Resolution of gauge ambiguities in ultrastrong-coupling cavity quantum electrodynamics


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.

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.


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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).

Author information

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.

Correspondence to Salvatore Savasta.

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