  We further examine a theory of phase contrast imaging (PCI) of cold atomic
gases, first introduced by us in Phys. Rev. Lett. {\bf 112}, 233602 (2014). We
model the PCI measurement by directly calculating the entangled state between
the light and the atoms due to the ac Stark shift, which induces a conditional
phase shift on the light depending upon the atomic state. By interfering the
light that passes through the BEC with the original light, one can obtain
information of the atomic state at a single shot level. We derive an exact
expression for a measurement operator that embodies the information obtained
from PCI, as well as the back-action on the atomic state. By the use of exact
expressions for the measurement process, we go beyond the continuous variables
approximation such that the non-Gaussian regime can be accessed for both the
measured state and the post-measurement state. Features such as the photon
probability density, signal, signal variance, Fisher information, error of the
measurement, and the backaction are calculated by applying the measurement
operator to an atomic two spin state system. For an atomic state that is
initially in a spin coherent state, we obtain analytical expression for these
quantities. There is an optimal atom-light interaction time that scales
inversely proportional to the number of atoms, which maximizes the information
readout.
