冷原子中的合成規範場及自旋軌道耦合｜國立清華大學博碩士論文庫

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研究生：	鄭俊輝 Zheng,Jun-Hui
論文名稱：	冷原子中的合成規範場及自旋軌道耦合 Synthetic Gauge Field and Spin-orbit Coupling in Cold Atoms
指導教授：	王道維 Wang, Daw-Wei
口試委員:	郭西川 Gou, Shih-Chuan 米格爾 Cazalilla, Miguel 牟中瑜 Mou, Chung-Yu 陳柏中 Chen, Po-Chung 郭光宇 Guo, Guang-Yu
學位類別：	博士 Doctor
系所名稱：	理學院 - 物理學系 Department of Physics
論文出版年：	2016
畢業學年度：	104
語文別：	英文
論文頁數：	102
中文關鍵詞：	冷原子、合成規範場、自旋軌道耦合、拓撲絕緣體
外文關鍵詞：	Cold atoms, Synthetic gauge field, Spin-orbit coupling, Topological insulator
相關次數：	點閱：3 下載：0
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本論文，我將討論不同系統中自旋軌道耦合的效應：旋量玻色-愛因斯坦凝聚系統，雙層費米系統，以及二維拓撲絕緣體系統。在旋量玻色-愛因斯坦凝聚系統中，我將主要討論自旋軌道耦合和兩體相互作用的混合效應。運用絕熱近似，我系統的研究了在合成規範場中旋量玻色-愛因斯坦凝聚的基態，激發態以及相關的效應。在雙層費米系統中，我主要考慮自旋-層耦合及超流的效應。通過映射到有效模型，我證明了在零溫下，加入層之間帶有自旋翻轉的隧穿項，可以大大提高超導配對的臨界磁場。在二維拓撲絕緣體中，我主要研究Kane-Mele模型，并考慮在鋸齒形邊界上單顆磁性雜質的效應。我得到Kane-Mele模型譜以及波函數的解析解并討論了它的電輸運性質。進一步構造了一維有效模型來描述該系統。

In this thesis, I will discuss the effect of spin-orbit coupling in different systems: spinor Bose-Einstein Condensate (BEC) system, bilayer Fermionic system, and 2D topological insulator. In the spinor BEC system, I focus on the hybrid effect of spin-orbit coupling and two-body interaction. By using adiabatic approximation, I systematically investigate the ground state, elementary excitations and related effects of a BEC within a
synthetic vector potential. In the bilayer Fermionic system, I consider the effect of spin-layer
coupling and superfluidity. By mapping to an effective model, I demonstrate that at zero temperature
the critical value of the magnetic field for pairing can be significantly increased by including a spin-flip tunnelling between layers. In the 2D topological insulator, I focus on the Kane-Mele (KM) Hubbard model and consider the effect of a single spin-flip impurity at the Zigzag edge. I analytically obtain the spectra and wavefunction of the KM model and then discuss its electronic transport property. Furthermore, I develop a low energy effective 1D model to describe the system.

Introduction 1
1 Two-level toy model . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Artificial gauge field . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1 Adiabatic approximation . . . . . . . . . . . . . . . . . . . . . 7
2.2 Validity of adiabatic approximation . . . . . . . . . . . . . . . 7
2.3 Simple example . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3 Spin-orbit coupling vs. topological property . . . . . . . . . . . . . . 11
4 Structure of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Topological Condensate in an Interaction-induced Gauge Potential 14
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2 System Hamiltonian and its mean-field equation . . . . . . . . . . . . 17
2.1 Two-component GP equation . . . . . . . . . . . . . . . . . . 17
2.2 Interaction induced synthetic gauge field (Adiabatic approximation)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3 Results in uniform space . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.1 Ferromagnetism in strong interaction limit . . . . . . . . . . . 22
3.2 Excitations and energetic instability of superfluid current (weak
interaction limit) . . . . . . . . . . . . . . . . . . . . . . . . . 23
4 Results in a harmonic trap . . . . . . . . . . . . . . . . . . . . . . . . 27
4.1 Coreless vortex-antivortex pair in the 2D trap . . . . . . . . . 29
4.2 Coreless vortex ring in the 3D trap . . . . . . . . . . . . . . . 31
5 Experiment measurement . . . . . . . . . . . . . . . . . . . . . . . . 32
6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Superconductivity Enhanced by Spin-flip Tunnelling in The Presence
of A Magnetic Field 35
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2 System and methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.1 System Hamiltonian in original basis . . . . . . . . . . . . . . 38
2.2 Equivalent description in a rotated basis . . . . . . . . . . . . 39
2.3 Single-particle spectrum . . . . . . . . . . . . . . . . . . . . . 40
2.4 General framework in meanfield theory . . . . . . . . . . . . . 42
3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.1 Single layer limit . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.2 Zero Raman coupling limit . . . . . . . . . . . . . . . . . . . 46
3.3 Raman coupling with ϕ = 0 . . . . . . . . . . . . . . . . . . . 48
3.4 Raman coupling with ϕ = π/2 . . . . . . . . . . . . . . . . . . 50
4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
Electric Transport Property for Kane-Mele Model with A Spin-flip
Impurity 56
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2 Hamiltonian and boundary conditions . . . . . . . . . . . . . . . . . . 57
3 Spectra for edge states . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4 Wavefunction for the Semi-infinite System . . . . . . . . . . . . . . . 67
4.1 Wavefunction for edge States . . . . . . . . . . . . . . . . . . 68
4.2 Wavefunction for Bulk States . . . . . . . . . . . . . . . . . . 69
5 Green function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
6 Transport properties with a single impurity . . . . . . . . . . . . . . . 74
7 Basis for constructing an effective model . . . . . . . . . . . . . . . . 77
8 Effective one dimensional model . . . . . . . . . . . . . . . . . . . . . 79
9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
Conclusions 85
Appendix-A 86
Appendix-B 93
                                

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