Resonance of minimizers for n-level quantum systems with an arbitrary cost

Ugo Boscain; Grégoire Charlot

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 10, Issue: 4, page 593-614
  • ISSN: 1292-8119

Abstract

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We consider an optimal control problem describing a laser-induced population transfer on a n-level quantum system. For a convex cost depending only on the moduli of controls (i.e. the lasers intensities), we prove that there always exists a minimizer in resonance. This permits to justify some strategies used in experimental physics. It is also quite important because it permits to reduce remarkably the complexity of the problem (and extend some of our previous results for n=2 and n=3): instead of looking for minimizers on the sphere S 2 n - 1 ⊂ ℂ n one is reduced to look just for minimizers on the sphere S n - 1 ⊂ ℝ n . Moreover, for the reduced problem, we investigate on the question of existence of strict abnormal minimizer.

How to cite

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Boscain, Ugo, and Charlot, Grégoire. "Resonance of minimizers for n-level quantum systems with an arbitrary cost." ESAIM: Control, Optimisation and Calculus of Variations 10.4 (2010): 593-614. <http://eudml.org/doc/90745>.

@article{Boscain2010,
abstract = { We consider an optimal control problem describing a laser-induced population transfer on a n-level quantum system. For a convex cost depending only on the moduli of controls (i.e. the lasers intensities), we prove that there always exists a minimizer in resonance. This permits to justify some strategies used in experimental physics. It is also quite important because it permits to reduce remarkably the complexity of the problem (and extend some of our previous results for n=2 and n=3): instead of looking for minimizers on the sphere $S^\{2n-1\}\subset\mathbb\{C\}^n$ one is reduced to look just for minimizers on the sphere $S^\{n-1\}\subset \mathbb\{R\}^n$. Moreover, for the reduced problem, we investigate on the question of existence of strict abnormal minimizer. },
author = {Boscain, Ugo, Charlot, Grégoire},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Control of quantum systems; optimal control; sub-Riemannian geometry; resonance; pontryagin maximum principle; abnormal extremals; rotating wave approximation.; atomic levels; molecular levels; control of quantum systems; sub-Riemannian geometry; resonance; Pontryagin maximum principle; rotating wave approximation},
language = {eng},
month = {3},
number = {4},
pages = {593-614},
publisher = {EDP Sciences},
title = {Resonance of minimizers for n-level quantum systems with an arbitrary cost},
url = {http://eudml.org/doc/90745},
volume = {10},
year = {2010},
}

TY - JOUR
AU - Boscain, Ugo
AU - Charlot, Grégoire
TI - Resonance of minimizers for n-level quantum systems with an arbitrary cost
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 10
IS - 4
SP - 593
EP - 614
AB - We consider an optimal control problem describing a laser-induced population transfer on a n-level quantum system. For a convex cost depending only on the moduli of controls (i.e. the lasers intensities), we prove that there always exists a minimizer in resonance. This permits to justify some strategies used in experimental physics. It is also quite important because it permits to reduce remarkably the complexity of the problem (and extend some of our previous results for n=2 and n=3): instead of looking for minimizers on the sphere $S^{2n-1}\subset\mathbb{C}^n$ one is reduced to look just for minimizers on the sphere $S^{n-1}\subset \mathbb{R}^n$. Moreover, for the reduced problem, we investigate on the question of existence of strict abnormal minimizer.
LA - eng
KW - Control of quantum systems; optimal control; sub-Riemannian geometry; resonance; pontryagin maximum principle; abnormal extremals; rotating wave approximation.; atomic levels; molecular levels; control of quantum systems; sub-Riemannian geometry; resonance; Pontryagin maximum principle; rotating wave approximation
UR - http://eudml.org/doc/90745
ER -

