Nonlocal interactions in a BEC: an Analogue Gravity perspective

We add a minimal correction term to the local Gross-Pitaevskii equation to represent non-locality in the interactions. We show that the effective minimal non-locality can make the healing length decrease more rapidly with the increase of $s$-wave scattering length leaving the expression of the velocity of sound unaltered. We discuss the implication of this result for a Bose-Einstein Condensate (BEC) being used as an analogue gravity system. The presented result is important in the context of condensed matter physics as well because one can considerably change the size of a quantized vortex at finite $s$-wave scattering length by tuning the healing length.