這篇論文分為兩個主題。第一個主題是「四波混頻的雙光子備制」，是個完全理論的研究，從本書之第一章開始討論至到第五章；第二個主題是「室溫原子的雷德堡-電磁波引發透明光譜」，是搭配著理論分析與模擬的實驗研究，由本書之第六章開始討論至到第十一章。
雙光子備制的研究由第一章的簡介與動機開始，此章將論述雙光子備制對量子資訊科學的重要性；第二章將藉由解光學布拉赫方程式(optical Bloch equation)與馬克斯威爾-薛丁格方程式(Maxwell-Schrödinger equation)來呈現雙光子個別的特性；我們會在第三章計算格勞貝爾相關性函數(Glauber correlation function)和雙光子產生的速率，這些物理量不但是實驗上可以測量到的，它們還把產生雙光子的物理機制完完全全展露無遺；在第四章中，我們把格勞貝爾相關性函數中的各部分分成幾種形式做數值模擬，以探討各部分對格勞貝爾相關性函數的影響，我們所計算出的勞貝爾相關性函數也將與杜勝望教授團隊的結果做比較；第五章總結雙光子備制的研究。
第六章開始是第二個主題，我們將從研究動機開始講起，比較Λ型EIT和階梯型EIT在冷原子系統與室溫原子系統的差異性；第七章將論述耦合光的躍遷頻率與拉比頻率；第八章是實驗架設；第九章是室溫銣原子雷德堡電磁波引發透明光譜的重點，都普勒效應、光學幫浦效應與暫態效應的理論將與實驗互相驗證，不同參數的雷德堡電磁波引發透明光譜的實驗結果也將呈現；第十章將利用室溫銣原子雷德堡電磁波引發透明光譜做出頻率調變光譜，不同的調頻範圍與鎖相放大器不同的時間常數對光譜的影響也在此章中探討；第十一章總結室溫銣原子雷德堡電磁波引發透明光譜的研究。

There are two topics in my thesis. The first one is “Biphoton Generation with Four-Wave Mixing Process”, which I will discuss from Chapter 1 to Chapter 5, and the second one is “Rydberg-EIT Spectra with Room-Temperature Atoms”, which I will discuss from Chapter 6 to Chapter 11. The first topic is a totally theoretical study, while the second topic is an experiment-based study with theoretical explanations.
In Chapter 1, I will introduce the motivation of this biphoton generation, which is important to quantum information science. Then I will present the properties of Stokes photon and anti-Stokes photon by solving the optical Bloch equation and Maxwell-Schrödinger equation in Chapter 2. In Chapter 3, I will calculate the Glauber correlation function and the generation rate, which are the quantities that can be measured, and they demonstrate all the physics behind biphoton generation process. In Chapter 4, I numerically simulate the Glauber correlation function which has been separated into different parts in the integral, in order to see the corresponding effects. The complete Glauber correlation function is also compared with the one in Prof. Shengwang Du’s Group. Finally, Chapter 5 concludes the discussions of biphoton generation.
The second topic starts from the motivation of the Rydberg EIT spectra in Chapter 6. In this chapter, both the Λ-type and cascade-type EIT system are discussed. The formulae of both types are modified from the cold-atom system to the hot-atom system as well. In Chapter 7, the transition frequency and the Rabi frequency of the coupling beam are estimated. In Chapter 8, the experimental setup is displayed. The core of this study is Chapter 9, in which the theories of Doppler Effect, optical pumping effect, and transient effect are verified by the experimental results. I also present the EIT experimental results with different parameters in this chapter. In Chapter 10, the frequency-modulated Rydberg EIT spectra with different sinusoidal modulation spans and different time constants of the lock-in amplifier are demonstrated. Last but not least, Chapter 11 summarizes the study of Rydberg EIT with room-temperature atoms.

