Nothing Special   »   [go: up one dir, main page]
Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Article
  • Published:

Evidence of dual Shapiro steps in a Josephson junction array

Nature Physics volume 19, pages 851–856 (2023)Cite this article

Subjects

Abstract

The modern primary voltage standard is based on the a.c. Josephson effect and the ensuing Shapiro steps, where a microwave tone applied to a Josephson junction yields a constant voltage hf/2e determined by only the microwave frequency f, Planck’s constant h and the electron charge e. Duality arguments for current and voltage have long suggested the possibility of dual Shapiro steps—that a Josephson junction device could produce current steps with heights determined only by the applied frequency. Here we embed an ultrasmall Josephson junction in a high impedance array of larger junctions to reveal dual Shapiro steps. For multiple frequencies, we detect that the a.c. response of the circuit is synchronized with the microwave tone at frequency f, and the corresponding emergence of flat steps in the d.c. response with current 2ef, equal to the transport of a Cooper pair per tone period. This work extends phase–charge duality to Josephson circuits, which opens a broad range of possibilities in the field of circuit quantum electrodynamics and is an important step towards the long-sought closure of the quantum metrology electrical triangle.

Access through your institution
Buy or subscribe

This is a preview of subscription content, access via your institution

Access options

Access through your institution

Buy this article

  • Purchase on SpringerLink
  • Instant access to full article PDF
Buy now

Prices may be subject to local taxes which are calculated during checkout

Fig. 1: The Shapiro steps and their dual.
Fig. 2: Overview of device and measurement setup.
Fig. 3: D.c. and RF characterization of the Bloch array.
Fig. 4: Emergence of Bloch oscillations and dual Shapiro steps.
Fig. 5: Current plateaux at different microwave tone frequency and power.

Similar content being viewed by others

Data availability

Raw data, analysis scripts and additional measurements and details are publicly available on Zenodo at https://zenodo.org/record/6913393.

Code availability

Raw data, analysis scripts and additional measurements and details are publicly available on Zenodo at https://zenodo.org/record/6913393.

References

  1. Tinkham, M. Introduction to Superconductivity, 2nd edn (Dover, 2004).

  2. Sonin, E. B. Quantum rotator and Josephson junction: compact vs. extended phase and dissipative quantum phase transition. Low Temperature Physics 48, 400 (2022); https://doi.org/10.1063/10.0010205

  3. Josephson, B. D. Possible new effects in superconductive tunnelling. Phys. Lett. 1, 251–253 (1962).

    Article  ADS  MATH  Google Scholar 

  4. Shapiro, S. Josephson currents in superconducting tunneling: the effect of microwaves and other observations. Phys. Rev. Lett. 11, 80–82 (1963).

    Article  ADS  Google Scholar 

  5. Schön, G. & Zaikin, A. D. Quantum coherent effects, phase transitions, and the dissipative dynamics of ultra small tunnel junctions. Phys. Rep. 198, 237–412 (1990).

    Article  ADS  Google Scholar 

  6. Spiller, T. P., Clark, T. D., Prance, R. J., Prance, H. & Poulton, D. A. Electromagnetic duality in quantum circuits. Il Nuovo Cim. B Ser. 11 105, 43–52 (1990).

    Article  Google Scholar 

  7. Ingold, G.-L. & Nazarov, Y. V. In Single Charge Tunneling: Coulomb Blockade Phenomena In Nanostructures (eds Grabert, H & Devoret, M. H.) pp 21–107 (Springer, 1992).

  8. Corlevi, S., Guichard, W., Hekking, F. W. J. & Haviland, D. B. Phase-charge duality of a Josephson junction in a fluctuating electromagnetic environment. Phys. Rev. Lett. 97, 096802 (2006).

