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Optimal control in laser-induced population transfer for two- and three-level quantum systems. (English) Zbl 1059.81195
Summary: We apply the techniques of control theory and of sub-Riemannian geometry to laser-induced population transfer in two- and three-level quantum systems. The aim is to induce complete population transfer by one or two laser pulses minimizing the pulse fluences. Sub-Riemannian geometry and singular-Riemannian geometry provide a natural framework for this minimization, where the optimal control is expressed in terms of geodesics. We first show that in two-level systems the well-known technique of “$$\pi$$-pulse transfer” in the rotating wave approximation emerges naturally from this minimization. In three-level systems driven by two resonant fields, we also find the counterpart of the “$$\pi$$-pulse transfer”. This geometrical picture also allows one to analyze the population transfer by adiabatic passage.

##### MSC:
 81V80 Quantum optics 49N90 Applications of optimal control and differential games 53C17 Sub-Riemannian geometry
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