Contents

Abstract i
Acknowledgements ii

Topic I.
Theoretical Study of the Biphoton Generation with the EIT-based Four-Wave Mixing Process
Introduction and Motivation 2
Four-Wave Mixing 4
Optical Bloch Equation of a Four-Level System 4
Self-susceptibility 9
Cross-susceptibility 11
Effect of Ω_p ρ_24 on Susceptibilities 14
Effect of Ω_p ρ_34 and Ω_c ρ_43 on Susceptibilities 17
Two-photon Glauber Correlation Function 19
Derivation of Biphoton Wave Packet and Two-photon Glauber Correlation Function 19
Raman Transition Regime 32
EIT Regime 35
Slow-light Approximation Employed on k_as Being Inadequate 39
Biphoton Generation Rate (Coincidence Rate) 42
Numerical Simulation—Contributions from different terms 46
Comparison with JOSA(B) results 47
|√(ω ̅_s ω ̅_as )/2c∙L/2π ∫_(-∞)^∞▒〖χ_(s,as) (ω) e^(-iωτ) dω〗|^2 48
|∫_(-∞)^∞▒〖sinc(ωL/(2V_g )) e^((i ωL/(2V_g )) ) e^(-iωτ) dω〗|^2 50
|∫_(-∞)^∞▒〖sinc(i αL/2) e^((- αL/2) ) e^(-iωτ) dω〗|^2 52
|∫_(-∞)^∞▒〖sinc(ωL/(2V_g )) e^((i ωL/(2V_g ) - αL/2) ) e^(-iωτ) dω〗|^2 56
|∫_(-∞)^∞▒〖sinc(ωL/(2V_g )+i αL/2) e^((i ωL/(2V_g ) - αL/2) ) e^(-iωτ) dω〗|^2 60
Conclusion 62

Topic II. Rydberg-EIT Spectra with Room-Temperature Atoms
Introduction and Motivation 64
Maxwell- Schrödinger and Optical Bloch Equations of a -Type Three-Level System for Cold Atoms 65
Maxwell- Schrödinger and Optical Bloch Equations of a Cascade-Type Three-Level System for Cold Atoms 67
Maxwell- Schrödinger and Optical Bloch Equations of -Type and Cascade-Type Systems for Hot Atoms 69
Theoretical Predictions of Transition Frequency and Rabi Frequency of the Coupling Field 74
Transition Frequency 74
Clebsch-Gordan Coefficients 75
Effective Rabi Frequency 88
Rabi Frequencies in Different Bases 91
Experimental Setup 96
Blue Laser System 96
Optical System 97
Different Effects Discovered from Experimental Results 99
Doppler Effect 99
Optical Pumping Effect (Open System versus Closed System) 102
Transient Effect 106
Rydberg S and D States 113
Temperature of the Rb-87 vapor cell 114
Counter-propagation versus Co-propagation 115
Probe Rabi Frequency 116
Coupling Rabi Frequency 118
Frequency-Modulation Rydberg-EIT Spectra 121
Experimental Setup 121
Experimental Results 122
Conclusion 129
Appendix 131
Relation between susceptibility χ and Optical Coherence 131
Transmission for a pulse input in a Two-Level System 132
Absorption Coefficients of EIT and non-EIT Room-Temperature Atoms 135
Blue Laser System 140
List of Blue Light Optics and Components and Their Test Results 143
Bibliography 148

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[3] Jr-Chium Yu, “Theoretical Studies of the Electromagnetically-Induced-Transparency Spectrum in Hot Atoms and the Phase Mismatch of Four-Wave Mixing Processes in Cold Atoms”, master thesis in National Tsing Hua University (2014)
[4] M. Mack, F. Karlewski, H. Hattermann, S. Höckh, F. Jessen, D. Cano, and J. Fortágh, “Measurement of absolute transition frequencies of 87Rb to nS and nD Rydberg states by means of electromagnetically induced transparency”, Phys. Rev. A83, 052515 (2011)
[5] Ya-Wen Chuang, “Experimental Studies of Rydberg EIT spectra”, master thesis in National Tsing Hua University (2015)
[6] Huub Ruties, “Electromagnetically induced transparency of cold rubidium-87 atoms”, bachelor project thesis in University of Amsterdam (2012)
[7] J. Deiglmayr, M. Reetz-Lamour*, T. Amthor, S. Westermann, A.L. de Oliveira, M. Weidemüller, “Coherent excitation of Rydberg atoms in an ultracold gas”, Optics Communication 264 (2006) 293-298
[8] P. C. Guan and Ite. A. Yu*, “Simplification of the electromagnetically induced transparency system with degenerate Zeeman state”, Phy. Rev. A76, 033817 (2007)