    Article  ADS  Google Scholar 

  9. Arutyunov, K. Y., Golubev, D. S. & Zaikin, A. D. Superconductivity in one dimension. Phys. Rep. 464, 1–70 (2008).

    Article  ADS  Google Scholar 

  10. Guichard, W. & Hekking, F. W. J. Phase–charge duality in Josephson junction circuits: role of inertia and effect of microwave irradiation. Phys. Rev. B 81, 064508 (2010).

    Article  ADS  Google Scholar 

  11. Kerman, A. J. Flux–charge duality and topological quantum phase fluctuations in quasi-one-dimensional superconductors. N. J. Phys. 15, 105017 (2013).

    Article  Google Scholar 

  12. Likharev, K. K. & Zorin, A. B. Theory of the Bloch-wave oscillations in small Josephson junctions. J. Low. Temp. Phys. 59, 347–382 (1985).

    Article  ADS  Google Scholar 

  13. Averin, D. V., Zorin, A. B. & Likharev, K. K. Bloch oscillations in small Josephson junctions. Sov. Phys. JETP 61, 7 (1985).

    Google Scholar 

  14. Averin, D. V. & Odintsov, A. A. Phase locking of the Bloch oscillations in ultrasmall Josephson junctions. Phys. B 165–166, 935–936 (1990). LT-19.

    Article  ADS  Google Scholar 

  15. Hu, G. Y. & O’Connell, R. F. Bloch oscillations in small-capacitance Josephson junctions. Phys. Rev. B 47, 8823–8830 (1993).

    Article  ADS  Google Scholar 

  16. Pekola, J. P. et al. Single-electron current sources: toward a refined definition of the ampere. Rev. Mod. Phys. 85, 1421–1472 (2013).

    Article  ADS  Google Scholar 

  17. Bylander, J., Duty, T. & Delsing, P. Current measurement by real-time counting of single electrons. Nature 434, 361–364 (2005).

    Article  ADS  Google Scholar 

  18. Carruthers, P. & Nieto, M. M. Phase and angle variables in quantum mechanics. Rev. Mod. Phys. 40, 411–440 (1968).

    Article  ADS  Google Scholar 

  19. Loss, D. & Mullen, K. Effect of dissipation on phase periodicity and the quantum dynamics of Josephson junctions. Phys. Rev. A 43, 2129–2138 (1991).

    Article  ADS  Google Scholar 

  20. Ashcroft, N. W. & Mermin, N. D. Solid State Physics (Holt, Rinehart, and Winston, 1976).

  21. Puertas Martínez, J. et al. A tunable Josephson platform to explore many-body quantum optics in circuit-QED. npj Quantum Inf. 5, 19 (2019).

    Article  ADS  Google Scholar 

  22. Léger, S. ébastien et al. Observation of quantum many-body effects due to zero point fluctuations in superconducting circuits. Nat. Commun. 10, 5259 (2019).

    Article  ADS  Google Scholar 

  23. Shahar, D., Tsui, D. C., Shayegan, M., Shimshoni, E. & Sondhi, S. L. Evidence for charge–flux duality near the quantum Hall liquid-to-insulator transition. Science 274, 589–592 (1996).

    Article  ADS  Google Scholar 

  24. Ovadia, M., Kalok, D., Sacépé, B. & Shahar, D. Duality symmetry and its breakdown in the vicinity of the superconductor–insulator transition. Nat. Phys. 9, 415–418 (2013).

    Article  Google Scholar 

  25. Kuzmin, L. & Haviland, D. Observation of the Bloch oscillations in an ultrasmall Josephson junction. Phys. Rev. Lett. 67, 2890–2893 (1991).

    Article  ADS  Google Scholar 

  26. Di Marco, A., Hekking, F. W. J. & Rastelli, G. Quantum phase-slip junction under microwave irradiation. Phys. Rev. B 91, 184512 (2015).

    Article  ADS  Google Scholar 

  27. Andersson, K., Delsing, P. & Haviland, D. Synchronous Cooper pair tunneling in a 1D-array of Josephson junctions. Phys. B 284–288, 1816–1817 (2000).

