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

TL;DR

This paper investigates how the maximum non-linear coupling constant for a Bose-Einstein condensate depends on defect shape and size across different dimensions, revealing a universal scaling law at the localization-delocalization threshold.

Contribution

It introduces an analytical Thomas-Fermi based model for g_max, demonstrating a universal scaling law for the non-linear coupling constant in BECs with finite-range defects.

Findings

g_max depends on R_0 sqrt(V_0) across defect shapes and dimensions

The model shows surprising agreement with explicit calculations over wide parameter ranges

A scaling rule relates localized states of different defects with the same R_0 sqrt(V_0) product

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 Schroedinger equation that describes a Bose-Einstein condensate was used to calculate the maximum non-linear coupling constant, g_max, and thus the maximum number of atoms, N_max, that the defect potential can localize. An analytical model, based on the Thomas-Fermi approximation, is introduced for the wavefunction. We show that g_max becomes a function of R_0 sqrt(V_0) for various one-, two-, and three-dimensional defect shapes with depths V_0 and characteristic lengths R_0. Our explicit calculations show surprising agreement with this crude model over a wide range of V_0 and R_0. 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 R_0 sqrt(V_0) will thus be related to each other with the same g_max and will have the same (reduced) density profile in the free-direction/s.