Main

The Josephson effect1,2 is the keystone of quantum information processing with superconducting hardware: it constitutes a unique source of low-loss nonlinearity, which is essential for the implementation of superconducting quantum bits, and it plays a similarly fundamental role as the nonlinear current–voltage relation of diodes in semiconductor circuitry. In particular, tunnel Josephson junctions (JJs), formed by two overlapping superconducting films separated by a thin insulating barrier, have enabled superconducting hardware to become one of the leading platforms for the realization of fault-tolerant quantum computers3,4,5,6. JJs are also at the heart of quantum limited amplification7, metrological applications8 such as the definition of the voltage9 and a possible future current standard10, and they enable quantum detectors such as the microwave photon counter11. With the advancement12,13,14 of superconducting artificial atom technology, the measurement and understanding of subtle features in the Josephson effect, similar to the fine structure discovered in natural atoms, is increasingly relevant in setting the accuracy of both circuit control and circuit models.

Although the mesoscopic dimensions of JJs imply the existence of many conduction channels, for tunnel junctions this complexity is usually condensed into a single effective parameter, the critical current Ic, in the well-known Josephson current–phase relation, CφR (grey line in Fig. 1):

$$I(\varphi )={I}_{{{{\rm{c}}}}}\sin \varphi \,,$$
(1)

where φ is the superconducting phase difference across the junction. This simplification is remarkable given the fact that other types of junctions, such as weak links, point contacts and ferromagnetic JJs, generally exhibit non-sinusoidal CφRs containing higher Josephson harmonics: \(\sin (2\varphi )\), \(\sin (3\varphi )\) and so on15,16,17,18,19,20,21. Here we show that Josephson harmonics are also relevant for tunnel JJs (Fig. 1).

Fig. 1: Josephson harmonics are relevant for the CφR of tunnel junctions.
figure 1

The nonlinear CφR is the fingerprint of a JJ, which relates the supercurrent I(φ) to the phase φ (inset). For tunnel JJs, the CφR has been considered to be purely sinusoidal (dashed grey line; equation (1)), with the maximum given by the critical current Ic. However, as we show in this work, even in tunnel JJs, the underlying microscopic complexity of the charge transport can manifest in the contribution of higher harmonics to the CφR. As an example, the red line shows a CφR consistent with measured data (CD1 of the KIT sample), which includes the harmonics expected from a mesoscopic model assuming an inhomogeneous AlOx barrier. The shaded red area shows the difference from the purely sinusoidal CφR. We provide CφRs for all other measured samples in Supplementary Fig. 7.

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To understand the limits of the approximation equation (1) for tunnel junctions, we have to take a closer look at commonly used Al–AlOx–Al JJs, fabricated by shadow evaporation22 and schematized in Fig. 2a–c, which reveals a complex microscopic reality. The CφR of the junction is obtained by summing the supercurrents of N conduction channels, \(I(\varphi )=\mathop{\sum }\nolimits_{n = 1}^{N}{I}_{n}(\varphi )\). Each channel (Fig. 2b) has a transparency-dependent CφR (refs. 16,23) that can be expressed as a Fourier series:

$${I}_{n}(\varphi )\propto \frac{{T}_{n}\sin \varphi }{\sqrt{1-{T}_{n}{\sin }^{2}(\varphi /2)}}=\,\hspace{2.22144pt}\mathop{\sum }\limits_{m=1}^{\infty }\,\hspace{2.22144pt}{c}_{m}({T}_{n})\sin (m\varphi )\,.$$
(2)

The conduction channel transparency Tn is defined as the tunnel probability for an electron impinging on the insulating barrier of channel n, and cm(Tn) are the order m Fourier coefficients for In(φ). These coefficients alternate in sign and decay in magnitude with increasing order m (Fig. 2d). The ratio ∣cm+1/cm∣ of successive coefficients increases with Tn (Supplementary Section IA): the more transparent a channel, the more relevant the contribution of higher harmonics. To put it simply, in higher-transparency channels, it is more likely for Cooper pairs to tunnel together in groups of m, which correspond to the \(\sin (m\varphi )\) terms in the CφR.

