  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 {for open
systems} to compute gauge-invariant emission spectra, photon flux rates, and
quantum correlation functions which show significant disagreement with previous
results obtained using the standard quantum Rabi model. Explicit examples are
shown using both the dipole gauge and the Coulomb gauge.
