多種氣體在玻色愛因斯坦凝聚現象分佈的探討

A Numerical Study of the Distribution of Multi-Component Bose-Einstein Condensates

江秉陽(Ping-Yang Chiang)

指導教授 ： 張書銘

國立交通大學/理學院/應用數學系暨研究所/碩士(2013年)

https://doi.org/10.6842/NCTU.2013.00331

摘要

本論文裡，我們運用數值模擬來研究多種種類氣體的玻色愛因斯坦冷凝現象的分佈情形，結果顯示玻色氣體在激發態時氣體們會因為夠大的排斥力而形成分離的區塊。格洛斯皮塔夫斯基方程式是一個描述多種種類氣體的玻色愛因斯坦冷凝現象的方程式，我們使用高斯賽德爾疊代法來計算它的能量狀態。離散格洛斯皮塔夫斯基方程後，導出了非線性特徵值的問題。我們用高斯賽德爾疊代法此方法來計算在基態時的非線性特徵值問題，以及找出其相對應的能量。此數值結果我們發現了分離區塊的分佈情形，以及能量隨著氣體個數增加而增加。

關鍵字

多種種類氣體的玻色愛因斯坦冷凝 ； 高斯賽德爾疊代法 ； 格洛斯皮塔夫斯基方程式 ； 非線性特徵值問題

並列摘要

In this thesis, we study the distribution of m segregated nodal domains of the m-mixture of Bose-Einstein condensates under positive and large repulsive scattering lengths. It
is shown that components of positive bound states may crowd out each other and form segregated nodal domains when the repulsive scattering is large enough. We use the iteration
method, Gauss-Seidel-type iteration (GSI), for the computation of energy states of the time-independent vector Gross-Pitaevskii equation (VGPE) which describes a multi-component Bose-Einstein condensate (BEC). A discretization of the VGPE leads to a nonlinear algebraic eigenvalue problem (NAEP). The GSI method can thus be used to compute ground states and positive bound states, as well as the corresponding energies of a multi-component BEC. We discover the distribution of multiple segregated nodal domains and the energy would increase with the number of Bose-Einstein condensates component increases.

並列關鍵字

Multi-Component Bose-Einstein Condensate ； vector Gross-Pitaevskii equation ； Gauss-Seidel-type iteration ； nonlinear eigenvalue problem

參考文獻

feedback schemes. Joumal of Physics B: Atomic, Molecular and Optical Physics,

46(8),085301, 2013.

[2] W. Z. Bao. Ground states and dynamics of multi-component bose-einstein condensates.

[3] W. Z. Bao and Q. Du. Computing the ground state solution of bose-einstein condensates

[4] W. Z. Bao and D. Jaksch. An explicit unconditionally stable numerical methods for