Fig. 2: Josephson harmonics result from junction barrier inhomogeneity.
figure 2

a, False-coloured scanning electron microscope image of a typical Al–AlOx–Al JJ fabricated at KIT. The bottom and top electrodes are coloured blue and yellow, respectively. Inset, circuit symbol for a JJ with phase difference φ across the barrier. b, Cross-section schematic of the superconductor–insulator–superconductor JJ at the location indicated by the dash-dotted line in a. The supercurrent In(φ) of each conduction channel n = 1, …, N depends on its transparency Tn (equation (2)). We sketch a distribution of multiple low and a few high transparencies T1, …, TN in green and red, respectively. c, False-coloured high-angle annular dark field STEM image centred on the AlOx tunnel barrier of a typical JJ fabricated at KIT, with average thickness d ≈ 2 nm as indicated by the white arrow. Individual columns of atoms of the Al grain in the top electrode are visible due to zone axis alignment, which is not the case for the bottom Al electrode (additional STEM images with thickness variations and structural defects such as grain boundaries are shown in Supplementary Fig. 27). d, Normalized Fourier coefficients cm(Tn) of the JJ CφR (equation (2)) for a low (10−6, green) and high (10−2, red) transparency channel. Note the alternating sign for even and odd order m and the fact that high-transparency channel coefficients (in red) remain relevant to higher order. e, Sketch of how the higher-order terms in the JJ Hamiltonian modulate the potential and shift the energy levels (red) of superconducting artificial atoms compared to a purely cosφ potential (grey). In this Article, we focus on transmon devices, which consist of a large capacitor in parallel to the JJ (refer to the circuit schematic inset). The discrepancy between the models generally increases at higher levels. f, The higher-order Josephson harmonics also influence the charge dispersion of the transmon levels versus offset charge ng. The two branches per energy level correspond to a change between even and odd charge parity (that is, quasiparticle tunnelling79,80; Supplementary Fig. 23 in Supplementary Section IIIC).

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In the limit Tn → 0, only the \(\sin \varphi\) term of equation (2) survives. If all channels in a JJ are in this limit, we recover the purely sinusoidal CφR of equation (1), with the critical current of the junction Ic proportional to the sum of transparencies. Assuming a perfectly homogeneous barrier, for a typical junction with ~μm2 area and resistance comparable to the resistance quantum, one expects N ≈ 106 and Tn ≈ 10−6 (refs. 24,25), leading to negligible (below 10−6) corrections to the purely sinusoidal CφR.

But is this the reality? Here we argue that in the presence of contaminants, atomic scale defects26 and random crystalline orientations of the grains in contact, evidenced by scanning transmission electron microscope (STEM) images and molecular dynamics simulations (Fig. 2c and Supplementary Section IV), we have reasons to doubt it. In fact, about two decades ago, AlOx barrier inhomogeneity motivated the transition in magnetic junctions to more uniform oxides such as MgO (refs. 27,28,29). Consequently, we expect a distribution of transparencies in AlOx (refs. 30,31) with possibly a few relatively high-transparency channels32,33 introducing measurable corrections to the CφR (Fig. 1). The microscopic structure of each barrier is therefore imprinted on the CφR of the JJ, and the challenge is how to experimentally access this information.

For our study of tunnel JJs, we use transmon devices34, in which a JJ is only shunted by a large capacitor to form a nonlinear oscillator with the potential energy defined by the CφR of the junction (Fig. 2e). The resulting individually addressable transition frequencies in the microwave regime can be measured using circuit quantum electrodynamics techniques35. We compare the spectra of multiple samples to the prediction of the standard transmon Hamiltonian based on a sinusoidal CφR (equation (1)) and find increasing deviations for the higher energy levels of all samples, as sketched in Fig. 2e,f. Only by accounting for higher harmonics in the CφR are we able to accurately describe the entire energy spectrum. A similar methodology was used in ref. 18 to reconstruct the CφR of a semiconductor nanowire Josephson element. While our study focuses on transmon qubits, the conclusions we draw regarding the CφR of tunnel junctions should trigger a re-evaluation of the current models for tunnel-JJ-based devices used in quantum technology and metrology35,36,37,38,39.

Since transmons are widely available in the community, we are able to measure and model the spectra of multiple samples from laboratories around the globe: fixed-frequency transmons fabricated and measured at the Karlsruhe Institute of Technology (KIT; Supplementary Fig. 18) in three cooldowns (CDs; Supplementary Fig. 19) and Ecole Normale Supérieure (ENS) Paris (same device as in ref. 40), a tunable transmon subject to an in-plane magnetic field at the University of Cologne (Köln; identical setup and similar device as in ref. 41; Supplementary Fig. 23) and 20 qubits from the IBM Hanoi processor (IBM). All transmons are based on standard Al–AlOx–Al tunnel junctions (Fig. 2) and are measured in either a three-dimensional architecture or a two-dimensional coplanar waveguide geometry (for detailed descriptions of each sample, see Supplementary Section III). The spectroscopy data consists of (1) transition frequencies f0j into transmon states j = 1, 2, … up to j = 6, each measured as j-photon transitions at frequencies f0j/j, and (2) the resonator frequencies \({f}_{{{{\rm{res}}}},\;j}\) depending on the transmon state j = 0, 1 (Methods).

