# Kinemon: inductively shunted transmon artificial atom

Daria Kalacheva<EMAIL_ADDRESS>Skolkovo Institute of Science and
Technology, Skolkovo Innovation Center, Moscow 121205, Russia  Moscow
Institute of Physics and Technology, 9 Institutsky Lane, Dolgoprudny 141700,
Russia  National University of Science and Technology MISIS, 119049 Moscow,
Russia  Gleb Fedorov  Moscow Institute of Physics and Technology, 9
Institutsky Lane, Dolgoprudny 141700, Russia  National University of Science
and Technology MISIS, 119049 Moscow, Russia  Russian Quantum Center, National
University of Science and Technology MISIS, 119049 Moscow, Russia  Julia
Zotova  Skolkovo Institute of Science and Technology, Skolkovo Innovation
Center, Moscow 121205, Russia  Moscow Institute of Physics and Technology, 9
Institutsky Lane, Dolgoprudny 141700, Russia  National University of Science
and Technology MISIS, 119049 Moscow, Russia  Shamil Kadyrmetov  Moscow
Institute of Physics and Technology, 9 Institutsky Lane, Dolgoprudny 141700,
Russia  Alexey Kirkovskii  Moscow Institute of Physics and Technology, 9
Institutsky Lane, Dolgoprudny 141700, Russia  Aleksei Dmitriev  Moscow
Institute of Physics and Technology, 9 Institutsky Lane, Dolgoprudny 141700,
Russia  Oleg Astafiev  Skolkovo Institute of Science and Technology, Skolkovo
Innovation Center, Moscow 121205, Russia  Moscow Institute of Physics and
Technology, 9 Institutsky Lane, Dolgoprudny 141700, Russia

###### Abstract

We experimentally investigate inductively shunted transmon-type artificial
atoms as an alternative to address the challenges of low anharmonicity and the
need for strong charge dispersion in superconducting quantum systems. We
characterize several devices with varying geometries and parameters (Josephson
energies and capacitances), and find a good agreement with calculations. Our
approach allows us to retain the benefits of transmon qubit engineering and
fabrication technology and high coherence, while potentially increasing
anharmonicity. The approach offers an alternative platform for the development
of scalable multi-qubit systems in quantum computing.

††preprint: APS/123-QED

Superconducting artificial quantum systems, such as the capacitively shunted
charge qubits (transmons and X-mons) are now commonly used to build prototypes
of quantum processors because of their simple design and low decoherence rates
[1, 2, 3, 4]. However, scaling up quantum registers composed of low
anharmonicity physical qubits faces challenges due to the uncontrolled
transitions to upper states and limitations in speed of quantum operations [5,
6, 7, 8, 9, 10]. Additionally, non-negligible charge dispersion of the higher
energy levels complicates the use of such artificial atoms as qudits [11].
These problems drive the search for alternative physical qubits and materials
[12, 13, 14, 15].

While retaining simplicity in fabrication and operation, together with charge
noise insensitivity, one can increase the non-linearity of a transmon by
decreasing its shunting capacitance and, at the same time, shunting it by a
linear inductance. Strictly speaking, this modification produces a flux qubit
[16, 17, 18, 19, 20], more specifically, an rf-SQUID or a fluxonium [21];
however, its parameters can be chosen so that the resulting eigenstates are
transmon-like, living in a single-well potential, not a two-well one. The
latter helps to avoid the exponential sensitivity of the transition
frequencies to the Josephson energy variations, which has so far limited the
applications of flux qubits in multi-qubit devices. Also, one can expect that
the inductive shunt will remove charge dispersion for arbitrarily high energy
states.

In this study, we explore a new hybrid design combining the transmon circuit
with a compact kinetic inductor – a kinemon (kinetic-inductance-shunted
transmon) artificial atom. We design and investigate experimentally a family
of such systems with various combinations of Josephson energy $E_{J}$,
inductive energy $E_{L}$ and charging energy $E_{C}$ [3]. We also show that
the inductive element can be placed inside an $\alpha$-SQUID [22] which, for a
correct ratio of resulting loop areas, opens a way to modulate the effective
Josephson energy, while keeping the parabolic potential contribution fixed.
Importantly, we find that, confirming our previous tests of coplanar
resonators [23], the kinetic inductor based on aluminum ultra-thin-film does
not cause any noticeable deterioration of the coherence times. Finally, as
this kind of inductor exhibits relatively good reproducibility in fabrication,
we find it a promising component for future quantum circuits.

