Scaling laws for the non-linear coupling constant of a Bose-Einstein condensate at the threshold of delocalization

Abstract

We explore the localization of a quasi-one-, quasi-two-, and three-dimensional ultra-cold gas by a finite-range defect along the corresponding 'free'-direction/s. The time-independent non-linear Schrödinger equation that describes a Bose-Einstein condensate was used to calculate the maximum non-linear coupling constant, gmax, and thus the maximum number of atoms, Nmax, that the defect potential can localize. An analytical model, based on the Thomas-Fermi approximation, is introduced for the wavefunction. We show that gmax becomes a function of R0 √ V0 for various one-, two-, and three-dimensional defect shapes with depths V0 and characteristic lengths R0. Our explicit calculations show surprising agreement with this crude model over a wide range of V0 and R0. A scaling rule is also found for the wavefunction for the ground state at the threshold at which the localized states approach delocalization. The implication is that two defects with the same product R0 √ V0 will thus be related to each other with the same gmax and will have the same (reduced) density profile in the free-direction/s.