In Fig. 3, we compare the measured transition frequencies to predictions \({f}_{0j}^{{{\;{\rm{model}}}}}\), obtained by exact diagonalization of two different model Hamiltonians. The first model is the standard transmon model, which has served the community for over 15 years34

$${H}_{{{{\rm{std}}}}}=4{E}_{{{{\rm{C}}}}}{(n-{n}_{{{{\rm{g}}}}})}^{2}-{E}_{{{{\rm{J}}}}}\cos \varphi +{H}_{{{{\rm{res}}}}}\,,$$
(3)

where EC is the charging energy, EJ is the Josephson energy, ng is the offset charge and the operators n and φ represent the charge normalized by twice the electron charge and the phase difference across the junction, respectively. All models include the readout resonator Hamiltonian given by \({H}_{{{{\rm{res}}}}}={{\varOmega }}{a}^{{\dagger} }a+Gn(a+{a}^{{\dagger} })\), where Ω is the bare resonator frequency, G is the electrostatic coupling strength and a† (a) is the bosonic creation (annihilation) operator. Including \({H}_{{{{\rm{res}}}}}\) ensures that dressing of the states due to transmon-resonator hybridization is taken into account34,35,42,43.

Fig. 3: Standard transmon model fails to describe the measured frequency spectra.
figure 3

a, Differences between the frequencies \({f}_{0j}^{{{\,{\rm{model}}}}}\) predicted by the standard transmon model in equation (3) and the measured transitions \({f}_{0j}^{{{\,{\rm{experiment}}}}}\). The markers indicate the different experiments at KIT (red stars), ENS (yellow squares), Köln (blue diamonds) and IBM (green circles). For the KIT experiment, we show results for three successive CDs of the same sample (CD1–3, dark red to bright red, respectively). For the Köln experiment, we chose a set of measured transitions at a fixed magnetic field (blue arrow in Fig. 4a). For the IBM experiment, we show results for 20 qubits in the IBM Hanoi device, using different marker sizes and shades of green. Measurement imprecisions are on the order of 1 MHz and not visible in the figure. Note that the scale on the vertical axis is linear between ±100 MHz and logarithmic onward. Dashed and dotted lines are guides to the eye. b, Same as a, with \({f}_{0j}^{{{\,{\rm{model}}}}}\) given by the Josephson harmonics Hamiltonian in equation (4). Top, model truncated at EJ4. Bottom, mesoscopic model of tunnelling through an inhomogeneous AlOx barrier, where \({E}_{{{{\rm{J}}}}m}(\bar{d},\sigma )\) is parameterized in terms of the average barrier thickness \(\bar{d}\) and the standard deviation σ (equation (5); the fit values are listed in the table inset). c, Ranges of the Josephson harmonics ratios ∣EJm/EJ1∣ that are consistent with the measured spectra. The ranges are represented by coloured vertical bars using the same colouring as in a. For the IBM Hanoi device, we show the ranges for qubits 0–2 from left to right (ranges for the other qubits are shown in Supplementary Section IC3). The shaded grey area highlights the region between two limiting cases: the fully open quantum point contact with unit transparency and a homogeneous barrier with Tn = 10−6 for all n. Turquoise markers on the vertical bars indicate the harmonics ratios calculated from the mesoscopic model, where the average thickness \(\bar{d}\) and the standard deviation σ are given in b. Inset, an Al–AlOx–Al junction obtained from molecular dynamics simulations (Supplementary Fig. 25) with average barrier thickness \(\bar{d}=1.5\,{{{\rm{nm}}}}\) (Fig. 2c).

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We obtain the parameter set (EC, EJ, Ω, G) of the standard transmon model in equation (3) by solving the inverse eigenvalue problem (IEP)44,45,46,47 for the measured spectroscopy data (Methods). For the Köln sample, these data include the offset charge dispersion (additional data for different magnetic fields are given in Supplementary Section IID). We note that the IEP is the very same science problem that was historically solved to model the energy spectra of natural atoms and molecules (see for example refs. 48,49,50), which led to the discovery of the fine structure.