Figure 1:  (a) The potential (solid black line) and the energy levels (gray
lines) for the conventional transmon (top) and for the inductivly shunted one
(bottom) for $\varphi_{\mathrm{e}}=0$. (b) Circuit model of the sample. SQUIDs
consists of Josephson junction and a kinetic wire with the energies $E_{J}$
and $E_{L}$ respectively and are shunted by a large capacitance with an energy
$E_{C}$ forming a kinemon atom. Group A (green color) represents assymetric
kinemons with different energy ratios (Kinemons I - VI). Group B (violet
color) represents symmetric kinemons with two JJs and a kinetic inductance
wire between them (Kinemons VII - VIII). (c) Overview optical image of the
fabricated sample. Microfabrication design includes two group of art. atoms
with different topologies and variations of $E_{C}$, $E_{L}$ and $E_{J}$. (d,
e) Enlarged optical image of two different kinemon modification.

Without the inductive shunt, a Josephson junction is characterized by a
periodic potential $U_{J}=-E_{J}\cos\varphi$, where $\varphi$ is the phase
across the junction [Fig. 1(a)]. Adding a small parallel capacitance to the
circuit results in the formation of energy bands of Bloch waves in the
periodic potential [24], which can be represented as
$\psi^{\prime}(\varphi)=e^{i\frac{q^{\prime}}{2e}\varphi}u(\varphi)$, so that
$\psi^{\prime}(\varphi)\neq\psi^{\prime}(\varphi+2\pi)$ and only
$u(\varphi)=u(\varphi+2\pi)$. Here, $q^{\prime}$ represents the quasicharge,
being the analogue of the crystal momentum in solids, and the energies inside
the band are periodic functions of $q^{\prime}$ [condensed matter book]. We
note that for a charge qubit with discrete number of Cooper pairs allowed on
the island, we can apply the rotor analogy [1, 25], so that the states after a
full rotation are indistinguishable. Then, the wave function is
$2\pi$-periodic in the $\varphi$ representation, and the bands seem to
disappear. However, as there is a mathematical correspondence between the
quasicharge $q^{\prime}$ and the induced charge $n_{g}$:
$n_{g}=q^{\prime}/2e$, the energy configuration of the system still exhibits
the same oscillatory behavior as a function of $n_{g}$. While a larger
capacitance localizes the lower energy states in distinct potential wells and
significantly reduces the widths of the lowest bands, higher lying bands
remain open, as shown in Fig. 1(a), and thus are still sensitive to the
induced charge. Moreover, increasing the capacitance always comes at the cost
of reducing the anharmonicity $\alpha$ [1].

To completely prevent the formation of energy bands (remove any energy
dependence on the induced charge $n_{g}$), it is necessary to disrupt the
periodicity. This can be achieved by implementing a shunting inductance, which
introduces a parabolic potential $U_{L}=E_{L}\varphi^{2}/2$ to $U_{J}$ as
exemplified in Fig. 1(a). As a result, the wavefunctions of the lowest states
become localized in the central well. Note that in the case of a small $E_{L}$
($\leq E_{J}$), the anharmonicity and the energy structure are predominantly
determined by $E_{J}$ and $E_{C}$. This is because the Josephson energy
$E_{J}$ governs the energy landscape of the system, with the smaller inductive
energy $E_{L}$ providing only a minor perturbation. Larger $E_{L}/E_{J}$
results in less anharmonicity but can be compensated by increasing $E_{C}$. In
the present work, though, we study inductively shunted artificial atoms
fabricated using the standard transmon technology ($E_{C}\ll E_{J}\lesssim
E_{L}$) and aim to verify their properties and coherence.

