  A quantum dipole interacting with an optical cavity is one of the key models
in cavity quantum electrodynamics (cavity-QED). To treat this system
theoretically, the typical approach is to truncate the dipole to two levels.
However, it has been shown that in the ultrastrong-coupling regime, this
truncation naively destroys gauge invariance. By truncating in a manner
consistent with the gauge principle, we introduce master equations to compute
gauge-invariant emission spectra and quantum correlation functions which show
significant disagreement with previous results obtained using the standard
quantum Rabi model, with quantitative differences already present in the strong
coupling regime. Explicit examples are shown using both the dipole gauge and
the Coulomb gauge.