In Fig. 3a, we show that the standard transmon model in equation (3) fails to describe the measured frequency spectra for all samples. The observed deviations are much larger than the measurement imprecision, for which we can set a conservative upper bound on the order of 1 MHz. While the standard transmon model with two parameters can trivially match the f01 and f02 transitions, the measured f03 can already deviate by more than 10 MHz. The deviations are positive for the KIT, ENS and Köln samples, while the IBM transmons mostly show negative deviations (Supplementary Section IC5). It is important to remark that other corrections, such as the stray inductance in the JJ leads, hidden modes coupled to the qubit, the coupling between qubits as present on the IBM multi-qubit device, or an asymmetry in the superconducting energy gaps, while being relevant, cannot fully account for the measured discrepancy (Supplementary Section ID). Notably, similar deviations can be found in previously published transmon spectra41,51,52,53, as we detail in Supplementary Fig. 4 and Supplementary Sections IC2 and IC4.

In Fig. 3b, we demonstrate that orders of magnitude better agreement with our measured spectra can be achieved by using the Josephson harmonics model:

$${H}_{{{{\rm{har}}}}}=4{E}_{{{{\rm{C}}}}}{(n-{n}_{{{{\rm{g}}}}})}^{2}-\mathop{\sum}\limits_{m\ge 1}{E}_{{{{\rm{J}}}}m}\cos (m\varphi )+{H}_{{{{\rm{res}}}}}\,.$$
(4)

In general, the values EJm are a fingerprint of each junction’s channel-transparency distribution ρ(T) with many degrees of freedom. Here we consider two simplified models (further models are discussed in Supplementary Section IC): (1) a phenomenological model truncated at EJ4 (top panel) and (2) a mesoscopic model of tunnelling through a non-uniform oxide barrier (bottom panel). We note that the phenomenological EJ4 model guarantees agreement for the lowest four transitions (Methods), and while many samples have physically reasonable EJm coefficients when truncating at EJ4, a few JJs require terms up to EJ6 (Supplementary Section IC3).

The mesoscopic model allows us to derive \(\rho (T;\bar{d},\sigma )\) based on a Gaussian thickness distribution with average thickness \(\bar{d}\) and standard deviation σ (Supplementary Section IB4). As a consequence, all Josephson harmonics for m ≥ 2 are parameterized in terms of the two parameters \(\bar{d}\) and σ according to

$${E}_{{{{\rm{J}}}}m}(\bar{d},\sigma )\propto \int\nolimits_{0}^{1}\,{c}_{m}(T)\,\rho (T;\bar{d},\sigma )\,{{{\rm{d}}}}T\,,$$
(5)

where the Fourier coefficients cm(T) (equation (2) and Fig. 2d) are weighted by the channel-transparency distribution \(\rho (T;\bar{d},\sigma )\). In this model, relatively large ratios ∣EJm/EJ1∣ originate from higher-transparency contributions from the narrower regions of the barrier (compare the STEM images in Supplementary Fig. 27). The model can describe the samples at KIT, ENS and Köln (Fig. 3b) but not the IBM device (Supplementary Section IB4). The model parameters \(\bar{d}\) and σ (Fig. 3b) are comparable to results from molecular dynamics simulation and STEM pictures of the oxide barrier (Supplementary Section IV).

In Fig. 3c, we indicate the ranges of EJm coefficients consistent with the measured spectra. The bars represent the lower and upper limits of Josephson harmonics ratios ∣EJm/EJ1∣. The corresponding \(\sin (m\varphi )\) contribution to the CφR is given by m∣EJm/EJ1∣ (see Fig. 1 for the KIT sample). The ratios lie between two limiting cases spanning the physical regime (shaded grey area): (1) the upper limit, ∣EJm/EJ1∣ = 3/(4m2 − 1), corresponds to an open quantum point contact—that is, one channel with T = 1—and (2) the lower limit, ∣EJm/EJ1∣ ≈ (T/4)m−1/m3/2, corresponds to a perfectly homogeneous low-transparency barrier (Tn = T = 10−6 for all n). For the scanning routine, we include harmonics up to EJ10 to obtain results within the physical regime and to see when truncation is possible (Methods). Remarkably, for all samples, the EJ2 contribution is in the few percent range even after considering additional corrections such as series inductance or gap asymmetry in the superconducting electrodes (Supplementary Section ID).

