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Monotonic time-discretized schemes in quantum control

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Abstract

Most of the numerical simulations in quantum (bilinear) control have used monotonically convergent algorithms of Krotov (introduced by Tannor et al. [15]), of Zhu & Rabitz [16] or their unified formulation in [17]. However, the properties of the discrete version of these procedures have not been yet tackled with. We present in this paper a stable time and space discretization which preserves the monotonic properties of the monotonic algorithms. Numerical results show that the newly derived algorithms are stable and enable various experimentations.

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References

  1. Rabitz, H.: Shaped laser pulses as reagents. Science 299, 525–526 (2003)

    Article  Google Scholar 

  2. Rabitz, H., Turinici, G., Brown, E.: Control of quantum dynamics: Concepts, procedures and future prospects. In: Ph. G. Ciarlet, editor, Computational Chemistry, Special Volume (C. Le Bris Editor) of Handbook of Numerical Analysis, vol X, 833–887. Elsevier Science B.V. (2003)

  3. Shi, S., Woody, A., Rabitz, H.: Optimal Control of Selective Vibrational Excitation in Harmonic Linear Chain Molecules. J. Chem. Phys., 88, 6870–6883 (1988)

    Article  Google Scholar 

  4. Salomon, J.: Limit points of the monotonic schemes for quantum control, to appear in the Proceedings of the 44th IEEE Conference on decision and Control, Sevilla, Spain, December 12–15 (2005)

  5. Elliott, D.L.: Bilinear systems, University of Maryland. J. Webster (ed.), Wiley Encyclopedia of Electrical and Electronics Engineering Online Copyright © 1999 by John Wiley & Sons

  6. Judson, R.S., Rabitz, H.: Teaching lasers to control molecules. Phys. Rev. Lett. 68, 1500–1503 (1992)

    Article  Google Scholar 

  7. Levis, R.J., Menkir, G., Rabitz, H.: Selective bond dissociation and rearrangement with optimally tailored, strong field laser pulses. Science 292, 709–713 (2001)

    Article  Google Scholar 

  8. Assion, A., Baumert, T., Bergt, M., Brixner, T., Kiefer, B., Seyfried, V., Strehle, M., Gerber, G.: Control of chemical reactions by feedback-optimized phase-shaped femtosecond laser pulses. Science 282, 919–922 (1998)

    Article  Google Scholar 

  9. Bergt, M., Brixner, T., Kiefer, B., Strehle, M., Gerber, G.: Controlling the femto-chemistry of Fe(CO)5. J. Phys. Chem. A. 103, 10381–10387 (1999)

    Article  Google Scholar 

  10. Weinacht, T., Ahn, J., Bucksbaum, P.: Controlling the shape of a quantum wavefunction. Nature 397, 233–235 (1999)

    Article  Google Scholar 

  11. Bardeen, C.J., Yakovlev, V.V., Wilson, K.R., Carpenter, S.D., Weber, P.M., Warren, W.S.: Feedback quantum control of molecular electronic population transfert. Chem. Phys. Lett. 280, 151–158 (1997)

    Article  Google Scholar 

  12. Bardeen, C.J., Yakovlev, V.V., Squier, J.A., Wilson, K.R.: Quantum control of population transfer in green flourescent protein by using chirped femtosecond pulses. J. Am. Chem. Soc. 120, 13023–13027 (1998)

    Article  Google Scholar 

  13. Strang, G.: Accurate partial difference methods I: Linear Cauchy problems. Arch. Rat. Mech. and An. 12, 392–402 (1963)

    Article  MATH  MathSciNet  Google Scholar 

  14. Hornung, T., Motzkus, M., de Vivie-Riedle, R.: Adapting optimal control theory and using learning loops to provide experimentally feasible shaping mask patterns. J. Chem.Phys. 115, 3105–3111 (2001)

    Article  Google Scholar 

  15. Tannor, D., Kazakov, V., Orlov, V.: Control of photochemical branching: Novel procedures for finding optimal pulses and global upper bounds. In Time Dependent Quantum Molecular Dynamics, edited by Broeckhove J. and Lathouwers L. Plenum, 347–360 (1992)

  16. Zhu, W., Rabitz, H.: A rapid monotonically convergent iteration algorithm for quantum optimal control over the expectation value of a positive definite operator. J. Chem. Phys. 109, 385–391 (1998)

    Article  Google Scholar 

  17. Maday, Y., Turinici, G.: New formulations of monotonically convergent quantum control algorithms. J. Chem. Phys. 118, 8191–8196 (2003)

    Article  Google Scholar 

  18. Bandrauk, A.D., Shen, H.: Exponential split operator methods for coupled Schrödinger equations. J. Chem. Phys. 99, 1185–1193 (1993)

    Article  Google Scholar 

  19. Salomon, J.: Phd. thesis. in progress

  20. Truong, T.N., Tanner, J.J., Bala, P., Andrew McCammon, J., Kouri, D.J., Lesyng, B., Hoffman, D.K.: A comparative study of time dependent quantum mechanical wave packet evolution methods. J. Chem. Phys. 96, 2077–2084 (1992)

    Article  Google Scholar 

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Authors and Affiliations

  1. Université Pierre et Marie Curie, Paris 6, Laboratoire Jacques-Louis Lions, 175, rue du Chevaleret, 75013, Paris, France

    Yvon Maday & Julien Salomon

  2. INRIA Rocquencourt, Domaine de Voluceau, Rocquencourt, B.P. 105, 78153, Le Chesnay Cedex, France

    Gabriel Turinici

  3. CERMICS-ENPC, Champs sur Marne, 77455, Marne la Vallée Cedex, France

    Gabriel Turinici

Authors
  1. Yvon Maday
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  2. Julien Salomon
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  3. Gabriel Turinici
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Correspondence to Yvon Maday.

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Maday, Y., Salomon, J. & Turinici, G. Monotonic time-discretized schemes in quantum control. Numer. Math. 103, 323–338 (2006). https://doi.org/10.1007/s00211-006-0678-x

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  • DOI: https://doi.org/10.1007/s00211-006-0678-x

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