    Article  ADS  Google Scholar 

  28. Weißl, T. et al. Bloch band dynamics of a Josephson junction in an inductive environment. Phys. Rev. B 91, 014507 (2015).

    Article  ADS  Google Scholar 

  29. Cedergren, K. et al. Insulating Josephson junction chains as pinned Luttinger liquids. Phys. Rev. Lett. 119, 167701 (2017).

    Article  ADS  Google Scholar 

  30. Lau, C. N., Markovic, N., Bockrath, M., Bezryadin, A. & Tinkham, M. Quantum phase slips in superconducting nanowires. Phys. Rev. Lett. 87, 217003 (2001).

    Article  ADS  Google Scholar 

  31. Lehtinen, J. S., Zakharov, K. & Arutyunov, K. Y. Coulomb blockade and Bloch oscillations in superconducting Ti nanowires. Phys. Rev. Lett. 109, 187001 (2012).

    Article  ADS  Google Scholar 

  32. Wang, Z. M., Lehtinen, J. S. & Arutyunov, K. Y. Towards quantum phase slip based standard of electric current. Appl. Phys. Lett. 114, 242601 (2019).

    Article  ADS  Google Scholar 

  33. Webster, C. H. et al. NbSi nanowire quantum phase-slip circuits: dc supercurrent blockade, microwave measurements, and thermal analysis. Phys. Rev. B 87, 144510 (2013).

    Article  ADS  Google Scholar 

  34. Peruzzo, M. et al. Geometric superinductance qubits: controlling phase delocalization across a single Josephson junction. PRX Quantum 2, 040341 (2021).

    Article  ADS  Google Scholar 

  35. Rolland, C. et al. Antibunched photons emitted by a dc-biased Josephson junction. Phys. Rev. Lett. 122, 186804 (2019).

    Article  ADS  Google Scholar 

  36. Yoshihiro, K. Observation of “Bloch oscillations” in granular tin films. Phys. B 152, 207–211 (1988).

    Article  ADS  Google Scholar 

  37. Grünhaupt, L. et al. Granular aluminium as a superconducting material for high-impedance quantum circuits. Nat. Mater. 18, 816–819 (2019).

    Article  ADS  Google Scholar 

  38. Bell, M., Sadovskyy, I., Ioffe, L., Kitaev, A. & Gershenson, M. Quantum superinductor with tunable nonlinearity. Phys. Rev. Lett. 109, 137003 (2012).

    Article  ADS  Google Scholar 

  39. Masluk, N., Pop, I., Kamal, A., Minev, Z. & Devoret, M. Microwave characterization of Josephson junction arrays: implementing a low loss superinductance. Phys. Rev. Lett. 109, 137002 (2012).

    Article  ADS  Google Scholar 

  40. Pechenezhskiy, I. V., Mencia, R. A., Nguyen, L. B., Lin, Yen-Hsiang & Manucharyan, V. E. The superconducting quasicharge qubit. Nature 585, 368–371 (2020).

    Article  ADS  Google Scholar 

  41. Blais, A., Grimsmo, A. L., Girvin, S. M. & Wallraff, A. Circuit quantum electrodynamics. Rev. Mod. Phys. 93, 025005 (2021).

    Article  ADS  MathSciNet  Google Scholar 

  42. Arndt, L., Roy, A. & Hassler, F. Dual Shapiro steps of a phase-slip junction in the presence of a parasitic capacitance. Phys. Rev. B 98, 014525 (2018).

    Article  ADS  Google Scholar 

  43. Haviland, D. B., Andersson, K. & Ågren, P. Superconducting and insulating behavior in one-dimensional Josephson junction arrays. J. Low. Temp. Phys. 118, 733–749 (2000).

    Article  ADS  Google Scholar 

  44. Dolata, R., Scherer, H., Zorin, A. B. & Niemeyer, J. Single-charge devices with ultrasmall Nb/AlOx/Nb trilayer Josephson junctions. J. Appl. Phys. 97, 054501 (2005).

