Higher-order harmonics in Josephson tunnel junctions due to series inductance

Nature Physics (2026) Cite this article Josephson tunnel junctions are essential elements of superconducting quantum circuits. The standard analysis of these circuits presumes a 2π-periodic sinusoidal potential for a tunnel junction, but higher-order corrections to this Josephson potential, often referred to as harmonics, cause deviations from the expected circuit behaviour. Two potential sources of these harmonics are the intrinsic current–phase relationship of the Josephson junction and the inductance of the metallic traces connecting the junction to other circuit elements. Here we introduce a method to distinguish the origin of the observed harmonics using nearly symmetric superconducting quantum interference devices. Spectroscopic measurements of level transitions in multiple devices reveal features that cannot be explained by a standard cosine potential, but are accurately reproduced when accounting for a second-harmonic contribution to the model. The observed scaling of the second harmonic with Josephson junction size indicates that it is due almost entirely to the metallic trace inductance. These results can inform the design of next-generation superconducting circuits for quantum information processing and investigations of the supercurrent diode effect.This is a preview of subscription content, access via your institution Access Nature and 54 other Nature Portfolio journals Get Nature+, our best-value online-access subscription $32.99 / 30 days cancel any timeSubscribe to this journal Receive 12 print issues and online access $259.00 per yearonly $21.58 per issueBuy this articleUSD 39.95Prices may be subject to local taxes which are calculated during checkoutSource data are provided with this paper. Additional data are available from the corresponding authors upon reasonable request.The code used for numerical simulations and data analysis is available from the corresponding authors upon reasonable request.Blais, A., Huang, R.-S., Wallraff, A., Girvin, S. M. & Schoelkopf, R. J. Cavity quantum electrodynamics for superconducting electrical circuits: an architecture for quantum computation. Phys. Rev. A 69, 062320 (2004).Article ADS Google Scholar Likharev, K. K. & Semenov, V. K. RSFQ logic/memory family: a new Josephson-junction technology for sub-terahertz-clock-frequency digital systems. IEEE Trans. Appl. Supercond. 1, 3 (1991).Article ADS Google Scholar Barone, A. & Paterno, G. Physics and Applications of the Josephson Effect (Wiley, 1982).Misaki, K. & Nagaosa, N. Theory of the nonreciprocal Josephson effect. Phys. Rev. B 103, 245302 (2021).Article ADS Google Scholar Peacock, A. et al. Single optical photon detection with a superconducting tunnel junction. Nature 381, 135 (1996).Article ADS Google Scholar Barzanjeh, S., Pirandola, S., Vitali, D. & Fink, J. M. Microwave quantum illumination using a digital receiver. Sci. Adv. 6, eabb0451 (2020).Article ADS Google Scholar Golubov, A. A., Kupriyanov, M. Y. & Il’ichev, E. The current-phase relation in Josephson junctions. Rev. Mod. Phys. 76, 411 (2004).Article ADS Google Scholar Willsch, D. et al. Observation of Josephson harmonics in tunnel junctions. Nat. Phys. 20, 815 (2024).Article Google Scholar Putterman, H. et al. Preserving phase coherence and linearity in cat qubits with exponential bit-flip suppression. Phys. Rev. X 15, 011070 (2025).

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This research is sponsored by US Army Research Office grant number W911NFF-23-1-0045 (Extensible and Modular Advanced Qubits), the US Department of Energy, Office of Science, National Quantum Information Science Research Centers, Co-design Center for Quantum Advantage (C2QA), under contract number DE-SC0012704 and under Air Force contract number FA8702-15-D-0001. J.K. and J.A. gratefully acknowledge support from the Korea Foundation for Advanced Studies (KFAS). M.H. is supported by an appointment to the Intelligence Community Postdoctoral Research Fellowship Program at the Massachusetts Institute of Technology administered by the Oak Ridge Institute for Science and Education (ORISE) through an interagency agreement between the US Department of Energy and the Office of the Director of National Intelligence (ODNI). The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the US Air Force or the US government.These authors contributed equally: Junghyun Kim, Max Hays.Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, MA, USAJunghyun Kim, Max Hays, Ilan T. Rosen, Junyoung An, Helin Zhang, Aranya Goswami, Kate Azar, Terry P. Orlando, Jeffrey A. Grover, Kyle Serniak & William D. OliverDepartment of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, MA, USAJunghyun Kim, Junyoung An, Kate Azar, Terry P. Orlando & William D. OliverLincoln Laboratory, Massachusetts Institute of Technology, Lexington, MA, USAKate Azar, Jeffrey M. Gertler, Bethany M. Niedzielski, Mollie E. Schwartz & Kyle SerniakDepartment of Physics, Massachusetts Institute of Technology, Cambridge, MA, USAWilliam D. OliverSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarJ.K. and M.H. planned the experiment, performed the theoretical calculations and analysed the data. M.H. conceived the initial concept of the experiment. J.K. conducted the measurements and performed numerical circuit simulations. J.A. and I.T.R. assisted with the experimental set-up. I.T.R., J.A., H.Z., A.G., K.A. and J.M.G. assisted in the analysis and interpretation of the data. K.S. designed the devices. B.M.N. fabricated the devices with coordination from K.S. and M.E.S. M.H., T.P.O., J.A.G., K.S. and W.D.O. supervised the project. J.K. and M.H. wrote the manuscript with assistance from I.T.R. and with feedback from all authors.Correspondence to Junghyun Kim, Max Hays or William D. Oliver.The authors declare no competing interests.Nature Physics thanks Çağlar Girit, Eli Levenson-Falk and Roman-Pascal Riwar for their contribution to the peer review of this work. Peer reviewer reports are available.Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.Supplementary Figs. 1–12, Tables 1–5 and text/discussion.Numerical data presented in Supplementary Fig. 4.Numerical data presented in Supplementary Fig. 5.Numerical data presented in Supplementary Fig. 6.Numerical and experimental data presented in Supplementary Fig. 7.Numerical data presented in Supplementary Fig. 8.Numerical data presented in Supplementary Fig. 9.Numerical and experimental data presented in Supplementary Fig. 10.Numerical data presented in Supplementary Fig. 11.Numerical data.Numerical and experimental data.Numerical data for the fit and experimental data for three chips.Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.Reprints and permissionsKim, J., Hays, M., Rosen, I.T. et al. Higher-order harmonics in Josephson tunnel junctions due to series inductance. Nat. Phys. (2026). https://doi.org/10.1038/s41567-026-03285-5Download citationReceived: 28 July 2025Accepted: 08 April 2026Published: 12 May 2026Version of record: 12 May 2026DOI: https://doi.org/10.1038/s41567-026-03285-5Anyone you share the following link with will be able to read this content:Sorry, a shareable link is not currently available for this article. Provided by the Springer Nature SharedIt content-sharing initiative