Two-mode squeezing and entanglement in atomic boson sampling

Recently, atomic boson sampling of excited atom occupations in an equilibrium interacting gas with a Bose-Einstein condensate (BEC) has been suggested as a process that could be #P-hard for classical computing. Here we consider this process within the simplest possible model of a BEC trap - the box with the periodic boundary conditions. Remarkably, this model remains pertained to #P-hardness and quantum supremacy. We evince two-mode squeezing and entanglement in the statistics of sampling from (i) a single eigen-squeeze mode or (ii) two counter-propagating waves. Although not complex enough on its own to show #P-hard behavior, such a sampling reveals the basic mechanism of the #P-hardness - an interplay between two intrinsic entities existing in the BEC gas, the eigen-energy quasiparticles and eigen-squeeze modes, that causes the interference and squeezing of the sampled excited atom states.