臺灣博碩士論文加值系統

論文第一部分我們提出一個可以在具有軸對稱陷阱中的玻色-愛因斯坦凝聚體 ( Bose-Einstein condensate, BEC ) 裡找到靜止量子化渦旋環解的新數值方案。根據量子化渦旋環在rz 平面的環流要量子化的條件我們可以
猜測出凝聚體波函數所具有的相位，而更進一步地求得有效的系統能量泛函以及所對應的Gross-Pitaevskii 方程。利用數值方法求解此對應 GP 方程的最低能量態波函數即可得到 BEC 中的量子化渦旋環解。應用此方法到二分量凝聚體系統也可以求得三維的 Skyrmion 結構。
推廣這個數值方案可以在 BEC 中產生任意的量子化渦旋結構，有助於 BEC 中量子化渦旋動力學的研究。而論文的第二個主題為研究在薄餅型凝聚體中不穩定量子化渦旋環的動力學。由於量子渦旋線的彎曲波擾動造成渦旋環結構的不穩定將導致量子渦旋線斷裂，而在經過長時間的演化之後將致使系統達到一個新的狀態─量子紊流態 ( quantum turbulence state ) 。我們證實量子紊流與古典紊流的能譜同樣遵守 Kolmogorov 的 −5/3 冪次定律。

In the first part, we propose a numerical scheme for obtaining the stationary vortex-ring solutions of the Gross-Pitaevskii (GP) equation for an axisymmetrically trapped Bose-Einstein condensate (BEC). The effective energy functional and the associated GP equation are derived by assuming a trial phase profile for the wavefunction that is subject to the condition of circulation quantization on the rz plane. The wavefunction of the vortex ring is determined by solving the ground state of the effective GP equation numerically. Application of our method to the formation of a three-dimensional Skyrmion in a trapped two-component BEC is demonstrated.
By generalizing the technique of generating vortex ring, complex vortex configurations composed of single vortex or many vortices can be obtained, and their dynamical evolutions can also be observed by integrating the GP equation numerically. As a result of this scheme, we investigate the dynamics of an unstable vortex ring in a pancake-shaped BEC by solving the GP equation numerically. It is found that a quasisteady turbulent state with long relaxation time can be achieved through the disruption of a perturbed vortex ring in the
condensate owing to the bending-wave instability. We verify that this quantum turbulent state is characterized by Kolmogorov energy spectrum.

《目次》
第一章 前言 1
第二章 理論
2.1 Gross-Pitaevskii ( GP ) 方程 10
2.2 Thomas-Fermi ( TF ) 近似 14
2.3 量子流體力學 ( quantum hydrodynamics ) 15
第三章數值方法
3.1 GP 方程式無因次化 17
3.2 虛數時間演化法 ( Imaginary Time Propagation Method ) 17
3.3 空間部分的處理：傅立葉擬譜法 ( Fourier Pseudospectral Method ) 18
3.4 時間部分的處理：阮格–卡達法 ( Runge-Kutta Method ) 20
第四章量子渦旋
4.1 環流量子化 23
4.2 長直量子化渦旋線 23
4.3 對稱軸指向量子化渦旋環 25
4.4 運動方程式 42
4.5 各式各樣的渦旋結構 50
第五章紊流與量子紊流 54
第六章結論 65
參考文獻 67
附圖
圖 1 26
圖 2 31
圖 3 33
圖 4 35
圖 5 36
圖 6 38
圖 7 39
圖 8 41
圖 9 42
圖 10 44
圖 11 46
圖 12 48
圖 13 49
圖 14 52
圖 15 57
圖 16 58
圖 17 60
圖 18 62
圖 19 63

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