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Journal of Mathematical Chemistry

, Volume 53, Issue 2, pp 718–736 | Cite as

Systematically altering the apparent topology of constrained quantum control landscapes

  • A. Donovan
  • H. Rabitz
Original Paper

Abstract

A quantum control experiment typically seeks a shaped electromagnetic field to drive a system towards a specified observable objective. The large number of successful experiments can be understood through an exploration of the underlying quantum control landscape, which maps the objective as a function of the control variables. Specifically, under certain assumptions, the control landscape lacks suboptimal traps that could prevent identification of an optimal control. One of these assumptions is that there are no restrictions on the control variables, however, in practice control resources are inevitably constrained. The associated constrained quantum control landscape may be difficult to freely traverse due to the presence of limited resource induced traps. This work develops algorithms to (1) seek optimal controls under restricted resources, (2) explore the nature of apparent suboptimal landscape topology, and (3) favorably alter trap topology through systematic relaxation of the constraints. A set of mathematical tools are introduced to meet these needs by working directly with dynamic controls, rather than the prior studies that employed intermediate so-called kinematic control variables. The new tools are illustrated using few-level systems showing the capability of systematically relaxing constraints to convert an isolated trap into a level set or saddle feature on the landscape, thereby opening up the ability to find new solutions including those of higher fidelity. The results indicate the richness and complexity of the constrained quantum control landscape upon considering the tradeoff between resources and freedom to move on the landscape.

Keywords

Quantum control Quantum theory Constrained quantum control Constrained quantum control landscapes 

Notes

Acknowledgments

A.D. acknowledges support from the Program in Plasma Science and Technology at Princeton University and the NSF (CHE-1058644), ARO (W911NF-13-1-0237), and ARO-MURI (W911NF-11-1-2068). H.R. acknowledges partial support from DOE (DE-FG02-02ER15344).

References

  1. 1.
    C. Brif, R. Chakrabarti, H. Rabitz, Control of quantum phenomena: past, present and future. New J. Phys. 12, 075008 (2010)CrossRefGoogle Scholar
  2. 2.
    H. Rabitz, M. Hsieh, C. Rosenthal, Quantum optimally controlled transition landscapes. Science 303, 1998 (2004)CrossRefGoogle Scholar
  3. 3.
    V. Ramakrishna, M. Salapaka, M. Dahleh, H. Rabitz, A. Pierce, Controllability of molecular systems. Phys. Rev. A 51, 960 (1995)CrossRefGoogle Scholar
  4. 4.
    S. Schirmer, H. Fu, A. Solomon, Complete controllability of quantum systems. Phys. Rev. A 63, 063410 (2001)CrossRefGoogle Scholar
  5. 5.
    J. Werschnik, E. Gross, Tailoring laser pulses with spectral and fluence constraints using optimal control theory. J. Opt. B 7, S300 (2005)CrossRefGoogle Scholar
  6. 6.
    M. Lapert, R. Tehini, G. Turinici, D. Sugny, Monotonically convergent optimal control theory of quantum systems with spectral constraints on the control field. Phys. Rev. A 79, 063411 (2009)CrossRefGoogle Scholar
  7. 7.
    P. von den Hoff, S. Thallmair, M. Kowalewski, R. Siemering, R. de Vivie-Riedle, Optimal control theory-closing the gap between theory and experiment. Phys. Chem. Chem. Phys. 14, 14460 (2012)CrossRefGoogle Scholar
  8. 8.
    K. Moore, H. Rabitz, Exploring constrained quantum control landscapes. J. Chem. Phys. 137, 134113 (2012)CrossRefGoogle Scholar
  9. 9.
    C.-C. Shu, N. Henriksen, Phase-only shaped laser pulses in optimal control theory: application to indirect photofragmentation dynamics in the weak-field limit. J. Chem. Phys. 136, 044303 (2012)CrossRefGoogle Scholar
  10. 10.
    A. Donovan, V. Beltrani, H. Rabitz, Exploring the impact of constraints in quantum optimal control through a kinematic formulation. Chem. Phys. 425, 46 (2013)CrossRefGoogle Scholar
  11. 11.
    A. Donovan, V. Beltrani, H. Rabitz, Local topology at limited resource induced suboptimal traps on the quantum control landscape. J. Math. Chem. 52, 407 (2014)CrossRefGoogle Scholar
  12. 12.
    A. Donovan, H. Rabitz, Investigating constrained quantum control through a kinematic-to-dynamic variable transformation. Phys. Rev. A 90, 013408 (2014)Google Scholar
  13. 13.
    A. Rothman, T.-S. Ho, H. Rabitz, Exploring the level sets of quantum control landscapes. Phys. Rev. A 73, 053401 (2006)CrossRefGoogle Scholar
  14. 14.
    H. Rabitz, T.-S. Ho, M. Hsieh, R. Kosut, M. Demiralp, Topology of optimally controlled quantum mechanical transition probability landscapes. Phys. Rev. A 74, 012721 (2006)CrossRefGoogle Scholar
  15. 15.
    V. Beltrani, J. Dominy, T.-S. Ho, H. Rabitz, Exploring the top and bottom of the quantum control landscape. J. Chem. Phys. 134, 194106 (2011)CrossRefGoogle Scholar
  16. 16.
    A. Persidis, High-throughput screening. Nat. Biotechnol. 5, 488 (1998)CrossRefGoogle Scholar
  17. 17.
    A. Donovan, V. Beltrani, H. Rabitz, Quantum control by means of hamiltonian structure manipulation. Phys. Chem. Chem. Phys. 13, 7348 (2011)CrossRefGoogle Scholar
  18. 18.
    C. Wedge, G. Timco, E. Spielberg, R. George, F. Tuna, S. Rigby, E. McInnes, R. Winpenny, S. Blundell, A. Ardavan, Chemical engineering of molecular qubits. Phys. Rev. Lett. 108, 107204 (2012)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Department of ChemistryPrinceton UniversityPrincetonUSA

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