Figure 2:  (a) Kinemon I: (Top) Transmission spectroscopy of the coupled
resonator showing feedline frequency response depending on the external flux
bias $\varphi_{\mathrm{e}}$. (Bottom) Experimental two-tone spectroscopy,
displaying the magnitude of the readout signal at a properly chosen readout
frequency as a function of flux bias $\varphi_{\mathrm{e}}$ and kinemon atom
excitation frequency. For Group A, the anharmonicity sign changes at the
$\varphi_{\mathrm{e}}=0.75\,\pi$ and gets back at
$\varphi_{\mathrm{e}}=1.25\,\pi$. Numerical simulation of the spectrum
reproducing experimental results with labeled transitions. (b) Kinemon VII:
(Top) Transmission spectroscopy of the coupled resonator. (Bottom)
Experimental two-tone spectroscopy, with insets (middle and right) showing
magnifications at the bottom and top flux sweet spots. For Group B, the
regimes of harmonic oscillator are noticeable near
$\varphi_{\mathrm{e}}=\pi+2\pi k,\ k\in Z$ (left and middle inset), while at
$\varphi_{\mathrm{e}}\approx 1.15\,\pi$ only the three lowest levels are
equidistant and shown in the left inset.

A schematic equivalent circuit of kinemon artificial atom is depicted in Fig.
1(b), illustrating two different configurations of qubits investigated in this
work. Qubits consist of two main parts:
$\text{Al}/\text{Al}\text{O}_{x}/\text{Al}$ Josephson junctions and a kinetic
inductance wire with energy $E_{L}=\Phi_{0}^{2}/4\pi^{2}L_{k}$ due to the wire
inductance $L_{k}$ to form a SQUID loop. The first circuit scheme operates
with an asymmetric topology, utilizing a small single loop formed by an
aluminum wire interrupted by a single Josephson junction (Group A in Fig.
1(b)). A key characteristic of this mode is the variation of shunting
capacitance values ($E_{C}=2e^{2}/C$) to examine coherence time. The second
circuit scheme employs a symmetric topology that merges the benefits of both
rf-SQUID and transmon designs (Group B in Fig. 1(b)). This approach aims to
enhance qubit performance by incorporating a double-loop architecture
connected by a shared kinetic inductance wire.

The common Hamiltonian, which covers both the considered geometries (single-
and double-loop circuits), experesses as follows

$\begin{split}\mathcal{\hat{H}}=-E_{C}\frac{\partial^{2}}{\partial\varphi^{2}}&+\frac{1}{2}E_{L}\varphi^{2}-E_{J1}\cos(\varphi+\kappa\varphi_{\mathrm{e}})\\\
&-E_{J2}\cos\left(\varphi-(1-\kappa)\varphi_{\mathrm{e}}\right),\end{split}$
(1)

where $\varphi_{\mathrm{e}}$ is the total flux phase induced by an external
magnetic field, $\kappa$ is the coefficient of $\varphi_{\mathrm{e}}$
distribution between two loops, $E_{J1}$ and $E_{J2}$ are the Josephson
energies of the junctions in each loop according to Fig. 1(b). The Hamiltonian
for Group A is obtained by putting $E_{J2}=0$.

An optical image of the sample is presented in Fig. 1(c), depicting the
microfabricated superconducting circuit containing eight kinemon artificial
atoms. SQUIDs are tiny loops of superconducting wire and Josephson junctions
connected to other circuit elements, such as capacitors and resonators. The
SQUIDs are highlighted in green and violet false colors depending on their
architecture Fig. 1(d, e).

The fabrication of the kinetic inductance wire is done as follows. A silicon
substrate is cooled down by liquid nitrogen during metal deposition to obtain
uniform films up to 3.5 nm thick. It is known that aluminum films, deposited
at room temperature are negatively impacted by formation of granules. Cold
film deposition allows us to fabricate long ultrathin wires with high degree
of homogeneity [23]. The detailed fabrication process described in App. B. To
achieve the necessary inductive energy $E_{L}$, 200 nm wide and 8 nm thick
aluminum wires with varying lengths are integrated into the circuit. Since the
kinetic inductance per square of such a film is about 0.03 nH/$\square$, the
wire length in the device ranges from 80 to 240 $\mu$m, depending on the
desired $E_{L}$.

