雙光子是一對具有時間關聯性的糾纏態光子對。實驗上藉由熱原子系統的自發四波混頻機制產生雙光子對。雙光子對的其中一個光子作為觸發光子並開啟一個時間窗口收另一個因慢光效應而延遲抵達偵測器probe光子。探測(probe)光子即是我們可以拿來進行量子操作的預報型單光子。然而除了產生預報型單光子之外，檢驗系統產生的單光子純度也是一件相當重要的工作。量測條件自相關函數可以讓我們知道系統的單光子純度。本論文將著重於從理論的觀點出發，了解雙光子的訊號背景比、光譜亮度和條件自相關函數之間的關係。
在這篇文章中我們進一步探討幫浦光功率對雙光子偵測率的關係，給出了偵測率飽和效應的公式，並且探討訊號背景比(SBR)跟光譜亮度(spectral brightness)的反比現象，過去的實驗結果也顯示訊號背景比高，光譜亮度反而下降。最後探討條件自相關函數和雙光子SBR的關係，並且推導得到條件自相關函數和SBR之間的關係式，並且得到兩個重要的參數 g_(p,b|t=1)^((2) )和 g_(b,b|t=1)^((2) )。這個公式除了顯示SBR越大，條件自相關函數值越小，預報型單光子越純外，還告訴我們在不量測條件自相關函數下，也可以直接從兩分鐘的雙光子實驗中的SBR知道單光子的純度。

Biphotons are a pair of entangled photons with time correlation. In the experiment, we generate the biphoton pairs from room temperature Rb vapor cell by spontaneous four-wave mixing process (SFWM). One of the photons from the biphoton pair can be a trigger photon and opens a time window, and we can receive the other photon(heralded photon or probe) in this time window. The probe photon is received later than the trigger photon due to the slow light effect. However, in addition to generating heralded single photons, examining the purity of single photons is also an important work. Measuring the conditional auto-correlation function is a way to understand the purity of our heralded single photons. In this thesis, we will focus on understanding the relationship between SBR, spectral brightness, and conditional auto-correlation function from the theoretical point of view.
In this thesis, we further discuss the relation between pump power and detection rate and derive a saturation effect formula, then we illustrate the signal-to-background ratio(SBR) is inverse proportion to the spectral brightness(SB), the past experimental results also show that the SBR increases, but the SB decreases instead. Then discuss the relation and derive the formula between conditional autocorrelation function g_c^((2) ) and SBR, and get two important parameters g_(p,b|t=1)^((2) ) and g_(b,b|t=1)^((2) ). The formula of g_c^((2) ) shows that when the SBR increases, the g_c^((2) ) decreases. In addition to decreasing the value of 〖 g〗_c^((2) ) can increase the purity of heralded single photon, the 〖 g〗_c^((2) ) formula also tells us that we can know the 〖 g〗_c^((2) ) minimum just by measuring the 2 minutes biphoton data and without time-consuming conditional auto-correlation measurement.

致謝......................................................i
摘要.....................................................ii
Abstract................................................iii
目錄......................................................iv
第一章 熱原子系統雙光子實驗.................................1
1-1　熱原子產生預報型單光子介紹與應用........................1
1-2　熱原子系統雙光子的實驗跟物理機制簡介....................2
1-3　條件自相關函數CACF介紹.................................6
1-4　條件自相關函數CACF學術文獻.............................7
第二章　雙光子偵測率的飽和效應..............................12
2-1　雙光子飽和效應成因與推導...............................12
2-1.1 飽和效應成因.......................................12
2-1.2 飽和效應公式推導....................................13
2-2 偵測率對幫浦光強度的實驗數據和理論預測的比較...........15
第三章　訊號背景比與 spectral brightness的關聯性............17
3-1　去相干率隨耦合光強度的修正.............................17
3-2　訊號背景比與 spectral brightness 的關係探討............20
第四章　條件自相關函數的物理模型與過去實驗結果的探討...........22
4-1　二階自相關函數與光子聚束及反聚束簡介.....................22
4-2 雙光子條件自相關函數的推導與物理探討...................24
4-2.1條件自相關函數的介紹跟探討............................. 24
4-1.2條件自相關函數跟SBR的關係推導...........................26
4-3 條件自相關函數(CACF)實驗結果的探討.....................29
4-4 訊號(probe)對背景光子及背景對背景的條件相關函數的探討.....31
4-5 條件自相關函數和訊號背景比及spectral brightness的關聯性....36
第五章　總結.................................................37
附錄A 相干光的飽和效應計算.................................38
附錄B 真值表..............................................40
參考資料.....................................................43

