Resonance of minimizers for n-level quantum systems with an arbitrary cost
ESAIM: Control, Optimisation and Calculus of Variations, Tome 10 (2004) no. 4, pp. 593-614.

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 S2n-1⊂ℂn one is reduced to look just for minimizers on the sphere Sn-1⊂ℝn. Moreover, for the reduced problem, we investigate on the question of existence of strict abnormal minimizer.

DOI : 10.1051/cocv:2004022
Classification : 49J15, 81V80, 53C17, 49N50
Mots-clés : control of quantum systems, optimal control, sub-riemannian geometry, resonance, pontryagin maximum principle, abnormal extremals, rotating wave approximation 
@article{COCV_2004__10_4_593_0,
     author = {Boscain, Ugo and Charlot, Gr\'egoire},
     title = {Resonance of minimizers for n-level quantum systems with an arbitrary cost},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {593--614},
     publisher = {EDP-Sciences},
     volume = {10},
     number = {4},
     year = {2004},
     doi = {10.1051/cocv:2004022},
     mrnumber = {2111082},
     zbl = {1072.49002},
     language = {en},
     url = {https://www.numdam.org/articles/10.1051/cocv:2004022/}
}
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
PY  - 2004
SP  - 593
EP  - 614
VL  - 10
IS  - 4
PB  - EDP-Sciences
UR  - https://www.numdam.org/articles/10.1051/cocv:2004022/
DO  - 10.1051/cocv:2004022
LA  - en
ID  - COCV_2004__10_4_593_0
ER  - 
%0 Journal Article
%A Boscain, Ugo
%A Charlot, Grégoire
%T Resonance of minimizers for n-level quantum systems with an arbitrary cost
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2004
%P 593-614
%V 10
%N 4
%I EDP-Sciences
%U https://www.numdam.org/articles/10.1051/cocv:2004022/
%R 10.1051/cocv:2004022
%G en
%F COCV_2004__10_4_593_0
Boscain, Ugo; Charlot, Grégoire. Resonance of minimizers for n-level quantum systems with an arbitrary cost. ESAIM: Control, Optimisation and Calculus of Variations, Tome 10 (2004) no. 4, pp. 593-614. doi : 10.1051/cocv:2004022. https://www.numdam.org/articles/10.1051/cocv:2004022/

[1] A.A. Agrachev and Yu.L. Sachkov, Control Theory from the Geometric Viewpoint. Springer-Verlag, EMS (2004) 1-410. | MR | Zbl

[2] A.A. Agrachev and A.V. Sarychev, Sub-Riemannian metrics: minimality of abnormal geodesics versus subanaliticity. ESAIM: COCV 2 (1997) 377-448. | Numdam | MR | Zbl

[3] C. Altafini, Controllability of quantum mechanical systems by root space decomposition of su(N). J. Math. Phys. 43 (2002) 2051-2062. | MR | Zbl

[4] R. El Assoudi, J.P. Gauthier and I.A.K. Kupka, On subsemigroups of semisimple Lie groups. Ann. Inst. H. Poincaré Anal. Non Linéaire 13 (1996) 117-133. | Numdam | MR | Zbl

[5] A. Bellaiche, The tangent space in sub-Riemannian geometry. Sub-Riemannian geometry. Progr. Math. 144 (1996) 1-78. | MR | Zbl

[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. | MR | Zbl

[8] B. Bonnard and M. Chyba, The Role of Singular Trajectories in Control Theory. Springer, SMAI, Vol. 40 (2003). | MR | Zbl

[9] U. Boscain and B Piccoli, Optimal Synthesis for Control Systems on 2-D Manifolds. Springer, SMAI, Vol. 43 (2004). | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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). | MR

[13] W.M. Boothby and E.N. Wilson, Determination of the transitivity of bilinear systems. SIAM J. Control Optim. 17 (1979) 212-221. | MR | Zbl

[14] P. Brunovsky, Existence of Regular Syntheses for General Problems. J. Differ. Equations 38 (1980) 317-343. | MR | Zbl

[15] P. Brunovsky, Every Normal Linear System Has a Regular Time-Optimal Synthesis. Math. Slovaca 28 (1978) 81-100. | MR | Zbl

[16] D. D'Alessandro and M. Dahleh, Optimal control of two-level quantum systems. IEEE Trans. Automat. Control 46 (2001) 866-876. | Zbl

[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. | MR | Zbl

[19] M. Gromov, Carnot-Carathéodory spaces seen from within. Sub-Riemannian geometry. Progr. Math. 144 (1996) 79-323. | MR | Zbl

[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). | MR | Zbl

[22] V. Jurdjevic and I.K. Kupka, Control Systems on Semisimple Lie Groups and Their Homogeneous Spaces. Ann. Inst. Fourier 31 (1981) 151-179. | Numdam | MR | Zbl

[23] V. Jurdjevic and H.J. Sussmann, Controllability of Non-Linear systems. J. Differ. Equation 12 95-116. | MR | Zbl

[24] N. Khaneja, R. Brockett and S.J. Glaser, Time optimal control in spin systems. Phys. Rev. A 63 (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). | MR | Zbl

[28] R. Montgomery, A survey of singular curves in sub-Riemannian geometry. J. Dyn. Control Syst. 1 (1995) 49-90. | MR | Zbl

[29] B. Piccoli, Classifications of Generic Singularities for the Planar Time-Optimal Synthesis. SIAM J. Control Optim. 34 (1996) 1914-1946. | MR | Zbl

[30] B. Piccoli and H.J. Sussmann, Regular Synthesis and Sufficiency Conditions for Optimality. SIAM. J. Control Optim. 39 (2000) 359-410. | MR | Zbl

[31] L.S. Pontryagin, V. Boltianski, R. Gamkrelidze and E. Mitchtchenko, The Mathematical Theory of Optimal Processes. John Wiley and Sons, Inc (1961). | Zbl

[32] M.A. Daleh, A.M. Peirce and H. Rabitz, Optimal control of quantum-mechanical systems: Existence, numerical approximation, and applications. Phys. Rev. A 37 (1988). | MR

[33] V. Ramakrishna, K.L. Flores, H. Rabitz and R. Ober, Quantum control by decomposition of su(2). Phys. Rev. A 62 (2000).

[34] Y. Sachkov, Controllability of Invariant Systems on Lie Groups and Homogeneous Spaces. J. Math. Sci. 100 (2000) 2355-2427. | MR | Zbl

[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. | MR | Zbl

Cité par Sources :