Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Localized Josephson hot spots due to two-level systems

Joshuah T. Heath1,2,*, Alexander C. Tyner1,2, Thue Christian Thann3, Vincent P. Michal3, Peter Krogstrup3, Mark Kamper Svendsen3, and Alexander V. Balatsky1,2,†

  • 1Nordita, Stockholm University, and KTH Royal Institute of Technology, Hannes Alfvéns väg 12, SE-106 91 Stockholm, Sweden
  • 2Department of Physics, University of Connecticut, Storrs, Connecticut 06269, USA
  • 3NNF Quantum Computing Programme, Niels Bohr Institute, University of Copenhagen, Denmark

  • *Contact author: joshuah.t.heath@su.se
  • †Contact author: balatsky@kth.se

Phys. Rev. Applied 25, 014022 – Published 9 January, 2026

DOI: https://doi.org/10.1103/ncn4-m48z

Abstract

Superconducting qubits are often adversely affected by two-level systems (TLSs) within the Josephson junction, which contribute to decoherence and subsequently limit the performance of the qubit. By treating the TLS as a soft (i.e., low-frequency) bosonic mode localized in real space, we find that a single TLS in either the amorphous-oxide surface or the superconducting bulk may result in a localized “hot spot” of amplified Josephson energy. Such amplification is shown to have a non-negligible effect on the T1 time of a simple phase qubit, regardless of whether or not the TLS is on resonance with the qubit frequency. With this study, we identify unique fingerprints of TLS defects in the Josephson current and qubit decoherence time.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (125)

  1. R. W. Simmonds, K. M. Lang, D. A. Hite, S. Nam, D. P. Pappas, and J. M. Martinis, Decoherence in Josephson phase qubits from junction resonators, Phys. Rev. Lett. 93, 077003 (2004).
  2. Clemens Müller, Jared H. Cole, and Jürgen Lisenfeld, Towards understanding two-level-systems in amorphous solids: Insights from quantum circuits, Rep. Prog. Phys. 82, 124501 (2019).
  3. Li-Chung Ku and Clare C. Yu, Decoherence of a Josephson qubit due to coupling to two-level systems, Phys. Rev. B 72, 024526 (2005).
  4. John M. Martinis, K. B. Cooper, R. McDermott, Matthias Steffen, Markus Ansmann, K. D. Osborn, K. Cicak, Seongshik Oh, D. P. Pappas, R. W. Simmonds, and Clare C. Yu, Decoherence in Josephson qubits from dielectric loss, Phys. Rev. Lett. 95, 210503 (2005).
  5. Alexander P. M. Place, et al., New material platform for superconducting transmon qubits with coherence times exceeding 0.3 ms, Nat. Commun. 12, 1 (2021).
  6. Zhe Wang, Clare C. Yu, and Ruqian Wu, Why superconducting Ta qubits have fewer tunneling two-level systems at the air-oxide interface than Nb qubits, Phys. Rev. Appl. 23, 024017 (2025).
  7. Chenlu Wang, et al., Towards practical quantum computers: Transmon qubit with a lifetime approaching 0.5 ms, npj Quantum Inf. 8, 1 (2022).
  8. Alexander C. Tyner, Joshuah T. Heath, Thue Christian Thann, Vincent P. Michal, Peter Krogstrup, Mark Kamper Svendsen, and Alexander V. Balatsky, Identification of soft modes in amorphous Al2O3 via first-principles, Adv. Quantum Technol. 8, e2500170 (2025).
  9. Timothy C. DuBois, Manolo C. Per, Salvy P. Russo, and Jared H. Cole, Delocalized oxygen as the origin of two-level defects in Josephson junctions, Phys. Rev. Lett. 110, 077002 (2013).
  10. Kartiek Agarwal, Ivar Martin, Mikhail D. Lukin, and Eugene Demler, Polaronic model of two-level systems in amorphous solids, Phys. Rev. B 87, 144201 (2013).
