The Josephson effect is fundamental to superconducting quantum information processing, providing low-loss nonlinearity crucial for superconducting qubits. Tunnel Josephson junctions (TJs), typically modeled with an idealized sinusoidal current-phase relation (CPR), *I(φ) = Ic sin φ*, form the core of these qubits. This simplification assumes vanishingly low-transparency channels in the AlOx barrier. However, real-world junctions exhibit microscopic inhomogeneities, suggesting that this simplified model might be inadequate. This research explores the limitations of the simplified sinusoidal CPR in describing the energy spectra of transmon qubits. Transmons are widely used superconducting artificial atoms, and their energy spectra are highly sensitive to the precise form of the CPR. Accurate modeling is essential for optimizing qubit performance and developing reliable quantum technologies based on these devices. The study investigates the impact of higher-order Josephson harmonics on transmon energy levels and charge dispersion, providing a more realistic model for improved design and control of superconducting qubits and other related devices. The significance lies in the potential for improved accuracy in circuit models and control, directly impacting the fidelity and scalability of superconducting quantum computers and other applications, including metrology and parametric amplifiers.

Previous research has extensively studied the Josephson effect and its applications in superconducting circuits. While the sinusoidal CPR accurately describes many aspects of Josephson junctions, particularly those with low-transparency barriers, deviations from this ideal behavior have been observed in various junction types, including weak links, point contacts, and ferromagnetic junctions. These deviations manifest as higher-order Josephson harmonics in the CPR. Studies have shown the importance of considering these harmonics in specific types of junctions, but their relevance for widely used Al-AlOx-Al tunnel junctions, the foundation of many superconducting qubits, has remained largely unexplored. This study builds upon previous work by investigating the impact of these harmonics in transmon qubits, which are highly sensitive to subtle features in the CPR. The authors build on a long history of work in modeling the energy spectra of atoms and molecules, adapting the techniques used to the specific case of transmon qubits.

The researchers experimentally measured the energy spectra of transmon qubits from various laboratories (Karlsruhe Institute of Technology (KIT), Ecole Normale Supérieure (ENS) Paris, University of Cologne, and IBM). These transmons utilized standard Al-AlOx-Al tunnel junctions fabricated using shadow evaporation. Spectroscopy data included transition frequencies and resonator frequencies, determined using circuit quantum electrodynamics techniques. The authors then compared the measured spectra to predictions generated by two models: a standard transmon model based on the sinusoidal CPR (*Hstd*) and a Josephson harmonics model (*Hhar*) that incorporated higher-order harmonics. The *Hstd* model, widely used for over 15 years, is given by: *Hstd = 4Ec(n − ng)2 − EJ cos φ + Hres*, where Ec is the charging energy, EJ is the Josephson energy, ng is the offset charge, and Hres accounts for the readout resonator. The *Hhar* model includes higher harmonics: *Hhar = 4Ec(n − ng)2 − Σm≥1 Em cos(mφ) + Hres*. The parameters for both models were extracted by solving the inverse eigenvalue problem using measured spectroscopy data. The authors also employed a mesoscopic model of tunneling through an inhomogeneous AlOx barrier to derive a distribution of channel transparencies, which was used to calculate the amplitudes of the higher harmonics. This allowed comparison between experimental results and theoretical models based on different assumptions about barrier homogeneity. The study compared the efficacy of the standard model against the developed Josephson harmonics model in accurately fitting and replicating experimental data. The parameters of the mesoscopic model (average barrier thickness and standard deviation) were linked to microscopic structural information obtained from scanning transmission electron microscopy (STEM) and molecular dynamics simulations.

The study revealed significant discrepancies between the measured transmon energy spectra and the predictions of the standard transmon model based on the sinusoidal CPR. These deviations were much larger than the measurement uncertainties and were observed across all samples from various laboratories. In contrast, the Josephson harmonics model, incorporating higher-order harmonics, yielded orders of magnitude better agreement with the measured spectra. The second harmonic (m=2) contribution was found to be in the few percent range for all samples, indicating a non-negligible influence on the system's behavior. The authors found that the inclusion of the harmonics improved the agreement of measured data with predicted values. The mesoscopic model provided a reasonable description of the harmonic amplitudes. This model, parameterized by the average barrier thickness and standard deviation, correlated with STEM images and molecular dynamics simulations, supporting the physical explanation of the observed Josephson harmonics. A direct consequence of the inclusion of these harmonics was also a more accurate description of charge dispersion. In particular, the authors observed that Josephson harmonics can significantly reduce charge dispersion compared to the predictions of the standard model, implying a path towards reducing charge noise decoherence. They observed a factor of 4 decrease in charge dispersion in some of the IBM qubits.

The findings demonstrate the inadequacy of the simplified sinusoidal CPR for accurately modeling the behavior of AlOx tunnel junctions in transmon qubits. The presence of higher Josephson harmonics, attributed to the inhomogeneity of the AlOx barrier, significantly affects the energy levels and charge dispersion of these devices. The improved agreement between the Josephson harmonics model and experimental data highlights the importance of considering microscopic details of the junction structure. The ability to accurately predict the energy spectra and charge dispersion is crucial for optimizing the design and control of superconducting qubits and achieving higher fidelity quantum operations. The significant reduction in charge dispersion observed in some samples with large Josephson harmonics suggests a potential optimization strategy for mitigating charge noise decoherence. This could lead to more robust and scalable quantum computers.

This work reveals the significant impact of higher-order Josephson harmonics on the energy spectra and charge dispersion of transmon qubits. The improved modeling achieved by incorporating these harmonics necessitates a reevaluation of current models for AlOx-based devices across quantum technology and metrology. Future research should explore the deliberate engineering of Josephson harmonics to optimize qubit performance and reduce errors. The techniques presented here could be adapted to characterize other tunnel-junction-based devices.

While the mesoscopic model provides a reasonable description of the observed harmonics, the simplified Gaussian distribution of barrier thickness may not fully capture the complexity of the actual barrier structure. Further research could investigate more sophisticated models of barrier inhomogeneity. The study primarily focused on transmon qubits; the extent to which these findings generalize to other superconducting circuit architectures requires further investigation. The analysis relied on a limited set of samples from different laboratories, suggesting that more extensive characterization across a broader range of fabrication parameters and measurement conditions would be beneficial for broader generalization.