我們探討雙 Λ 四波混頻在不同近似下的理論模型，和其產生光子對的物理行為。延伸 Kolchin 的論文討論三種模型分別是遠離共振模型、遠離共振-基態假設模型、和完整模型。對三種模型進行理論計算及模擬，探討其物理行為，研究各種參數和四大係數對四波混頻的影響，探討頻譜和三種關聯性函數。
藉由三種關聯性函數也討論出調製原子密度大小使得光子對會有時間關聯性和頻率關聯性的量子糾纏現象，以及藉由耦合光和原子密度可調製出關聯性高、同調時間長、頻寬窄、和產生率高的光子對，接著利用遠離共振模型的近似解找出在目前實驗條件下的優化參數。也藉由三種模型的比較，定義出符合遠離共振模型的適用條件，以及完整模型下會有的光偏移和居量轉移影響產生率等物理現象。
為了更貼合實際情形，我們在完整模型加入相位失配，發現相位失配效應大小會有不同的物理現象，大相位失配會有頻譜分裂的情形，而產生非同調性四波混頻；小相位失配會有電磁波引發透明延遲相位補足相位失配，產生同調性四波混頻。我們研究出相位失配效應的界限，跟電磁波引發透明的頻寬內的相位變化有關。
最後得出四波混頻產生光子對的行為與產生、耦合和濾波組合而成。

We discuss the theoretical models of double Λ four-wave mixing in different approximations, and discuss the physical behavior of photon pairs generated. According to Kolchin’s paper extended three models ,such as far-detuning model, far-detuning model under the ground-state approximation, and full model.Theoretical calculations and simulations by three models explain their physical behavior, and study the influence of various parameters and four major coefficients on four-wave mixing with the frequency spectrum and three types of correlation functions.
We discuss the quantum-entangled phenomenon by three types of correlation functions.Modulated the atomic density have generated correlation of photon pairs,thus photon pairs have time-entangled correlation and frequency-entangled correlation.Furthermore, the coupling light and atomic density can be modulated to achieve high correlation, long coherence time, and narrow-band frequency. Therefore, Using far-detuning model to find the optimal parameters under the current experimental conditions. Based on the comparison of the three models, we defined the applicable conditions for the far-detuning model.In the full model, there will be physical phenomena such as light shift and occupancy transferance.
In order to fit the actual situation, we added phase mismatch to the full model and found that the magnitude of the phase mismatch effect will cause different physical phenomena.For instance, condition in large phase mismatch have split the spectrum, and this is non-coherent four-wave mixing. In addition, condition in small phase mismatch have caused delay phase by electromagnetically inducing transparency for making up the phase mismatch, and this is coherent four-wave mixing. Moreover, We study the boundary of the phase mismatch effect,related to the phase variance in electromagnetically inducing transparency. Finally, the behavior of four-wave mixing generated photon pairs have been composed of generation,coupling and filtering.

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