我們研究 Λ 型原子系綜在三種理論模型下的量子糾纏效應。分別是半古典模型、 微擾全量子模型與非微擾全量子模型。對於前兩者，我們進行基於弱場近似與一階 微擾展開的理論計算，並提供探測場光子在電磁波引發透明 (EIT) 中傳播行為的解析 解。考慮到原子群在介質中的演化會受到真空環境的影響，我們捨棄古典模型常用 的光布拉赫方程，反而用探討開放系統時引入的海森堡–朗之萬方程式。另一方面， 對於光場在介質中的演化，我們保留古典模型也使用過的馬克士威–薛丁格方程。藉 由全量子模型的解析解，我們可以探討探測場與耦合場之間形成糾纏的物理圖像， 發現電磁波引發透明的量子行為與耦合場的量子漲落有密切關聯。由於不同量子態 表現出不同的不確定性，狀態演化可以用正交方差 (quadrature variance) 表示。此外， 糾纏的定義是透過 Duan 發表的定理來建構，分別由兩連續變量的總方差和相關性組 成。解析解得到的糾纏受限於居量假設與不確定性，類愛因斯坦–波多爾斯基–羅森 (Einstein–Podolsky–Rosen) 算符的方差極限值大約是 0.75。為了讓 Λ 型的理論架構更 完整，以及消除一階微擾造成的失真，我們研究非微擾全量子模型包含電磁波引發 透明與同調居量捕獲下的量子效應。數值解得到的糾纏受限於不確定性，類愛因斯 坦–波多爾斯基–羅森算符的方差極限值大約是 0.5。雖然解析解存在失真，但對於理 解糾纏背後的物理背景功不可沒。數值解讓我們突破電磁波引發透明對光場強度的 限制，得以研究同調居量捕獲配置的量子行為。

We investigate the quantum entanglement effects in a Λ–type atomic ensemble under three theoretical models: the semi–classical model, the perturbation full–quantum model, and the non–perturbation full–quantum model. For the first two models, we perform theoretical analysis based on the weak–field approximation and first–order perturbation, providing analytical solutions for the propagation behavior of probe field photons in electromagnetically induced transparency (EIT). Considering that the evolution of atomic ensembles in the medium is influenced by the vacuum reservoir, we abandon the optical Bloch equations (OBEs) commonly used in classical models and instead employ the Heisenberg–Langevin equations (HLEs) which are often introduced when studying open quantum systems. On the other hand, for the evolution of the probe field in the EIT medium, we retain the Maxwell–Schrödinger equation (MSE) used in classical models. Through the analytical solutions provided by the full– quantum model, we can reveal the physical picture of entanglement construction between the probe and coupling field, discovering that the quantum behavior of EIT is closely related to the coupling field fluctuations. Since different quantum states have different uncertainty properties, state evolution can be represented by quadrature variance. Furthermore, the criterion of entanglement is built through Duan’s theorem, consisting of the total variance and correlations of two continuous variables. The entanglement obtained from analytical solution is limited by the population assumption and uncertainty, with a variance of EPR (Einstein– Podolsky–Rosen)–like operators limit of approximately 0.75. To make the Λ–type theoretical framework more complete and to eliminate distortions caused by first–order perturbation, we study the non–perturbation full–quantum model, including the quantum effects under EIT and coherent population trapping (CPT). The entanglement obtained from numerical solutions is limited by the uncertainty, with a variance of EPR–like operators limit of approximately 0.5. Although analytical solutions may have distortions, they are invaluable for understanding the physical backgrounds behind entanglement. Numerical solutions allow us to overcome the intensity limitations imposed by EIT on the probe field, enabling the study of quantum behavior in CPT configurations.

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