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Explicit approximate controllability of the Schrödinger equation with a polarizability term

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Abstract

We consider a controlled Schrödinger equation with a dipolar and a polarizability term, used when the dipolar approximation is not valid. The control is the amplitude of the external electric field, it acts nonlinearly on the state. We extend in this infinite dimensional framework previous techniques used by Coron, Grigoriu, Lefter and Turinici for stabilization in finite dimension. We consider a highly oscillating control and prove the semi-global weak \(H^2\) stabilization of the averaged system using a Lyapunov function introduced by Nersesyan. Then it is proved that the solutions of the Schrödinger equation and of the averaged equation stay close on every finite time horizon provided that the control is oscillating enough. Combining these two results, we get approximate controllability to the ground state for the polarizability system with explicit controls. Numerical simulations are presented to illustrate those theoretical results.

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References

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

    Google Scholar 

  2. Boussaid N, Caponigro M, Chambrion T (2011) Weakly-coupled systems in quantum control. INRIA Nancy-Grand Est “CUPIDSE” Color program, September

  3. Boscain U, Caponigro M, Chambrion T, Sigalotti M (2012) 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. 311:423–455

    Google Scholar 

  4. Beauchard K, Laurent C (2010) Local controllability of 1D linear and nonlinear Schrödinger equations with bilinear control. J Math Pures Appl (9) 94(5):520–554

    Google Scholar 

  5. Beauchard K, Mirrahimi M (2009) Practical stabilization of a quantum particle in a one-dimensional infinite square potential well. SIAM J. Control Optim 48(2):1179–1205

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  8. Cazenave T (2003) Semilinear Schrödinger equations. In: Courant Lecture Notes in Mathematics, vol 10. New York University Courant Institute of Mathematical Sciences, New York

  9. Coron J-M (2007) Control and nonlinearity. In: Mathematical Surveys and Monographs, vol 136. American Mathematical Society, Providence

  10. Couchouron J-F (2002) Compactness theorems for abstract evolution problems. J. Evol. Equ. 2(2):151–175

    Article  MathSciNet  MATH  Google Scholar 

  11. Couchouron J-F (2010) Strong stabilization of controlled vibrating systems. ESAIM : COCV. Torino. doi:10.1051/cocv/2010041

  12. Coron J-M, d’ Andréa-Novel B (1998) Stabilization of a rotating body beam without damping. IEEE Trans. Automat. Control 43(5):608–618

    Article  MathSciNet  MATH  Google Scholar 

  13. Coron J-M, d’ Andréa-Novel B, Bastin G (2007) A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws. IEEE Trans Automat Control 52(1):2–11

  14. Coron J-M, Grigoriu A, Lefter C, Turinici G (2009) Quantum control design by Lyapunov trajectory tracking for dipole and polarizability coupling. New J Phys 11(10)

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

    Article  MathSciNet  MATH  Google Scholar 

  16. Dion CM, Bandrauk AD, Atabek O, Keller A, Umeda H, Fujimura Y (1999) Two-frequency IR laser orientation of polar molecules. Numerical simulations for hcn. Chem Phys Lett 302:215–223

    Article  Google Scholar 

  17. Dion CM, Keller A, Atabek O, Bandrauk AD (1999) Laser-induced alignment dynamics of HCN: roles of the permanent dipole moment and the polarizability. Phys Rev 59:1382

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  19. Grigoriu A, Lefter C, Turinici G (2009) Lyapunov control of Schrödinger equation: beyond the dipole approximations. In: Proceedings of the 28th IASTED international conference on modelling, identification and control, Innsbruck, Austria, pp 119–123

  20. Hale JK, Lunel SM (1990) Averaging in infinite dimensions. J Integral Equ Appl 2(4):463–494

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  22. Mirrahimi M, Sarlette A, Rouchon P (2010) Real-time synchronization feedbacks for single-atom frequency standards: V- and lambda-structure systems. In: Proceedings of the 49th IEEE conference on decision and control, Atlanta, pp 5031–5036

  23. Nersesyan V (2009) Growth of Sobolev norms and controllability of the Schrödinger equation. Comm Math Phys 290(1):371–387

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  25. Rouchon P (2003) Control of a quantum particle in a moving potential well. In: Lagrangian and Hamiltonian Methods for Nonlinear Control. IFAC, Laxenburg, pp 287–290

  26. Sanders JA, Verhulst F, Murdock J (2007) Averaging methods in nonlinear dynamical systems. In: Applied Mathematical Sciences, vol 59, 2nd edn. Springer, New York

  27. Turinici G (2007) Beyond bilinear controllability: applications to quantum control. In: Control of coupled partial differential equations. In: Internat Ser Numer Math, vol 155. Birkhäuser, Basel, pp 293–309

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Acknowledgments

The author would like to thank K. Beauchard for having interested him in this problem and for fruitful discussions.

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Authors and Affiliations

  1. CMLA UMR 8536, ENS Cachan, 61 avenue du Président Wilson, 94235, Cachan, France

    Morgan Morancey

  2. CMLS UMR 7640, Ecole Polytechnique, 91128, Palaiseau, France

    Morgan Morancey

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  1. Morgan Morancey
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Correspondence to Morgan Morancey.

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The author was partially supported by the “Agence Nationale de la Recherche” (ANR), Projet Blanc EMAQS number ANR-2011-BS01-017-01.

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Morancey, M. Explicit approximate controllability of the Schrödinger equation with a polarizability term. Math. Control Signals Syst. 25, 407–432 (2013). https://doi.org/10.1007/s00498-012-0102-2

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  • DOI: https://doi.org/10.1007/s00498-012-0102-2

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