Abstract
摘要：自從1995年物理學家在實驗室做出玻色-愛因斯坦凝聚現象後，有關玻色-愛因斯坦凝聚現象的研究發展，就一直持續的蓬勃發展。不論在理論或實驗方面，都有許多豐富的成果。然而，也有許多困難的問題，亟待解決。本計畫針對描述玻色-愛因斯坦凝聚現象的複合非線性薛丁格方程式，發展兼具高度效能、精度、與穩定性的計算方法，藉以求得所對應的數值解與其行為與特性。由此我們將可藉數值模擬的方式，探討玻色-愛因斯坦凝聚現象的物理特性，進而提供協助解決實際工業界應用問題的計算工具。由於這個複合非線性薛丁格偏微分方程系統具有相當的複雜度，目前仍然缺乏有效的解析或數值方法。
本計劃將針對下列幾點作深入的研究：
- 探討在雙組份凝態中具尖端型態的最小能量解的分佈
- 位能阱與外加磁場對尖端型態的最小能量解的影響
- 將尖端型態的最小能量解行為的研究擴展到多組份凝態，高維度結構，與各種型態的位能阱
- 發展複合非線性薛丁格方程組上的高效率的離散方法與大型特徵植問題解法，以求得離散能階與波函數，並將數值結果視覺化
- 與物理實驗或其他理論分法比較，以驗證計算結果的正確性，以及改進數學模型與演算法
- 進行不同問題架構下的動態分歧分析
- 在循序與平行電腦架構下實作所提出的演算法，並進行大規模的數值實驗，並將所發展軟體與實際應用結合，以探討並解決實際物理與工程問題。
我們相信透過此計畫的執行，將可對以上具有相當複雜度與挑戰性的計算問題做一深入的探討，並獲致具體成果。相關研究人員可獲得有關玻色-愛因斯坦凝聚計算的寶貴經驗與嚴格訓練。我們也將驗證本專題計畫所產出的演算法與計算軟體，在進行凝聚現像物理特性的定性與定量分析時，亦將具有相當的學術貢獻與應用價值。
Abstract: We propose studying physical characteristics of Bose-Einstein condensates by numerical computations. Efficient, high accuracy, and robust numerical schemes will be developed to solve the coupled nonlinear Schrödinger equations that model the Bose-Einstein condensates. The numerical schemes will be used to solve the eigenvalue problem system and to explore the solution behaviors. The computational results are further expected to numerically simulate the real world situations, so that we may explore the physical phenomena and provide tools for assisting applications designs. The topics we are planning to investigate are quite complicated and little numerical methods exist for the target problems.
To conquer the challenges, this project will focus on the following points.
- Exploring the minimal energy spike configurations in a binary-mixture condensate.
- Investigating the effect of trap potential and magnetic field to the spikes of binary-mixture condensate.
- Expanding the studies to multiple-component condensates and higher-dimension configurations, and various shapes of potentials.
- Developing efficient discretization schemes to the coupled nonlinear Schrödinger equations and eigenvalue problem system solvers for the resulting large scale eigenvalue problems.
- Justifying the numerical schemes by comparing the computational results with the analytically and experimental results.
- Conducting bifurcation analysis on the various problem settings.
- Implementing the proposed algorithms on both sequential and parallel computers to conduct numerical experiments.
- Applying the developed schemes and software to explore the physics and engineering applications.
By conducting the project, we expect the involving researchers will gain valuable experience and rigorous training in the computation of the coupled nonlinear Schrödinger equations. We also expect the resulting numerical schemes and software will provide useful assistances while conducting quantitative and qualitative numerical simulation to the Bose-Einstein condensates
Keyword(s)
玻色-愛因斯坦凝聚
複合非線性薛丁格方程組
尖端型態解
離散能階
超大型特徵值問題
疊代法
數值模擬
計算方法與軟體
Bose-Einstein condensates
coupled nonlinear Schrö
dinger equations
energy states
large scale eigenvalue problems
iterative methods
numerical simulations
computational methods and software
