Bogoliubov theory for atom scattering into separate regions

We review the Bogoliubov theory in the context of recent experiments, where atoms are scattered from a Bose-Einstein Condensate into two well-separated regions. We find the full dynamics of the pair-production process, calculate the first and second order correlation functions and show that the system is ideally number-squeezed. We calculate the Fisher information to show how the entanglement between the atoms from the two regions changes in time. We also provide a simple expression for the lower bound of the useful entanglement in the system in terms of the average number of scattered atoms and the number of modes they occupy. We then apply our theory to a recent "twin-beam" experiment \[R. Bücker {\it et al.}, Nat. Phys. {\bf 7}, 608 (2011)\]. The only numerical step of our semi-analytical description can be easily solved and does not require implementation of any stochastic methods.