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.
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