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Journal of Mathematical Physics / Volume 51 / Issue 9 / ARTICLES / Dynamical Systems

J. Math. Phys. 51, 092705 (2010); http://dx.doi.org/10.1063/1.3479390 (44 pages)

The energy minimization problem for two-level dissipative quantum systems

B. Bonnard1, O. Cots2, N. Shcherbakova1, and D. Sugny3

1Institut de Mathématiques de Bourgogne, UMR CNRS 5584, BP 47870, 21078 Dijon, France
2IRIT, ENSEEIHT, UMR CNRS 5055, 31062 Toulouse, France
3Laboratoire Interdisciplinaire Carnot de Bourgogne (ICB), UMR 5209 CNRS-Université de Bourgogne, 9 Av. A. Savary, BP 47 870, F-21078 Dijon Cedex, France

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(Received 19 November 2009; accepted 23 July 2010; published online 15 September 2010)

In this article, we study the energy minimization problem of dissipative two-level quantum systems whose dynamics is governed by the Kossakowski–Lindblad equations. In the first part, we classify the extremal curve solutions of the Pontryagin maximum principle. The optimality properties are analyzed using the concept of conjugate points and the Hamilton–Jacobi–Bellman equation. This analysis completed by numerical simulations based on adapted algorithms allows a computation of the optimal control law whose robustness with respect to the initial conditions and dissipative parameters is also detailed. In the final section, an application in nuclear magnetic resonance is presented.

© 2010 American Institute of Physics

Article Outline

  1. INTRODUCTION
  2. GEOMETRIC ANALYSIS OF THE EXTREMAL CURVES
    1. Maximum principle
    2. Geometric computations of the extremals
      1. Normal extremals in spherical coordinates
      2. Abnormal extremals in spherical coordinates
    3. The analysis in the normal case
    4. Normal extremals in meridian planes
      1. The integrable case γ− = 0
      2. The general case γ− ≠ 0
    5. Normal extremals in nonmeridian planes
      1. The integrable case γ− = 0
      2. Analysis in the case γ− ≠ 0
  3. THE OPTIMALITY PROBLEM
    1. Existence theorem
    2. Optimality concepts
    3. Symmetries and optimality
      1. The integrable case
      2. The general case
    4. The geometric properties of the variational equation and estimation of conjugate points
      1. Preliminaries
      2. The integrable case
      3. Computation of the conjugate locus for short periodic orbits in the meridian case
    5. The value function and Hamilton–Jacobi–Bellman equation
      1. The abnormal case
      2. The Hamilton–Jacobi–Bellman theory in the normal case
      3. The global Hamilton–Jacobi–Bellman equation
  4. GEOMETRIC ALGORITHMS AND NUMERICAL SIMULATIONS
    1. The COTCOT code
    2. The smooth continuation method
    3. Robustness issues
    4. Numerical simulations
      1. Application in nuclear magnetic resonance
      2. Extremals and conjugate points in the integrable case
      3. Conjugate loci, spheres, and wave fronts
      4. Extremals and conjugate points in the nonintegrable case
      5. Continuation results

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KEYWORDS and PACS

Keywords

Bell theorem, master equation, numerical analysis, optimal control, quantum theory

PACS

  • 03.65.-w

    Quantum mechanics

  • 03.65.Ud

    Entanglement and quantum nonlocality (e.g. EPR paradox, Bell's inequalities, GHZ states, etc.)

  • 02.60.-x

    Numerical approximation and analysis

ARTICLE DATA

Digital Object Identifier
http://dx.doi.org/10.1063/1.3479390

PUBLICATION DATA

ISSN

0022-2488 (print)  
1089-7658 (online)

Publisher

$abstract.society.value is a Member of CrossRef American Institute of Physics

For access to fully linked references, you need to log in.
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    Assémat, E., Lapert, M., Zhang, Y., Braun, M., Glaser, S. J., and Sugny, D., “Simultaneous time-optimal control of the inversion of two spin-1/2 particles,” Phys. Rev. A 82, 013415 (2010).

    Ramakrishna, S. and Seideman, T., “Intense laser alignment in dissipative media as a route to solvent dynamics,” Phys. Rev. Lett. 95, 113001 (2005).

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