dorsal/arxiv
View SchemaAnalytic Controllability of Time-Dependent Quantum Control Systems
| Authors | Chunhua Lan, Tzyh-Jong Tarn, Quo-Shin Chi, John W. Clark |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0409147 |
| URL | https://arxiv.org/abs/quant-ph/0409147 |
| DOI | 10.1063/1.1867979 |
Abstract
The question of controllability is investigated for a quantum control system in which the Hamiltonian operator components carry explicit time dependence which is not under the control of an external agent. We consider the general situation in which the state moves in an infinite-dimensional Hilbert space, a drift term is present, and the operators driving the state evolution may be unbounded. However, considerations are restricted by the assumption that there exists an analytic domain, dense in the state space, on which solutions of the controlled Schrodinger equation may be expressed globally in exponential form. The issue of controllability then naturally focuses on the ability to steer the quantum state on a finite-dimensional submanifold of the unit sphere in Hilbert space -- and thus on analytic controllability. A relatively straightforward strategy allows the extension of Lie-algebraic conditions for strong analytic controllability derived earlier for the simpler, time-independent system in which the drift Hamiltonian and the interaction Hamiltonia have no intrinsic time dependence. Enlarging the state space by one dimension corresponding to the time variable, we construct an augmented control system that can be treated as time-independent. Methods developed by Kunita can then be implemented to establish controllability conditions for the one-dimension-reduced system defined by the original time-dependent Schrodinger control problem. The applicability of the resulting theorem is illustrated with selected examples.
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"abstract": "The question of controllability is investigated for a quantum control system\nin which the Hamiltonian operator components carry explicit time dependence\nwhich is not under the control of an external agent. We consider the general\nsituation in which the state moves in an infinite-dimensional Hilbert space, a\ndrift term is present, and the operators driving the state evolution may be\nunbounded. However, considerations are restricted by the assumption that there\nexists an analytic domain, dense in the state space, on which solutions of the\ncontrolled Schrodinger equation may be expressed globally in exponential form.\nThe issue of controllability then naturally focuses on the ability to steer the\nquantum state on a finite-dimensional submanifold of the unit sphere in Hilbert\nspace -- and thus on analytic controllability. A relatively straightforward\nstrategy allows the extension of Lie-algebraic conditions for strong analytic\ncontrollability derived earlier for the simpler, time-independent system in\nwhich the drift Hamiltonian and the interaction Hamiltonia have no intrinsic\ntime dependence. Enlarging the state space by one dimension corresponding to\nthe time variable, we construct an augmented control system that can be treated\nas time-independent. Methods developed by Kunita can then be implemented to\nestablish controllability conditions for the one-dimension-reduced system\ndefined by the original time-dependent Schrodinger control problem. The\napplicability of the resulting theorem is illustrated with selected examples.",
"arxiv_id": "quant-ph/0409147",
"authors": [
"Chunhua Lan",
"Tzyh-Jong Tarn",
"Quo-Shin Chi",
"John W. Clark"
],
"categories": [
"quant-ph"
],
"doi": "10.1063/1.1867979",
"title": "Analytic Controllability of Time-Dependent Quantum Control Systems",
"url": "https://arxiv.org/abs/quant-ph/0409147"
},
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