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Resolution of gauge ambiguities in ultrastrong-coupling cavity quantum electrodynamics

Nature Physics volume 15, pages 803–808 (2019)Cite this article

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Matters Arising to this article was published on 22 December 2023

Abstract

In quantum electrodynamics, the choice of gauge influences the form of light–matter interactions. However, gauge invariance implies that all physical results should be independent of this formal choice. The Rabi model, a widespread description for the dipolar coupling between a two-level atom and a quantized electromagnetic field, seemingly violates this principle in the presence of ultrastrong light–matter coupling, a regime that is now experimentally accessible in many physical systems. This failure is attributed to the finite-level truncation of the matter system, an approximation that enters the derivation of the Rabi model. Here, we identify the source of gauge violation and provide a general method for the derivation of light–matter Hamiltonians in truncated Hilbert spaces that produces gauge-invariant physical results, even for extreme light–matter interaction regimes. This is achieved by compensating the non-localities introduced in the construction of the effective Hamiltonians. The resulting quantum Rabi Hamiltonian in the Coulomb gauge differs significantly in form from the standard one, but provides the same physical results obtained by using the dipole gauge. These results shed light on gauge invariance in the non-perturbative and extreme-interaction regimes, and solve long-lasting controversies arising from gauge ambiguities in the quantum Rabi and Dicke models.

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Fig. 1: Numerical comparisons between different gauges.
Fig. 2: Breakdown of gauge invariance.

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Data availability

The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

References

  1. Kockum, A. F., Miranowicz, A., Liberato, S. D., Savasta, S. & Nori, F. Ultrastrong coupling between light and matter. Nat. Rev. Phys. 1, 19–40 (2019).

    Article  Google Scholar 

  2. Forn-Díaz, P., Lamata, L., Rico, E., Kono, J. & Solano, E. Ultrastrong coupling regimes of light–matter interaction. Preprint at https://arxiv.org/abs/1804.09275 (2018).

  3. Yoshihara, F. et al. Superconducting qubit–oscillator circuit beyond the ultrastrong-coupling regime. Nat. Phys. 13, 44–47 (2017).

    Article  Google Scholar 

  4. Yoshihara, F. et al. Inversion of qubit energy levels in qubit-oscillator circuits in the deep-strong-coupling regime. Phys. Rev. Lett. 120, 183601 (2018).

    Article  ADS  Google Scholar 

  5. Ashhab, S. & Nori, F. Qubit-oscillator systems in the ultrastrong-coupling regime and their potential for preparing nonclassical states. Phys. Rev. A 81, 042311 (2010).

    Article  ADS  Google Scholar 

  6. Casanova, J., Romero, G., Lizuain, I., García-Ripoll, J. J. & Solano, E. Deep strong coupling regime of the Jaynes–Cummings model. Phys. Rev. Lett. 105, 263603 (2010).

    Article  ADS  Google Scholar 

  7. Niemczyk, T. et al. Circuit quantum electrodynamics in the ultrastrong-coupling regime. Nat. Phys. 6, 772–776 (2010).

    Article  Google Scholar 

  8. Zaks, B. et al. THz-driven quantum wells: coulomb interactions and Stark shifts in the ultrastrong coupling regime. New J. Phys. 13, 083009 (2011).

    Article  ADS  Google Scholar 

  9. Stassi, R., Ridolfo, A., Di Stefano, O., Hartmann, M. J. & Savasta, S. Spontaneous conversion from virtual to real photons in the ultrastrong-coupling regime. Phys. Rev. Lett. 110, 243601 (2013).

    Article  ADS  Google Scholar 

  10. Cirio, M., De Liberato, S., Lambert, N. & Nori, F. Ground state electroluminescence. Phys. Rev. Lett. 116, 113601 (2016).

    Article  ADS  Google Scholar 

  11. De Liberato, S., Gerace, D., Carusotto, I. & Ciuti, C. Extracavity quantum vacuum radiation from a single qubit. Phys. Rev. A 80, 053810 (2009).

    Article  ADS  Google Scholar 

  12. De Liberato, S. Light–matter decoupling in the deep strong coupling regime: the breakdown of the Purcell effect. Phys. Rev. Lett. 112, 016401 (2014).