    Article  ADS  Google Scholar 

  45. Lenz, G., Talanina, I. & de Sterke, C. M. Bloch oscillations in an array of curved optical waveguides. Phys. Rev. Lett. 83, 963–966 (1999).

    Article  ADS  Google Scholar 

  46. Levinsen, M. T., Chiao, R. Y., Feldman, M. J. & Tucker, B. A. An inverse ac Josephson effect voltage standard. Appl. Phys. Lett. 31, 776–778 (1977).

    Article  ADS  Google Scholar 

  47. Shaikhaidarov, R. S. et al. Quantized current steps due to the a.c. coherent quantum phase-slip effect. Nature 608, 45–49 (2022).

    Article  ADS  Google Scholar 

  48. Göbel, E. O. & Siegner, U. The New International System of Units (SI) – Quantum Metrology and Quantum Standards (Wiley, 2019).

  49. Sailer, T. et al. Measurement of the bound-electron g-factor difference in coupled ions. Nature 606, 479–483 (2022).

    Article  ADS  Google Scholar 

  50. Hamilton, C. A. Josephson voltage standards. Rev. Sci. Instrum. 71, 3611–3623 (2000).

    Article  ADS  Google Scholar 

  51. Koch, J. et al. Charge-insensitive qubit design derived from the Cooper pair box. Phys. Rev. A 76, 042319 (2007).

    Article  ADS  Google Scholar 

  52. Schreier, J. A. et al. Suppressing charge noise decoherence in superconducting charge qubits. Phys. Rev. B 77, 180502 (2008).

    Article  ADS  Google Scholar 

  53. Crescini, N., Cailleaux, S., Guichard, W., Naud, C., Buisson, O., Murch, K. & Roch, N. Evidence of dual Shapiro steps in a Josephson junctions array. Zenodo https://doi.org/10.5281/zenodo.6913393 (2022).

  54. Ergül, A. et al. Phase sticking in one-dimensional Josephson junction chains. Phys. Rev. B 88, 104501 (2013).

    Article  ADS  Google Scholar 

  55. Leppäkangas, J. et al. Antibunched photons from inelastic Cooper-pair tunneling. Phys. Rev. Lett. 115, 027004 (2015).

    Article  ADS  Google Scholar 

  56. Hofheinz, M. et al. Bright side of the Coulomb blockade. Phys. Rev. Lett. 106, 217005 (2011).

    Article  ADS  Google Scholar 

  57. Krupko, Y. et al. Kerr nonlinearity in a superconducting Josephson metamaterial. Phys. Rev. B 98, 094516 (2018).

    Article  ADS  Google Scholar 

Download references

Acknowledgements

The help of D. Basko is deeply acknowledged and appreciated. The support of the superconducting circuits team of Institut Néel is warmly acknowledged. We are also grateful to J. Aumentado, M. Devoret, T. Duty, S. Florens, D. Haviland, J. Renard, B. Sacepe and I. Snyman for their fruitful discussion and comments on our work. Furthermore, our gratitude goes to the members of the Triangle consortium, namely, P. Joyez, Ç. Girit, C. Ciuti, H. Le Sueur and A. Wagner, for their valuable discussion and insights. The samples were fabricated in the clean room facility of Institute Néel, Grenoble; we would like to thank all the staff for help with device fabrication. We would like to acknowledge E. Eyraud for his help in the installation of the experimental setup. This work was supported by the French National Research Agency (ANR) in the framework of the TRIANGLE project (ANR-20-CE47-0011). N.C. is supported by the European Union’s Horizon 2020 research and innovation programme under Marie Skłodowska-Curie grant agreement QMET no. 101029189. K.W.M. acknowledges support from NSF grant no. PHY-1752844 (CAREER), AFOSR MURI grant no. FA9550-21-1-0202 and ONR grant no. N00014- 21-1-2630.

