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研究生：	高至遠 Gao, Jhih-Yuan
論文名稱：	以耦合諧振子模型初步探討巨觀量子現象之糾纏動力學 A Study of Coupled Harmonic Oscillator Models toward Quantum Entanglement Dynamics in Macroscopic Quantum Phenomena
指導教授：	周忠憲 Chou, Chung-Hsien
學位類別：	碩士 Master
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：	90
中文關鍵詞：	量子糾纏、糾纏分離、糾纏度量、可分性判別原則
外文關鍵詞：	quantum entanglement, disentanglement, entanglement measure, separability criterion
相關次數：	點閱：129 下載：3
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Peres-Horodecki-Simon判別原則，以及logarithmic negativity是用來判斷以及度量高斯態量子糾纏的有效方式。在本文中，我們準備了數個由彼此間有交互作用的諧振子構成的模型，藉由上述的判別方式以及度量，我們可以解析地計算出每一對諧振子間的糾纏狀態，而由此可發現一些有趣的現象，包含了有限時間內的糾纏分離。此外，我們比較了質心座標之間，以及其構成粒子之間的糾纏狀態，藉此理解其於複合系統中扮演的角色。我們希望這些可以解析計算的模型能夠幫助我們理解彼此間有交互作用的系統，以及大型系統的糾纏現象。

Peres-Horodecki-Simon criterion and logarithmic negativity are very powerful tools to determine the separability and to measure the entanglement of Gaussian states. In this thesis, we set up several models, all of which comprise a number of coupled oscillators, and by facilitating the separability criterion and measure we're able to calculate the entanglement between each pair of oscillators at any time analytically, which reveals several interesting phenomena, including entanglement sudden death and revival of entanglement. Also, we compare the entanglement between center of mass coordinates and that of their member oscillators, and thereby understand the role of it in a composite system. Lastly, we'll make an attempt at appreciating the effects of particle numbers on entanglement. We hope these analytically solvable models can help us understand more about the entanglement of interacting systems and of large systems.

Introduction 9
1 Entanglement  . . . . . . . 9
2 Death and Revival of Entanglement  . . . . . 9
3 Macroscopic Quantum Phenomenon and the Center of Mass . . . . . . 9
4 Separability Criterion and Entanglement Measure. . . . . . . . 10
4.1 Peres-Horodecki-Simon Separability Criterion  . . . . . . 10
4.2 Negativity and Logarithmic Negativity  . . . . . . . . 12
5 Our Models and Methods . . . . . . . 12
Model 1: Two Oscillators 14
1 The Model . . . . . . . . . . . 14
2 Entanglement Dynamics. . . . . 14
3 Entanglement Measure and the Periodicity of States . . . . . . . . 16
4 Effect of Each Parameter  . . . . . . . . . 18
Model 2: Four Oscillators with a Symmetric Setup 21
1 The Model  . . . . . . . . . 21
2 Entanglement Dynamics  . . . . . . 22
3 Center of Mass Coordinates . . . . . . . 24
3.1 More Discussion on EX12X34 and E13  . . . . . . 25
4 Effect of Each Parameter . . . . . . . . . . 25
4.1 The Competition between alpha, beta and gamma 
 . . . . . . . . . . . . . 25
4.2 The Role of the Interaction Strength for General Coordinates . . . . . . . . . . 28
4.3 Mass and Frequency  . . . . . . . 31
5 Phase Diagram . . . . . . . . . . . 31
Model 3: A Two-to-One Three-Oscillator System 37
1 The Model . . . . . . . . . . . . 37
2 Entanglement Dynamics . . . . . . . . . 38
3 Center of Mass Coordinate . . . . . . . . . . 38
4 Effect of Each Parameter . . . . . . . . . 39
4.1 Entanglement between 1 and 2. . . . . . . . 40
4.2 Entanglement between 2 and 3 . . . . . . . . . . . 41
5 Phase Diagram . . . . . . . . . . 41
Model 4: A Three-to-One Four-Oscillator System 50
1 The Model . . . . . . . . . . . 50
2 Entanglement Dynamics . . . . . . . 51
2.1 Center of Mass Coordinate . . . . . . . . . 51
3 Effect of Each Parameter . . . . . . . . . . 52
Groups of Identical Particles 55
1 Hamiltonian. . . . . . . . . . 55
2 Wave Function . . . . . . . . . 57
3 Expectation Value . . . . . . . 58
3.1 Single Position Operator . . . . . . . . . 58
3.2 Single Momentum Operator . . . . . . . 58
3.3 Multiplication of Operators of the Same Group. . . . . . . . 58
3.4 Multiplication of Operators from Different Groups  . . . . . . . 59
4 Covariance Matrix . . . . . . . . 59
Model 5: Entanglement between Identical Particles 60
1 Entanglement Dynamics . . . . . . 61
2 In
uence of Each Parameter . . . . . 63
2.1 Ground State as the Initial State: beta, omega, and m  . . . . . . . . . . 63
2.2 Varied Initial States: a and b  . . . . . . . . 63
2.3 Particle Number . . . . . 63
2.4 Fixed Initial States: Under Different Hamiltonians . . . . . . . . 66
Discussion and Summary 70
1 Parameters . . . . . . . . . . 70
2 Entanglement Strength and Lasting Disentanglement. . . . . . . . . . . 70
3 Comparison with a Previous Work . . . . . . . . . 71
4 Center of Mass Coordinate . . . . . . . 71
5 Identical Particles . . . . . . . . 72
6 Conclusion and Prospect . . . . . . . . . 73
Appendices 76
Appendix A State Evolution of Coupled Oscillators 76
Appendix B Coordinate Transformation in Quantum Mechanics: Wave Function and
Basis 78
Appendix C Coordinate Transformation in Quantum Mechanics: Hamiltonian and
Operators 80
Appendix D Wave Function of Two Coupled Oscillators 82
Appendix E Periodicity 84
E.1 Analytical Method  . . . . . . . . 84
E.1.1 Rational Eigenvalues . . . . . . 85
E.1.2 Rational Ratio . . . . . . . . 85
E.2 Numerical Method. . . . . . . . 85
E.2.1 By the Eigenvalues . . . . 85
E.2.2 Inner Product  . . . . . . . 86
Appendix F Some Properties of Identical Particles 88
Appendix G The Equation of Motion of the CM Coordinate 89
                                    

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校內：2015-07-18公開
校外：2015-07-18公開

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