非馬可夫性質是系統動力學中描述系統與環境交互作用的重要特徵，因此非馬可夫動力學的研究在量子資訊、量子通訊等量子工程領域中扮演了重要的角色。在這篇論文中，我們以一個客觀且有組織的方法研究了光子在雙折射晶體中的動力學。藉由一項以全光學實驗討論光子在雙折射晶體動力學中非馬可夫性質的重要工作 [B.-H. Liu et al., Nature Phys. 7, 931 (2011)]，我們利用了其中有限的實驗數據，準確地描述該實驗中光子的動力學，並藉此比較目前主要幾種非馬可夫量度對於分析光子偏振在雙折射晶體中動力學之結果；我們展現了以量化量子過程為基礎的非馬可夫量度，可以成功鑑別出在該實驗中以其他非馬可夫量度偵測不到的非馬可夫性質。與目前其他基於分析狀態特性的非馬可量度相比，量化光子動力學中量子過程的方法對於鑑別光子在雙折射晶體中非馬可夫動力學能具有更好的解析度。另外，我們也展示了在這樣高解析度的非馬可夫量度下，如何透過光學實驗辨別光子的非馬可夫動力學，並且能對於需要討論系統與環境作用的量子工程提供更精確的設計。

Non-Markovianity of a dynamical process characterizes the interaction between a principal system and an environment. Hence, the study in the non-Markovian dynamics plays an important role for quantum engineering. In this thesis, we investigate the non-Markovian dynamics of photon polarization in a birefringent crystal with an objective and organized approach. Following the limited experimental data of the seminal experiment on the non-Markovian effect [B.-H. Liu et al., Nature Phys. 7, 931 (2011)], we precisely describe the photon dynamics in that experiment in the form of process matrix. With it, we then compare different non-Markovian criteria by identifying the non-Markovianity of the constructed processes. In this thesis, we show that all the processes identified as Markovian in the experiment of Liu et al. can be indeed certified as non-Markovian dynamics. Compared with the existing verifications based on the analysis of state properties, the method of quantifying quantum composition of the photon dynamics has finer resolution in identifying non-Markovian processes for photons in the birefringent crystal. Our result demonstrates an experimentally feasible characterization of photon dynamics using an all-optical set-up and could promote an accurate design for system-environment quantum engineering.

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