _et al._ [2020] P. Harvey-Collard, G. Zheng, J. Dijkema, N. Samkharadze, A. Sammak, G. Scappucci, and L. M. Vandersypen, On-chip microwave filters for high-impedance resonators with gate-defined quantum dots, Physical Review Applied 14, 034025 (2020).
  * Yu  _et al._ [2021] C. X. Yu, S. Zihlmann, G. Troncoso Fernández-Bada, J.-L. Thomassin, F. Gustavo, É. Dumur, and R. Maurand, Magnetic field resilient high kinetic inductance superconducting niobium nitride coplanar waveguide resonators, Applied Physics Letters 118 (2021).
  * Frasca  _et al._ [2023] S. Frasca, I. N. Arabadzhiev, S. B. de Puechredon, F. Oppliger, V. Jouanny, R. Musio, M. Scigliuzzo, F. Minganti, P. Scarlino, and E. Charbon, NbN films with high kinetic inductance for high-quality compact superconducting resonators, Physical Review Applied 20, 044021 (2023).
  * Zhang  _et al._ [2024] X. Zhang, Z. Zhu, N. Ong, and J. Petta, High-impedance superconducting resonators and on-chip filters for circuit quantum electrodynamics with semiconductor quantum dots, Physical Review Applied 21, 014019 (2024).
  * Kang  _et al._ [2023] Y. Kang, Z.-H. Li, Z.-Z. Kong, F.-G. Li, T.-Y. Hao, Z.-C. Wei, S.-Y. Deng, B.-C. Wang, H.-O. Li, G.-L. Wang, _et al._ , Coupling of hole double quantum dot in planar germanium to a microwave cavity, arXiv preprint arXiv:2310.08145 (2023).
  * Pechenezhskiy  _et al._ [2020] I. V. Pechenezhskiy, R. A. Mencia, L. B. Nguyen, Y.-H. Lin, and V. E. Manucharyan, The superconducting quasicharge qubit, Nature 585, 368 (2020).
  * Peruzzo  _et al._ [2020] M. Peruzzo, A. Trioni, F. Hassani, M. Zemlicka, and J. M. Fink, Surpassing the resistance quantum with a geometric superinductor, Physical Review Applied 14, 044055 (2020).
  * Shearrow  _et al._ [2018] A. Shearrow, G. Koolstra, S. J. Whiteley, N. Earnest, P. S. Barry, F. J. Heremans, D. D. Awschalom, E. Shirokoff, and D. I. Schuster, Atomic layer deposition of titanium nitride for quantum circuits, Applied Physics Letters 113 (2018).
  * Borisov  _et al._ [2020] K. Borisov, D. Rieger, P. Winkel, F. Henriques, F. Valenti, A. Ionita, M. Wessbecher, M. Spiecker, D. Gusenkova, I. Pop, _et al._ , Superconducting granular aluminum resonators resilient to magnetic fields up to 1 Tesla, Applied Physics Letters 117 (2020).
  * Moshe  _et al._ [2020] A. G. Moshe, E. Farber, and G. Deutscher, Granular superconductors for high kinetic inductance and low loss quantum devices, Applied Physics Letters 117 (2020).
  * Grünhaupt  _et al._ [2018] L. Grünhaupt, N. Maleeva, S. T. Skacel, M. Calvo, F. Levy-Bertrand, A. V. Ustinov, H. Rotzinger, A. Monfardini, G. Catelani, and I. M. Pop, Loss mechanisms and quasiparticle dynamics in superconducting microwave resonators made of thin-film granular aluminum, Physical review letters 121, 117001 (2018).
  * Maleeva  _et al._ [2018] N. Maleeva, L. Grünhaupt, T. Klein, F. Levy-Bertrand, O. Dupre, M. Calvo, F. Valenti, P. Winkel, F. Friedrich, W. Wernsdorfer, _et al._ , Circuit quantum electrodynamics of granular aluminum resonators, Nature communications 9, 3889 (2018).
  * Zhang  _et al._ [2019] W. Zhang, K. Kalashnikov, W.-S. Lu, P. Kamenov, T. DiNapoli, and M. Gershenson, Microresonators fabricated from high-kinetic-inductance aluminum films, Physical Review Applied 11, 011003 (2019).
  * Rotzinger  _et al._ [2016] H. Rotzinger, S. Skacel, M. Pfirrmann, J. Voss, J. Münzberg, S. Probst, P. Bushev, M. Weides, A. Ustinov, and J. Mooij, Aluminium-oxide wires for superconducting high kinetic inductance circuits, Superconductor Science and Technology 30, 025002 (2016).
  * Scappucci  _et al._ [2021] G. Scappucci, C. Kloeffel, F. A. Zwanenburg, D. Loss, M. Myronov, J.-J. Zhang, S. De Franceschi, G. Katsaros, and M. Veldhorst, The germanium quantum information route, Nature Reviews Materials 6, 926 (2021).
  * Cohen and Abeles [1968] R. W. Cohen and B. Abeles, Superconductivity in granular aluminum films, Physical Review 168, 444 (1968).
  * Deutscher  _et al._ [1973] G. Deutscher, H. Fenichel, M. Gershenson, E. Grünbaum, and Z. Ovadyahu, Transition to zero dimensionality in granular aluminum superconducting films, Journal of Low Temperature Physics 10, 231 (1973).
