Letter | Published:

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.

Access optionsAccess options

Rent or Buy article

Get time limited or full article access on ReadCube.

from$8.99

All prices are NET prices.

Data availability

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

References

  1. 1.

    Sachdev, S. Quantum Phase Transitions (Cambridge University Press, 2011).

  2. 2.

    Bradley, R. M. & Doniach, S. Quantum fluctuations in chains of Josephson junctions. Phys. Rev. B 30, 1138–1147 (1984).

  3. 3.

    Fazio, R., Wagenblast, K.-H., Winkelholz, C. & Schön, G. Tunneling into one-dimensional Josephson chains and Luttinger liquids. Phys. B 222, 364–369 (1996).

  4. 4.

    Basko, D. M. & Hekking, F. W. J. Disordered Josephson junction chains: Anderson localization of normal modes and impedance fluctuations. Phys. Rev. B 88, 094507 (2013).

  5. 5.

    Sondhi, S., Girvin, S., Carini, J. & Shahar, D. Continuous quantum phase transitions. Rev. Mod. Phys. 69, 315–333 (1997).

  6. 6.

    Wu, H.-K. & Sau, J. D. Theory of coherent phase modes in insulating Josephson junction arrays. Preprint at https://arxiv.org/abs/1811.07941 (2018).

  7. 7.

    Bard, M., Protopopov, I. & Mirlin, A. Decay of plasmonic waves in Josephson junction chains. Phys. Rev. B 98, 224513 (2018).

  8. 8.

    Houzet, M. & Glazman, L. I. Microwave spectroscopy of a weakly-pinned charge density wave in a superinductor. Preprint at https://arxiv.org/abs/1901.01515 (2019).

  9. 9.

    Giamarchi, T. & Schulz, H. J. Anderson localization and interactions in one-dimensional metals. Phys. Rev. B 37, 325–340 (1988).

  10. 10.

    Korshunov, S. Effect of dissipation on the low-temperature properties of a tunnel-junction chain. Zh. Eksp. Teor. Fiz. 95, 1058–1075 (1989).

  11. 11.

    Choi, M.-S., Yi, J., Choi, M. Y., Choi, J. & Lee, S.-I. Quantum phase transitions in Josephson-junction chains. Phys. Rev. B 57, 716–719 (1998).

  12. 12.

    Giamarchi, T. Quantum Physics In One Dimension (Oxford University Press, 2004).

  13. 13.

    Bard, M., Protopopov, I., Gornyi, I., Shnirman, A. & Mirlin, A. Superconductor–insulator transition in disordered Josephson-junction chains. Phys. Rev. B 96, 064514 (2017).

  14. 14.

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

  15. 15.

    Devoret, M. H. Quantum fluctuations in electrical circuits. In Fluctuations Quantiques/Quantum Fluctuations: Les Houches Session LXIII (ed. Reynaud, S. et al.) 351–386 (Elsevier, 1997).

  16. 16.

    Lin, Y.-H., Nelson, J. & Goldman, A. M. Superconductivity of very thin films: the superconductor–insulator transition. Phys. C 514, 130–141 (2015).

  17. 17.

    Chow, E., Delsing, P. & Haviland, D. B. Length-scale dependence of the superconductor-to-insulator quantum phase transition in one dimension. Phys. Rev. Lett. 81, 204–207 (1998).

  18. 18.

    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).

  19. 19.

    Ergül, A. et al. Localizing quantum phase slips in one-dimensional Josephson junction chains. New J. Phys. 15, 095014 (2013).

  20. 20.

    Choi, M.-S., Choi, M., Choi, T. & Lee, S.-I. Cotunneling transport and quantum phase transitions in coupled Josephson-junction chains with charge frustration. Phys. Rev. Lett. 81, 4240–4243 (1998).

  21. 21.

    Weißl, T. et al. Kerr coefficients of plasma resonances in Josephson junction chains. Phys. Rev. B 92, 104508 (2015).

  22. 22.

    Fukuyama, H. & Lee, P. A. Dynamics of the charge-density wave. I. Impurity pinning in a single chain. Phys. Rev. B 17, 535–541 (1978).

  23. 23.

    Vogt, N. et al. One-dimensional Josephson junction arrays: lifting the Coulomb blockade by depinning. Phys. Rev. B 92, 045435 (2015).

  24. 24.

    Matveev, K. A., Larkin, A. I. & Glazman, L. I. Persistent current in superconducting nanorings. Phys. Rev. Lett. 89, 096802 (2002).

  25. 25.

    Rastelli, G., Pop, I. M. & Hekking, F. W. J. Quantum phase slips in Josephson junction rings. Phys. Rev. B 87, 174513 (2013).

  26. 26.

    Crane, R. et al. Survival of superconducting correlations across the two-dimensional superconductor–insulator transition: a finite-frequency study. Phys. Rev. B 75, 184530 (2007).

  27. 27.

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

  28. 28.

    Sacépé, B. et al. Localization of preformed Cooper pairs in disordered superconductors. Nat. Phys. 7, 239–244 (2011).

  29. 29.

    André, A. et al. A coherent all-electrical interface between polar molecules and mesoscopic superconducting resonators. Nat. Phys. 2, 636–642 (2006).

  30. 30.

    Schuster, D., Fragner, A., Dykman, M., Lyon, S. & Schoelkopf, R. Proposal for manipulating and detecting spin and orbital states of trapped electrons on helium using cavity quantum electrodynamics. Phys. Rev. Lett. 105, 040503 (2010).

  31. 31.

    Stockklauser, A. et al. Strong coupling cavity QED with gate-defined double quantum dots enabled by a high impedance resonator. Phys. Rev. X 7, 011030 (2017).

  32. 32.

    Samkharadze, N. et al. Strong spin–photon coupling in silicon. Science 359, 1123–1127 (2018).

  33. 33.

    Arrangoiz-Arriola, P. et al. Coupling a superconducting quantum circuit to a phononic crystal defect cavity. Phys. Rev. X 8, 031007 (2018).

Download references

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.

Additional information

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

Supplementary information

Supplementary Information

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

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Further reading

Fig. 1: The Josephson transmission line and wireless radio-frequency spectroscopy set-up.
Fig. 2: Collective modes in the Josephson transmission line.
Fig. 3: Broadening and random frequency shifts of collective mode resonances.
Fig. 4: The reversible transition in the Q-factor frequency dependence.