  Cavity and circuit QED study light-matter interaction at its most fundamental
level. Yet, this interaction is most often neglected when considering the
coupling of this system with an environment. In this paper, we show how this
simplification, which leads to the standard quantum optics master equation, is
at the root of unphysical effects. Including qubit relaxation and dephasing,
and cavity relaxation, we derive a master equation that takes into account the
qubit-resonator coupling. Special attention is given to the ultrastrong
coupling regime, where the failure of the quantum optical master equation is
manifest. In this situation, our model predicts an asymmetry in the vacuum Rabi
splitting that could be used to probe dephasing noise at unexplored
frequencies. We also show how fluctuations in the qubit frequency can cause
sideband transitions, squeezing, and Casimir-like photon generation.