Table 1:  Kinemons parameters extracted by fitting. Coherence times are given at $\varphi_{\mathrm{e}}=0$. | I | II | III | IV | V | VI | VII | VIII
---|---|---|---|---|---|---|---|---
$E_{J}/h$, GHz | $5.38$ | $6.00$ | $4.00$ | $2.92$ | $2.44$ | $5.90$ | $8.61$ | $14.00$
$E_{C}/h$, GHz | $0.90$ | $1.10$ | $1.50$ | $1.95$ | $1.80$ | $0.7$ | $0.47$ | $0.32$
$E_{L}/h$, GHz | $8.59$ | $8.75$ | $7.40$ | $8.40$ | $9.07$ | $14.65$ | $8.11$ | $12.2$
$\omega^{(t)}_{01}/2\pi$, GHz | $4.947$ | $5.596$ | $5.719$ | $6.508$ | $6.359$ | $5.312$ | $4.769$ | $5.008$
$\omega_{r}/2\pi$, GHz | $7.185$ | $7.284$ | $7.341$ | $7.433$ | $7.495$ | $7.608$ | $7.688$ | $7.779$
$g_{s}/2\pi$, MHz | $64$ | $44$ | $34$ | $35$ | $34$ | $90$ | $83$ | $68$
$\alpha^{(t)}/h$, MHz | $-86$ | $-118$ | $-131$ | $-116$ | $-87$ | $-49$ | $-84$ | $-80$
$\alpha^{(b)}/h$, MHz | $219$ | $301$ | $257$ | $182$ | $124$ | $96$ | $-$ | $-$
$T_{1}$, $\mu s$ | $17.92\pm 0.95$ | $17.56\pm 0.86$ | $19.45\pm 1.62$ | $8.95\pm 0.33$ | $8.61\pm 0.29$ | $20.39\pm 0.93$ | $19.18\pm 0.78$ | $14.83\pm 0.87$
$T_{2}$, $\mu s$ | $11.45\pm 0.95$ | $17.30\pm 1.65$ | $11.63\pm 1.20$ | $7.80\pm 0.85$ | $9.69\pm 0.42$ | $13.98\pm 0.75$ | $6.59\pm 0.30$ | $12.28\pm 0.65$
$T_{2E}$, $\mu s$ | $7.92\pm 2.35$ | $20.84\pm 1.11$ | $-$ | $8.05\pm 0.83$ | $14.87\pm 0.56$ | $18.25\pm 1.06$ | $12.73\pm 0.50$ | $13.32\pm 1.02$
Figure 3:  Anharmonicity as a function of the external flux bias
$\varphi_{\mathrm{e}}$ for kinemon I, V an VII. For kinemons I and V, a zero
anharmonicity implies that the first three energy levels are equidistant. As
for kinemon VII, the anharmonicity becomes exactly zero
$\varphi_{\mathrm{e}}=\pi$ at as it transitions into the harmonic regime,
while at $\varphi_{\mathrm{e}}\approx 1.15\,\pi$ the behavior is similar to
Group A.

In Fig. 2(a) top, we display the data of transmission spectroscopy of the
sample via the feedline, showing the microwave response of the readout
resonator I as a function of the flux bias $\varphi_{\mathrm{e}}$ through the
SQUID of kinemon I (Group A). See all resonators spectra in Fig. 6 in App. D.
The pattern is a combination of a smooth dependence, formed by the kinemon
first excited state located below the resonator frequency, and an avoided
crossing pattern with the second excited state [26]. Direct observation of
these transitions is enabled by the cross-Kerr dispersive spectroscopy, for
which the data are displayed in Fig. 2(a) bottom. The minimum frequency (one-
half flux quantum) and the maximum frequency (zero flux) are considered the
flux ‘sweet spots’. Fig. 2(a) bottom, also presents the fits to the
experimental transition frequencies based on the circuit Hamiltonian (Eq. 1).
The fits align well with the experimental transition frequencies near the
sweet spots $\varphi_{\mathrm{e}}=0$ and $\varphi_{\mathrm{e}}=\pi/2$. The
single-photon transition between the ground $|0\rangle$ and excited
$|1\rangle$ states occurs at $f_{01}=4.947\,\text{GHz}$ at
$\varphi_{\mathrm{e}}=0$. Furthermore, the bottom sweet spot of the transition
from $|0\rangle$ to $|2\rangle$ is at $f_{02}=5.6\,\text{GHz}$ and not visible
at $\varphi_{\mathrm{e}}=\pi$ and corresponds to anti-crossings in the
transmission spectroscopy. However, the two-photon transition can be observed
and is associated with the spectroscopic line at $f_{02/2}=4.8,\text{GHz}$.
Also, at $\varphi_{\mathrm{e}}\approx 0.75\,\pi$ and
$\varphi_{\mathrm{e}}\approx 1.25\,\pi$ there are regimes when the first three
energy levels are equidistant and are the pivot points in the anharmonicity
sign. The atom with zero anharmonicity for the first two transitions [27]
($\ket{0}\to\ket{1}$ and $\ket{1}\to\ket{2}$), which is intermediate between a
two-level system (TLS) and a harmonic oscillator, may find interesting
applications in the generation of non-classical light [28]. All extracted
parameters from the fits are summarised in Table 1. The extracted parameters
are also used to make predictions and test the validity of the theoretical
model employed in the analysis. The experimental setup and measurement
equipment are described in App. C.