  11. Luke Gordon, Hazem Abu-Farsakh, Anderson Janotti, and Chris G. Van de Walle, Hydrogen bonds in Al2O3 as dissipative two-level systems in superconducting qubits, Sci. Rep. 4, 1 (2014).
  12. Sebastian Zanker, Michael Marthaler, and Gerd Schön, Decoherence and decay of two-level systems due to nonequilibrium quasiparticles, IEEE Trans. Appl. Supercond. 26, 1700204 (2016).
  13. S. E. de Graaf, L. Faoro, L. B. Ioffe, S. Mahashabde, J. J. Burnett, T. Lindström, S. E. Kubatkin, A. V. Danilov, and A. Ya. Tzalenchuk, Two-level systems in superconducting quantum devices due to trapped quasiparticles, Sci. Adv.6, eabc5055 (2020).
  14. P. W. Anderson, B. I. Halperin, and C. M. Varma, Anomalous low-temperature thermal properties of glasses and spin glasses, Philos. Mag. 25, 1 (1972).
  15. W. A. Phillips, Tunneling states in amorphous solids, J. Low Temp. Phys. 7, 351 (1972).
  16. W. Andrews Phillips, Amorphous Solids (Springer, Berlin, Germany, 1981).
  17. Lijin Wang, Andrea Ninarello, Pengfei Guan, Ludovic Berthier, Grzegorz Szamel, and Elijah Flenner, Low-frequency vibrational modes of stable glasses, Nat. Commun. 10, 1 (2019).
  18. Lunjie Zeng, Dung Trung Tran, Cheuk-Wai Tai, Gunnar Svensson, and Eva Olsson, Atomic structure and oxygen deficiency of the ultrathin aluminium oxide barrier in Al/AlOx/Al Josephson junctions, Sci. Rep. 6, 1 (2016).
  19. Hyuntae Jung, Yongmin Kim, Kyooho Jung, Hyunsik Im, Yu. A. Pashkin, O. Astafiev, Y. Nakamura, Hosik Lee, Y. Miyamoto, and J. S. Tsai, Potential barrier modification and interface states formation in metal-oxide-metal tunnel junctions, Phys. Rev. B 80, 125413 (2009).
  20. Junguang He, Wei-Ting Lin, and J. A. Sauls, Theory of two-level tunneling systems in superconductors, Prog. Theor. Exp. Phys. 2025, 063I01 (2025).
  21. F. Marsiglio, Coherence effects in electromagnetic absorption in superconductors, Phys. Rev. B 44, 5373 (1991).
  22. J. M. Ziman, Electrons and Phonons: The Theory of Transport Phenomena in Solids (OUP, Oxford, England, UK, 2001).
  23. Andrey V. Chubukov, Artem Abanov, Ilya Esterlis, and Steven A. Kivelson, Eliashberg theory of phonon-mediated superconductivity—When it is valid and how it breaks down, Ann. Phys. 417, 168190 (2020).
  24. Joshuah T. Heath and Rufus Boyack, Universal scaling relations in electron-phonon superconductors, Phys. Rev. Lett. 134, 216002 (2025).
  25. B. C. Stipe, M. A. Rezaei, and W. Ho, Single-molecule vibrational spectroscopy and microscopy, Science 280, 1732 (1998).
  26. Jian-Xin Zhu, K. O. Rasmussen, A. R. Bishop, and A. V. Balatsky, Detecting breather excitations with inelastic tunneling spectroscopy, arXiv:cond-mat/0407215.
  27. A. V. Balatsky, I. Vekhter, and J.-X. Zhu, Impurity-induced states in conventional and unconventional superconductors, Rev. Mod. Phys. 78, 373 (2006).
  28. R. J. Behm, N. García, and H. Rohrer, Scanning Tunneling Microscopy and Related Methods (Springer Netherlands, Dordrecht, The Netherlands, 2010).
