從理想脈衝序到高溫超導體的弱馬約拉納態看量子資訊的保護

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研究生：	高潤婷 Kao, Jun-Ting
論文名稱：	從理想脈衝序到高溫超導體的弱馬約拉納態看量子資訊的保護 Two aspects of protecting quantum information:from the optimal pulse sequence to weak Majorana modes in high-Tc superconductors
指導教授：	牟中瑜 Mou, Chung-Yu
口試委員:	林秀豪 Hsiu-Hau Lin 陳柏中 Po-Chung Chen 崔章琪 Chang-Chyi Tsuei 仲崇厚 Chung-Hou Chung 張明哲 Ming-Che Chang
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：	2014
畢業學年度：	102
語文別：	英文
論文頁數：	97
中文關鍵詞：	量子資訊、馬約拉納費米子、高溫超導
外文關鍵詞：	quantum information, Majorana fermion, high-Tc superconductor
相關次數：	點閱：1 下載：0
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此論文包含了兩個部分：研究量子位元的保存。第一部分，處理以自旋為主的量子位元之退相干問題。特別的是，在動力學的架構下，我們發展了一套高度有效率的方式對抗外界的擾動。這樣的方式，主要施針對由於海神堡類型的自旋交互作用引起的退相干問題。第二部分，我們研究了相對來說比較頑強的馬約拉納態。由於實驗上的啓發，我們研究存在拉什巴自旋軌道交互作用下的高溫超導相圖。此外，我們發現一個由於來自自旋三重態庫柏對的d向量和自旋軌道交乎作用的g向量不平行引起的拓普相變。這樣一個現象在過去的研究裡頭都被忽略了。雖然這樣一個相變不在平均場論的範疇裡頭，但是在不考慮任何非有序，兩個粒子交互作用和外場的情況下，這樣一個最小的模型揭示了拓普相變不需要一個完整能隙的塌縮或著重建的過程。這和一般拓普相變的理論有所抵觸。此外，我們也研究了傳輸性質，包含了金屬和超導態的晶體結構，和約瑟芬晶體。特別的是，我們指出和晶體動量相關的臨界晶體長度對於在高溫超導體當中偵測馬約拉納費米子扮演了一個重要的角色。

The thesis consists of two parts for investigating protection of quantum bits(qubits). For this purpose, the first part deals with the ubiquitous decoherence problem for the spin-based qubits. Spin as a building block of qubit(s) in open systems is explored. In particular, we develop a highly efficient method within the dynamical decoupling scenario against the disturbance from the outside world. This method aims at solving the general decoherence problem arising from the Heisenberg-like spin-spin interaction. In the second part, we investigate the Majorana modes which are known to be robust. Motivated by experimental indications, we study the phase diagram of high-Tc superconductors in the presence of the Rashba spin-orbit interaction. Besides, we find a topological phase transition driven by the fact: the dk vector from the spin-triplet pairing is nonparallel to the gk vector in the Rashba spin-orbit interaction. This effect was overlooked in the previous studies. Though the tran- sition point is beyond the mean-field parameter space, this single-particle Hamiltonian without any disorders, two-particle interactions and even external field is the minimal model which reveals that topological phase transition does not need a full gap closing and reopening process. This is contradictory to the generic concept of topological phase transition. We also study the transport properties in N-S junctions and Josephson junctions. Particularly, we point out that the momentum dependent critical length lc(ky) plays a crucial role for the detection of Majorana fermions in high-Tc cuprate superconductors.

Contents
Abstract.......................................... i Acknowledgements .................................... ii
Introduction 1
1 Overview....................................... 1
Qubit(s) Dynamics 3
1 Introduction .................................... 3
2 TheoreticalFormulationandGeneralConsideration . . . . . . . . . . . . . . . 6
3 Short Time and Long time Dynamics of Single Qubit Decoherence . . . . . . . 11
3.1 Ising-likecoupling.............................. 11 2.3.2 Short-time behavior of Heisenberg-like coupling . . . . . . . . . . . . . 13 2.3.3 LongtimedynamicsofHeisenberg-likecoupling . . . . . . . . . . . . . 18
4 ShortTimeandlongtimeDynamicsofTwoQubits . . . . . . . . . . . . . . . 19
5 Summary ...................................... 23
Phase diagram of high-Tc cuprate in the presence of the Rashba spin-orbit interaction 24
1 Introduction .................................... 24
2 Canonicaltransformation.............................. 26
3 Phasediagram ................................... 28
3.1 Gutzwiller coefficients:Renormalized mean-field theory . . . . . . . . . 29
3.2 Phase diagram with renormalized mean-field theory . . . . . . . . . . 30
4 Summary ...................................... 38
Edge states and Topological phase 39
1 Introduction .................................... 39
iii
1.1 Majorana fermion: Non-local and Non-Abelian Statistics . . . . . . . . 40
2 Excitation of ribbon and bulk-boundary correspondence . . . . . . . . . . . . 44 4.2.1 (110)orientation .............................. 47 4.2.2 (120)orientation .............................. 54
3 Semi-infiniteanalysis................................ 56
4 Topologicalinvariants................................ 59 4.4.1 Flatbandwindingnumber......................... 59 4.4.2 Z2invariant................................. 64
5 Summary ...................................... 68
Transport Property 70
1 Introduction .................................... 70
2 PlanarJunction:AndreevBoundState...................... 71
3 Josephson Junction: Anomalous current phase relation . . . . . . . . . . . . . 76
3.1 JosephsonJunctioninthephaseη>ηc .................. 79
3.2 JosephsonJunctioninthephaseη<ηc .................. 85
4 Summary ...................................... 85
Appendix A DIII class and the flat band winding number NBDI 87 A.1 Particle-holesymmetry............................... 87 A.2 Time-reversalsymmetry .............................. 88 A.3 Chiralsymmetry .................................. 88 A.4 Z2invariant..................................... 89
A.4.1 Z2inDIIIclass............................... 90
                                

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