An analytical study of resonant transport of Bose-Einstein condensates

TL;DR

This paper analytically investigates the transport properties and eigenstate behaviors of Bose-Einstein condensates in a one-dimensional square well, revealing bistability, crossing scenarios, and state transformations due to nonlinearity.

Contribution

It provides an analytical study of resonant transport, bound states, and eigenstate transformations in a nonlinear Schrödinger equation model for Bose-Einstein condensates.

Findings

Resonances and bound states are obtained analytically.

Transformed resonances to bound states due to nonlinearity.

Identified critical nonlinearity for state transformation.

Abstract

We study the stationary nonlinear Schr\"odinger equation, or Gross-Pitaevskii equation, for a one--dimensional finite square well potential. By neglecting the mean--field interaction outside the potential well it is possible to discuss the transport properties of the system analytically in terms of ingoing and outgoing waves. Resonances and bound states are obtained analytically. The transmitted flux shows a bistable behaviour. Novel crossing scenarios of eigenstates similar to beak--to--beak structures are observed for a repulsive mean-field interaction. It is proven that resonances transform to bound states due to an attractive nonlinearity and vice versa for a repulsive nonlinearity, and the critical nonlinearity for the transformation is calculated analytically. The bound state wavefunctions of the system satisfy an oscillation theorem as in the case of linear quantum mechanics. Furthermore, the implications of the eigenstates on the dymamics of the system are discussed.