  29. Vinay Ambegaokar and Alexis Baratoff, Tunneling between superconductors, Phys. Rev. Lett. 10, 486 (1963).
  30. T. A. Fulton and D. E. McCumber, dc Josephson effect for strong-coupling superconductors, Phys. Rev. 175, 585 (1968).
  31. Anton Bespalov, Manuel Houzet, Julia S. Meyer, and Yuli V. Nazarov, Theoretical model to explain excess of quasiparticles in superconductors, Phys. Rev. Lett. 117, 117002 (2016).
  32. Zi-Qing Huang, Shu-Kun Ye, Yong-Qiang Xu, Tian-Yi Jiang, Tian-Yue Hao, Bao-Chuan Wang, Xiang-Xiang Song, Hai-Ou Li, Guang-Can Guo, Gang Cao, and Guo-Ping Guo, Revealing spin-flip two-level systems using ultra-thin film superconducting resonators, arXiv:2412.15856.
  33. Lukas Grünhaupt, Martin Spiecker, Daria Gusenkova, Nataliya Maleeva, Sebastian T. Skacel, Ivan Takmakov, Francesco Valenti, Patrick Winkel, Hannes Rotzinger, Wolfgang Wernsdorfer, Alexey V. Ustinov, and Ioan M. Pop, Granular aluminium as a superconducting material for high-impedance quantum circuits, Nat. Mater. 18, 816 (2019).
  34. M. Kristen, J. N. Voss, M. Wildermuth, A. Bilmes, J. Lisenfeld, H. Rotzinger, and A. V. Ustinov, Giant two-level systems in a granular superconductor, Phys. Rev. Lett. 132, 217002 (2024).
  35. A. J. Millis, S. Sachdev, and C. M. Varma, Inelastic scattering and pair breaking in anisotropic and isotropic superconductors, Phys. Rev. B 37, 4975 (1988).
  36. D. K. Morr and R. H. Nyberg, Localized bosonic modes in superconductors, Phys. Rev. B 68, 060505(R) (2003).
  37. A. V. Balatsky, Ar. Abanov, and J.-X. Zhu, Inelastic tunneling spectroscopy in a d-wave superconductor, Phys. Rev. B 68, 214506 (2003).
  38. G. M. Eliashberg, Interactions between electrons and lattice vibrations in a superconductor, Sov. Phys. JETP 11, 696 (1960).
  39. G. M. Eliashberg, Interactions between electrons and lattice vibrations in a superconductor, Sov. Phys. JETP 12, 1437 (1961).
  40. G. Rickayzen, Theory of Superconductivity (John Wiley and Sons Inc., New York, 1965).
  41. D. J. Scalapino, in Superconductivity: Part 1 (In Two Parts), edited by R.D. Parks (Marcel Dekker Inc., New York, 1969), p. 449.
  42. John Bardeen, Electron-phonon interactions and superconductivity, Phys. Today 26, 41 (1973).
  43. F. Marsiglio, Eliashberg theory: A short review, Ann. Phys. 417, 168102 (2020).
  44. B. D. Josephson, Possible new effects in superconductive tunnelling, Phys. Lett. 1, 251 (1962).
  45. B. D. Josephson, The discovery of tunnelling supercurrents, Rev. Mod. Phys. 46, 251 (1974).
  46. Aaron M. Holder, Kevin D. Osborn, C. J. Lobb, and Charles B. Musgrave, Bulk and surface tunneling hydrogen defects in alumina, Phys. Rev. Lett. 111, 065901 (2013).
  47. I. M. Pop, T. Fournier, T. Crozes, F. Lecocq, I. Matei, B. Pannetier, O. Buisson, and W. Guichard, Fabrication of stable and reproducible submicron tunnel junctions, J. Vac. Sci. Technol. B 30, 010607 (2012).
  48. C. M. Quintana, et al., Characterization and reduction of microfabrication-induced decoherence in superconducting quantum circuits, Appl. Phys. Lett. 105, 062601 (2014).
