Advertisement

Conditions for the Absence of Local Extrema in Problems of Quantum Coherent Control

  • N. B. Il’in
  • A. N. Pechen
Article
  • 16 Downloads

Abstract

We consider a terminal control problem for quantum systems which is formulated as the problem of maximizing the objective functional at some fixed finite time. Within the framework of this problem, we discuss known results on the local maxima of the objective functional that are not global. This question is important for quantum control, since such local maxima could make it difficult to find the global maximum by local search in numerical optimization or under laboratory conditions.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    L. Accardi, Y. G. Lu, and I. Volovich, Quantum Theory and Its Stochastic Limit (Springer, Berlin, 2002).CrossRefzbMATHGoogle Scholar
  2. 2.
    H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford Univ. Press, Oxford, 2002).zbMATHGoogle Scholar
  3. 3.
    C. Brif, R. Chakrabarti, and H. Rabitz, “Control of quantum phenomena,” in Advances in Chemical Physics, Ed. by S. A. Rice and A. R. Dinner (J. Wiley & Sons, New York, 2012), Vol. 148, pp. 1–76.Google Scholar
  4. 4.
    R. W. Brockett, “Least squares matching problems,” Linear Algebra Appl. 122–124, 761–777 (1989).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    P. Brumer and M. Shapiro, Principles of the Quantum Control of Molecular Processes (J.Wiley & Sons, Hoboken, NJ, 2003).zbMATHGoogle Scholar
  6. 6.
    D. D’Alessandro, Introduction to Quantum Control and Dynamics (Chapman & Hall, Boca Raton, FL, 2008).zbMATHGoogle Scholar
  7. 7.
    P. de Fouquieres and S. G. Schirmer, “A closer look at quantum control landscapes and their implication for control optimization,” Infin. Dimens. Anal. Quantum Probab. Relat. Top. 16 (3), 1350021 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    S. J. Glaser, U. Boscain, T. Calarco, C. P. Koch, W. Köckenberger, R. Kosloff, I. Kuprov, B. Luy, S. Schirmer, T. Schulte-Herbrüggen, D. Sugny, and F. K. Wilhelm, “Training Schrödinger’s cat: quantum optimal control— Strategic report on current status, visions and goals for research in Europe,” Eur. Phys. J. D 69 (12), 279 (2015).CrossRefGoogle Scholar
  9. 9.
    S. J. Glaser, T. Schulte-Herbrüggen, M. Sieveking, O. Schedletzky, N. C. Nielsen, O. W. Sørensen, and C. Griesinger, “Unitary control in quantum ensembles: Maximizing signal intensity in coherent spectroscopy,” Science 280 (5362), 421–424 (1998).CrossRefGoogle Scholar
  10. 10.
    T.-S. Ho and H. Rabitz, “Why do effective quantum controls appear easy to find?,” J. Photochem. Photobiol. A 180 (3), 226–240 (2006).CrossRefGoogle Scholar
  11. 11.
    A. S. Holevo, Statistical Structure of Quantum Theory (Springer, Berlin, 2011), Lect. Notes Phys. Monogr. 67.zbMATHGoogle Scholar
  12. 12.
    V. S. Letokhov, Laser Control of Atoms and Molecules (Oxford Univ. Press, Oxford, 2007).Google Scholar
  13. 13.
    K. Lyakhov, H.-J. Lee, and A. Pechen, “Some features of Boron isotopes separation by the laser-assisted retardation of condensation method in multipass irradiation cell implemented as a resonator,” IEEE J. Quantum Electron. 52 (12), 1400208 (2016).CrossRefGoogle Scholar
  14. 14.
    J. Neumann, “Some matrix-inequalities and metrization of matric-space,” Izv. Nauchno-Issled. Inst. Mat. Mekh. Tomsk. Gos. Univ. im. V.V. Kuibysheva 1 (3), 286–300 (1937).zbMATHGoogle Scholar
  15. 15.
    Open Quantum Systems II: The Markovian Approach, Ed. by S. Attal, A. Joye, C.-A. Pillet (Springer, Berlin, 2006), Lect. Notes Math. 1881.Google Scholar
  16. 16.