    Article  ADS  Google Scholar 

  13. Garziano, L. et al. Multiphoton quantum Rabi oscillations in ultrastrong cavity QED. Phys. Rev. A 92, 063830 (2015).

    Article  ADS  Google Scholar 

  14. Garziano, L. et al. One photon can simultaneously excite two or more atoms. Phys. Rev. Lett. 117, 043601 (2016).

    Article  ADS  Google Scholar 

  15. Garziano, L., Ridolfo, A., De Liberato, S. & Savasta, S. Cavity QED in the ultrastrong coupling regime: photon bunching from the emission of individual dressed qubits. ACS Photon. 4, 2345–2351 (2017).

    Article  Google Scholar 

  16. Di Stefano, O. et al. Feynman-diagrams approach to the quantum Rabi model for ultrastrong cavity QED: stimulated emission and reabsorption of virtual particles dressing a physical excitation. New J. Phys. 19, 053010 (2017).

    Article  Google Scholar 

  17. Kockum, A. F., Miranowicz, A., Macrì, V., Savasta, S. & Nori, F. Deterministic quantum nonlinear optics with single atoms and virtual photons. Phys. Rev. A 95, 063849 (2017).

    Article  ADS  Google Scholar 

  18. Felicetti, S., Rossatto, D. Z., Rico, E., Solano, E. & Forn-Daz, P. Two-photon quantum Rabi model with superconducting circuits. Phys. Rev. A 97, 013851 (2018).

    Article  ADS  Google Scholar 

  19. Nataf, P. & Ciuti, C. Vacuum degeneracy of a circuit QED system in the ultrastrong coupling regime. Phys. Rev. Lett. 104, 023601 (2010).

    Article  ADS  Google Scholar 

  20. Nataf, P. & Ciuti, C. Protected quantum computation with multiple resonators in ultrastrong coupling circuit QED. Phys. Rev. Lett. 107, 190402 (2011).

    Article  ADS  Google Scholar 

  21. Romero, G., Ballester, D., Wang, Y. M., Scarani, V. & Solano, E. Ultrafast quantum gates in circuit QED. Phys. Rev. Lett. 108, 120501 (2012).

    Article  ADS  Google Scholar 

  22. Kyaw, T. H., Felicetti, S., Romero, G., Solano, E. & Kwek, L.-C. Scalable quantum memory in the ultrastrong coupling regime. Sci. Rep. 5, 8621 (2015).

    Article  ADS  Google Scholar 

  23. Wang, Y., Zhang, J., Wu, C., You, J. Q. & Romero, G. Holonomic quantum computation in the ultrastrong-coupling regime of circuit QED. Phys. Rev. A 94, 012328 (2016).

    Article  ADS  Google Scholar 

  24. Stassi, R. et al. Quantum nonlinear optics without photons. Phys. Rev. A 96, 023818 (2017).

    Article  ADS  Google Scholar 

  25. Kockum, A. F., Macrì, V., Garziano, L., Savasta, S. & Nori, F. Frequency conversion in ultrastrong cavity QED. Sci. Rep. 7, 5313 (2017).

    Article  ADS  Google Scholar 

  26. Armata, F., Calajo, G., Jaako, T., Kim, M. S. & Rabl, P. Harvesting multiqubit entanglement from ultrastrong interactions in circuit quantum electrodynamics. Phys. Rev. Lett. 119, 183602 (2017).

    Article  ADS  Google Scholar 

  27. Babiker, M. & Loudon, R. Derivation of the Power–Zienau–Woolley Hamiltonian in quantum electrodynamics by gauge transformation. Proc. R. Soc. Lond. A 385, 439–460 (1983).

    Article  ADS  MathSciNet  Google Scholar 

  28. Maggiore, M. A Modern Introduction to Quantum Field Theory (Oxford Series in Physics no. 12, Oxford University Press, 2005).

  29. Lamb, W. E. Fine structure of the hydrogen atom. III. Phys. Rev. 85, 259–276 (1952).

    Article  ADS  Google Scholar 

  30. Lamb, W. E., Schlicher, R. R. & Scully, M. O. Matter–field interaction in atomic physics and quantum optics. Phys. Rev. A 36, 2763–2772 (1987).