The Josephson harmonics ratios computed from the mesoscopic model in equation (5) are shown with turquoise markers. Notice that the barrier evolved between CDs of the KIT sample due to ageing (CD1 to CD2) and thermal annealing (CD2 to CD3) (Supplementary Section IIIA). Even for the most homogeneous barrier (CD3), the second-harmonic contribution is EJ2/EJ1 ≈ −2.4%, implying that there would be at least one conduction channel with a transparency T ≥ 0.29 (Supplementary Section IA). The methodology presented in Fig. 3 can serve as a tool to characterize Josephson harmonics and tunnel barrier homogeneity, independent of circuit design.

Since the charge dispersion increases for higher transmon levels (even for the standard transmon Hamiltonian34; Fig. 2f) and is exponentially sensitive to the shape of the JJ potential (Fig. 2e), a natural question arises: what are the consequences of the Josephson harmonics on the transmon’s susceptibility to offset charges? In Fig. 4a, we show the measured charge dispersion δf0j of the Köln device for states j = 1, 2, 3 versus the first transition frequency f01, which is tuned by an in-plane magnetic field B∥ of up to 0.4 T (see Supplementary Section IIIC for details). The charge dispersion predicted by the standard model (dashed grey lines) falls short of the measurements by a factor of 2–7 for the three measured transitions. In contrast, when using the Josephson harmonics model, the computed charge dispersion matches the data (blue lines). We emphasize that for both models, we use the same parameters as in the Fig. 3 analysis (that is, the standard model and the EJ4 model) and vary the first Josephson energy to match the qubit frequency f01 for different magnetic fields while keeping the EJm/EJ1 ratios constant.

Fig. 4: Influence of Josephson harmonics on the charge dispersion.
figure 4

a, Measured charge dispersion δf0j (blue diamonds) of states j = 1, 2, 3 for the experiment in Köln, plotted as a function of the f01 frequency. All transition frequencies are tuned, as the Josephson energy is suppressed by up to 35% by means of an in-plane magnetic field B∥ swept to 0.4 T. The standard model in equation (3), shown in dashed grey lines, underestimates the charge dispersion by a factor of 2–7 (grey arrows), while the Josephson harmonics model in equation (4) plotted in solid blue overlaps the measured data. Note that both are computed with the same parameters used for Fig. 3; the Josephson energy is reduced with increasing magnetic field, and the other parameters such as the EJm/EJ1 ratios are kept constant. The blue arrow indicates f01 = 5.079 GHz, corresponding to the dataset shown in Fig. 3. b, Evidence that Josephson harmonics can reduce the charge dispersion by an order of magnitude (grey arrows). The dashed grey lines represent the standard model predictions. In contrast, the green bars show results from all Josephson harmonics models. The data correspond to IBM qubits 0–2 (green bars in Fig. 3c) for the levels j = 1, 2, 3, 4; results for all other samples are shown in Supplementary Fig. 6. c, The values of EJ1/EC change compared to the standard model EJ/EC, which constitutes the main correction to the predicted charge dispersions in a and b. The bars represent the range of suitable ratios EJ1/EC (Fig. 3c) for the successive CDs of the KIT sample (red bars), the ENS sample (yellow bar), the Köln sample (blue diamonds, using the same colour coding as in s) and the IBM Hanoi device (green bars). The dashed diagonal line indicates the case in which the ratios EJ1/EC of the harmonics model and EJ/EC of the standard model are equal. Inset, correction (εhar − εstd)/εstd to the relative charge dispersion ε = δf0j/f01 for fixed \({E}_{{{{\rm{J}}}}}^{{{{\rm{std}}}}}/{E}_{{{{\rm{C}}}}}^{{{{\rm{std}}}}}={E}_{{{{\rm{J}}}}1}^{{{{\rm{har}}}}}/{E}_{{{{\rm{C}}}}}^{{{{\rm{har}}}}}\) for the Köln sample, where εstd is given by the standard charge dispersion34 and εhar is computed using the Josephson harmonics model.

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Interestingly, the presence of large Josephson harmonics, as in the case of the IBM qubits (Fig. 3c), can also reduce the charge dispersion, which directly decreases charge noise decoherence. We show evidence for this in Fig. 4b, on the first three IBM qubits, for which the charge dispersion of the qubit transition can be a factor of 4 lower than expected from the standard transmon model. This observation indicates a possible optimization route in which Josephson harmonics are engineered (for example, by shaping the channel transparencies or adding inductive elements in series) and the spectrum is steered towards regions of reduced charge dispersion and increased anharmonicity (Supplementary Fig. 8). A recent work54 proposes a similar approach to engineer arbitrary-shaped CφRs using networks of effective high-transparency JJs, each of which is a series of tunnel JJs.

