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Linear Algebra and its Applications

The Lie algebra structure and controllability of spin systems

Francesca Albertini, Domenico D’Alessandro

Published Jul 1, 2002• 60 citations
Publisher
DOI
10.1016/s0024-3795(02)00290-2

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In this paper, we provide a complete analysis of the Lie algebra structure of a system of n interacting spin 1/2 particles with different gyromagnetic ratios in an electro-magnetic field. We relate the structure of this Lie algebra 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. We prove that for these systems all the controllability notions, including the possibility of driving the state or the evolution operator of the system, are equivalent. We also give a necessary and sufficient condition for controllability in terms of the properties of the above described graph. We provide extensions to the case of possibly equal gyromagnetic ratios.

IEEE Conference on Decision and Control (CDC)•Jan 1, 2001•1 citation
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.
In this paper, we provide a complete analysis of the Lie algebra structure of a system of n interacting spin 1/2 particles with different gyromagnetic ratios in an electro-magnetic field. We relate the structure of this Lie algebra 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. We prove that for these systems all the controllability notions, including the possibility of driving the state or the evolution operator of the system, are equivalent. We also give a necessary and sufficient condition for controllability in terms of the properties of the above described graph. We provide extensions to the case of possibly equal gyromagnetic ratios.
In this paper, we provide a complete analysis of the Lie algebra structure of a system of n interacting spin 1/2 particles with different gyromagnetic ratios in an electro-magnetic field. We relate the structure of this Lie algebra 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. We prove that for these systems all the controllability notions, including the possibility of driving the state or the evolution operator of the system, are equivalent. We also give a necessary and sufficient condition for controllability in terms of the properties of the above described graph. We provide extensions to the case of possibly equal gyromagnetic ratios.