The Gross‐Pitaevskii Model of Spinor BEC

Dongming Yan,Tian Ma

doi:10.1155/2012/757616

Abstract

The Gross‐Pitaevskii model of spinor Bose‐Einstein condensates is studied. Using the abstract results obtained for infinite‐dimensional Hamilton system, we establish the mathematical theory for the model of spinor BEC. Furthermore, three conservative quantities of spinor BEC, that is, the energy, total particle number, and magnetization intensity, are also proved.

Highlights

After the first remarkable experiments concerning the observation of Bose-Einstein condensate BEC in dilute gases of alkali atoms such as 87Rb 1, 23Na 2, and 7Li 3 the interest in this phenomenon has revived 4, 5
In 17, Pu et al discuss the energy eigenstates, ground and spin mixing dynamics of a spin-1 spinor BEC for a dilute atomic vapor confined in an optical trap
From 2.3 and 2.7 , we have ψ0 2 − ψ−1 2 ψ12 gs 2ψ−11ψ01ψ02 − ψ−21 ψ01 2 ψ−21 ψ02 2 , ψ0 2 − ψ−1 2 ψ11 gs 2ψ−21ψ01ψ02 ψ−11 ψ01 2 − ψ−11 ψ02 2 , Combining 6.5 , 6.6 with 3.1 , 3.2 , we can get following equalities: dM dt δM ψ1, ψ2 δψ[1]

Summary

Introduction

After the first remarkable experiments concerning the observation of Bose-Einstein condensate BEC in dilute gases of alkali atoms such as 87Rb 1 , 23Na 2 , and 7Li 3 the interest in this phenomenon has revived 4, 5. In 17 , Pu et al discuss the energy eigenstates, ground and spin mixing dynamics of a spin-1 spinor BEC for a dilute atomic vapor confined in an optical trap. Their results go beyond the mean field picture and are developed within a fully quantized framework. In 19 , Zou and Mathis propose a three-step scheme for generating the maximally entangled atomic Greenberger-Horne-Zeilinger GHZ states in a spinor BEC by using strong classical laser fields to shift atom level and drive single-atom Raman transition Their scheme can be directly used to generate the maximally entangled states between atoms with hyperfine spin 0 and 1. Three conservative quantities of spinor BEC, that is, the energy, total particle number, and magnetization intensity are proved

Gross-Pitaevskii Model for Spinor BEC

Equivalent Form of Spinor BEC

Infinite Dimensional Hamilton System

The Existence of Global Solution of Spinor BEC

The Conservative Quantities of Spinor BEC