dorsal/arxiv
View SchemaThe Lie algebra Structure and Nonlinear Controllability of Spin Systems
| Authors | F. Albertini, D. D'Alessandro |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0106115 |
| URL | https://arxiv.org/abs/quant-ph/0106115 |
Abstract
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.
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"abstract": "In this paper, we study the controllability properties and the Lie algebra\nstructure of networks of particles with spin immersed in an electro-magnetic\nfield. We relate the Lie algebra structure to the properties of a graph whose\nnodes represent the particles and an edge connects two nodes if and only if the\ninteraction between the two corresponding particles is active. For networks\nwith different gyromagnetic ratios, we provide a necessary and sufficient\ncondition of controllability in terms of the properties of the above mentioned\ngraph and describe the Lie algebra structure in every case. For these systems\nall the controllability notions, including the possibility of driving the\nevolution operator and/or the state, are equivalent. For general networks (with\npossibly equal gyromagnetic ratios), we give a sufficient condition of\ncontrollability. A general form of interaction among the particles is assumed\nwhich includes both Ising and Heisenberg models as special cases.\n Assuming Heisenberg interaction we provide an analysis of low dimensional\ncases (number of particles less then or equal to three) which include necessary\nand sufficient controllability conditions as well as a study of their Lie\nalgebra structure. This also, provides an example of quantum mechanical systems\nwhere controllability of the state is verified while controllability of the\nevolution operator is not.",
"arxiv_id": "quant-ph/0106115",
"authors": [
"F. Albertini",
"D. D\u0027Alessandro"
],
"categories": [
"quant-ph"
],
"title": "The Lie algebra Structure and Nonlinear Controllability of Spin Systems",
"url": "https://arxiv.org/abs/quant-ph/0106115"
},
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