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

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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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References

  1. Adami, R., Boscain, U.: Controllability of the Schrödinger equation via intersection of eigenvalues. In: Proceedings of the 44th IEEE Conference on Decision and Control, pp. 1080–1085 (2005)

  2. Albertini F., D’Alessandro D.: Notions of controllability for bilinear multilevel quantum systems. IEEE Trans. Automat. Control 48(8), 1399–1403 (2003)

    Article  MathSciNet  Google Scholar 

  3. Ball J.M., Marsden J.E., Slemrod M.: Controllability for distributed bilinear systems. SIAM J. Control Optim. 20(4), 575–597 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  4. Barut A.O., Rączka R.: Theory of Group Representations and Applications, 2nd edn. World Scientific Publishing Co., Singapore (1986)

    Book  MATH  Google Scholar 

  5. Beauchard K.: Local controllability of a 1-D Schrödinger equation. J. Math. Pures Appl. (9) 84(7), 851–956 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  6. Beauchard K., Coron J.-M.: Controllability of a quantum particle in a moving potential well. J. Funct. Anal. 232(2), 328–389 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  7. Beauchard K., Laurent C.: Local controllability of 1D linear and nonlinear Schrödinger equations with bilinear control. J. Math. Pures Appl. 94(5), 520–554 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  8. Beauchard K., Nersesyan V.: Semi-global weak stabilization of bilinear Schrödinger equations. C. R. Math. Acad. Sci. Paris 348(19–20), 1073–1078 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  9. Bloch A.M., Brockett R.W., Rangan C.: Finite controllability of infinite-dimensional quantum systems. IEEE Trans. Automat. Control 55(8), 1797–1805 (2010)

    Article  MathSciNet  Google Scholar 

  10. Born M., Fock V.: Beweis des adiabatensatzes. Zeitschrift für Physik A Hadrons Nuclei 51(3–4), 165–180 (1928)

    ADS  MATH  Google Scholar 

  11. Boscain U., Caponigro M., Chambrion T., Sigalotti M.: A weak spectral condition for the controllability of the bilinear Schrödinger equation with application to the control of a rotating planar molecule. Comm. Math. Phys. 311(2), 423–455 (2012)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  12. Boscain U., Caponigro M., Sigalotti M.: Multi-input Schrödinger equation: controllability, tracking, and application to the quantum angular momentum. J. Differ. Eq. 256(11), 3524–3551 (2014)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  13. Boscain U., Charlot G., Gauthier J.-P., Guérin S., Jauslin H.-R.: Optimal control in laser-induced population transfer for two- and three-level quantum systems. J. Math. Phys. 43(5), 2107–2132 (2002)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  14. Boscain, U., Chertovskih, R.A., Gauthier, J.P., Remizov, A.O.: Hypoelliptic diffusion and human vision: a semidiscrete new twist. SIAM J. Imaging Sci. 7(2), 669–695 (2014)

  15. Boscain U., Chittaro F., Mason P., Sigalotti M.: Adiabatic control of the Schroedinger equation via conical intersections of the eigenvalues. IEEE Trans. Automat. Control 57(8), 1970–1983 (2012)

    Article  MathSciNet  Google Scholar 

  16. Boussaid N., Caponigro M., Chambrion T.: Weakly-coupled systems in quantum control. IEEE Trans. Automat. Control. 58(9), 2205–2216 (2013)

    Article  MathSciNet  Google Scholar 

  17. Brockett, R., Rangan, C., Bloch, A.: The controllability of infinite quantum systems. In: Proceedings of the 42nd IEEE Conference on Decision and Control, pp. 428–433 (2003)

  18. Brockett, R.W.: Lie theory and control systems defined on spheres. SIAM J. Appl. Math., 25, 213–225, 1973. Lie algebras: applications and computational methods (Conf., Drexel Univ., Philadelphia, Pa., 1972)

  19. Burgarth D., Yuasa K.: Quantum system identification. Phys. Rev. Lett. 108(8), 080502 (2012)

    Article  ADS  Google Scholar 

  20. Chambrion T.: Periodic excitations of bilinear quantum systems. Automatica 48(9), 2040–2046 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  21. Chambrion T., Mason P., Sigalotti M., Boscain U.: Controllability of the discrete-spectrum Schrödinger equation driven by an external field. Ann. Inst. H. Poincaré Anal. Non Linéaire 26(1), 329–349 (2009)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  22. Chittaro, F., Mason, P.: Adiabatic control of quantum control systems with three inputs, Preprint

  23. Cole J.H., Schirmer S.G., Greentree A.D., Wellard C.J., Oi D.K., Hollenberg L.C.: Identifying an experimental two-state hamiltonian to arbitrary accuracy. Phys. Rev. A 71(6), 062312 (2005)

    Article  ADS  Google Scholar 

  24. D’Alessandro, D.: Introduction to quantum control and dynamics. In: Applied Mathematics and Nonlinear Science Series. Chapman, Hall/CRC, Boca Raton (2008)

  25. Dixmier J.: Les C*-algèbres et leurs représentations. Cahiers Scientifiques, Fasc. XXIX. Gauthier-Villars & Cie, Éditeur-Imprimeur, Paris (1964)

  26. El Assoudi R., Gauthier J.P., Kupka I.A.K.: On subsemigroups of semisimple Lie groups. Ann. Inst. H. Poincaré Anal. Non Linéaire 13(1), 117–133 (1996)

    MATH  MathSciNet  Google Scholar 

  27. Ervedoza S., Puel J.-P.: Approximate controllability for a system of Schrödinger equations modeling a single trapped ion. Ann. Inst. H. Poincaré Anal. Non Linéaire 26(6), 2111–2136 (2009)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  28. Guérin, S., Jauslin, H.: Control of quantum dynamics by laser pulses: Adiabatic Floquet theory. In: Advances in Chemical Physics, vol. 125 (2003)

  29. Hilgert, J., Hofmann, K.H., Lawson, J.D.: Lie groups, convex cones, and semigroups. In: Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, Oxford Science Publications (1989)

  30. Illner R., Lange H., Teismann H.: Limitations on the control of Schrödinger equations. ESAIM Control Optim. Calc. Var. 12(4), 615–635 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  31. Jurdjevic, V.: Geometric control theory. In: Cambridge Studies in Advanced Mathematics, vol. 52. Cambridge University Press, Cambridge (1997)

  32. Jurdjevic V., Sussmann H.J.: Control systems on Lie groups. J. Differ. Equ. 12, 313–329 (1972)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  33. Keyl, M., Zeier, R., Schulte-Herbrueggen, T.: Controlling several atoms in a cavity. New J. Phys. 16, 065010 (2014). doi:10.1088/1367-2630/16/6/065010

  34. Law C.K., Eberly J.H.: Arbitrary control of a quantum electro-magnetic field. Phys. Rev. Lett. 76(7), 1055–1058 (1996)

    Article  ADS  Google Scholar 

  35. Leghtas Z., Turinici G., Rabitz H., Rouchon P.: Hamiltonian identification through enhanced observability utilizing quantum control. IEEE Trans. Autom. Control 57(10), 2679–2683 (2012)

    Article  MathSciNet  Google Scholar 

  36. Mirrahimi M.: Lyapunov control of a quantum particle in a decaying potential. Ann. Inst. H. Poincaré Anal. Non Linéaire 26(5), 1743–1765 (2009)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  37. Nersesyan V.: Global approximate controllability for Schrödinger equation in higher Sobolev norms and applications. Ann. Inst. H. Poincaré Anal. Non Linéaire 27(3), 901–915 (2010)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  38. Nersesyan V., Nersisyan H.: Global exact controllability in infinite time of Schrödinger equation. J. Math. Pures Appl. (9) 97(4), 295–317 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  39. Pazy, A.: Semigroups of linear operators and applications to partial differential equations. In: Applied Mathematical Sciences Series, Springer, New York (1983)

  40. Rellich, F.: Perturbation theory of eigenvalue problems. Assisted by J. Berkowitz. With a preface by Jacob T. Schwartz. Gordon and Breach Science Publishers, New York (1969)

  41. Shore, B.W.: The Theory of Coherent Atomic Excitation, vol. 2 Set. Wiley-VCH, New York (1996)

  42. Smith P.A.: Everywhere dense subgroups of Lie groups. Bull. Amer. Math. Soc. 48, 309–312 (1942)

    Article  MATH  MathSciNet  Google Scholar 

  43. Teufel, S.: Adiabatic perturbation theory in quantum dynamics. In: Lecture Notes in Mathematics, vol. 1821. Springer, Berlin (2003)

  44. Turinici, G.: On the controllability of bilinear quantum systems. In: Defranceschi, M., Le Bris, C. (eds.) Mathematical Models and Methods for ab Initio Quantum Chemistry. Lecture Notes in Chemistry, vol. 74. Springer, Berlin (2000)

  45. von Neumann, J., Wigner, E.: überdas Verhalten von Eigenwerten bei Adiabatischen Prozessen. Z. Phys. 30 ,467–470 (1929)

    MATH  Google Scholar 

  46. Weil, A.: L’intégration dans les groupes topologiques et ses applications. Actual. Sci. Ind., no. 869. Hermann et Cie., Paris, (1940) (This book has been republished by the author at Princeton, N. J., 1941.)

  47. Yuan, H., Lloyd, S.: Controllability of the coupled spin-\({\frac{1}{2}}\) harmonic oscillator system. Phys. Rev. A (3) 75(5), 052331 (2007)

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Authors and Affiliations

  1. CNRS, CMAP, École Polytechnique, Palaiseau, France

    Ugo Boscain

  2. INRIA Saclay, Team GECO, Palaiseau, France

    Ugo Boscain & Mario Sigalotti

  3. Laboratoire LSIS, Université de Toulon, La Garde Cedex, France

    Jean-Paul Gauthier

  4. Aix-Marseille Univ, LSIS, 13013, Marseille, France

    Francesco Rossi

Authors
  1. Ugo Boscain
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  2. Jean-Paul Gauthier
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  3. Francesco Rossi
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  4. Mario Sigalotti
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Correspondence to Ugo Boscain.

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Communicated by A. Winter

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Boscain, U., Gauthier, JP., Rossi, F. et al. Approximate Controllability, Exact Controllability, and Conical Eigenvalue Intersections for Quantum Mechanical Systems. Commun. Math. Phys. 333, 1225–1239 (2015). https://doi.org/10.1007/s00220-014-2195-6

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  • DOI: https://doi.org/10.1007/s00220-014-2195-6

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