  Superconducting electrical circuits can be used to study the physics of
cavity quantum electrodynamics (QED) in new regimes, therefore realizing
circuit QED. For quantum information processing and quantum optics, an
interesting regime of circuit QED is the dispersive regime, where the detuning
between the qubit transition frequency and the resonator frequency is much
larger than the interaction strength. In this paper, we investigate how
non-linear corrections to the dispersive regime affect the measurement process.
We find that in the presence of pure qubit dephasing, photon population of the
resonator used for the measurement of the qubit act as an effective heat bath,
inducing incoherent relaxation and excitation of the qubit. Measurement thus
induces both dephasing and mixing of the qubit, something that can reduce the
quantum non-demolition aspect of the readout. Using quantum trajectory theory,
we show that this heat bath can induce quantum jumps in the qubit state and
reduce the achievable signal-to-noise ratio of a homodyne measurement of the
voltage.
