Conjecture on BEC of Lévy-flight particles and its fractional Gross–Pitaevskii description

Determine whether an ultracold gas of particles undergoing Lévy-flight dynamics can form a Bose–Einstein condensate that, in the mean-field approximation, is governed by a fractional Gross–Pitaevskii equation whose kinetic-energy operator is the Riesz fractional derivative of order α and whose interaction term is cubic, and derive a consistent microscopic foundation for this fractional Gross–Pitaevskii equation.

Background

Fractional quantum mechanics replaces the standard kinetic-energy operator with a nonlocal Riesz fractional derivative, leading to the fractional Schrödinger equation. Motivated by this framework and by optical emulation proposals, the authors paper nonlinear localized states using a fractional nonlinear Schrödinger/Gross–Pitaevskii model in quasiperiodic lattices.

They explicitly state a conjecture that an ultracold gas of particles performing Lévy flights may form a Bose–Einstein condensate described at the mean-field level by a Gross–Pitaevskii equation built from the fractional Schrödinger operator plus a cubic contact interaction. They further note that a consistent microscopic derivation of such a fractional Gross–Pitaevskii equation has not yet been reported, and thus the model is currently adopted phenomenologically.