在這篇論文中，我們將會利用含有共振腔結構的非線性晶體以及其自發性參量降頻轉換(Spontaneous parametric down-conversion, SPDC)過程產生出能量-時間糾纏的窄頻雙光子，並且利用後選擇的方法以及Hong-Ou-Mandel干涉對頻率簡併的優化，我們能製備出偏振形式上的糾纏光子，即為偏振糾纏光子，接著我們會利用CHSH不等式、量子態斷層掃描、併發度(concurrence)的計算與測量來證實我們實驗上所製造的雙光子確實為偏振糾纏光子，最後我們還提供了一個純化糾纏光子的技術，我們利用了調製波包的方法，使頻率不為簡併態的糾纏光子提升了其糾纏上的性質，並且一樣利用CHSH不等式以及併發度的計算來驗證其純化效果。

In this thesis, we generate energy-time entangled narrow-band photon pairs by using nonlinear crystal with monolithic cavity and spontaneous parametric down-conversion processes. We generate the polarization entangled photons with post selection and optimize the polarization entanglement with Hong-Ou-Mandel Interferometer.
Finally, we confirm the polarization entanglement with the CHSH inequality, quantum state tomography and concurrence. We also develop a technique for purifying the entangled photons by modulating the photons.

摘要 i
致謝 iii
目錄 iv
圖表目錄 vi
第一章 實驗目的 1
第二章 理論介紹 2
2-1 雙光子製備機制 2
2-1-1 自發性參量降頻轉換(Spontaneous parametric down-conversion, SPDC) 2
2-1-2二階強度關聯函數G2τ(Second-order Intensity Correlation Function) 3
2-2 Hong-Ou-Mandel干涉 6
2-3量子力學試驗 8
2-3-1 EPR悖論(EPR Paradox) 8
2-3-2 CHSH型貝爾不等式(CHSC inequality) 12
2-3-3 CHSH不等式最大破壞(Maximum S-Parameter) 17
2-4 量子態斷層掃描(Quantum state Tomography) 19
2-4-1 布洛赫球(Bloch Sphere) 19
2-4-2 單一量子位元斷層掃描(Single Qubit Tomography) 23
2-4-3 多體量子位元斷層掃描(Multi-Qubit Tomography) 24
2-5 糾纏度量(Measurement of Entanglement) 26
2-6 糾纏光子之純化(Purification of Entanglement) 27
第三章 系統源架設 31
3-1 雷射系統 31
3-2 鎖頻系統 32
3-3 窄頻光子晶體設計 33
第四章 實驗架設及量測 36
4-1 雙光子測量與調控 36
4-1-1 雙光子測量 36
4-1-2 調控單光子 38
4-2 Hong-Ou-Mandel干涉 40
4-2-1 頻率簡併雙光子製備 40
4-2-2 同路徑HOM干涉 43
4-3 CHSH不等式測量 46
4-3-1 偏振糾纏光子製備 46
4-3-2 糾纏光子關聯性測量 47
4-3-3 CHSH不等式之破壞 49
4-4 量子態斷層掃描(Quantum State Tomography) 51
4-5 純化糾纏光子 53
4-5-1 EOM單邊調製之純化 53
4-5-2 EOM 雙邊調製之純化 56
第五章 總結 57
附錄 58
附錄A 非線性光學 58
A1 非線性光學(Nonlinear Optics) 58
A2 非線性波動方程式(Nonlinear Wave Equation) 59
A3 二階非線性效應(Second-order Nonlinear Effect) 61
A4 相位匹配與準相位匹配(Phase Matching Condition) 63
A5耦合波動方程式(Coupled Wave Equation) 66
附錄B S值與雙光子干涉對比度關係 69
參考文獻 72

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