本論文主要是從理論觀點研究一個被限制在簡諧位能井中的BEC原子團，受到一無序位能井（disorder potential）的作用影響而產生的動力學行為。在實驗結果方面，我們參考Chen et al. [Phys. Rev. A 77, 033632 (2008)] 及Dries et al. [Phys. Rev. A 82, 033603 (2010)]兩篇論文。這裡的實驗環境是先準備好BEC於簡諧位能井中，BEC在簡諧位能井中的形狀被設計成柱狀雪茄型，接著將無序位能井加入系統中，隨後將簡諧位能井瞬間拉動產生位移，則BEC原子團將因為簡諧位能井的不平衡而開始來回震盪。因受到無序位能井的影響，BEC原子團振盪的幅度會漸漸變小，類似於古典的阻尼震盪，我們將此現象稱為「消散現象」 （dissipation）。

第一章主要是簡介近期發現的BEC冷原子團的相關物理現象並介紹在第三章要使用到的Gross-Pitaevskii Equation（GPE）。第二章主要介紹Wu和Zaremba [Phys. Rev. Lett. 106, 165301 (2011)] 所提出的微擾理論，從微觀的角度來研究BEC原子團的振盪消散行為；我們將指出此微擾理論不足以描述實驗的地方。第三章代表本篇論文的主要工作，我們使用GPE方法來理論摸擬BEC原子團的振盪消散行為。摸擬發現BEC原子團的振盪消散呈現兩段式的遞減行為，結果與Dries et al.的實驗結果一致。另外摸擬結果也顯示BEC原子團振盪後期，當BEC原子團速度變很小時，其空間分佈的行為與安德森侷域化(Anderson localization)的行為一致。

This thesis theoretically studies the dissipation of a harmonically trapped Bose-Einstein condensate (BEC) due to an external disorder potential. Our results are aimed to explain the experimental results done by Chen et al. [Phys. Rev. A 77, 033632 (2008)] and Dries et al. [Phys. Rev. A 82, 033603 (2010)]. Experimentally the BEC is prepared in a cylindrically harmonic trap of the cigar shape. By abruptly shifting the harmonic trap, center mass of the BEC will undergo a simple harmonic oscillation (SHO). If the system is imposed in a disorder potential, the system will then undergo a dissipated oscillation, somewhat similar to the oscillation of a classical damped SHO.

Chapter 1 briefly reviews some key physics for the BEC. Chapter 2 introduces a perturbation theory recently proposed by Wu and Zaremba [Phys. Rev. Lett. 106, 165301 (2011)] to explain the behavior of the dissipated oscillating BEC. While it provides a microscopic theory for the dissipated oscillating BEC, it is lack of the origin to explain an important phenomenon observed by the experiment, namely a two-regime crossover dissipation for the oscillating BEC. Chapter 3 represents the major work of this thesis. Based on the approach of the Gross-Pitaevskii Equation (GPE), we numerically study the dissipation of the oscillating BEC. It is shown that the damping associated with the elementary excitation and the collective excitation built in the GPE naturally explains the crossover behavior mentioned above. Our simulating results are shown to be in good agreement with the experiment. Moreover, it is also shown that the oscillating BEC tends to localize when time is long enough, to which its density profile is intimately related to the phenomenon of Anderson localization.

[1] S. N. Bose, Plancks Law and Light Quantum Hypothesis, Z. phys. 26, 178 (1924)
[2] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Weiman, and E. A. Cornell Science 269, 198 (1995)
[3] C. J. Pethick, H. Smith, Bose-Einstein Condensation in Dilute Gases, Cambridge University Press (2002)
[4] L. Pitaevskii, and S. Stringari, Bose-Einstein Condensation, Oxford Science Publications (2003).
[5] Stephen T. Thornton, and Jerry B. Marion, Classical Dynamics of Particles and System, Book/Cole-Thomson Learning (2004).
[6] J. E. Ley, L. Fallani, M. Modugno, D. S. Wiersma, C. Fort, and M. Inguscio, Bose-Einstein Condensation in a Random Potential, Phys. Rev. Lett. 95, 070401 (2005).
[7] Y. P. Chen et al., Phase coherence and superfluid-insulator in a disordered Bose-Einstein condensate, Phys. Rev. A 77, 033632 (2008).
[8] M. von Hase, Bose-Einstein Condensation in Weak and Strong Disorder Potentials, Bachelor Thesis, Freie Universität Berlin (2010), Section 2.2.
[9] Z. Wu, and E. Zaremba, Dissipative Dynamics of a Harmoniclly Confined Bose-Einstein Condensation , Phys. Rev. Lett. 106, 165301 (2011).
[10] John F. Dobson, Harmonic-Potential Theorem: Implications for Approximate Many-Body Theories , Phys. Rev. Lett. 73, 2244 (1994)
[11] E. Zaremba, Sound propagation in a cylindrical Bose-condensed gas, Phys. Rev. A 57, 158 (1998).
[12] A. Brunello et al., Momentum transferred to a trapped Bose-Einstein condensate by stimulated light scattering, Phys. Rev. A 64, 063614 (2001)
[13] D. Dries et al., Dissipative transport of a Bose-Einstein condensate, Phys. Rev. A 82, 033603 (2010)
[14] Yuan Quing-Xin, and Ding Guo-Hui, Computing Ground State Solution of Bose-Einstein Condensates Trapped in One-Dimensional Harmonic Potential, Commun. Theor. Phys. 46 (2006) pp. 873-878
[15] P. W. Anderson, Absence of Diffusion in Certain Random Lattices, Phys. Rev. 109, 1492(1958)
[16] L. Sanchez-Palencia et al., Anderson Localization of Expanding Bose-Einstein Condensates in Random Potentials, Phys. Rev. Lett. 98, 210410 (2007)
[17] T. –L. Horng et al., Stationary wave patterns generated by impurity moving with supersonic velocity through a Bose-Einstein condensate, Phys. Rev. A 79, 053619 (2009)