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研究生： 黃超羣
Chao-Chun Huang
論文名稱： 在一個和二個組成的玻色-愛因斯坦凝結系統的研究主題
TOPICS ON ONE AND TWO-COMPONENT BOSE-EINSTEIN CONDENSED SYSTEMS
指導教授： 吳文欽
Wu, Wen-Chin
學位類別： 博士
Doctor
系所名稱： 物理學系
Department of Physics
論文出版年： 2010
畢業學年度： 98
論文頁數： 95
中文關鍵詞： 玻色愛因斯坦凝結 、光晶格 、旋渦
英文關鍵詞： BOSE-EINSTEIN CONDENSATION, BEC, DIPOLAR, VORTEX, OPTICAL LATTICE
論文種類： 學術論文
相關次數： 點閱：139  下載：8 
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    • 這是一篇探討在一個和二個混合玻色-愛因斯坦凝結系統中基態波函數和動力學性質在各種環境中的物理. 變分法和差分法是主要的數學方法. 一開始主要討探能量穩定和 GP 方程在二個混合系統中的物理. 我們利用modified Gaussian (MG) 去探討在混合系統中相分離的現象和動力學的性質. 在作用力不是很強的情況下,我們發現MG 是一個很好的變分函數. 另外我們也利用MG去研究有很強的極化系統,並考慮高次項,相對於first Born approximation (FBA)的影響.發現在系統接近不穩定時,高次項的影響變的更重要. 接著我們討論超晶格的光晶格, 發現有類似在固態系統中OPTICAL PHONON 和ACOUSTIC PHONON的行為發生.並給出了解析解當動量很小的時候. 最後我們考慮在旋轉玻色-愛因斯坦凝結系統四次項位能井和非對稱位態井旋渦的物理行為. 發現在有交互作用力下比在沒有交互作用力下有更豐富的旋渦晶格的產生. 當旋轉的很快時,這些旋渦晶格最後都消失了

    This thesis concerns the ground-state property and dynamics of one- and two- component trapped Bose-Einstein condensates (BEC) in a variety of states and regimes. Variational method and finite-difference numerical method are applied throughout this thesis. Starting from the conditions of energetic sta-bility, coupled time independent and dependent Gross-Pitaveskii (GP) equa-tions are re-derived for a two-component system. With the phenomenon of phase separation built in, we introduce a trial wavefunction, called "modified Gaussian (MG) function". MG function is shown to be more suitable for a
    two-component as well as one-component system, providing that the (nonlin-ear) interaction effect is not too strong. Using MG trial wavefunction, the equilibrium and dynamical properties of a two-component system are studied in details. With the MG trial wavefunction in hand, we then study a BEC system of strong dipolar interaction. Since dipolar interaction is long-range and can be tuned to be resonant, a more realistic treatment for scattering
    should go beyond the first Born approximation (FBA). It is shown that the effect going beyond FBA is significantly enhanced when the system is close the phase boundary of collapse. To simulate the environment of a real crystalline
    solid, we also consider a one-dimensional optical lattice with a basis, i.e., a superlattice. Analytical results of acoustic and optical phonons are reported.
    Measurements of these modes can give unambiguous evidence to see whether the system is in the superfluid or Mott insulting regimes. Finally, we consider the effect of anharmonic trap on vortex arrays of a one-dimensional rapid ro-tating BEC. It is shown that due to the anharmonic quartic trap, the system remains stable at high rotating velocity and vortex lattices form even in the absence of the repulsive s-wave interaction (g). When g is present, the in-terplay between g and the quartic trap potential can lead to rich vortex lattice transition states as a function of ­, to which vortex lattices vanish eventually at some higher ­.

    Table of Contents vi List of Figures vii 1 Introduction 1 2 Theory of one and two-component BEC systems 5 2.1 Introduction . . . . . . . . . . . . . 5 2.2 Coupled GP energy functionals . . . . . . .. . . . . . . . . . . . . 6 2.3 Energetic stability and ground-state wavefunction . . . . . . . . . . . 8 2.4 Properties of two-component BEC system . . . . . . 11 2.4.1 Motivation . . . . . . . . . . . . . . . . . . . 11 2.4.2 Remarks on Gaussian function . . . . . . . . . 11 2.4.3 Modi¯ed Gaussian function . . . . . . . . . . . . . . . . . . . 12 2.4.4 Equilibrium and dynamical properties . . . . . . 16 2.5 Summary . . . . . . . . . . . . . . . . . . . . . . 23 3 BEC in optical superlattice 25 3.1 Introduction . . . . . . . . . . . . . . . . . . . . 25 3.1.1 Optical lattice . . . . . . . . . . . . . . 26 3.1.2 Relevant energy scales . . . . . 27 3.2 Tight-binding model . . . . . . . . . . . . . 28 3.3 Oscillations in a 1D optical superlattice . . . . . . . . . . . . . . . . . 30 3.3.1 Motivation . . 30 3.3.2 An optical superlattice . . . . . . . . . . 31 3.3.3 Phonon modes in optical superlattices . . . . . . . . . . . . . 32 3.4 Summary . . . . . . . . . . . . . . . . . . . . . 39 4 BEC with dipolar interaction 41 4.1 Introduction . . . . . . . . . . . . . . . . . . 41 4.1.1 Pseudopotential of dipole interaction . . . . . . . . . . . . . . 42 4.2 Generalized scattering amplitude . . . . . . . . . . . . . . . . . . . . 44 4.3 Beyond the first Born approximation . . . . . . . . . 46 4.3.1 Motivation . . . . . . . . . . . . . 46 4.3.2 E®ective Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . 47 4.3.3 Ground-state properties . . . . . . . . . . . . . . . . . . . . . 50 4.3.4 Elementary excitations . . . . . . . . . . . . . . . . . . . . . . 53 4.4 Summary . . . . . . . . . . . . . . . . . . . . . 57 5 Vortex state of fast rotating BEC 59 5.1 Introduction . . . . .. . . . . . . . . . . . . . . . . 59 5.1.1 Quantized Vortices . . . . . . . . . . . . . . . . . . . . . . . . 60 5.1.2 Slow and fast rotation . . . . . . . . . . . . . . . . . . . . . . 60 5.1.3 Lowest Landau levels (LLL) . . . . . . . . . . . . . . . . . . . 61 5.2 More topics of rapidly rotating system . . . . . . . . . . . . . . . . . 63 5.3 Vortices under extreme elongation in a harmonic plus quartic trap . . 64 5.3.1 Motivation . . . . . . . . . . . . . . . . . . . . 64 5.3.2 Energy functionals . . . . . . . . . . . . . . 65 5.3.3 Noninteracting system . . . . . . . . . . . . . . . . . . . . . . 68 5.3.4 Effect of interaction . . . . . . . . . 72 5.4 Summary . . . . . . . . . . . . . . . . . . . . 74 A Conditions of energetic stability 77 B Variational method 81 Bibliography 85

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