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Reply to: Gauge non-invariance due to material truncation in ultrastrong-coupling quantum electrodynamics

Nature Physics volume 20, pages 379–380 (2024)Cite this article

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The Original Article was published on 22 December 2023

replying to Stokes, A. & Nazir, A. Nature Physics https://doi.org/10.1038/s41567-023-02155-8 (2023)

In their Matters Arising1, Stokes and Nazir point out that the main text of ref. 2 did not explicitly define the unitary operator \(\hat{{{{\mathcal{U}}}}}\) as: \(\hat{{{{\mathcal{U}}}}}=\hat{U}(\hat{P}x\hat{P})\), where x is the coordinate of the effective particle and

$$\hat{P}={\hat{I}}_{2}=\left\vert 0\right\rangle \left\langle 0\right\vert +\left\vert 1\right\rangle \left\langle 1\right\vert ,$$
(1)

is the identity for the truncated Hilbert space (see the text below equation (5) in ref. 2). As can be inferred from both equation (9) and Section I of the Supplementary Information of ref. 2, all our results in the correct Coulomb gauge have been obviously obtained using the unitary operator \(\hat{{{{\mathcal{U}}}}}=\hat{U}(\hat{P}x\hat{P})\).

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References

  1. Stokes, A. & Nazir, A. Gauge non-invariance due to material truncation in ultrastrong-coupling quantum electrodynamics. Nat. Phys. https://doi.org/10.1038/s41567-023-02155-8 (2023).

  2. Di Stefano, O. et al. Resolution of gauge ambiguities in ultrastrong-coupling cavity QED. Nat. Phys. 15, 803 (2019).

    Article  Google Scholar 

  3. Wilson, K. G. Confinement of quarks. Phys. Rev. D 10, 2445–2459 (1974).

    Article  CAS  ADS  Google Scholar 

  4. Gattringer, C. & Lang, C. B. Quantum Chromodynamics On the Lattice: An Introductory Presentation (Springer, 2010).

  5. Wiese, U.-J. Ultracold quantum gases and lattice systems: quantum simulation of lattice gauge theories. Ann. Phys. 525, 777–796 (2013).

    Article  MathSciNet  CAS  Google Scholar 

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

    Article  CAS  PubMed  PubMed Central  ADS  Google Scholar 

  7. 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 

  8. Maggiore, M. A Modern Introduction to Quantum Field Theory Vol. 12 (Oxford Univ. Press, 2005).

  9. Savasta, S. et al. Gauge principle and gauge invariance in two-level systems. Phys. Rev. A 103, 053703 (2021).

    Article  MathSciNet  CAS  ADS  Google Scholar 

  10. Peierls, R. Zur Theorie des Diamagnetismus von Leitungselektronen. Z. Phys. 80, 763–790 (1933).

    Article  CAS  ADS  Google Scholar 

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Authors and Affiliations

  1. MIFT Department, Università di Messina, Messina, Italy

    Omar Di Stefano, Alessio Settineri, Roberto Stassi & Salvatore Savasta

  2. Center for Quantum Computing and Cluster for Pioneering Research, RIKEN, Wako-shi, Japan

    Vincenzo Macrì, Luigi Garziano & Franco Nori

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

    Franco Nori

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Stefano, O.D., Settineri, A., Macrì, V. et al. Reply to: Gauge non-invariance due to material truncation in ultrastrong-coupling quantum electrodynamics. Nat. Phys. 20, 379–380 (2024). https://doi.org/10.1038/s41567-023-02178-1

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