  Systems in the dispersive regime of cavity quantum electrodynamics (QED) are
approaching the limits of validity of the dispersive approximation. We present
a model which takes into account nonlinear corrections to the dressing of the
atom by the field. We find that in the presence of pure dephasing, photons
populating the cavity act as a heat bath on the atom, inducing incoherent
relaxation and excitation. These effects are shown to reduce the achievable
signal-to-noise ratio in cavity QED realizations where the atom is measured
indirectly through cavity transmission and in particular in circuit QED.