References

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  1. A.A. Agrachev and Yu.L. Sachkov, Control Theory from the Geometric Viewpoint. Springer-Verlag, EMS (2004) 1-410.  
  2. A.A. Agrachev and A.V. Sarychev, Sub-Riemannian metrics: minimality of abnormal geodesics versus subanaliticity. ESAIM: COCV2 (1997) 377-448.  
  3. C. Altafini, Controllability of quantum mechanical systems by root space decomposition of s u ( N ) . J. Math. Phys.43 (2002) 2051-2062.  
  4. R. El Assoudi, J.P. Gauthier and I.A.K. Kupka, On subsemigroups of semisimple Lie groups. Ann. Inst. H. Poincaré Anal. Non Linéaire13 (1996) 117-133.  
  5. A. Bellaiche, The tangent space in sub-Riemannian geometry. Sub-Riemannian geometry. Progr. Math.144 (1996) 1-78.  
  6. K. Bergmann, H. Theuer and B.W. Shore, Coerent population transfer among quantum states of atomes and molecules. Rev. Mod. Phys. 70 (1998) 1003-1025.  
  7. V.G. Boltyanskii, Sufficient Conditions for Optimality and the Justification of the Dynamics Programming Principle. SIAM J. Control Optim.4 (1996) 326-361.  
  8. B. Bonnard and M. Chyba, The Role of Singular Trajectories in Control Theory. Springer, SMAI, Vol. 40 (2003).  
  9. U. Boscain and B Piccoli, Optimal Synthesis for Control Systems on 2-D Manifolds. Springer, SMAI, Vol. 43 (2004).  
  10. U. Boscain, G. Charlot, J.-P. Gauthier, S. Guérin and H.-R. Jauslin, Optimal Control in laser-induced population transfer for two- and three-level quantum systems. J. Math. Phys.43 (2002) 2107-2132.  
  11. U. Boscain, T. Chambrion and J.-P. Gauthier, On the K+P problem for a three-level quantum system: Optimality implies resonance. J. Dyn. Control Syst.8 (2002) 547-572.  
  12. U. Boscain, T. Chambrion and J.-P. Gauthier, Optimal Control on a n-level Quantum System, in Proc. of the 2nd IFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control, Astolfi, Gordillo and van der Schaft Eds., Elsevier (2003).  
  13. W.M. Boothby and E.N. Wilson, Determination of the transitivity of bilinear systems. SIAM J. Control Optim. 17 (1979) 212-221.  
  14. P. Brunovsky, Existence of Regular Syntheses for General Problems. J. Differ. Equations38 (1980) 317-343.  
  15. P. Brunovsky, Every Normal Linear System Has a Regular Time-Optimal Synthesis. Math. Slovaca28 (1978) 81-100.  
  16. D. D'Alessandro and M. Dahleh, Optimal control of two-level quantum systems. IEEE Trans. Automat. Control46 (2001) 866-876.  
  17. U. Gaubatz, P. Rudecki, M. Becker, S. Schiemann, M. Kulz and K. Bergmann, Population switching between vibrational levels in molecular beams. Chem. Phys. Lett. 149 (1988) 463.  
  18. J.P. Gauthier and G. Bornard, Controlabilite des sytemes bilineaires. SIAM J. Control Optim.20 (1982) 377-384.  
  19. M. Gromov, Carnot-Carathéodory spaces seen from within. Sub-Riemannian geometry. Progr. Math.144 (1996) 79-323.  
  20. R.G. Hulet and D. Kleppner, Rydberg Atoms in “Circular” states. Phys. Rev. Lett.51 (1983) 1430-1433.  
  21. V. Jurdjevic, Geometric Control Theory. Cambridge University Press (1997).  
  22. V. Jurdjevic and I.K. Kupka, Control Systems on Semisimple Lie Groups and Their Homogeneous Spaces. Ann. Inst. Fourier31 (1981) 151-179.  
  23. V. Jurdjevic and H.J. Sussmann, Controllability of Non-Linear systems. J. Differ. Equation12 95-116.  
  24. N. Khaneja, R. Brockett and S.J. Glaser, Time optimal control in spin systems. Phys. Rev. A63 (2001).  
  25. N. Khaneja and S.J. Glaser, Cartan decomposition of SU(n) and Control of Spin Systems. J. Chem. Phys.267 (2001) 11-23.  
  26. C. Liedenbaum, S. Stolte and J. Reuss, Inversion produced and reversed by adiabatic passage. Phys. Rep.178 (1989) 1-24.  
  27. R. Montgomery, A Tour of Subriemannian Geometry. American Mathematical Society, Mathematical Surveys and Monographs (2002).  
  28. R. Montgomery, A survey of singular curves in sub-Riemannian geometry. J. Dyn. Control Syst.1 (1995) 49-90.  
  29. B. Piccoli, Classifications of Generic Singularities for the Planar Time-Optimal Synthesis. SIAM J. Control Optim.34 (1996) 1914-1946.  
  30. B. Piccoli and H.J. Sussmann, Regular Synthesis and Sufficiency Conditions for Optimality. SIAM. J. Control Optim. 39 (2000) 359-410.  
  31. L.S. Pontryagin, V. Boltianski, R. Gamkrelidze and E. Mitchtchenko, The Mathematical Theory of Optimal Processes. John Wiley and Sons, Inc (1961).  
  32. M.A. Daleh, A.M. Peirce and H. Rabitz, Optimal control of quantum-mechanical systems: Existence, numerical approximation, and applications. Phys. Rev. A37 (1988).  
  33. V. Ramakrishna, K.L. Flores, H. Rabitz and R.Ober, Quantum control by decomposition of su(2). Phys. Rev. A62 (2000).  
  34. Y. Sachkov, Controllability of Invariant Systems on Lie Groups and Homogeneous Spaces. J. Math. Sci.100 (2000) 2355-2427.  
  35. B.W. Shore, The theory of coherent atomic excitation. New York, NY, Wiley (1990).  
  36. H.J. Sussmann, The Structure of Time-Optimal Trajectories for Single-Input Systems in the Plane: the C ∞ Nonsingular Case. SIAM J. Control Optim.25 (1987) 433-465.  

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