Projecting an ultra-strongly-coupled system in a non-energy-eigenbasis with a driven nonlinear resonator

We explore the problem of projecting the ground-state of an ultra-strong-coupled circuit-QED system into a non-energy-eigenstate. As a measurement apparatus we consider a nonlinear driven resonator. We find that the post-measurement state of the nonlinear resonator exhibits a large correlation with the post-measurement state of the ultra-strongly coupled system even when the coupling between measurement device and system is much smaller than the energy scales of the system itself. While the projection is imperfect, we argue that because of the strong nonlinear response of the resonator it works in a practical regime where a linear measurement apparatus would fail.