在本論文中，我們藉由冷原子的四波混頻效應，實現了頻率域中的 Hong-Ou-Mandel干涉實驗。利用四波混頻機制，我們可將進入原子團的780奈米(或795奈米)波長入射光轉換成780奈米及795波長奈米出射光；而藉由改變四波混頻機制中的單光子失諧(one-photon detuning)，我們可以進一步控制入射光被轉換成此二種不同頻率的轉換效率。因此，本論文中所使用的四波混頻機制，可以被視為頻率域中的可調變分光器。
傳統上，Hong-Ou-Mandel干涉實驗使用兩道相同頻率的入射光。而為了進行頻率域中的Hong-Ou-Mandel干涉實驗，我們在入射光被轉換成780奈米和795奈米波長出射光的轉換效率相同時，同時送入780奈米和795奈米波長的入射光。此時兩道入射光的量子態，皆為多光子或單光子等級的coherent state。藉由分析second order correlation function g (2), 我們觀察到兩道不同頻率的出射光之間，的確藉由Hong-Ou-Mandel干涉建立了相干性。
我們藉由四波混頻機制，量測到了頻率域的Hong-Ou-Mandel干涉現象，也代表著四波混頻頻率轉換的過程，具備了高相干性─入射與出射光之間存在穩定相位關係，過程中損失的能量也甚微。因此，藉由測量頻率域的Hong-Ou-Mandel干涉現象，我們可以驗證四波混頻轉換過程，對入射訊號的高保真度。

We perform the frequency domain Hong-Ou-Mandel interference based on the four-wave mixing process in a cold rubidium system. While the four-wave mixing process converts 780 nm (or 795 nm) input light into 795 nm and 780 nm output lights, adjusting the one-photon detuning in the process allows us to change the conversion efficiencies, thus making the four-wave mixing process an adjustable frequency domain beam splitter. When the conversion efficiencies from 780 nm (or 795 nm) input light into 780 nm and 795 nm output lights are equal, we send lights with the two different frequencies into the four-wave mixing process simultaneously and conduct the frequency domain Hong-Ou-Mandel interference. Examining the second order correlation function g (2), we show that the Hong-Ou-Mandel interference does build a correlation between the two output lights with different frequencies when the input lights are either many-photon level or single-photon level coherent states. The frequency domain Hong-Ou-Mandel interference measured indicates the four-wave mixing being a coherent process with a constant input-output phase relation and low loss, suggesting a high fidelity in the frequency conversion process.
In this thesis, we illustrate the motivation in Chapter 1. The idea and theory of the normal and frequency domain Hong-Ou-Mandel interferences are shown in Chapter 2. In Chapter 3, the principle of four-wave mixing and EIT is shown. In Chapter 4, we demonstrate the principle of using four-wave mixing process to perform the frequency domain Hong-Ou-Mandel interference. In Chapter 5, the experimental setup of the four-wave mixing process is presented. In Chapter 6, we show the experimental results of frequency conversion via four-wave mixing. In Chapter 7, the optimization of the Hong-Ou-Mandel experiment along with the theoretical calculation of g (2) is demonstrated. The experimental results of the frequency domain Hong-Ou-Mandel interference are presented in Chapter 8. Finally, the conclusion and outlook are drawn in Chapter 9.