    Article  ADS  Google Scholar 

  31. Starace, A. F. Length and velocity formulas in approximate oscillator-strength calculations. Phys. Rev. A 3, 1242–1245 (1971).

    Article  ADS  Google Scholar 

  32. Girlanda, R., Quattropani, A. & Schwendimann, P. Two-photon transitions to exciton states in semiconductors. Application to CuCl. Phys. Rev. B 24, 2009–2017 (1981).

    Article  ADS  Google Scholar 

  33. Ismail-Beigi, S., Chang, E. K. & Louie, S. G. Coupling of nonlocal potentials to electromagnetic fields. Phys. Rev. Lett. 87, 087402 (2001).

    Article  ADS  Google Scholar 

  34. Bassani, F., Forney, J. J. & Quattropani, A. Choice of gauge in two-photon transitions: 1s–2s transition in atomic hydrogen. Phys. Rev. Lett. 39, 1070–1073 (1977).

    Article  ADS  Google Scholar 

  35. Hepp, K. & Lieb, E. H. On the superradiant phase transition for molecules in a quantized radiation field: the Dicke maser model. Ann. Phys. 76, 360–404 (1973).

    Article  ADS  MathSciNet  Google Scholar 

  36. Wang, Y. K. & Hioe, F. T. Phase transition in the Dicke model of superradiance. Phys. Rev. A 7, 831–836 (1973).

    Article  ADS  Google Scholar 

  37. Rzażewski, K., Wódkiewicz, K. & Żakowicz, W. Phase transitions, two-level atoms, and the A 2 term. Phys. Rev. Lett. 35, 432–434 (1975).

    Article  ADS  Google Scholar 

  38. Lambert, N., Emary, C. & Brandes, T. Entanglement and the phase transition in single-mode superradiance. Phys. Rev. Lett. 92, 073602 (2004).

    Article  ADS  Google Scholar 

  39. Keeling, J. Coulomb interactions, gauge invariance and phase transitions of the Dicke model. J. Phys. Condens. Matter 19, 295213 (2007).

    Article  Google Scholar 

  40. Nataf, P. & Ciuti, C. No-go theorem for superradiant quantum phase transitions in cavity QED and counter-example in circuit QED. Nat. Commun. 1, 72 (2010).

    Article  ADS  Google Scholar 

  41. Vukics, A., Grießer, T. & Domokos, P. Elimination of the a-square problem from cavity QED. Phys. Rev. Lett. 112, 073601 (2014).

    Article  ADS  Google Scholar 

  42. Grießer, T., Vukics, A. & Domokos, P. Depolarization shift of the superradiant phase transition. Phys. Rev. A 94, 033815 (2016).

    Article  ADS  Google Scholar 

  43. De Bernardis, D., Jaako, T. & Rabl, P. Cavity quantum electrodynamics in the nonperturbative regime. Phys. Rev. A 97, 043820 (2018).

    Article  ADS  MathSciNet  Google Scholar 

  44. Dicke, R. H. Coherence in spontaneous radiation processes. Phys. Rev. 93, 99–110 (1954).

    Article  ADS  Google Scholar 

  45. De Bernardis, D., Pilar, P., Jaako, T., De Liberato, S. & Rabl, P. Breakdown of gauge invariance in ultrastrong-coupling cavity QED. Phys. Rev. A 98, 053819 (2018).

    Article  ADS  Google Scholar 

  46. Stokes, A. & Nazir, A. Gauge ambiguities imply Jaynes–Cummings physics remains valid in ultrastrong coupling QED. Nat. Commun. 10, 499 (2019).

    Article  ADS  Google Scholar 

  47. Savasta, S. & Girlanda, R. The particle–photon interaction in systems described by model Hamiltonians in second quantization. Solid State Commun. 96, 517–522 (1995).

    Article  ADS  Google Scholar 

  48. Savasta, S. & Girlanda, R. Quantum description of the input and output electromagnetic fields in a polarizable confined system. Phys. Rev. A 53, 2716–2726 (1996).

