Approximate controllability, exact controllability, and conical eigenvalue intersections for quantum mechanical systems

Ugo Boscain 1, 2 Jean-Paul Gauthier 2, 3 Francesco Rossi 3 Mario Sigalotti 1, 2
2 GECO - Geometric Control Design
Inria Saclay - Ile de France, X - École polytechnique, CNRS - Centre National de la Recherche Scientifique : UMR7641
Abstract : We study the controllability of a closed control-affine quantum system driven by two or more external fields. We provide a sufficient condition for controllability in terms of existence of conical intersections between eigenvalues of the Hamiltonian in dependence of the controls seen as parameters. Such spectral condition is structurally stable in the case of three controls or in the case of two controls when the Hamiltonian is real. The spectral condition appears naturally in the adiabatic control framework and yields approximate controllability in the infinite-dimensional case. In the finite-dimensional case it implies that the system is Lie-bracket generating when lifted to the group of unitary transformations, and in particular that it is exactly controllable. Hence, Lie algebraic conditions are deduced from purely spectral properties. We conclude the article by proving that approximate and exact controllability are equivalent properties for general finite-dimensional quantum systems.
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https://hal.archives-ouvertes.fr/hal-00869706
Contributor : Francesco Rossi <>
Submitted on : Friday, October 4, 2013 - 7:44:34 AM
Last modification on : Tuesday, April 2, 2019 - 2:03:42 AM

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Ugo Boscain, Jean-Paul Gauthier, Francesco Rossi, Mario Sigalotti. Approximate controllability, exact controllability, and conical eigenvalue intersections for quantum mechanical systems. Communications in Mathematical Physics, Springer Verlag, 2015, 333 (3), pp.1225-1239. ⟨10.1007/s00220-014-2195-6⟩. ⟨hal-00869706⟩

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