Author information

Author notes

  1. These authors contributed equally: Nicolò Crescini and Samuel Cailleaux.

Authors and Affiliations

  1. Grenoble INP, Institut Néel, Univ. Grenoble Alpes, CNRS, Grenoble, France

    Nicolò Crescini, Samuel Cailleaux, Wiebke Guichard, Cécile Naud, Olivier Buisson & Nicolas Roch

  2. Department of Physics, Washington University, St. Louis, MO, USA

    Kater W. Murch

Authors
  1. Nicolò Crescini
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Samuel Cailleaux
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Wiebke Guichard
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Cécile Naud
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Olivier Buisson
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Kater W. Murch
    View author publications

    You can also search for this author in PubMed Google Scholar

  7. Nicolas Roch
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

N.C., S.C. and N.R. designed the device and the experiment. N.C. fabricated the samples. N.C. and S.C. set up the experimental apparatus and performed the measurements. N.C. analysed the data. S.C. ran the numerical simulations and fabricated the control samples. All the authors discussed and interpreted the data and the results. N.C. and K.M. drafted the manuscript, which was discussed and improved by S.C. and N.R., before being proof-read and commented on by all authors.

Corresponding author

Correspondence to Nicolas Roch.

Ethics declarations

Competing interests

The authors declare no competing interests.

Peer review

Peer review information

Nature Physics thanks Samuel Benz and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Additional information

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Extended data

Extended Data Fig. 1 Simulation of the CJLR circuit under microwave irradiation with experimental noise.

This IV curves are used to determine how clear are the steps for different parameters combinations and do not resemble the actual steps observed in our device. The inset is the simulated circuit, where the diamond is the non-linear capacitor (\(\cos (\frac{\pi }{e}Q)\) element).

Extended Data Fig. 2 Simulated IV curve of a chain of four junctions in series with an inductance, a resistor, and a non-linear capacitor with and without an AC drive of frequency f.

Current plateaux of height 2ef are observed before each 2Δ current peak in qualitative agreement with the experimental observations.

Extended Data Fig. 3 Experimental setup used in this work.

The yellow boxes show the different stages of the dilution refrigerator with their temperatures on the right side, while the part of the apparatus at room temperature is in green. The device is in grey at the bottom of the figure, the reader is referred to the main text for its description.

Extended Data Fig. 4 Full IV curve of the Bloch array.

Plots (a) to (d) show the IV curve at different scales, anbd the black (grey) line indicates increasing (decreasing) ∣V0∣, as shown by the arrows in (a). The voltage scale is gradually reduced from (a) to (d); discrepancies between the current amplitude in the plots are related to measurement conditions. Finally, (e) shows the flux dependence of the Bloch array’s IV characteristic at low voltages.

Extended Data Fig. 5 Combined microwave and DC measurements.

Plot (a) is the variation of the transmission with respect to V0, where the IV was superimposed to the plot as a dashed line for illustrative purposes. In (b) we show slices of (a) at different voltages, where a 1 dB offset for each curve is added for clarity.

Extended Data Fig. 6 Power dependence of the microwave transmission of the odd modes of the Bloch array.

Power sweep of the odd modes identified in the Bloch array, ranging from (a) at 3.2 GHz to (f) at 7.5 GHz. For each plot we observe the same power-dependent behavior, analogue to the one of Fig. 4b (see the main text for more details). These measurements are used to find the correct power to obtain the locking, and thus the steps. Such power is signaled with a dashed grey line for the modes where it could be identified, and the resulting current plateaux are shown in Fig. 5c.

Rights and permissions

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 permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Crescini, N., Cailleaux, S., Guichard, W. et al. Evidence of dual Shapiro steps in a Josephson junction array. Nat. Phys. 19, 851–856 (2023). https://doi.org/10.1038/s41567-023-01961-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s41567-023-01961-4

This article is cited by

Search

Advanced search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing