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Communications in Mathematical Physics

, Volume 333, Issue 3, pp 1225–1239 | Cite as

Approximate Controllability, Exact Controllability, and Conical Eigenvalue Intersections for Quantum Mechanical Systems

  • Ugo BoscainEmail author
  • Jean-Paul Gauthier
  • Francesco Rossi
  • Mario Sigalotti
Article

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.

Keywords

Compact Group Conical Intersection Spectral Condition Exact Controllability Approximate Controllability 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Ugo Boscain
    • 1
    • 2
    Email author
  • Jean-Paul Gauthier
    • 3
  • Francesco Rossi
    • 4
  • Mario Sigalotti
    • 2
  1. 1.CNRS, CMAP, École PolytechniquePalaiseauFrance
  2. 2.INRIA Saclay, Team GECOPalaiseauFrance
  3. 3.Laboratoire LSISUniversité de ToulonLa Garde CedexFrance
  4. 4.Aix-Marseille UnivLSISMarseilleFrance

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