| Summary: | 碩士 === 國立交通大學 === 物理研究所 === 104 === 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.
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