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Theoretical methods for ultrastrong light-matter interactions

This article reviews theoretical methods developed in the last decade to understand cavity quantum electrodynamics in the ultrastrong-coupling regime, where the strength of the light-matter interaction becomes comparable to the photon frequency. Along with profound modifications of fundamental quantum optical effects giving rise to a rich phenomenology, this regime introduces significant theoretical challenges. One of the most important is the break-down of the rotating-wave approximation which neglects all non-resonant terms in light-matter interaction Hamiltonians. Consequently, a large part of the quantum optical theoretical framework has to be revisited in order to accurately account for all interaction terms in this regime. We give in this article a broad overview of the recent progress, ranging from analytical estimate of ground-state properties to proper derivations of master equations and computation of photodetection signals. For each aspect of the theory, the basic principles of the methods are illustrated on paradigmatic models such as quantum Rabi and spin-boson models. In this spirit, most of the article is devoted to effective models, relevant for the various experimental platforms in which the ultrastrong coupling has been reached, such as semiconductor microcavities and superconducting circuits. The validity of these models is discussed in the last part of the article, where we address recent debates on fundamental issues related to gauge invariance in the ultrastrong-coupling regime.