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# Quantum electrodynamics of a superconductor–insulator phase transition

## Abstract

A chain of Josephson junctions represents one of the simplest many-body models undergoing a superconductor–insulator quantum phase transition1,2. Apart from zero resistance, the superconducting state is necessarily accompanied by a sound-like mode due to collective oscillations of the phase of the complex-valued order parameter3,4. Little is known about the fate of this mode on entering the insulating state, where the order parameter’s amplitude remains non-zero, but the phase ordering is ‘melted’ by quantum fluctuations5. Here, we show that the phase mode survives far into the insulating regime, such that megaohm-resistance chains can carry gigahertz-frequency alternating currents as nearly ideal superconductors. The insulator reveals itself through interaction-induced broadening and random frequency shifts of collective mode resonances. Our spectroscopic experiment puts forward the problem of quantum electrodynamics of a Bose glass for both theory and experiment6,7,8. By pushing the chain parameters deeper into the insulating state, we achieved a wave impedance of the phase mode exceeding the predicted critical value by an order of magnitude9,10,11,12,13,14. The effective fine structure constant of such a one-dimensional electromagnetic vacuum exceeds unity, promising transformative applications to quantum science and technology.

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## Data availability

All datasets that support the findings of this study are available from the corresponding author upon reasonable request.

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## Acknowledgements

We acknowledge discussions with L. Glazman, M. Goldstein, M. Houzet, I. Protopopov, J. Sau and A. Shnirman. The work was supported by US NSF (DMR 1455261), US-Israel BSF (2016224), NSF PFC at JQI (1430094), and ARO-MURI (W911NF-15-1-0397) ‘Exotic states of light in superconducting circuits’.

## Author information

R.K., aided by N.G. and N.M., performed measurements and data analysis. R.M. and N.G. fabricated devices, Y.-H.L. developed the wireless waveguide interface. V.E.M. managed the project. All authors participated in extensive discussions of the experimental results and contributed to writing the manuscript.

Correspondence to V. E. Manucharyan.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

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## Supplementary information

### Supplementary Information

Supplementary Figs. 1–7, Table 1 and refs. 1–3.

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