  49. L. J. Zeng, P. Krantz, S. Nik, P. Delsing, and E. Olsson, The atomic details of the interfacial interaction between the bottom electrode of Al/AlOx/Al Josephson junctions and HF-treated Si substrates, J. Appl. Phys. 117, 163915 (2015).
  50. S. Fritz, A. Seiler, L. Radtke, R. Schneider, M. Weides, G. Weiß, and D. Gerthsen, Correlating the nanostructure of Al-oxide with deposition conditions and dielectric contributions of two-level systems in perspective of superconducting quantum circuits, Sci. Rep. 8, 1 (2018).
  51. A. A. Abrikosov, L. P. Gor’kov, and I. Y. Dzyaloshinskii, Quantum Field Theoretical Methods in Statistical Physics (Pergamon Press Ltd, Oxford, 1965), 2nd ed.
  52. Gerald D. Mahan, Many-Particle Physics (Springer US, 2000).
  53. F. Marsiglio and J. P. Carbotte, in Superconductivity, Conventional and Unconventional Superconductors, edited by K. H. Bennemann and J. B. Ketterson (Springer, Berlin, 2008), p. 73.
  54. J. P. Carbotte, Properties of boson-exchange superconductors, Rev. Mod. Phys. 62, 1027 (1990).
  55. Jinho Lee, K. Fujita, K. McElroy, J. A. Slezak, M. Wang, Y. Aiura, H. Bando, M. Ishikado, T. Masui, J.-X. Zhu, A. V. Balatsky, H. Eisaki, S. Uchida, and J. C. Davis, Interplay of electron–lattice interactions and superconductivity in Bi2Sr2CaCu2O8+δ, Nature 442, 546 (2006).
  56. A. V. Balatsky and J.-X. Zhu, Local strong-coupling pairing in d-wave superconductors with inhomogeneous bosonic modes, Phys. Rev. B 74, 094517 (2006).
  57. F. C. Niestemski, S. Kunwar, S. Zhou, Shiliang Li, H. Ding, Ziqiang Wang, Pengcheng Dai, and V. Madhavan, A distinct bosonic mode in an electron-doped high-transition-temperature superconductor, Nature 450, 1058 (2007).
  58. N. Jenkins, Y. Fasano, C. Berthod, I. Maggio-Aprile, A. Piriou, E. Giannini, B. W. Hoogenboom, C. Hess, T. Cren, and Ø. Fischer, Imaging the essential role of spin fluctuations in high-Tc superconductivity, Phys. Rev. Lett. 103, 227001 (2009).
  59. Y. Fasano, I. Maggio-Aprile, N. D. Zhigadlo, S. Katrych, J. Karpinski, and Ø. Fischer, Local quasiparticle density of states of superconducting SmFeAsO1−xFx single crystals: Evidence for spin-mediated pairing, Phys. Rev. Lett. 105, 167005 (2010).
  60. Lei Shan, Jing Gong, Yong-Lei Wang, Bing Shen, Xingyuan Hou, Cong Ren, Chunhong Li, Huan Yang, Hai-Hu Wen, Shiliang Li, and Pengcheng Dai, Evidence of a spin resonance mode in the iron-based superconductor Ba0.6K0.4Fe2As2 from scanning tunneling spectroscopy, Phys. Rev. Lett. 108, 227002 (2012).
  61. N. M. Plakida, V. L. Aksenov, and S. L. Drechsler, Anharmonic model for high-Tc superconductors, Europhys. Lett. 4, 1309 (1987).
  62. Baruch Rosenstein and B. Ya. Shapiro, High-temperature superconductivity in single unit cell layer FeSe due to soft phonons in the interface layer of the SrTiO3 substrate, Phys. Rev. B 100, 054514 (2019).
