絕熱過程一詞多用於熱力學範疇中。熱力學中考量的是系統之間的能量交換，在這裡提到的系統可以為粒子亦可為複數粒子的集合。因此在熱力學中，絕熱過程的定義為能量僅以功的方式呈現，意即沒有熱能散失至環境中。然而絕熱過程在量子力學中有相異於熱力學中的解釋。在量子力學中，絕熱過程是以一隨時變化的厄米特系統作為考量，其哈密頓算符在隨時間的變化中，其量子態幾乎不發生變化的過程。簡而言之，我們可以認為其物理意義重視的是量子態的維持度而非能量多寡。此敘述意味著即便是在開放系統，即非厄米特系統中，絕熱過程也應當適用。甚者，我們可以預期在適當的條件下，開放系統中的絕熱系統相較封閉系統可能藉由與外界的能量交換而具備總量子態維持度的差異。
本文基於1928年Dr. Born在絕熱過程的推導上，簡單回顧絕熱捷徑之求得方式。而絕熱捷徑一詞，乃參考自2010年 Dr. Chen基於絕熱過程的量子力學計算得出。相異於Dr. Born對於能量最小化的方法，該篇文章以數學方式求解出絕熱過程的替代方案，也因為該方式以加入能量的方式取代所需經歷的時間變化，故稱之為捷徑。其後，藉由加入2011年Dr.Ibáñez對於非厄米特系統的量子力學規範，並完備非厄米特系統下絕熱過程的計算。我們將會透過Dr. Ibáñez的計算加以數學分析與討論在非厄米特系統下，絕熱捷徑方法應用在 PT 對稱性系統上的影響。

Adiabatic process is a well-known noun in thermodynamics. In thermodynamics, we always consider the energy transition among different systems. Therefore, adiabatic process means that the energy output is work only without loss in environment. However, in quantum mechanics, adiabatic process is referred as the invariant of quantum state in a time-dependent Hamiltonian system. Difference from the definition in thermodynamics, we consider more in the quantum state in
stead of how much energy cost it in time evolution. It means we could even adopt a open system which is as known as non-Hermitian system to achieve the adiabatic process. Moreover, we could even expect the total transition probability could be manipulated by different non-Hermitian system.
Here we would give a overview through the adiabatic theorem and how the shortcuts to be adopt to adiabatic process. Rather than the adiabatic theorem from Dr. Born, 1928, the term, shortcut, is named by Dr. Chen, 2010, as another approach to adiabatic process. The difference between the both method is that shortcuts always have additional energy to deal with, rather the adiabatic theorem is a method only consider the minimized energy evolution. Thus the both way still follow the uncertainty principle, but have different trade off as time or energy.Besides the background knowledge related adiabatic process, we will adopt the research from Dr. Ibáñez to develop another non-Hermitian system, called PT-symmetric system to check the effect that shortcuts is applied on.

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