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.

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