    A. Pechen and N. Il’in, “Trap-free manipulation in the Landau–Zener system,” Phys. Rev. A 86 (5), 052117 (2012).CrossRefGoogle Scholar
  17. 17.
    A. N. Pechen and N. B. Il’in, “Coherent control of a qubit is trap-free,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 285, 244–252 (2014) [Proc. Steklov Inst. Math. 285, 233–240 (2014)].MathSciNetzbMATHGoogle Scholar
  18. 18.
    A. N. Pechen and N. B. Il’in, “On critical points of the objective functional for maximization of qubit observables,” Usp. Mat. Nauk 70 (4), 211–212 (2015) [Russ. Math. Surv. 70, 782–784 (2015)].CrossRefzbMATHGoogle Scholar
  19. 19.
    A. N. Pechen and N. B. Il’in, “On the problem of maximizing the transition probability in an n-level quantum system using nonselective measurements,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 294, 248–255 (2016) [Proc. Steklov Inst. Math. 294, 233–240 (2016)].MathSciNetzbMATHGoogle Scholar
  20. 20.
    A. N. Pechen and N. B. Il’in, “On extrema of the objective functional for short-time generation of single-qubit quantum gates,” Izv. Ross. Akad. Nauk, Ser. Mat. 80 (6), 217–229 (2016) [Izv. Math. 80, 1200–1212 (2016)].MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    A. Pechen and N. Il’in, “Control landscape for ultrafast manipulation by a qubit,” J. Phys. A: Math. Theor. 50 (7), 075301 (2017).MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    A. N. Pechen and D. J. Tannor, “Control of quantum transmission is trap free,” Can. J. Chem. 92 (2), 157–159 (2014).CrossRefGoogle Scholar
  23. 23.
    H. Rabitz, M. Hsieh, and C. Rosenthal, “Landscape for optimal control of quantum-mechanical unitary transformations,” Phys. Rev. A 72 (5), 052337 (2005).CrossRefGoogle Scholar
  24. 24.
    H. A. Rabitz, M. M. Hsieh, and C. M. Rosenthal, “Quantum optimally controlled transition landscapes,” Science 303 (5666), 1998–2001 (2004).CrossRefGoogle Scholar
  25. 25.
    N. Rach, M. M. Müller, T. Calarco, and S. Montangero, “Dressing the chopped-random-basis optimization: A bandwidth-limited access to the trap-free landscape,” Phys. Rev. A 92 (6), 062343 (2015).CrossRefGoogle Scholar
  26. 26.
    S. A. Rice and M. Zhao, Optical Control of Molecular Dynamics (J. Wiley & Sons, New York, 2000).Google Scholar
  27. 27.
    D. J. Tannor, Introduction to Quantum Mechanics: A Time-Dependent Perspective (Univ. Sci. Books, Sausalito, CA, 2007).Google Scholar
  28. 28.
    A. Trushechkin, “Semiclassical evolution of quantum wave packets on the torus beyond the Ehrenfest time in terms of Husimi distributions,” J. Math. Phys. 58 (6), 062102 (2017).MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    A. S. Trushechkin and I. V. Volovich, “Perturbative treatment of inter-site couplings in the local description of open quantum networks,” Europhys. Lett. 113 (3), 30005 (2016).CrossRefGoogle Scholar
  30. 30.
    I. V. Volovich, “Cauchy–Schwarz inequality-based criteria for the non-classicality of sub-Poisson and antibunched light,” Phys. Lett. A 380 (1–2), 56–58 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    I. V. Volovich and S. V. Kozyrev, “Manipulation of states of a degenerate quantum system,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 294, 256–267 (2016) [Proc. Steklov Inst. Math. 294, 241–251 (2016)].MathSciNetzbMATHGoogle Scholar
  32. 32.
    R. Wu, A. Pechen, H. Rabitz, M. Hsieh, and B. Tsou, “Control landscapes for observable preparation with open quantum systems,” J. Math. Phys. 49 (2), 022108 (2008).MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Steklov Mathematical Institute of Russian Academy of SciencesMoscowRussia
  2. 2.National University of Science and Technology MISiSMoscowRussia

Personalised recommendations