Theory of single-shot phase contrast imaging in spinor Bose-Einstein condensates

We introduce a theoretical framework for single-shot phase contrast imaging (PCI) measurements of spinor Bose-Einstein condensates. Our model allows for the simple calculation of the quantum backaction resulting from the measurement, and the amount of information that is read out. We find that there is an optimum time $G\tau \sim 1/N$ for the light-matter interaction ($G$ is the ac Stark shift frequency, $N$ is the number of particles in the BEC), where the maximum amount of information can be read out from the BEC. A universal information-disturbance tradeoff law $\epsilon_F \epsilon_G \propto 1/N^2$ is found where $\epsilon_F$ is the amount of backaction and $\epsilon_G$ is the estimation error. The PCI measurement can also be found to be a direct probe of the quantum fluctuations of the BEC, via the noise of the PCI signal.