Abstract:
To account for the non-local interactions in a Bose-Einstein Condensate (BEC), an addition
of a minimal correction term to the standard Gross-Pitaevskii model effectively can make the
healing length or characteristic length ( ) decrease more rapidly with the increase of s-wave
scattering length (a). From emergent gravity perspectives, this shrinking of via tuning a
through Feshbach resonance, in principle, does make the short-wavelength (i.e. high energy)
regime more accessible experimentally by pushing the Lorentz-breaking dispersion even more
towards the UV side [1]. The effects of the Lorentz-breaking quantum potential term in the
BEC-dynamics on independent multiple scales can be captured through a UV-IR coupling of
the phonon-excitation-modes with the emergence of the massive minimally coupled free Klein-
Gordon field. The analysis was argued on a (3+1)D flat spacetime where the presence of the
mass term gives a hint to cure the infrared divergences through a nonzero threshold for the
large-wavelength phonon excitations (say, Kz , 0 modes as perceived by an observer sitting
on x y plane) on lower dimensional models of emergent spacetimes [2]. The analysis was
extended for a canonical acoustic black hole on a (3+1)D curved spacetime through presenting
an emergent gravity model up to O( 2) accuracy. In our formalism, the growth rate of the
large-wavelength ‘secondary’ ! modes is found to hold the clue to extract the lost information
regarding the short-wavelength ‘primary’ !1 modes. And hence this can actually reveal the
relative abundance of the originally Hawking radiated quanta in a (3+1)D canonically curved
background [3]. Thiswaywe could address the trans-Planckian issues in Physics via introducing
the analogue Planck length (the healing length of the underlying cond.mat system, say BEC
in our case, is the analogue of the Planck length `p 1035m. in Gravity) and shrinking this
to an arbitrary small value by experimentally (Feshbach resonances) increasing the s-wave
scattering length a. We showed that the UV-IR coupling between two different bands of phonon
excitations is inevitable as perceived by an observer in the emergent gravity experiments. And
finally we tackled the way to retrieve the “information loss” in Hawking radiation within the
scope emergent spacetimes.