  63. Rolf Heid, in Correlations and Phase Transitions, edited by Eva Pavarini and Erik Koch (Forschungszentrum Jülich GmbH Institute for Advanced Simulation, Jülich, Germany, 2024), p. 6.1.
  64. Shreya Kumbhakar, Tuhin Kumar Maji, Binita Tongbram, Shinjan Mandal, Shri Hari Soundararaj, Banashree Debnath, T. Phanindra Sai, Manish Jain, H. R. Krishnamurthy, Anshu Pandey, and Arindam Ghosh, Engineering ultra-strong electron-phonon coupling and nonclassical electron transport in crystalline gold with nanoscale interfaces, Nat. Commun. 16, 61 (2025).
  65. Joshuah T. Heath, Alexander C. Tyner, S. Pamir Alpay, Peter Krogstrup, and Alexander V. Balatsky, Tailoring superconductivity with two-level systems, arXiv:2510.23710.
  66. F. Marsiglio, Eliashberg theory in the weak-coupling limit, Phys. Rev. B 98, 024523 (2018).
  67. F. Marsiglio, M. Schossmann, and J. P. Carbotte, Iterative analytic continuation of the electron self-energy to the real axis, Phys. Rev. B 37, 4965 (1988).
  68. I. O. Kulik, Magnitude of the critical Josephson tunnel current, Sov. Phys. JETP 22, 841 (1966).
  69. Because of the relative thinness of the amorphous oxide bulk [70, 123, 124], the influence of normal quasiparticles tunneling through the oxide near high-transmission regions (such as defects) [125], and the re-emergence of crystalline order in materials such as Ta [6, 8], we assume that there is a finite overlap of N1(ri,ω) and N2(ri,ω) in close proximity to the oxide interface. Therefore, a TLS on or near the interface will influence the LDOS of both electron populations.
  70. Fang Yang, Thomas Gozlinski, Tim Storbeck, Lukas Grünhaupt, Ioan M. Pop, and Wulf Wulfhekel, Microscopic charging and in-gap states in superconducting granular aluminum, Phys. Rev. B 102, 104502 (2020).
  71. Radoslaw C. Bialczak, R. McDermott, M. Ansmann, M. Hofheinz, N. Katz, Erik Lucero, Matthew Neeley, A. D. O’Connell, H. Wang, A. N. Cleland, and John M. Martinis, 1/f flux noise in Josephson phase qubits, Phys. Rev. Lett. 99, 187006 (2007).
  72. Evan Sheridan, Thomas F. Harrelson, Eric Sivonxay, Kristin A. Persson, M. Virginia P. Altoé, Irfan Siddiqi, D. Frank Ogletree, David I. Santiago, and Sinéad M. Griffin, Microscopic theory of magnetic disorder-induced decoherence in superconducting Nb films, arXiv:2111.11684.
  73. P. W. Anderson, Theory of dirty superconductors, J. Phys. Chem. Solids 11, 26 (1959).
  74. Maki Kazumi, in Superconductivity: Part 2 (In Two Parts), edited by R.D. Parks (Marcel Dekker Inc., New York, 1969), p. 1035.
  75. G. Lemarié, A. Kamlapure, D. Bucheli, L. Benfatto, J. Lorenzana, G. Seibold, S. C. Ganguli, P. Raychaudhuri, and C. Castellani, Universal scaling of the order-parameter distribution in strongly disordered superconductors, Phys. Rev. B 87, 184509 (2013).
  76. S. E. de Graaf, A. A. Adamyan, T. Lindström, D. Erts, S. E. Kubatkin, A. Ya. Tzalenchuk, and A. V. Danilov, Direct identification of dilute surface spins on Al2O3: Origin of flux noise in quantum circuits, Phys. Rev. Lett. 118, 057703 (2017).
  77. N. A. Saveskul, N. A. Titova, E. M. Baeva, A. V. Semenov, A. V. Lubenchenko, S. Saha, H. Reddy, S. I. Bogdanov, E. E. Marinero, V. M. Shalaev, A. Boltasseva, V. S. Khrapai, A. I. Kardakova, and G. N. Goltsman, Superconductivity behavior in epitaxial TiN films points to surface magnetic disorder, Phys. Rev. Appl. 12, 054001 (2019).
