The Lie algebra Structure and Nonlinear Controllability of Spin Systems
The Lie algebra Structure and Nonlinear Controllability of Spin Systems
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自旋系统的李代数结构和非线性可控性
DOI：
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发表时间：
2001
期刊：
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通讯作者：
D. D’Alessandro
中科院分区：
文献类型：
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作者：
F. Albertini;D. D’Alessandro
本文研究了电磁场中自旋粒子网络的能控性和李代数结构。我们将李代数结构与图的性质联系起来，图的节点代表粒子，边连接两个节点当且仅当两个相应粒子之间的相互作用是活跃的。对于具有不同旋磁比的网络，我们根据上述图的性质给出了能控的充要条件，并描述了每种情况下的李代数结构。对于这些系统，所有的可控性概念，包括驱动演化算子和/或状态的可能性，都是等价的。对于一般的网络（具有可能相等的旋磁比），我们给出了可控性的一个充分条件。粒子之间的相互作用的一般形式的假设，其中包括伊辛和海森堡模型作为特殊情况。
假设海森堡相互作用，我们提供了一个分析的低维情况下（粒子数小于或等于三个），其中包括必要和充分的可控性条件，以及他们的李代数结构的研究。这也提供了量子力学系统的一个例子，其中状态的可控性得到验证，而演化算子的可控性没有得到验证。
In this paper, we study the controllability properties and the Lie algebra structure of networks of particles with spin immersed in an electro-magnetic field. We relate the Lie algebra structure to the properties of a graph whose nodes represent the particles and an edge connects two nodes if and only if the interaction between the two corresponding particles is active. For networks with different gyromagnetic ratios, we provide a necessary and sufficient condition of controllability in terms of the properties of the above mentioned graph and describe the Lie algebra structure in every case. For these systems all the controllability notions, including the possibility of driving the evolution operator and/or the state, are equivalent. For general networks (with possibly equal gyromagnetic ratios), we give a sufficient condition of controllability. A general form of interaction among the particles is assumed which includes both Ising and Heisenberg models as special cases.
Assuming Heisenberg interaction we provide an analysis of low dimensional cases (number of particles less then or equal to three) which include necessary and sufficient controllability conditions as well as a study of their Lie algebra structure. This also, provides an example of quantum mechanical systems where controllability of the state is verified while controllability of the evolution operator is not.