  * Levy-Bertrand  _et al._ [2019] F. Levy-Bertrand, T. Klein, T. Grenet, O. Dupré, A. Benoît, A. Bideaud, O. Bourrion, M. Calvo, A. Catalano, A. Gomez, _et al._ , Electrodynamics of granular aluminum from superconductor to insulator: Observation of collective superconducting modes, Physical Review B 99, 094506 (2019).
  * Kroll  _et al._ [2019] J. G. Kroll, F. Borsoi, K. Van Der Enden, W. Uilhoorn, D. De Jong, M. Quintero-Pérez, D. Van Woerkom, A. Bruno, S. Plissard, D. Car, _et al._ , Magnetic-field-resilient superconducting coplanar-waveguide resonators for hybrid circuit quantum electrodynamics experiments, Physical Review Applied 11, 064053 (2019).
  * Mi  _et al._ [2017b] X. Mi, J. Cady, D. Zajac, J. Stehlik, L. Edge, and J. R. Petta, Circuit quantum electrodynamics architecture for gate-defined quantum dots in silicon, Applied Physics Letters 110 (2017b).
  * Valentini  _et al._ [2024] M. Valentini, O. Sagi, L. Baghumyan, T. de Gijsel, J. Jung, S. Calcaterra, A. Ballabio, J. Aguilera Servin, K. Aggarwal, M. Janik, _et al._ , Parity-conserving Cooper-pair transport and ideal superconducting diode in planar germanium, Nature Communications 15, 169 (2024).
  * Jirovec  _et al._ [2022] D. Jirovec, P. M. Mutter, A. Hofmann, A. Crippa, M. Rychetsky, D. L. Craig, J. Kukucka, F. Martins, A. Ballabio, N. Ares, _et al._ , Dynamics of hole singlet-triplet qubits with large g-factor differences, Physical Review Letters 128, 126803 (2022).
  * Kuit  _et al._ [2008] K. H. Kuit, J. Kirtley, W. Van Der Veur, C. Molenaar, F. Roesthuis, A. Troeman, J. Clem, H. Hilgenkamp, H. Rogalla, and J. Flokstra, Vortex trapping and expulsion in thin-film YBa 2 Cu 3 O 7- $\delta$ strips, Physical Review B 77, 134504 (2008).
  * Hendrickx  _et al._ [2024] N. Hendrickx, L. Massai, M. Mergenthaler, F. Schupp, S. Paredes, S. Bedell, G. Salis, and A. Fuhrer, Sweet-spot operation of a germanium hole spin qubit with highly anisotropic noise sensitivity, Nature Materials , 1 (2024).
  * Jirovec  _et al._ [2021] D. Jirovec, A. Hofmann, A. Ballabio, P. M. Mutter, G. Tavani, M. Botifoll, A. Crippa, J. Kukucka, O. Sagi, F. Martins, _et al._ , A singlet-triplet hole spin qubit in planar Ge, Nature Materials 20, 1106 (2021).
  * Wang  _et al._ [2021] H. Wang, S. Singh, C. McRae, J. C. Bardin, S. Lin, N. Messaoudi, A. Castelli, Y. Rosen, E. Holland, D. Pappas, _et al._ , Cryogenic single-port calibration for superconducting microwave resonator measurements, Quantum Science and Technology 6, 035015 (2021).
  * Wallraff  _et al._ [2013] A. Wallraff, A. Stockklauser, T. Ihn, J. R. Petta, and A. Blais, Comment on “Vacuum Rabi splitting in a semiconductor circuit QED system”, Physical review letters 111, 249701 (2013).
  * Clerk  _et al._ [2020] A. Clerk, K. Lehnert, P. Bertet, J. Petta, and Y. Nakamura, Hybrid quantum systems with circuit quantum electrodynamics, Nature Physics 16, 257 (2020).
  * Benito  _et al._ [2019] M. Benito, J. R. Petta, and G. Burkard, Optimized cavity-mediated dispersive two-qubit gates between spin qubits, Physical Review B 100, 081412 (2019).
  * Harvey  _et al._ [2018] S. Harvey, C. Bøttcher, L. Orona, S. Bartlett, A. Doherty, and A. Yacoby, Coupling two spin qubits with a high-impedance resonator, Physical Review B 97, 235409 (2018).
  * Forn-Díaz  _et al._ [2019] P. Forn-Díaz, L. Lamata, E. Rico, J. Kono, and E. Solano, Ultrastrong coupling regimes of light-matter interaction, Reviews of Modern Physics 91, 025005 (2019).
  * Burkard  _et al._ [2020] G. Burkard, M. J. Gullans, X. Mi, and J. R. Petta, Superconductor–semiconductor hybrid-circuit quantum electrodynamics, Nature Reviews Physics 2, 129 (2020).
  * House  _et al._ [2011] M. House, H. Pan, M. Xiao, and H. Jiang, Non-equilibrium charge stability diagrams of a silicon double quantum dot, Applied Physics Letters 99 (2011).
  * Ciancio  _et al._ [2022] R. Ciancio, R. E. Dunin-Borkowski, E. Snoeck, M. Kociak, R. Holmestad, J. Verbeeck, A. I. Kirkland, G. Kothleitner, and J. Arbiol, e-DREAM: the European Distributed Research Infrastructure for Advanced Electron Microscopy, Microscopy and Microanalysis 28, 2900 (2022).