The symmetric kinemons (Group B) are also characterized using spectroscopic
measurements. The kinemon VII transmission and two-tone spectra are presented
in Fig. 2(b), where in insets we highlight several distinct features at and
near the sweet spots. In the current scheme, we operate in the regime where
$E_{J1}=E_{J2}$, leading to zero anharmonicity at bottom sweet spots
($\alpha=0$ for $\varphi_{\mathrm{e}}=\pi+2\pi k,\ k\in Z$). In other words,
the Josephson energy is cancelled, when a half flux quantum $\Phi_{0}$
penetrates through the SQUID, causing a transition into the harmonic regime.
The modulation of the frequency at the top sweet spots corresponds to the non-
identical areas of the SQUIDs. By fitting the spectrum, we evaluate $\kappa$
to 0.35 and 0.37 for kinemon VII and VIII, respectively, which is in a good
agreement with the design areas. These values determine the locations of non-
periodic spots of three-level equidistance (similar to Group A) at
$\varphi_{\mathrm{e}}\approx 1.15\,\pi$ and $\varphi_{\mathrm{e}}\approx
4.9\,\pi$, respectively.

To illustrate better the behavior of the spectra, we plot the anharmonicity of
the kinemons I, V and VII vs. the magnetic flux in Fig. 3(). For all devices,
we calculate the anharmonicity as $\alpha/h=2\times(f_{02}/2-f_{01})$. For
kinemon I this yields -86 MHz and 219 MHz for the top and bottom sweet spots,
respectively. Kinemon V shows a lower peak anharmonicity, but in overall
demonstrates a flatter dependence on $\varphi_{\mathrm{e}}$ due to an
increased $E_{C}$ value. For kinemon VII, zero anharmonicity at
$\varphi=\pi+2\pi k,\ k\in Z$ is associated with a transition into a harmonic
oscillator; the additional feature at $\varphi\approx 1.15\,\pi$ mirrors the
effect observed in kinemons I and V, corresponding to the three-level
equidistance. Interestingly, for this qubit the anharmonicity is mostly
negative and is higher in absolute value in the top sweet spot near
$\varphi_{\mathrm{e}}=2\,\pi$ than at $\varphi_{\mathrm{e}}=0$. Another
peculiarity is that due to the non-trivial $\kappa$-dependence of potential
one can find a sign change of anharmonicity at $\varphi=\pi+2\pi k,\ k\in Z$.
However, if the flux is simultaneously near one of the other set of special
points $\varphi_{\mathrm{e}}/\pi=\frac{1+2k^{\prime}}{2\kappa-1},\
k^{\prime}\in Z$ (for our case, this is near $\varphi_{\mathrm{e}}\approx
3\,\pi,\ k^{\prime}=-1$), the sign of $\alpha$ remains the same.

Figure 4:  Energy relaxation and coherence times for kinemon VI measured at
different flux bias. The analysis considers two flux bias conditions,
$\varphi_{\mathrm{e}}=0$ (navy circles) and $\varphi_{\mathrm{e}}=\pi/2$ (red
triangles). The top left subplot reveals the $T_{1}$ time at the top sweet
spot, demonstrating stability with a value of $20.39\,\pm\,0.93\,\mu\text{s}$.
Triangles on the red line represent higher $T_{1}$ values,
$23.74\,\pm\,0.85\,\mu\text{s}$, suggesting slower relaxation at
$\varphi_{\mathrm{e}}=\pi/2$. The echo coherence time ($T^{E}_{2}$) is
depicted in the bottom left subplot. Measurements display times of
$18.25\,\pm\,1.06\,\mu\text{s}$ and $19.28\,\pm\,0.50\,\mu\text{s}$ for
$\varphi_{\mathrm{e}}=0$ and $\varphi_{\mathrm{e}}=\pi/2$ respectively.
Lastly, the remaining subplot illustrates Ramsey coherence times ($T_{2}$),
yielding values of $13.98\,\pm\,0.75\,\mu\text{s}$ and
$14.53\,\pm\,0.35\,\mu\text{s}$ at the respective sweet spots.