The main reason for the failure of the standard transmon model in describing the charge dispersion (when fitted to f01 and f02) is that it misjudges the value of EJ/EC. To quantify this effect, in Fig. 4c we plot the values of EJ1/EC from the Josephson harmonics model against the value of EJ/EC from the standard model. Indeed, the EJ1/EC ranges for many of our measurements are not compatible with the standard model EJ/EC ratio (dashed diagonal). We note that when evaluated for the same EJ/EC, the Josephson harmonics correction to the charge dispersion is relatively small (inset of Fig. 4c).

In summary, we have shown that for ubiquitous AlOx tunnel junctions, the microscopic structure, currently underappreciated in its complexity, causes level shifts and modifies the charge dispersion in superconducting artificial atoms. In order to fully describe the measured transmon energy spectra, we amend the standard \(\sin \varphi\) Josephson CφR for tunnel junctions to include higher-order \(\sin (m\varphi )\) harmonics, with the relative amplitude of the m = 2 term in the few percent range. We confirm this finding in various sample geometries from four different laboratories, and we argue that the source of the Josephson harmonics is the presence of relatively higher-transparency channels with T ≫ 10−6 in the AlOx tunnel barrier. The methodology shown here can reveal percent-level deviations from a sinusoidal CφR, which are hard to detect in more standard measurements based on asymmetric direct current superconducting quantum interference devices55.

The observation of Josephson harmonics in tunnel junctions highlights the need to revisit established models for superconducting circuits. Our work directly impacts the design and measurement of transmon qubits and processors: As an illustration, we show that by engineering Josephson harmonics, the dephasing due to charge noise can be reduced by an order of magnitude without sacrificing anharmonicity. These results ask for future research studying the implications of Josephson harmonics and associated Andreev bound states in other tunnel-JJ-based circuits, for example fluxonium or generalized flux qubits56.

In general, we expect the inclusion of the harmonics will refine the understanding of superconducting artificial atoms and will directly benefit, among others, quantum gate and computation schemes relying on higher levels57,58,59,60,61,62,63, quantum-non-demolition readout fidelities64,65,66 and frequency crowding mitigation in quantum processors67. Josephson harmonics will probably also have to be accounted for in topological JJ circuits68,69,70, parametric pumping schemes employed in microwave amplifiers and bosonic codes71,72, amplification and mixing7,73,74, JJ metrological devices8,9,10, Floquet qubits75,76, protected Josephson qubits68,70,77 and so on, and they can be harnessed to realize Josephson diodes78. As devices become increasingly sophisticated with progressively smaller error margins, higher-order Josephson harmonics will need to be either suppressed via the development of highly uniform and low-transparency barriers or engineered and included as an integral part of the device physics.

