Abstract

We consider a two-level open quantum system whose dynamics is governed by the Gorini–Kossakowski–Sudarshan–Lindblad equation with Hamiltonian and dissipation superoperator depending, respectively, on coherent and incoherent controls. Results about reachability, controllability, and minimum-time control are obtained in terms of the Bloch parametrization. First, we consider the case when the zero coherent and incoherent controls satisfy the Pontryagin maximum principle in the class of piecewise continuous controls. Second, for zero coherent control and for incoherent control lying in the class of constant functions, the reachability and controllability sets of the system are exactly described and some analytical results on the minimum-time control are found. Third, we consider a series of increasing values of the final time and the corresponding classes of controls with zero incoherent control and with coherent control equal to zero until a switching time instant and to a cosine function after it. The corresponding reachable points in the Bloch ball are numerically obtained and visualized. Fourth, a known method for estimating reachable sets is adapted and used to analyze the situation where the zero coherent and incoherent controls satisfy the Pontryagin maximum principle in the class of piecewise continuous controls while, as shown numerically, are not optimal.

中文翻译：

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开放两能级量子系统最小时间控制的可达可控集

摘要

我们考虑一个两级开放量子系统，其动力学由 Gorini-Kossakowski-Sudarshan-Lindblad 方程控制，哈密顿量和耗散超级算子分别取决于相干和非相干控制。根据 Bloch 参数化获得关于可达性、可控性和最小时间控制的结果。首先，我们考虑零相干和非相干控制在分段连续控制类中满足庞特里亚金最大值原理的情况。其次，对于常函数类的零相干控制和非相干控制，准确描述了系统的可达性和可控性集，并得到了一些最小时间控制的分析结果。第三，我们考虑最终时间的一系列增加值和相应的控制类别，其中零非相干控制和相干控制为零，直到切换时间瞬间和之后的余弦函数。Bloch球中相应的可达点通过数值获得并可视化。第四，一种已知的估计可达集的方法被改编并用于分析零相干和非相干控制在分段连续控制类中满足庞特里亚金最大值原理而如数字所示不是最优的情况。