Squeezing 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 [V.V. Kocharovsky et al., PRA 106, 063312 (2022),

Entropy 24, 1771 (2022)] 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

calculate, analytically and numerically, the effect of single- and two-mode squeezing on the statistics of

sampling from a single eigen-squeeze mode and two counter-propagating waves. Although not complex

enough on its own to show ♯P-hard behavior, single- or two-mode sampling reveals the basic

mechanism for the emergence of the computational ♯P-hardness in a many-body quantum system of

interacting atoms.