Methods

Diagonalizing the Hamiltonians to obtain model predictions

We construct the matrices of Hstd in equation (3) and Hhar in equation (4) by first diagonalizing the bare transmon matrix (excluding \({H}_{{{{\rm{res}}}}}\)) in the charge basis \(\{\left\vert n\right\rangle \}\), where \(4{E}_{{{{\rm{C}}}}}{(n-{n}_{{{{\rm{g}}}}})}^{2}={\sum }_{n}4{E}_{{{{\rm{C}}}}}{(n-{n}_{{{{\rm{g}}}}})}^{2}\left\vert n\right\rangle \,\left\langle n\right\vert\) is diagonal and \(-{E}_{{{{\rm{J}}}}m}\cos (m\varphi )=-{\sum }_{n}{E}_{{{{\rm{J}}}}m}/2\,(\left\vert n\right\rangle \,\left\langle n+m\right\vert +\left\vert n+m\right\rangle \,\left\langle n\right\vert )\) has constant entries −EJm/2 on the mth subdiagonal (we ensure enough terms by generally verifying that the predictions do not change if more terms are included). This yields the transmon eigenenergies Ej and eigenstates \(\left\vert\, j\right\rangle\). Then we diagonalize the joint transmon-resonator Hamiltonian \({H}_{{{{\rm{std}}}}/{{{\rm{har}}}}}={\sum }_{j}{E}_{j}\left\vert\, j\right\rangle \,\left\langle\, j\right\vert +{{\varOmega }}{a}^{{\dagger} }a+{\sum }_{j,{j}^{{\prime} }}G\left\vert\, j\right\rangle \,\left\langle\, j\right\vert \,n\,\left\vert\, {j}^{{\prime} }\right\rangle \,\left\langle\, {j}^{{\prime} }\right\vert (a+{a}^{{\dagger} })\), where \(a={\sum }_{k}\sqrt{k+1}\left\vert k\right\rangle \,\left\langle k+1\right\vert\). To each resulting eigenenergy \({E}_{\overline{l}}\) and eigenstate \(\left\vert \overline{l}\,\right\rangle\), we assign a photon label k and a transmon label j based on the largest overlap \(\mathrm{max}_{k,\;j}| \left\langle kj| \overline{l}\,\right\rangle |\) (this only works for small k; Supplementary Section IIC), which yields the dressed energies \({E}_{\overline{kj}}\) and states \(\left\vert \overline{kj}\,\right\rangle\). This procedure is done for both ng = 0 and ng = 1/2. From the resulting dressed energies \({E}_{\overline{kj}}({n}_{{{{\rm{g}}}}})\), we compute the transmon transition frequencies \({f}_{0j}^{{{\;{\rm{model}}}}}({n}_{{{{\rm{g}}}}})=({E}_{\overline{0j}}({n}_{{{{\rm{g}}}}})-{E}_{\overline{00}}({n}_{{{{\rm{g}}}}}))/2\uppi\) and the resonator frequencies \({f}_{{{{\rm{res}}}},\,j}^{{{\,{\rm{model}}}}}({n}_{{{{\rm{g}}}}})=({E}_{\overline{1j}}({n}_{{{{\rm{g}}}}})-{E}_{\overline{0j}}({n}_{{{{\rm{g}}}}}))/2\uppi\) (setting ℏ = 1). The predicted frequencies are then given by \({f}_{0j}^{{{\,{\rm{model}}}}}=({f}_{0j}^{{{\,{\rm{model}}}}}(0)+{f}_{0j}^{{{\,{\rm{model}}}}}(1/2))/2\), \({f}_{{{{\rm{res}}}},j}^{{{\,{\rm{model}}}}}=({f}_{{{{\rm{res}}}},j}^{{{\,{\rm{model}}}}}(0)+{f}_{{{{\rm{res}}}},j}^{{{\,{\rm{model}}}}}(1/2))/2\), and the charge dispersion is \(\delta {f}_{0j}^{{{\,{\rm{model}}}}}=|\, {f}_{0j}^{{{\,{\rm{model}}}}}(0)-{f}_{0j}^{{{\,{\rm{model}}}}}(1/2)|\). We consistently use n = −N, …, N with N = 14 and thus 2N + 1 = 29 charge states, j = 0, …, M − 1 with M = 12 transmon states and k = 0, …, K − 1 with K = 9 resonator states, where N, M and K have been chosen by verifying that the model predictions change by less than a few kHz when adding more states.

