Weak Qubit Measurement with a Nonlinear Cavity: Beyond Perturbation Theory

We analyze the use of a driven nonlinear cavity to make a weak continuous measurement of a dispersively-coupled qubit. We calculate the backaction dephasing rate and measurement rate beyond leading-order perturbation theory using a phase-space approach which accounts for cavity noise squeezing. Surprisingly, we find that increasing the coupling strength beyond the regime describable by leading-order perturbation theory (i.e. linear response) allows one to come significantly closer to the quantum limit on the measurement efficiency. We interpret this behaviour in terms of the non-Gaussian photon number fluctuations of the nonlinear cavity. Our results are relevant to recent experiments using superconducting microwave circuits to study quantum measurement.