能階躍遷的量子控制–控制時間與強度的關係

Abstract

量子控制問題的困難點在於，它是一個藉由尋找合適的外加場來達成控制目標的逆問題。我們利用具有回饋性質的數值計算法與變分法，得到一個能夠有效率的尋找合適的外加場的方法，這個方法即為最佳控制理論(The optimal control theory)。在最佳控制理論中，有兩個參數是由我們自行設定，它們分別是控制時間與自懲因子(penalty factor)，這兩個參數分別限制完成控制目標的時間與可使用的外加場能量之大小。 這兩個參數的變動會影響我們對於系統的可控制性，但是我們無法從最佳化的過程中找到合適的參數值，因此在這篇論文中，我們利用最佳控制理論去控制二、三、四階系統的能階躍遷，我們把這兩個參數設視作變數，藉由改變這些兩個參數去計算控制結果，來討論這兩個參數如何影響系統的可控制性。 從計算的結果來看，我們得到達成成功控制的控制時間(T)與兩能階間的能隙(∆E)，有一個相似於測不準原理的關係等式 T=γℏ/∆E，在這個等式中的γ是一個參數；大致上，∆E被選擇為能階系統中最小的能隙，而γ則由我們的控制目標來決定。當控制時間小於這個等式所給予的時間時，我們仍可找到成功的控制例子，這是因為高強度的外加場能夠讓能階產生位移，讓控制能在短時間完成；這種在短時間內達成成功控制的例子，都發生在自懲因子及小的情況中。

Quantum control is a difficult problem, because it is an inverse problem, in which we search for the external field to obtain a given target state at a final time. The optimal control theory is a powerful tool to obtain the control field utilizing an iterative numerical calculation together with the variational principle. In this theory, we have to set two parameters, the control time T and the penalty factor α. The control time specifies the final time at which we desire to have the target state. The penalty factor limit the energy of the external field. These parameters are not optimized but given by us. These parameters determine the controllability of the system. In this thesis, to discuss how the parameters are related with the controllability, we repeat the optimal control simulation many times with various values of the control time and the penalty factor for simple problems of transitions in two, three, and four level systems. We found a general relation similar to the uncertainty principle, which relates the sufficient control time and energy gaps, namely =γℏ/∆E , where T is the control time, longer than which allows high quality of controls, ∆E is the energy gap between two states, and γ is a parameter. The choice of γ and two states to determine ∆E depend on the problem of interests. Generally speaking, ∆E is the smallest energy gap in the system, and γ depend on the target state. Because the energy gap can be modified by an intense field (DC Stark shift), a fast control is made possible by intense field. We found the optimal control theory employs this type of control when a smallα is chosen.