在本論文中，我們以零溫下，有阻尼的Jaynes-Cummings系統為例，比較三種描述量子開放系統的不同方法。我們首先從Liouville-von Neumann方程出發，得到玻恩近似下，非局域的主方程，接著，將玻恩近似下的主方程做更進一步的後馬可夫近似，得到局域的主方程。第三種方法則是利用凱爾迪西的非平衡態格林函數方法對非平衡態格林函數做玻恩近似。此外，我們也求出零溫下，有阻尼的Jaynes-Cummings模型的嚴格解。最後，我們利用三種方法以及嚴格解的結果分別求出有阻尼的Jaynes-Cummings系統處在激發態的機率隨時間的演化情形。藉由分析嚴格解的結果，我們了解耦合強度以及頻譜寬窄對於系統的影響，並以此為基準分析三種方法所取的各種不同近似，對結果所造成的影響為何。未來可以此分析為基礎，進一步推廣到有限溫度下。

In this thesis, we use three different methods to describe open quantum systems, in particular, to study the damped Jaynes-Cummings model at zero temperature. We begin with Liouville-von Neumann equation and Born approximation to obtain a time-convolution Master Equation. Then, we make Post-Markov approximation further and obtain a time-convolutionless Master Equation. The third method is based on Keldysh’s non-equilibrium Green function method plus Born approximation. Besides, we also solve exactly the damped Jaynes-Cummings model at zero temperature. We use these methods mentioned above to study the evolution of the probability of the excited state in damped Jaynes-Cummings model. We analyze the result of the exact solution to understand the effect of the coupling strength and spectrum width on the system. By compared the results obtained from the three different approaches with the exact solution, we find the validity of each approach. We also find that the non-equilibrium Green function approach is the most suitable one to study the dynamics of open systems, and may be extended to investigate the open systems dynamics in strong couplings and finite temperature in the future.

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