Abstract i
摘要 ii
誌謝 iii
Contents iv
Chapter 1 Motivation 1
Chapter 2 Background 2
2.1 Second Order Correlation Function g(2) 2
2.2 Review on Hong-Ou-Mandel Interference 4
2.2.1 General Description of Scattering Matrices 4
2.2.2 Hong-Ou-Mandel Interference with Single Photon Input 7
2.2.3 Hong-Ou-Mandel Interference with Phase Correlated Coherent State Inputs 11
2.2.4 Hong-Ou-Mandel Interference with Phase Un-correlated Coherent State 16
2.2.5 Hong-Ou-Mandel Interference in a General Scattering Matrix with Coherent State 18
2.3 Review on Frequency Domain Hong-Ou-Mandel Interference 20
Chapter 3 Principle of Frequency Conversion via Four-Wave Mixing 21
3.1 Principle of Electromagnetically Induced Transparency (EIT) 21
3.2 Principle of Slow Light 30
3.2.1 Group Velocity Calculation 30
3.2.2 Slow Light with Gaussian Pulse Input 33
3.3 Principle of Four-wave Mixing Process 35
Chapter 4 Principle of Hong-Ou-Mandel Interference via Four-Wave Mixing 41
4.1 Scattering Matrix of a Four-Wave Mixing Process 41
4.2 Property of the Scattering Matrix of Four-wave Mixing Process 45
4.2.1 Conversion Efficiency 45
4.2.2 Contrast phase 46
4.2.3 The Scattering Matrix of a Beam Splitter-like FWM Process 47
4.3 Four-Wave Mixing with Two Phase-Correlated Input Light 49
4.4 Four-Wave Mixing with Two Phase-Uncorrelated Input Light 51
4.4.1 Generation of the Randomized Relative Phase 51
4.4.2 g(2) without De-Coherent Rate and Two-Photon Detunings 52
4.4.3 g(2) with De-coherent Rate and Two-Photon Detunings 54
4.5 Discussion on Relation between g(2) and Coherence in the FWM Process 55
4.5.1 Normal Modes of the FWM Process 55
4.5.2 Ideal Process and Fidelity 56
4.5.3 g(2) in the Condition of Equal Conversion Efficiency 57
4.5.4 g(2) in a General Four-Wave Mixing Process 60
Chapter 5 Experimental Setup 62
5.1 Laser Frequencies and Injection Lock Scheme 62
5.2 Time Sequence in the Experiment 68
5.2.1 Normal MOT Stage 69
5.2.2 Dark MOT & Optical Pumping Stage 70
5.2.3 Four-Wave Mixing Stage 71
Chapter 6 Four-Wave Mixing Frequency Conversion Efficiency 73
6.1 Experimental Result of Slow Light 73
6.2 Conversion Efficiency to One-Photon Detuning with 780 nm Input 75
6.3 Conversion Efficiency to Two-Photon Detuning with 780 nm Input 76
6.4 Highest Conversion Efficiency with 780 nm Input 77
6.5 Beam Splitter-Like Four-Wave Mixing with 780 nm Input 79
6.6 Beam Splitter-Like Four-Wave Mixing with 795 nm Input 81
6.7 Corrlation between Output Lights of the BS-like FWM 82
Chapter 7 Optimization of Hong-Ou-Mandel Interference based on Four-Wave Mixing 84
7.1 Hong-Ou-Mandel Interference with a Normal Beam Splitter 84
7.2 Observation on Hong-Ou-Mandel Interference based on FWM 86
7.2.1 Two-Photon Detuning Difference = 0 kHz 87
7.2.2 Two-Photon Detuning Difference ~ 400 kHz 88
7.3 Experimental Study on g(2) versus Two-Photon Detuning Difference 89
7.4 Experimental Study on g(2) versus Time Window Widths 90
7.5 Experimental Study on g(2) versus Input Pulse Widths 91
7.6 Theoritical Calaulation of g(2) with Different Affecting Factors 92
7.6.1 Background Noise 93
7.6.2 Split Ratio Difference 94
7.6.3 Input Power Difference 95
7.6.4 SPCM Collection Efficiency Drop 96
7.6.5 Phase Modulation Speed and the Time Resolution Limit 98
Chapter 8 Hong-Ou-Mandel Interference via Four-Wave Mixing 99
8.1 g(2) Value Dip to 795 nm Pulse Input Delay Time 99
8.2 g(2) Value with Single-Photon Level Coherent State Input 102
Chapter 9 Discussion and Outlook 103
Appendices... 104
Appendix A. Derivation of g (2) Values in the HMT Experiment 105
Appendix B. Four-Wave Mixing Process with Two Input Fields 107
Appendix C. Derivation of Approximated Formula in Section 7.6 111
Appendix D. Photomultiplier Modules (PMTs) Calibration 114
Appendix E. SPCM Efficiency to Different Frequencies 115
Appendix F. Calibration of Efficiencies of Two SPCMs 118
References… 119

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