Nonisotropic 3-level quantum systems: complete solutions for minimum time and minimum energy. (English) Zbl 1084.81083

Boscain, Ugo; Chambrion, Thomas; Charlot, Grégoire

Discrete Contin. Dyn. Syst., Ser. B 5, No. 4, 957-990 (2005).

Summary: We apply techniques of subriemannian geometry on Lie groups and of optimal synthesis on 2-D manifolds to the population transfer problem in a three-level quantum system driven by two laser pulses, of arbitrary shape and frequency. In the rotating wave approximation, we consider a nonisotropic model, i.e., a model in which the two coupling constants of the lasers are different. The aim is to induce transitions from the first to the third level, minimizing (1) the time of the transition (with hounded laser amplitudes), (2) the energy transferred by lasers to the system (with fixed final time). After reducing the problem to real variables, for the purpose (1) we develop a theory of time optimal syntheses for distributional problem on 2-D manifolds, while for the purpose (2) we use techniques of subriemannian geometry on 3-D Lie groups. The complete optimal syntheses are computed.

MSC:

81V80	Quantum optics
49J15	Existence theories for optimal control problems involving ordinary differential equations
53C17	Sub-Riemannian geometry
78A60	Lasers, masers, optical bistability, nonlinear optics

Keywords:

control of quantum systems; optimal control; optimal synthesis; subriemannian geometry; minimum time; Hamiltonian systems on Lie groups; Pontryagin maximum principle