  In the ultra-strong coupling regime of a light-matter system, the ground
state exhibits non-trivial entanglement between the atom and photons. For the
purposes of exploring the measurement and control of this ground state, here we
analyze the dynamics of such an ultra-strongly-coupled system interacting with
a driven nonlinear resonator acting as a measurement apparatus. Interestingly,
although the coupling between the atom and the nonlinear resonator is much
smaller than the typical energy scales of the ultra-strongly-coupled system, we
show that we can generate a strong correlation between the nonlinear resonator
and the light-matter system. A subsequent coarse- grained measurement on the
nonlinear resonator significantly affects the light-matter system, and the
phase of the light changes depending on the measurement results. Also, we
investigate the conditions for when the nonlinear resonator can be entangled
with the ultra-strongly coupled system, which is the mechanism that allows us
to project the ground state of the ultra-strongly coupled system into a
non-energy eigenstate.