    Article  ADS  Google Scholar 

  49. Sundaresan, N. M. et al. Beyond strong coupling in a multimode cavity. Phys. Rev. X 5, 021035 (2015).

    Google Scholar 

  50. Malekakhlagh, M., Petrescu, A. & Türeci, H. E. Cutoff-free circuit quantum electrodynamics. Phys. Rev. Lett. 119, 073601 (2017).

    Article  ADS  Google Scholar 

  51. Gely, M. F. et al. Convergence of the multimode quantum Rabi model of circuit quantum electrodynamics. Phys. Rev. B 95, 245115 (2017).

    Article  ADS  Google Scholar 

  52. Bosman, S. J. et al. Multi-mode ultra-strong coupling in circuit quantum electrodynamics. npj Quantum Inf. 3, 46 (2017).

    Article  ADS  Google Scholar 

  53. Muñoz, C. S., Nori, F. & De Liberato, S. Resolution of superluminal signalling in non-perturbative cavity quantum electrodynamics. Nat. Commun. 9, 1924 (2018).

    Article  ADS  Google Scholar 

  54. Hughes, S. Breakdown of the area theorem: carrier-wave Rabi flopping of femtosecond optical pulses. Phys. Rev. Lett. 81, 3363–3366 (1998).

    Article  ADS  Google Scholar 

  55. Ciappina, M. F. et al. Carrier-wave Rabi-flopping signatures in high-order harmonic generation for alkali atoms. Phys. Rev. Lett. 114, 143902 (2015).

    Article  ADS  Google Scholar 

  56. Brabec, T. & Krausz, F. Intense few-cycle laser fields: frontiers of nonlinear optics. Rev. Mod. Phys. 72, 545–591 (2000).

    Article  ADS  Google Scholar 

  57. Manucharyan, V. E., Koch, J., Glazman, L. I. & Devoret, M. H. Fluxonium: single Cooper-pair circuit free of charge offsets. Science 326, 113–116 (2009).

    Article  ADS  Google Scholar 

  58. Louisell, W. H. Quantum Statistical Properties of Radiation (Wiley, 1990).

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Acknowledgements

The authors acknowledge discussions with S. De Liberato, A. Nazir and P. Rabl. F.N. is supported in part by the MURI Center for Dynamic Magneto-Optics via the Air Force Office of Scientific Research (AFOSR) (FA9550-14-1-0040), the Army Research Office (ARO) (grant no. W911NF-18-1-0358), the Asian Office of Aerospace Research and Development (AOARD) (grant no. FA2386-18-1-4045), the Japan Science and Technology Agency (JST) (via the Q-LEAP programme and CREST grant no. JPMJCR1676), the Japan Society for the Promotion of Science (JSPS) (JSPS-RFBR grant no. 17-52-50023 and JSPSFWO grant no. VS.059.18N), the RIKEN-AIST Challenge Research Fund and the John Templeton Foundation. S.S. acknowledges support from the Army Research Office (ARO) (grant no. W911NF1910065).

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Authors and Affiliations

  1. Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering Research, Wako-shi, Saitama, Japan

    Omar Di Stefano, Vincenzo Macrì, Luigi Garziano, Roberto Stassi, Salvatore Savasta & Franco Nori

  2. Dipartimento di Scienze Matematiche e Informatiche, Scienze Fisiche e Scienze della Terra, Università di Messina, Messina, Italy

    Alessio Settineri & Salvatore Savasta

  3. Physics Department, The University of Michigan, Ann Arbor, MI, USA

    Franco Nori

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Contributions

S.S. conceived the main idea and F.N. supervised the work. S.S., O.D. and F.N. designed the study. O.D. and L.G. performed analytical calculations. O.D. and A.S. performed numerical calculations. V.M. and R.S. numerically studied the full Rabi model and the Dicke model. O.D., S.S., L.G. and F.N. contributed to writing the manuscript. All authors were involved in the preparation and discussion of the manuscript.

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Correspondence to Salvatore Savasta.

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Di Stefano, O., Settineri, A., Macrì, V. et al. Resolution of gauge ambiguities in ultrastrong-coupling cavity quantum electrodynamics. Nat. Phys. 15, 803–808 (2019). https://doi.org/10.1038/s41567-019-0534-4

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