[1] 林義閔, ‘‘熱原子產生的預報型單光子之自相關函數研究” , 國立清華大學, 碩士論文，民國111年
[2] C.-Y. Hsu, Y.-S. Wang, J.-M. Chen, F.-C. Huang, Y.-T. Ke, E. K. Huang, W. Hung, K.-L. Chao, S.-S. Hsiao, Y.-H. Chen, C.-S. Chuu, Y.-C. Chen, Y.-F. Chen, and I. A. Yu, “Generation of sub-MHz and spectrally-bright biphotons from hot atomic vapors with a phase mismatch-free scheme,” Opt. Express 29, 4632 (2021).
[3] S.-S. Hsiao,1, W.-K. Huang, Y.-M. Lin, J.-M. Chen, C.-Y. Hsu, and I. A. Yu, “Temporal profile of biphotons generated from a hot atomic vapor and spectrum of electromagnetically induced transparency,” Phys. Rev. A 106, 023709 (2022).
[4] J.-M. Chen, C.-Y. Hsu, W.-K. Huang, S.-S. Hsiao, F.-C. Huang, Y.-H. Chen, C.-S. Chuu, Y.-C. Chen, Y.-F. Chen, and I. A. Yu, “Room-temperature biphoton source with a spectral brightness near the ultimate limit,” Phys. Rev. Research 4, 023132 (2022).
[5] Chi Shu, Peng Chen, Tsz Kiu Aaron Chow, Lingbang Zhu, Yanhong Xiao, M.M.T. Loy & Shengwang Du, ‘‘Subnatural-linewidth biphotons from a Doppler-broadened hot atomic vapour cell”, Nature Communications volume 7, 12783 (2016)
[6] O. Davidson, R. Finkelstein, E. Poem, and O. Firstenberg, “Bright multiplexed source of indistinguishable single photons with tunable GHz-bandwidth at room temperature”, New J. Phys. 23, 073050 (2021)
[7] L. Podhora, P. Obšil, I. Straka, M. Ježek, and L. Slodička, ‘‘Nonclassical photon pairs from warm atomic vapor using a single driving laser”, Opt. Express 25, 31230 (2017).
[8] Kai-Hong Luo and Harald Herrmann and Stephan Krapick and Benjamin Brecht and Raimund Ricken and Viktor Quiring and Hubertus Suche and Wolfgang Sohler and Christine Silberhorn, ‘‘Direct generation of genuine single-longitudinal-mode narrowband photon pairs”, New Journal of Physics, Volume 17, 073039, (2015)
[9] Yoon-Seok Lee, Sang Min Lee, Heonoh Kim, and Han Seb Moon, ‘‘Highly bright photon-pair generation in Doppler-broadened ladder-type atomic system” ,Opt. Express 24, 28083 (2016)
[10] 王大邦, ‘‘四波混頻機制下的相干態Hong-Ou-Mandel干涉之理論與實驗研究” , 國立清華大學, 碩士論文，民國107年。
[11] Christopher Gerry and Peter Knight, ‘‘ Introductory Quantum Optics”, Chapter 2.5 and Chapter 6.2,( Cambridge University Press, 2004)
[12] Marcelo Davanço, Jun Rong Ong, Andrea Bahgat Shehata, Alberto Tosi, Imad Agha,Solomon Assefa, Fengnian Xia, William M. J. Green, Shayan Mookherjea,s Kartik Srinivasan. ‘‘Telecommunications-band heralded single photons from a silicon nanophotonic chip”, Appl. Phys. Lett. 100, 261104 (2012)
[13] Alessandro Seri, Dario Lago-Rivera, Andreas Lenhard, Giacomo Corrielli, Roberto Osellame, Margherita Mazzera, and Hugues de Riedmatten. ‘‘Quantum Storage of Frequency-Multiplexed Heralded Single Photons”, Phys. Rev. Lett. 123, 080502 (2019)
[14] Sunil Mittal, Elizabeth A. Goldschmidt & Mohammad Hafezi, ‘‘A topological source of quantum light” ,Nature volume 561, 502–506 (2018)
[15] Xiyuan Lu, Wei C. Jiang, Jidong Zhang, and Qiang Lin, ‘‘Biphoton Statistics of Quantum Light Generated on a Silicon Chip”, ACS Photonics , 3, 9, 1626–1636(2016)
[16] Matthias Scholz, Lars Koch, and Oliver Benson, ‘‘Statistics of Narrow-Band Single Photons for Quantum Memories Generated by Ultrabright Cavity-Enhanced Parametric Down-Conversion”, Phys. Rev. Lett ,102, 063603(2009)
[17] Zhu, Lingbang & Guo, Xianxin & Shu, Chi & Jeong, Heejeong & Du, Shengwang, ‘‘Bright narrowband biphoton generation from a hot rubidium atomic vapor cell”, Appl. Phys. Lett. 110, 161101 (2017)
[18] Ce Yang, Zhenjie Gu, Peng Chen, Zhongzhong Qin, J. F. Chen, and Weiping Zhang, ‘‘Tomography of the Temporal-Spectral State of Subnatural-Linewidth Single Photons from Atomic Ensembles” , Phys. Rev. Applied 10, 054011(2018)
[19] Yefeng Mei, Yiru Zhou, Shanchao Zhang, Jianfeng Li, Kaiyu Liao, Hui Yan, Shi-Liang Zhu, and Shengwang Du,‘‘Einstein-Podolsky-Rosen Energy-Time Entanglement of Narrow-Band Biphotons”, Phys. Rev. Lett. 124, 010509(2020)
[20] R. T. Willis, F. E. Becerra, L. A. Orozco, and S. L. Rolston, ‘‘Photon statistics and polarization correlations at telecommunications wavelengths from a warm atomic ensemble,” Opt. Express 19, 14632-14641 (2011)
[21] Luwei Zhao, Xianxin Guo, Chang Liu, Yuan Sun, M. M. T. Loy, and Shengwang Du, ‘‘Photon pairs with coherence time exceeding 1  μs”, Optica Vol. 1, 84-88 (2014)
 
