  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.
