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