[22] Yujie Zhang, Ryan Spiniolas, Kai Shinbrough, Bin Fang, Offir Cohen, and V. O. Lorenz, ‘‘Dual-pump approach to photon-pair generation: demonstration of enhanced characterization and engineering capabilities”, Opt. Express 27, 19050-19061 (2019)
[23] Chaoxuan Ma, Xiaoxi Wang, and Shayan Mookherjea, ‘‘Photon-pair and heralded single photon generation initiated by a fraction of a 10 Gbps data stream,” Opt. Express 26, 22904-22915 (2018)
[24] Luwei Zhao, Xianxin Guo, Yuan Sun, Yumian Su, M. M. T. Loy, and Shengwang Du, ‘‘Shaping the Biphoton Temporal Waveform with Spatial Light Modulation”, Phys. Rev. Lett. 115, 193601(2015)
[25] Shanchao Zhang, J. F. Chen, Chang Liu, M. M. T. Loy, G. K. L. Wong, and Shengwang Du, ‘‘Optical Precursor of a Single Photon”, Phys. Rev. Lett. 106, 243602(2011)
[26] Chang Liu, Shanchao Zhang, Luwei Zhao, Peng Chen, C. -H. F. Fung, H. F. Chau, M. M. T. Loy, and Shengwang Du,‘‘Differential-phase-shift quantum key distribution using heralded narrow-band single photons”,Opt. Express 21, 9505-9513 (2013)
[27] Pau Farrera, Georg Heinze, Boris Albrecht, Melvyn Ho, Matías Chávez, Colin Teo, Nicolas Sangouard & Hugues de Riedmatten ,‘‘Generation of single photons with highly tunable wave shape from a cold atomic ensemble”, Nature Communications 7(1), 13556 (2016)
[28] Peng Chen, Chi Shu, Xianxin Guo, M. M. T. Loy, and Shengwang Du, ‘‘Measuring the Biphoton Temporal Wave Function with Polarization-Dependent and Time-Resolved Two-Photon Interference”, Phys. Rev. Lett. 114, 010401(2015)
[29] Luwei Zhao, Yumian Su, and Shengwang Du, ‘‘Narrowband biphoton generation in the group delay regime”, Phys. Rev. A 93, 033815(2016)
[30] A. A. Shukhin, J. Keloth, K. Hakuta, and A. A. Kalachev, ‘‘Heralded single-photon and correlated-photon-pair generation via spontaneous four-wave mixing in tapered optical fibers” ,Phys. Rev. A 101, 053822(2020)
[31] Xiyuan Lu, Steven Rogers, Thomas Gerrits, Wei C. Jiang, Sae Woo Nam, and Qiang Lin, ‘‘Heralding single photons from a high-Q silicon microdisk,” Optica 3, 1331-1338 (2016)
 
[32] Daniel Llewellyn, Yunhong Ding, Imad I. Faruque, Stefano Paesani, Davide Bacco, Raffaele Santagati, Yan-Jun Qian, Yan Li, Yun-Feng Xiao, Marcus Huber, Mehul Malik, Gary F. Sinclair, Xiaoqi Zhou, Karsten Rottwitt, Jeremy L. O’Brien, John G. Rarity, Qihuang Gong, Leif K. Oxenlowe, Jianwei Wang & Mark G. Thompson, ‘‘Chip-to-chip quantum teleportation and multi-photon entanglement in silicon”, Nature Physics volume 16, pages148–153 (2020)
[33] Shanchao Zhang, Chang Liu, Shuyu Zhou, Chih-Sung Chuu, M. M. T. Loy, and Shengwang Du, ‘‘Coherent Control of Single-Photon Absorption and Reemission in a Two-Level Atomic Ensemble” ,Phys. Rev. Lett. 109, 263601(2012)
[34] Chang Liu, Yuan Sun, Luwei Zhao, Shanchao Zhang, M. M. T. Loy, and Shengwang Du, ‘‘Efficiently Loading a Single Photon into a Single-Sided Fabry-Perot Cavity”, Phys. Rev. Lett. 113, 133601(2014)
[35] Kai-Yu Liao, Xin-Ding Zhang, Guang-Zhou Guo, Bao-Quan Ai, Hui Yan & Shi-Liang Zhu, ‘‘Experimental test of the no-go theorem for continuous ψ-epistemic models”, Scientific Reports volume 6, Article number: 26519 (2016)