  78. Wan-Ting Liao, T. P. Kohler, K. D. Osborn, R. E. Butera, C. J. Lobb, F. C. Wellstood, and M. Dreyer, Scanning tunneling Andreev microscopy of titanium nitride thin films, Phys. Rev. B 100, 214505 (2019).
  79. C. Carbillet, V. Cherkez, M. A. Skvortsov, M. V. Feigel’man, F. Debontridder, L. B. Ioffe, V. S. Stolyarov, K. Ilin, M. Siegel, D. Roditchev, T. Cren, and C. Brun, Spectroscopic evidence for strong correlations between local superconducting gap and local Altshuler-Aronov density of states suppression in ultrathin NbN films, Phys. Rev. B 102, 024504 (2020).
  80. F. Marsiglio, R. Akis, and J. P. Carbotte, Phonon self-energy effects due to superconductivity: A real-axis formulation, Phys. Rev. B 45, 9865 (1992).
  81. Jian-Xin Zhu, Bogoliubov-de Gennes Method and Its Applications (Springer International Publishing, Cham, Switzerland, 2016).
  82. P. Graham Pritchard and James M. Rondinelli, Suppressed paramagnetism in amorphous Ta2O5−x oxides and its link to superconducting qubit performance, Phys. Rev. Appl. 23, 064062 (2025).
  83. Olli Mansikkamäki, Alexander Tyner, Alexander Bilmes, Ilya Drozdov, and Alexander Balatsky, Two-tone spectroscopy for the detection of two-level systems in superconducting qubits, arXiv:2404.14039.
  84. John M. Martinis, S. Nam, J. Aumentado, and C. Urbina, Rabi oscillations in a large Josephson-junction qubit, Phys. Rev. Lett. 89, 117901 (2002).
  85. John M. Martinis, S. Nam, J. Aumentado, K. M. Lang, and C. Urbina, Decoherence of a superconducting qubit due to bias noise, Phys. Rev. B 67, 094510 (2003).
  86. Yoni Shalibo, Ya’ara Rofe, David Shwa, Felix Zeides, Matthew Neeley, John M. Martinis, and Nadav Katz, Lifetime and coherence of two-level defects in a Josephson junction, Phys. Rev. Lett. 105, 177001 (2010).
  87. A. A. Houck, J. A. Schreier, B. R. Johnson, J. M. Chow, Jens Koch, J. M. Gambetta, D. I. Schuster, L. Frunzio, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, Controlling the spontaneous emission of a superconducting transmon qubit, Phys. Rev. Lett. 101, 080502 (2008).
  88. G. Catelani, J. Koch, L. Frunzio, R. J. Schoelkopf, M. H. Devoret, and L. I. Glazman, Quasiparticle relaxation of superconducting qubits in the presence of flux, Phys. Rev. Lett. 106, 077002 (2011).
  89. C. Song, T. W. Heitmann, M. P. DeFeo, K. Yu, R. McDermott, M. Neeley, J. M. Martinis, and B. L. T. Plourde, Microwave response of vortices in superconducting thin films of Re and Al, Phys. Rev. B 79, 174512 (2009).
  90. S. E. de Graaf, L. Faoro, J. Burnett, A. A. Adamyan, A. Ya. Tzalenchuk, S. E. Kubatkin, T. Lindström, and A. V. Danilov, Suppression of low-frequency charge noise in superconducting resonators by surface spin desorption, Nat. Commun. 9, 1 (2018).
  91. K. B. Cooper, M. Steffen, R. McDermott, R. W. Simmonds, S. Oh, D. A. Hite, D. P. Pappas, and J. M. Martinis, Observation of quantum oscillations between a Josephson phase qubit and a microscopic resonator using fast readout, Phys. Rev. Lett. 93, 180401 (2004).
