Renormalization and Low-Energy Effective Models in Cavity and Circuit QED

The quantum Rabi model (QRM) is a cornerstone in the study of light-matter interactions within cavity quantum electrodynamics (CQED). It effectively captures the dynamics of a two-level system coupled to a single-mode resonator, serving as a foundation for understanding phenomena in various fields, such as superconducting qubits, quantum dots, and trapped ions. However, at sufficiently large couplings, its predictions deviate significantly from those of microscopic models which account for multiple energy levels for the matter degree of freedom, and issues of gauge invariance in truncated Hilbert spaces further undermine its reliability. In this work, we introduce a renormalized QRM, which incorporates the influence of higher energy levels, resulting in a highly accurate quantitative description of the system even under very strong coupling conditions. This model corrects the deviations observed in the standard QRM, thereby enhancing the accuracy in the prediction of energy eigenvalues and physical observables. Additionally, we propose a gauge-invariant formulation, ensuring consistent and accurate results across different gauges. To illustrate its broad applicability, we apply this approach to two specific cases: an atom in a double-well potential, and the fluxonium qubit. The renormalized QRM allows efficient simulation of CQED systems with finite anharmonicity within a computationally simple framework, thereby emerging as a versatile tool for the accurate modeling of quantum technology devices.