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# Steady Floquet–Andreev states in graphene Josephson junctions

## Abstract

Engineering quantum states through light–matter interaction has created a paradigm in condensed-matter physics. A representative example is the Floquet–Bloch state, which is generated by time-periodically driving the Bloch wavefunctions in crystals. Previous attempts to realize such states in condensed-matter systems have been limited by the transient nature of the Floquet states produced by optical pulses1,2,3, which masks the universal properties of non-equilibrium physics. Here we report the generation of steady Floquet–Andreev states in graphene Josephson junctions by continuous microwave application and direct measurement of their spectra by superconducting tunnelling spectroscopy. We present quantitative analysis of the spectral characteristics of the Floquet–Andreev states while varying the phase difference of the superconductors, the temperature, the microwave frequency and the power. The oscillations of the Floquet–Andreev-state spectrum with phase difference agreed with our theoretical calculations. Moreover, we confirmed the steady nature of the Floquet–Andreev states by establishing a sum rule of tunnelling conductance4, and analysed the spectral density of Floquet states depending on Floquet interaction strength. This study provides a basis for understanding and engineering non-equilibrium quantum states in nanodevices.

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## Data availability

The data supporting the findings of this study are available from the corresponding authors upon reasonable request.

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## Acknowledgements

We thank H.-J. Lee, K. W. Kim, J. C. W. Song, D. Cho, K. C. Fong and C. Lee for reading the manuscript, and C. B. Winkelmann for a discussion about the Tien–Gordon model. S.P., S.J., Y.-B.C. and G.-H.L. acknowledge the support of the Samsung Science and Technology Foundation (project number SSTF-BA1702-05) for device fabrications and low-temperature measurements. W.J. and G.-H.L. acknowledge the support of National Research Foundation of Korea (NRF) funded by the Korean Government (grant numbers 2016R1A5A1008184, 2020R1C1C1013241 and 2020M3H3A1100839) for data analysis. W.L. and G.Y.C. acknowledge the support of the National Research Foundation of Korea (NRF) funded by the Korean Government (grant numbers 2020R1C1C1006048 and 2020R1A4A3079707), as well as grant number IBS-R014-D1. W.L. and G.Y.C. are also supported by the Air Force Office of Scientific Research under award number FA2386-20-1-4029. K.W. and T.T. acknowledge support from the Elemental Strategy Initiative conducted by the MEXT, Japan, grant number JPMXP0112101001 and JSPS KAKENHI grant number JP20H00354.

## Author information

Authors

### Contributions

G.-H.L. and G.Y.C. conceived and supervised the project. S.P. designed and fabricated the devices. T.T. and K.W. provided the hBN crystal. S.P., S.J. and Y.-B.C. performed the measurements. W.L. and G.Y.C. carried out theoretical calculations. S.P., W.L., S.J., J.P., W.J., G.Y.C. and G.-H.L. performed the data analysis. S.P., W.L., S.J., G.Y.C. and G.-H.L. wrote the paper.

### Corresponding authors

Correspondence to Gil Young Cho or Gil-Ho Lee.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

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### Peer review information

Nature thanks Romain Danneau, James McIver and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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## Supplementary information

### Supplementary Information

This file contains Supplementary text, figures, tables, equations and references.

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Park, S., Lee, W., Jang, S. et al. Steady Floquet–Andreev states in graphene Josephson junctions. Nature 603, 421–426 (2022). https://doi.org/10.1038/s41586-021-04364-8

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• DOI: https://doi.org/10.1038/s41586-021-04364-8