Bose-Einstein condensates in a one-dimensional double square well: Analytical solutions of the Nonlinear Schr\"odinger equation and tunneling splittings

TL;DR

This paper derives analytical stationary solutions of the nonlinear Schrödinger equation for a symmetric double square well, revealing symmetry-breaking states and their role in forming macroscopic quantum superpositions and tunneling splittings.

Contribution

It provides explicit analytical solutions including symmetry-breaking states for the nonlinear Schrödinger equation in a double well, expanding understanding of nonlinear quantum phenomena.

Findings

Existence of symmetry-breaking localized solutions

Formation of macroscopic quantum superpositions

Analysis of tunneling splittings in nonlinear regimes

Abstract

We present a representative set of analytic stationary state solutions of the Nonlinear Schr\"odinger equation for a symmetric double square well potential for both attractive and repulsive nonlinearity. In addition to the usual symmetry preserving even and odd states, nonlinearity introduces quite exotic symmetry breaking solutions - among them are trains of solitons with different number and sizes of density lumps in the two wells. We use the symmetry breaking localized solutions to form macroscopic quantum superpositions states and explore a simple model for the exponentially small tunneling splitting.