Optimal control in laser-induced population transfer for two- and three-level quantum systems

We apply the techniques of control theory and of sub-Riemannian geometry to laser-induced population transfer in twoand 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 wellknown technique of ‘‘p-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 ‘‘p-pulse transfer.’’ This geometrical picture also allows one to analyze the population transfer by adiabatic passage. © 2002 American Institute of Physics. @DOI: 10.1063/1.1465516#