Finally, we analyse the energy relaxation time $T_{1}$, the Ramsey coherence
time $T_{2}$, and the echo coherence time $T^{E}_{2}$ (See in Table 1). The
experimental results for kinemon VI are presented in Fig. 4, which provides a
comparative analysis of relaxation times under two distinct flux bias
conditions, specifically at $\varphi_{\mathrm{e}}=0$ (represented in navy
color) and $\varphi_{\mathrm{e}}=\pi/2$ (represented in red color).
Corresponding measurement protocols are presented on each subplot. We also
measure the coherence outside the sweet spot and find the characteristic times
to be about 600 ns and 200 ns for $T^{E}_{2}$ and $T_{2}$, respectively; the
reduction is probably caused by the flux noise due to the insufficient
filtering. The relaxation time remains the same with respect to the flux
value. The experimental data for $T_{1}$ are presented in the top left
subplot. At the top sweet spot, a specific value noted as
$20.39\,\pm\,0.93\,\mu\text{s}$. The values represented by triangles on the
red line seem slightly higher than at $\varphi_{\mathrm{e}}=0$, indicating a
slower relaxation rate at $\varphi_{\mathrm{e}}=\pi/2$ with a particular time
noted as $23.74\,\pm 0.85\,\mu\text{s}$. The echo coherence time is plotted in
the bottom left subplot, with measurement results of $18.25\pm
1.06\,\mu\text{s}$ and $19.28\,\pm\,0.50\,\mu\text{s}$ at flux biases of
$\varphi_{\mathrm{e}}=0$ and $\varphi_{\mathrm{e}}=\pi/2$, respectively. The
Ramsey coherence times are presented in the remaining subplot and yield
$13.98\,\pm\,0.75\,\mu\text{s}$ and $14.53\,\pm\,0.35\,\mu\text{s}$ at both
sweet spots, respectively. Also, additional measurements of conventional
transmons fabricated with the same technological process give about
$14\,\mu\text{s}$, $8\,\mu\text{s}$ and $9\,\mu\text{s}$ for $T_{1}$, $T_{2}$,
$T^{E}_{2}$, respectively; however, observed performance improvement of
kinemons could be caused by differences between the measurement setups.

The use of low-loss material, such as ultra-thin aluminum film inductors, not
only improves device scalability but also enhances performance compared to
other materials [20, 29, 30]. These observations indicate that the shunting
capacitance may be further reduced to make the kinemon design even more
compact compared to the transmons. For example, the typical value of
$E_{c}=1.5$ GHz for transmons [31, 32] could be increased up to 5-6 GHz for
kinemons which will scale down the capacitors, which is promising for the
scalability of multi-qubit systems. Additionally, the advantage of having a
sign-changing anharmonicity attracts interest in waveguide quantum optics
[33], for example, allowing to emit pairs of correlated photons [34] and
observe nonlinear intermodulation processes [35] or could help to optimise
gate errors caused by a parasitic partial CPHASE operation induced by high-
order coupling [36, 37], or be useful for new regimes of Bose-Hubbard model
simulators [38, 39].

In conclusion, this study demonstrates that inductively shunted transmon
qubits, utilizing ultra-thin aluminum film inductors, provide a promising
platform for scalable quantum computing applications. To increase
anharmonicity and strike a balance between high anharmonicity and low charge
noise sensitivity, a reduction in capacitance should be considered. Thus,
introducing a non-zero kinetic inductance component can address this issue.
While this study represents a significant step towards realizing large-scale,
practical quantum computing systems, further research is required to fully
explore and validate the potential of this approach in the field of quantum
computing.

## Data Availability

The raw data that support the findings of this study are available on a
reasonable request from the corresponding author.

###### Acknowledgements.

The authors are grateful to Russian Science Foundation Project Grant No.
21-42-00025 for financial support. The sample was fabricated using equipment
of MIPT Shared Facilities Center.

##  Appendix A Design of the qubit samples

A layer-by-layer design was generated using the Klayout-python library [40],
which automates the design of superconducting quantum circuits. This library
utilizes the KLayout layout design program API and enables the execution of
arbitrary Python code through an embedded interpreter. The library specializes
in designing microwave and superconducting qubit planar designs, including
drawing patterns, simulation, and domain-specific design rule checkers.