Solving the IEP to obtain model parameters

The inverse problem47,81 to obtain the parameters xstd of the standard model Hamiltonian in equation (3) and xhar of the harmonics model Hamiltonian in equation (4), such that the linear combinations of eigenvalues \({{{\bf{f}}}}=(\,{f}_{01}^{{{\,{\rm{model}}}}},{f}_{02}^{{{\,{\rm{model}}}}},\ldots ,{f}_{0{N}_{f}}^{{{\,{\rm{model}}}}},{f}_{{{{\rm{res}}}},0}^{{{\,{\rm{model}}}}},{f}_{{{{\rm{res}}}},1}^{{{\,{\rm{model}}}}})\) agree with the measured data, is an instance of the Hamiltonian parameterized IEP (HamPIEP; Supplementary Section IIA2). We solve the HamPIEP using the globally convergent Newton method82 with cubic line search and backtracking83 and the Broyden–Fletcher–Goldfarb–Shanno algorithm84 as implemented in TensorFlow Probability85,86. The Jacobian ∂f/∂x is obtained by performing automatic differentiation through the diagonalization with TensorFlow. For the EJ4 model shown in Fig. 3b, the IEP is solved unambiguously for x = (EJ1, EJ2, EJ3, EJ4, Ω, G) using the lowest four transmon transition frequencies, and we fix the values \({E}_{{{{\rm{C}}}}}^{{{{\rm{KIT}}}}}/h=\) 242 MHz, \({E}_{{{{\rm{C}}}}}^{{{{\rm{ENS}}}}}/h=\) 180 MHz and \({E}_{{{{\rm{C}}}}}^{{{{\rm{IBM}}}}}/h=\) 300 MHz, respectively, to make the models consistent with further available information such as accurate finite-element simulations (Supplementary Section IIIA) or knowledge of the transmon capacitance. For the mesoscopic model (Supplementary Section IB4), the parameters \({{{\bf{x}}}}=(\bar{d},\sigma ,{E}_{{{{\rm{C}}}}},{E}_{{{{\rm{J}}}}},{{\varOmega }},G)\) are found by minimizing the function \(\mathop{\sum }\nolimits_{j = 1}^{{N}_{f}}|\;{f}_{0j}^{{{\,{\rm{model}}}}}/j-{f}_{0j}^{{{\,{\rm{experiment}}}}}/j| +\mathop{\sum }\nolimits_{j = 0}^{1}|\, {f}_{{{{\rm{res}}}},\,j}^{{{\,{\rm{model}}}}}-{f}_{{{{\rm{res}}}},\,j}^{{{\,{\rm{experiment}}}}}|\) using the Broyden–Fletcher–Goldfarb–Shanno algorithm. The initial values for the minimization are given by \(\bar{d}=1.64\,{{{\rm{nm}}}}\) (taken from the molecular dynamics result in Supplementary Section IV), \(\sigma =\bar{d}/4\) (also Supplementary Table 2) and (EC, EJ, Ω, G) from the standard transmon model. For the Köln data, where 288 data points have to be described by the same model parameters x (Fig. 4a) and only the Josephson energy is varied, we use cubic interpolation as a function of \({f}_{01}^{{{\,{\rm{model}}}}}\) and include only a few central points for the available frequencies in the solution of the IEP (the residuals are given in Supplementary Fig. 17). All model parameters are provided in the repository87 accompanying this manuscript.

Scanning the Josephson energies

To obtain the range of suitable Josephson energies {EJm} (shown in Fig. 3c) that are consistent with a measured spectrum, we use an exhaustive scanning procedure. A spectroscopy dataset of Nf measured transition frequencies f0j, j = 1, …, Nf and two resonator frequencies \({f}_{{{{\rm{res}}}},0}\) and \({f}_{{{{\rm{res}}}},1}\) uniquely determines—via the HamPIEP—the values \({{{\bf{x}}}}=({E}_{{{{\rm{J1}}}}},\ldots ,{E}_{{{{\rm{J}}}}{N}_{f}},{{\varOmega }},G)\). We then scan the values of four additional ratios \({{{\bf{y}}}}=({E}_{{{{\rm{J}}}}{N}_{f}+1}/{E}_{{{{\rm{J}}}}1},\ldots ,{E}_{{{{\rm{J}}}}{N}_{f}+4}/{E}_{{{{\rm{J}}}}1})\) for each of these four EJm/EJ1 over 16 geometrically spaced values between the point contact limit 3(−1)m+1/(4m2 − 1) and \({(-1)}^{m+1}\min \{1{0}^{-7},| {E}_{{{{\rm{J}}}}m+1}/{E}_{{{{\rm{J}}}}1}| \}\) (always skipping the first to ensure ∣EJm/EJ1∣ > ∣EJm+1/EJ1∣). Additionally, we include y = (0, 0, 0, 0) to see if truncation at \({E}_{{{{\rm{J}}}}{N}_{f}}\) is allowed. For each combination y, we solve the HamPIEP for the spectroscopy data to obtain the unique solution x. We call the ratios \({{{\bf{e}}}}=(1,{E}_{{{{\rm{J}}}}2}/{E}_{{{{\rm{J}}}}1},\ldots ,{E}_{{{{\rm{J}}}}{N}_{f}+4}/{E}_{{{{\rm{J}}}}1})\) a trajectory that can reproduce the spectrum. However, the trajectory e may not be physical, since (1) some of the leading ratios EJm/EJ for m ≤ Nf might be beyond the quantum point-contact limit, (2) the Josephson energies might not be strictly decreasing in absolute value for increasing order m, or (3) the signs might not be alternating. Note that this can also happen when the Josephson harmonics model in equation (4) is truncated at too-low orders (Supplementary Section IC3). For all EJm, the maximum and minimum possible ratios ∣EJm/EJ∣ define the vertical bars in Fig. 3c.