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Evidence of dual Shapiro steps in a Josephson junction array

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

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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.

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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.

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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.

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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.

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  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

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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.

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Correspondence to Nicolas Roch.

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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.

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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

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