Discrete bright solitons in Bose-Einstein condensates and dimensional reduction in quantum field theory

TL;DR

This paper derives effective 1D models for Bose-Einstein condensates and quantum fields, studying bright solitons, collapse phenomena, and dimensional reduction, connecting continuous and lattice descriptions in quantum many-body systems.

Contribution

It introduces a discrete nonpolynomial Schrödinger equation and a nonpolynomial Heisenberg equation, linking 3D and 1D models, and explores their implications for solitons and Bose-Hubbard models.

Findings

Bright solitons lead to condensate collapse above a critical attraction.

Derived a generalized Bose-Hubbard model from 3D quantum field theory.

Revealed the connection to the Lieb-Liniger model in the absence of periodic potential.

Abstract

We first review the derivation of an effective one-dimensional (1D) discrete nonpolynomial Schr\"odinger equation from the continuous 3D Gross-Pitaevskii equation with transverse harmonic confinement and axial periodic potential. Then we study the bright solitons obtained from this discrete nonpolynomial equation showing that they give rise to the collapse of the condensate above a critical attractive strength. We also investigate the dimensional reduction of a bosonic quantum field theory, deriving an effective 1D nonpolynomial Heisenberg equation from the 3D Heisenberg equation of the continuous bosonic field operator under the action of transverse harmonic confinement. Moreover, by taking into account the presence of an axial periodic potential we find a generalized Bose-Hubbard model which reduces to the familiar 1D Bose-Hubbard Hamiltonian only if a strong inequality is satisfied. Remarkably, in the absence of axial periodic potential our 1D nonpolynomial Heisenberg equation gives the generalized Lieb-Liniger theory we obtained some years ago.