  92. Jürgen Lisenfeld, Grigorij J. Grabovskij, Clemens Müller, Jared H. Cole, Georg Weiss, and Alexey V. Ustinov, Observation of directly interacting coherent two-level systems in an amorphous material, Nat. Commun. 6, 1 (2015).
  93. Alexander Bilmes, Serhii Volosheniuk, Jan David Brehm, Alexey V. Ustinov, and Jürgen Lisenfeld, Quantum sensors for microscopic tunneling systems, npj Quantum Inf. 7, 1 (2021).
  94. Jürgen Lisenfeld, Alexander Bilmes, Anthony Megrant, Rami Barends, Julian Kelly, Paul Klimov, Georg Weiss, John M. Martinis, and Alexey V. Ustinov, Electric field spectroscopy of material defects in transmon qubits, npj Quantum Inf. 5, 1 (2019).
  95. Alexander Bilmes, Serhii Volosheniuk, Alexey V. Ustinov, and Jürgen Lisenfeld, Probing defect densities at the edges and inside Josephson junctions of superconducting qubits, npj Quantum Inf. 8, 24 (2022).
  96. Mustafa Bal, et al., Systematic improvements in transmon qubit coherence enabled by niobium surface encapsulation, npj Quantum Inf. 10, 1 (2024).
  97. It is also important to note that both Nb and Ta share a common issue of oxidation (as discussed in the former case by Ref. [96]), which is a general experimental problem that must be appropriately managed regardless of e-p coupling or the breakdown of amorphous structure.
  98. Riccardo Manenti, Mario Motta, Riccardo Manenti, and Mario Motta, Quantum Information Science (Oxford University Press, Oxford, England, UK, 2023).
  99. Dante Colao Zanuz, Quentin Ficheux, Laurent Michaud, Alexei Orekhov, Kilian Hanke, Alexander Flasby, Mohsen Bahrami Panah, Graham J. Norris, Michael Kerschbaum, Ants Remm, François Swiadek, Christoph Hellings, Stefania Lazăr, Colin Scarato, Nathan Lacroix, Sebastian Krinner, Christopher Eichler, Andreas Wallraff, and Jean-Claude Besse, Mitigating losses of superconducting qubits strongly coupled to defect modes, Phys. Rev. Appl. 23, 044054 (2025).
  100. S. Engelsberg and J. R. Schrieffer, Coupled electron-phonon system, Phys. Rev. 131, 993 (1963).
  101. F. Doğan and F. Marsiglio, Self-consistent modification to the electron density of states due to electron-phonon coupling in metals, Phys. Rev. B 68, 165102 (2003).
  102. P. R. Weiss and Elihu Abrahams, Correlation of electron amplitudes in impure metals, Phys. Rev. 111, 722 (1958).
  103. Walter Metzner and Dieter Vollhardt, Correlated lattice fermions in d=∞ dimensions, Phys. Rev. Lett. 62, 324 (1989).
  104. D. Vollhardt, K. Byczuk, and M. Kollar, Dynamical mean-field theory, arXiv:1109.4833.
  105. Michel Devoret, in Les Houches Session LXIII, edited by E. Giacombino. S. Reynaud and J. Zinn-Justin (Elsevier Science, Amsterdam, 1997).
  106. L. D. Landau and E. M. Lifshitz, Quantum Mechanics Non-Relativistic Theory (Butterworth-Heinemann, Oxford, England, UK, 1981), 3rd ed., Vol. 3.
  107. A. S. Alexandrov, Theory of Superconductivity From Weak to Strong Coupling (IOP Publishing, 2003).
  108. Giovanni A. C. Ummarino, in Emergent Phenomena in Correlated Matter, edited by Eva Pavarini, Erik Koch, and Ulrich Schollwöck (2013), p. 13.31.
