Information and backaction due to phase contrast imaging measurements of cold atomic gases: beyond Gaussian states

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.