##  Appendix B Device fabrication

The device fabrication steps consist of five main stages: ground plane
construction, nanofabrication of Josephson junction and kinetic wire, bandage
deposition, and air-bridges construction.

We start with silicon substrate treatment, which includes piranha etching and
BHF dipping [41, 42]. The substrate is then immediately placed in the Plassys
e-beam evaporation system, and a 100 nm 99.999% aluminum film is evaporated.
The metallized substrate is spin-coated with optical resist AZ1517. The
coplanar waveguide feedline, resonators, qubit capacitors, and ground plane
hole array are patterned using a laser maskless optical lithography system,
followed by dry etching of the optical resist mask structure in BCl3/Cl2
inductively coupled plasma. Residual resist is then removed in
N-methyl-2-pyrrolidone (NMP) and cleaned in O2 plasma.

The next step includes hard-mask preparation [43]. The substrate is spin-
coated with polymer resist PMGI SF9. Then, a 30 nm tungsten nanolayer is
deposited in a Torr magnetron sputtering system, followed by ARP-04 resist
coating. Josephson junctions are patterned by electron lithography and
evaporated using the Dolan bridge technique [44], followed by lift-off in NMP.
To form the tunnel barrier, the first 25 nm aluminum junction electrode is
oxidized at 40 mBar. Then, a 45 nm electrode is evaporated and preventively
oxidized at 10 mBar. Residual resist is removed in NMP and cleaned in O2
plasma.

The kinetic part is fabricated during an additional cycle of e-beam
lithography. We use a single layer of ARP-04 e-beam resist to construct the
pattern. After development, an 8 nm aluminum film is evaporated at the Plassys
stage temperature of 170 K at a normal angle. Residual resist is then removed
in NMP and cleaned in O2 plasma.

Good galvanic contact between the layers is obtained by aluminum bandages
[45]. We use a similar process of single layer mask as for the kinetic part
above, but without cooling. A 150 nm aluminum film is evaporated with in-situ
Ar ion milling. Residual resist is removed during the lift-off process in NMP
and cleaned in O2 plasma.

Due to the presence of coplanar lines on the ground plane, we need to achieve
a uniform electrical potential; therefore, the final stage of sample
fabrication is the implementation of aluminum free-standing air-bridges [46].
A 7 µm layer of SPR220 photoresist is spin-coated, and the base layer is
patterned using a laser maskless optical lithography system. After
development, the substrate is heated to create a height gradient on the resist
edges, followed by a 600 nm aluminum evaporation with in-situ Ar ion etching.
A second layer of SPR220 photoresist is used to form a bridge structure.
Finally, the excess metal is dry-etched in BCl3/Cl2 inductively coupled
plasma. Residual resist is then removed in NMP and cleaned in O2 plasma.

Figure 5:  The schematic of experimental setup to measure the sample depicting
room temperature equipment and line configuration inside the BlueFors dilution
refrigerator, with the base temperature of 10 mK.

##  Appendix C Microwave experimental setup

Figure 6:  Transmission spectroscopy for all resonators coupled to kinemons
with varying energy ratios. $|S_{21}|$ includes the attenuation and
amplification in the measurement chain. Avoided crossings occur when a qubit’s
$\ket{0}\to\ket{2}$ transition (or even higher levels for kinemons VII and
VIII) intersects with its readout cavity frequency. Some very sharp avoided
crossings are just barely resolved in the simulations. Also, some of the
predicted features are smeared in experiment due to the power broadening
effects (located around 1.1 mA for VII and 1.4 mA for VIII).

The device under investigation is measured in a dilution refrigerator at 10 mK
(Fig. 5()). Signals are generated using an arbitrary waveform generator
(Keysight M3202A) and RF-synthesizers (SignalCore 5502A), followed by
upconversion in IQ-mixers (Marki IQ4509 and IQ0307). Excitation and dispersive
readout signals are combined using a directional coupler and sent to the
fridge, where they are attenuated by 60 dB to reduce thermal noise reaching
the sample. The response from the readout resonators located on the sample is
amplified using a HEMT- and room-temperature amplifiers. Finally, the readout
signal is downconverted and digitized at a 100 MHz IF frequency using a
Spectrum Instruments m4x PXI card. Using this scheme, qubit spectra are
obtained under continuous excitation and readout. Artificial atoms coherence
times are characterised using conventional time-domain techniques [47, 3].

