Gauge freedom, quantum measurements, and time-dependent interactions in cavity and circuit QED

Recently, new regimes of cavity QED, where the interaction strength is comparable (ultrastrong) or even higher (deep-strong) than the transition frequencies in the system, have been explored in several settings. It has been shown that, in these regimes, the quantum Rabi model violates gauge invariance.
We discuss a modified quantum Rabi model able to provide gauge-invariant physical results in any interaction regime [Nat Phys 15, 803 (2019)]. When the interaction strength is high, fundamental issues like the proper definition of subsystems and of their quantum measurements, the structure of light-matter ground states, or the analysis of time-dependent interactions are subject to ambiguities leading to even qualitatively distinct predictions. The resolution of these ambiguities is also important for understanding and designing next-generation quantum devices that will exploit the ultrastrong coupling regime. Here we discuss solutions to these issues. We also show that the adopted procedures are closely connected to lattice gauge theories, the most advanced and commonly used tool for describing gauge theories in the presence of a truncated infinite-dimensional Hilbert space.