  109. Mason Protter, Rufus Boyack, and Frank Marsiglio, Functional-integral approach to Gaussian fluctuations in Eliashberg theory, Phys. Rev. B 104, 014513 (2021).
  110. Shang-Shun Zhang, Yi-Ming Wu, Artem Abanov, and Andrey V. Chubukov, Superconductivity out of a non-Fermi liquid: Free energy analysis, Phys. Rev. B 106, 144513 (2022).
  111. P. B. Allen and R. C. Dynes, Transition temperature of strong-coupled superconductors reanalyzed, Phys. Rev. B 12, 905 (1975).
  112. F. Marsiglio, R. Akis, and J. P. Carbotte, Eliashberg theory in the very strong coupling regime, Physica C 153-155, 223 (1988).
  113. R. Combescot, Strong-coupling limit of Eliashberg theory, Phys. Rev. B 51, 11625 (1995).
  114. Emil A. Yuzbashyan, Michael K.-H. Kiessling, and Boris L. Altshuler, Superconductivity near a quantum critical point in the extreme retardation regime, Phys. Rev. B 106, 064502 (2022).
  115. Emil A. Yuzbashyan and Boris L. Altshuler, Migdal-Eliashberg theory as a classical spin chain, Phys. Rev. B 106, 014512 (2022).
  116. Emil A. Yuzbashyan and Boris L. Altshuler, Breakdown of the Migdal-Eliashberg theory and a theory of lattice-fermionic superfluidity, Phys. Rev. B 106, 054518 (2022).
  117. I. Esterlis, B. Nosarzewski, E. W. Huang, B. Moritz, T. P. Devereaux, D. J. Scalapino, and S. A. Kivelson, Breakdown of the Migdal-Eliashberg theory: A determinant quantum Monte Carlo study, Phys. Rev. B 97, 140501(R) (2018).
  118. Rufus Boyack, Sepideh Mirabi, and F. Marsiglio, Electrical conductivity and nuclear magnetic resonance relaxation rate of Eliashberg superconductors in the weak-coupling limit, Commun. Phys. 6, 1 (2023).
  119. H. J. Vidberg and J. W. Serene, Solving the Eliashberg equations by means of N-point Padé approximants, J. Low Temp. Phys. 29, 179 (1977).
  120. R. Blaschke and R. Blocksdorf, Influence of the inelastic electron-phonon scattering on the superconducting surface resistance, Z. Phys. B: Condens. Matter 49, 99 (1982).
  121. C. R. Leavens and D. S. Ritchie, Extension of the N-point Padé approximants solution of the Eliashberg equations to T∼Tc, Solid State Commun. 53, 137 (1985).
  122. M. Fibich, Phonon effects on nuclear spin relaxation in superconductors, Phys. Rev. Lett. 14, 561 (1965).
  123. W. H. Rippard, A. C. Perrella, F. J. Albert, and R. A. Buhrman, Ultrathin Aluminum Oxide Tunnel Barriers, Phys. Rev. Lett. 88, 046805 (2002).
  124. T. Connolly, P. D. Kurilovich, S. Diamond, H. Nho, C. G. L. Bøttcher, L. I. Glazman, V. Fatemi, and M. H. Devoret, Coexistence of nonequilibrium density and equilibrium energy distribution of quasiparticles in a superconducting qubit, Phys. Rev. Lett. 132, 217001 (2024).
  125. C. Kurter, C. E. Murray, R. T. Gordon, B. B. Wymore, M. Sandberg, R. M. Shelby, A. Eddins, V. P. Adiga, A. D. K. Finck, E. Rivera, A. A. Stabile, B. Trimm, B. Wacaser, K. Balakrishnan, A. Pyzyna, J. Sleight, M. Steffen, and K. Rodbell, Quasiparticle tunneling as a probe of Josephson junction barrier and capacitor material in superconducting qubits, npj Quantum Inf. 8, 1 (2022).

Outline

Information

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation