圍攪玻色 - 愛因斯坦凝結，用於描述稀薄氣體(玻色子)於磁場阱內透過雷射光圍攪因而產生角動量的量子態。求其穩定能量階圖，可以公式化為目標函數參數化之最佳問題。
在固定參數的情況下，許多最佳化問題之演算法可以用來找出一個局部穩定解。然而，窮盡使用這些方法很難決定參數化穩定能量區間的邊界。更糟的是，此種費時的做法可能會錯失一些非常關鍵的穩定解，尤其是存在多穩態現象的區間。在這篇論文中，我們結合了延拓法與梯度流方法去追蹤當角速度很大時的局部穩定曲線。

The rotating Bose-Einstein condensates describe the quantum state of dilute atomic gases imposed by a laser beam rotating with an angular velocity applied to the magnetic trap. Its stable energy diagram can be formulated as a cost parameterized optimization problem. When the parameter is fixed, various optimization algorithms have been well developed to find a local optimal solution. However, applying these methods for a large number of given parameters is difficult to determine the end points of the maximal parameterized interval. Even worse, this exhausted approach may miss some critical optimal solutions especially when multi-stable regions exist. In this thesis, we propose a procedure that combines the continuation method and the gradient flow method to track local optimal curves for larger angular velocity.

論文審定書------------------------------------------------------i
摘要----------------------------------------------------------------ii
Abstract----------------------------------------------------------iii
1 Introduction---------------------------------------------------1
2 Preliminary---------------------------------------------------4
2.1 The steepest descent method------------------4
3 Discretization-----------------------------------------------7
3.1 The magnetic trapping potentials--------------8
3.2 The Laplacian operator----------------------------9
3.3 The operator of Lz =i (y∂x-x∂y)-----------------11
3.4 The nonlinear term μ|u|^2u-----------------------13
3.5 The mass conservation condition-------------13
3.6 The Gross-Pitaevskii equation------------------14
3.7 The energy functional----------------------------- 15
4 Numerical methods---------------------------------------16
4.1 The gradient flow method-------------------------16
4.2 Initial points for the gradient flow method---17
4.3 The magnetic trapping potentials--------------18
5 Numerical results------------------------------------------19
5.1 The stable energy diagram---------------------- 19
5.2 Conclusions--------------------------------------------28
References------------------------------------------------------29

[1] J. R. Abo-Shaeer, W. Ketterle, C. Rarnan and J. M. Vogets (2001), Observation of vortex lattices in Bose-Einstein condensates, Science, 292 (5516), 476.

[2] M. H. Anderson, E. A. Cornell, J. R. Ensher, M. R. Matthews and C. E. Wieman (1995), Observation of Bose-Einstein condensation in a dilute atomic vapor, Science, 269, 198.

[3] B. P. Anderson, E. A. Cornell, P. C. Haljan, D. S. Hall, M. R. Matthews and C. E. Wieman (1999), Vortices in a Bose-Einstein condensate,
extit{Phys. Rev. Lett.}, 83, 2498.

[4] W. Bao(2004), Ground states and dynamics of multi-component Bose-Einstein condensates, SIAM Multiscale Model. Simul.,2 (2), 210-236.

[5] W. Bao, Q. Du (2004), Computing the ground state solution of Bose-Einstein condensates by a normalized gradient flow, SIAM J. Sci. Comput., 25 (5), 1674-1697.

[6] W. Bao, D. Jaksch, P. A. Markowich (2003), Numerical solution of the Gross-Pitaevskii equation for Bose-Einstein condensation, Journal of Computational Physics, 187, 318-342.

[7] W. Bao, P. A. Markowich and H. Wang (2005), Ground, symmetric and central vortex states in rotating Bose-Einstein condensates, Commun. Math. Sci, 3(1), 57-88.

[8] W. Bao, W. Tang (2003), Ground-state solution of Bose-Einstein
condensate by directly minimizing the energy functional, Journal of Computational Physics, 187, 230-254.

[9] C. C. Bradley, R. G. Hulet and C. A. Sackett (1995), Evidence of Bose-Einstein condensation in an atomic gas with attractive interactions, Phys. Rev. Lett., 75, 1687.

[10] S. -L. Chang, C. -S. Chien (2007), Adaptive continuation algorithms for computing energy levels of rotating Bose-Einstein condensate, Comput. Phys. Comm., 177, 707-719.

[11] F. Chevy, J. Dalibard, K. W. Madison and W. Wohlleben (2000), Vortex formation in a stirred Bose-Einstein condensate, Phys. Rev. Lett., 84 (5), 806-809.

[12] F. Chevy, J. Dalibard, K. W. Madison, W. Wohlleben, (2000), Vortices in a stirred Bose-Einstein condensate, J. Modern Opt., 47 (14-15), 2715-2724.

[13] E. A. Cornell, J. R. Ensher, D. S. Hall, M. R. Matthews and C. E. Wieman (1998), Dynamics of component separation in a binary mixture of Bose-Einstein condensates, Phys. Rev. Lett., 81, 1539-1542.

[14] E. A. Cornell, J. R. Ensher, D. S. Jin, M. R. Matthews and C. E. Wieman (1996),
Bose-Einstein condensation in a dilute gas: measurement of energy and ground-state occupation, Phys. Rev. Lett., 77, 4984.

[15] F. Dalfovo and S. Giorgini (1999), Theory of Bose-Einstein condensation in trapped gases, Rev. Mod. Phys., 71, 463.

[16] Y. -C. Kuo, W. -W. Lin, S. -F. Shieh, W. Wang (2011), Exploring bistability in rotating Bose-Einstein condensates by a quotient transformation invariant continuation method, Physica D, 240, 78-88.

[17] Y. -T. Shih, C. -C. Tsai (2013), A two-parameter continuation algorithm using radial basis function collocation method for rotating Bose-Einstein condensates, Journal of Computational Physics, 252, 37-51.