Figure 7: Simulated spectra for the kinemon I, obtained from the master
equation solution. Color shows the expectation value of the cavity field
$\lvert\left\langle\hat{a}\right\rangle\lvert$ (top) and the depopulation of
the ground state, $1-P(\ket{g})$, (bottom). Simulation parameters:
$\Omega/2\pi=200$ MHz for the kinemon spectrum ($=0.1$ MHz for the cavity
spectrum), $\omega_{r}/2\pi=7.1851$ GHz, $g/2\pi=64$ MHz, $\kappa=2.5\
\mu$s-1, $\gamma=10\,\mu$s-1.

##  Appendix D Cavity coupling

To plot the model curves describing the dependence of the resonator frequency
on the magnetic flux (the upper panels of Fig. 2()), we solve the following
Hamiltonian in the $\varphi$-basis for the kinemon and in the Fock basis for
the cavity:

$\hat{\mathcal{H}}_{\mathrm{cQED}}=\hat{\mathcal{H}}+\hbar\omega_{r}(\hat{a}^{\dagger}\hat{a}+1/2)+\hbar
g(\hat{a}^{\dagger}-\hat{a})\otimes\frac{\partial}{\partial\varphi},$

where $\hat{\mathcal{H}}$ is defined by Eq. (1), $\hat{a}$ is the bosonic
annihilation operator, and the capacitive coupling term is derived using the
canonical quantization expression for the $\hat{n}$ operator,
$[\hat{n},\hat{\varphi}]=i$. The corresponding coupling coefficient $g$ is
defined by fitting the model to the data.

For numerical solution of the stationary Shroedinger equation for
$\hat{\mathcal{H}}_{\text{cQED}}$, we use separate finite difference formulas
of the 6th order to approximate the first and the second derivatives. This
allows to achieve convergence of the necessary low-lying eigenenergies on a
rough $\varphi$ grid of 30 to 50 nodes, which then facilitates execution of
the computationally-demanding fitting algorithms.

To reproduce the multiphoton transitions which can be observed in the lower
panels of Fig. 2(), we also perform a time-domain simulation by solving the
GKSL equation based on the $\hat{\mathcal{H}}_{\mathrm{cQED}}$-defined unitary
evolution and dissipative energy relaxation dynamics characterized by the
collapse operators $\sqrt{\kappa}\hat{a}$ and $\sqrt{\gamma}\hat{b}$,
$\kappa,\gamma$ being the corresponding decay rates. Here, the kinemon
lowering operator $\hat{b}$ for each value of $\varphi_{e}$ is constructed
from the lowest eigenstates $\ket{E_{n}}$,

$\hat{b}=\hat{\mathbbm{1}}_{\text{r}}\otimes\sum_{n}\ket{E_{n}}\bra{E_{n+1}}.$

Then, at a given $\varphi_{e}$, the master equation is solved with an addition
of a resonator driving term to $\hat{\mathcal{H}}_{\mathrm{cQED}}$ of the form
$\hbar\Omega(\hat{a}^{\dagger}+\hat{a})\cos{\omega_{\text{d}}t}$. The driving
frequency $\omega_{\text{d}}/2\pi$ is scanned through the same range that is
used in the spectroscopy, i.e., from 2.5 to 5.5 GHz, and the resulting steady-
state density matrices are saved for further calculation of the observables.

We show the results for both the transmission spectroscopy and the two-tone
spectroscopy in Fig. 7(). We note in overall a good agreement with the
experimental data, but find an extra spectral line in Fig. 2()(a), having its
minimum at around 4.7 GHz, which is not reproduced in the simulation. We
attribute it to a sideband two-photon process taking one thermal photon from
the cavity and exciting the kinemon to its 4-th excited state located around
12 GHz above the ground state at the lower sweet spot (in the modeling, the
cavity it at zero temperature, so such a process can not be observed). The
upside-down-looking spectral line having a maximum frequency of 4.5 GHz is a
vice-versa process, depopulating the kinemon from $\ket{e}$ to $\ket{g}$ and
exciting the cavity. It is observable in the simulation as $\ket{e}$ is
slighlty populated for $\varphi\in[-1.2,-0.8]$ due to the numerical errors.

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