Hamiltonian identification for quantum systems : well-posedness and numerical approaches
ESAIM: Control, Optimisation and Calculus of Variations, Tome 13 (2007) no. 2, pp. 378-395.

This paper considers the inversion problem related to the manipulation of quantum systems using laser-matter interactions. The focus is on the identification of the field free hamiltonian and/or the dipole moment of a quantum system. The evolution of the system is given by the Schrödinger equation. The available data are observations as a function of time corresponding to dynamics generated by electric fields. The well-posedness of the problem is proved, mainly focusing on the uniqueness of the solution. A numerical approach is also introduced with an illustration of its efficiency on a test problem.

DOI : https://doi.org/10.1051/cocv:2007013
Classification : 93B30,  65K10
Mots clés : inverse problem, quantum systems, hamiltonian identification, optimal identification
@article{COCV_2007__13_2_378_0,
author = {Bris, Claude Le and Mirrahimi, Mazyar and Rabitz, Herschel and Turinici, Gabriel},
title = {Hamiltonian identification for quantum systems : well-posedness and numerical approaches},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {378--395},
publisher = {EDP-Sciences},
volume = {13},
number = {2},
year = {2007},
doi = {10.1051/cocv:2007013},
zbl = {1123.93040},
mrnumber = {2306642},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv:2007013/}
}
Bris, Claude Le; Mirrahimi, Mazyar; Rabitz, Herschel; Turinici, Gabriel. Hamiltonian identification for quantum systems : well-posedness and numerical approaches. ESAIM: Control, Optimisation and Calculus of Variations, Tome 13 (2007) no. 2, pp. 378-395. doi : 10.1051/cocv:2007013. http://www.numdam.org/articles/10.1051/cocv:2007013/

[1] F. Albertini and D. D'Alessandro, Notions of controllability for multilevel bilinear quantum mechanical systems. IEEE Trans. Automatic Control 48 (2003) 1399-1403.

[2] O.F. Alis, H. Rabitz, M.Q. Phan, C. Rosenthal and M. Pence, On the inversion of quantum mechanical systems: Determining the amount and type of data for a unique solution. J. Math. Chem. 35 (2004) 65-78. | Zbl 1045.81014

[3] Claudio Altafini, Controllability of quantum mechanical systems by root space decomposition of $su\left(N\right)$. J. Math. Phys. 43 (2002) 2051-2062. | Zbl 1059.93016

[4] A. Assion, T. Baumert, M. Bergt, T. Brixner, B. Kiefer, V. Seyfried, M. Strehle and G. Gerber, Control of chemical reactions by feedback-optimized phase-shaped femtosecond laser pulses. Science 282 (1998) 919-922.

[5] C. Bardeen, V.V. Yakovlev, K.R. Wilson, S.D. Carpenter, P.M. Weber and W.S. Warren, Feedback quantum control of molecular electronic population transfer. Chem. Phys. Lett. 280 (1997) 151.

[6] C.J. Bardeen, V.V. Yakovlev, J.A. Squier and K.R. Wilson, Quantum control of population transfer in green fluorescent protein by using chirped femtosecond pulses. J. Am. Chem. Soc. 120 (1998) 13023-13027.

[7] R.R. Barton and J.S. Jr. Ivey, Nelder-Mead simplex modifications for simulation optimization. Manage. Sci. 42 (1996) 954-973. | Zbl 0884.90118

[8] Y. Chen, P. Gross, V. Ramakrishna, H. Rabitz and K. Mease, Competitive tracking of molecular objectives described by quantum mechanics. J. Chem. Phys. 102 (1995) 8001-8010.

[9] C. Cohen-Tannoudji, B. Diu and F. Laloë, Mécanique Quantique, Volumes I & II. Hermann, Paris (1977).

[10] J.M. Geremia and H. Rabitz, Optimal hamiltonian identification: The synthesis of quantum optimal control and quantum inversion. J. Chem. Phys 118 (2003) 5369-5382.

[11] R.S. Judson and H. Rabitz, Teaching lasers to control molecules. Phys. Rev. Lett. 68 (1992) 1500.

[12] R.L. Kosut and H. Rabitz, Identification of quantum systems. In Proceedings of the 15th IFAC World Congress (2002).

[13] S. Kullback, Information Theory and Statistics. Wiley, New York (1959). | MR 103557 | Zbl 0088.10406

[14] S. Kullback and R.A. Leibler, On information and sufficiency. Ann. Math. Stat. 22 (1951) 79-86. | Zbl 0042.38403

[15] C. Le Bris, Y. Maday and G. Turinici, Towards efficient numerical approaches for quantum control. In Quantum Control: mathematical and numerical challenges, A. Bandrauk, M.C. Delfour and C. Le Bris Eds., CRM Proc. Lect. Notes Ser., AMS Publications, Providence, R.I. (2003) 127-142.

[16] R.J. Levis, G. Menkir and H. Rabitz, Selective bond dissociation and rearrangement with optimally tailored, strong-field laser pulses. Science 292 (2001) 709.

[17] B. Li, G. Turinici, V. Ramakrishna and H. Rabitz, Optimal dynamic discrimination of similar molecules through quantum learning control. J. Phys. Chem. B. 106 (2002) 8125-8131.

[18] Y. Maday and G. Turinici, New formulations of monotonically convergent quantum control algorithms. J. Chem. Phys 118 (18) (2003).

[19] M. Mirrahimi, P. Rouchon and G. Turinici, Lyapunov control of bilinear Schrödinger equations. Automatica 41 (2005) 1987-1994. | Zbl 1125.93466

[20] M. Mirrahimi, G. Turinici and P. Rouchon, Reference trajectory tracking for locally designed coherent quantum controls. J. Phys. Chem. A 109 (2005) 2631-2637.

[21] M.Q. Phan and H. Rabitz, Learning control of quantum-mechanical systems by laboratory identification of effective input-output maps. Chem. Phys. 217 (1997) 389-400.

[22] H. Rabitz, Perspective. Shaped laser pulses as reagents. Science 299 (2003) 525-527.

[23] V. Ramakrishna, M. Salapaka, M. Dahleh and H. Rabitz, Controllability of molecular systems. Phys. Rev. A 51 (1995) 960-966.

[24] S. Rice and M. Zhao, Optimal Control of Quatum Dynamics. Wiley (2000) (many additional references to the subjects of this paper may also be found here).

[25] N. Shenvi, J.M. Geremia and H. Rabitz, Nonlinear kinetic parameter identification through map inversion. J. Phys. Chem. A 106 (2002) 12315-12323.

[26] M. Tadi and H. Rabitz, Explicit method for parameter identification. J. Guid. Control Dyn. 20 (1997) 486-491. | Zbl 0884.93020

[27] G. Turinici and H. Rabitz, Quantum wavefunction controllability. Chem. Phys. 267 (2001) 1-9.

[28] G. Turinici and H. Rabitz, Wavefunction controllability in quantum systems. J. Phys. A 36 (2003) 2565-2576. | Zbl 1064.81558

[29] T Weinacht, J. Ahn and P. Bucksbaum, Controlling the shape of a quantum wavefunction. Nature 397 (1999) 233.

[30] W. Zhu and H. Rabitz, A rapid monotonically convergent iteration algorithm for quantum optimal control over the expectation value of a positive definite operator. J. Chem. Phys. 109 (1998) 385-391.

[31] W. Zhu and H. Rabitz, Potential surfaces from the inversion of time dependent probability density data. J. Chem. Phys. 111 (1999) 472-480.