Variance of an anisotropic Bose-Einstein condensate

The anisotropy of a trap potential can impact the density and variance of a Bose-Einstein condensate (BEC) in an opposite manner. We exemplify this effect for both the ground state and out-of-equilibrium dynamics of structureless bosons interacting by a long-range inter-particle interaction and trapped in a two-dimensional single-well potential. We demonstrate that even when the density of the BEC is, say, wider along the $y$ direction and narrower along the $x$ direction, its position variance can actually be smaller and momentum variance larger in the $y$ direction than in the $x$ direction. This behavior of the variance in a many-particle system is counterintuitive. It suggests using the variance as a tool to characterize the strength of correlations along the $y$ and $x$ directions in a trapped BEC.