# American Institute of Mathematical Sciences

November  2005, 5(4): 957-990. doi: 10.3934/dcdsb.2005.5.957

## Nonisotropic 3-level quantum systems: complete solutions for minimum time and minimum energy

 1 SISSA, via Beirut 2-4 34014 Trieste, Italy 2 SYSTeMS Research Group, University of Ghent, Technologiepark - Zwijnaarde 9, 9052 Zwijnaarde, Belgium 3 ACSIOM, I3M, CC51, Université Montpellier II, Place Eugène Bataillon, 34095 Montpellier Cedex 5, France

Received  October 2004 Revised  April 2005 Published  August 2005

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 bounded 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.
Citation: Ugo Boscain, Thomas Chambrion, Grégoire Charlot. Nonisotropic 3-level quantum systems: complete solutions for minimum time and minimum energy. Discrete & Continuous Dynamical Systems - B, 2005, 5 (4) : 957-990. doi: 10.3934/dcdsb.2005.5.957
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