The aim of this contribution is to rene some of the computations
of . The Lindblad equation modelling a two-level dissipative quantum system is investigated. The control can be interpretated as the action of a laser to rotate a molecule in gas phase, or as the eect of a magnetic eld on a spin 1=2 particle. For the energy cost, normal extremals of the maximum principle are solution to a three-dimensional Hamiltonian with parameters. The analysis is focussed on an integrable submodel which denes outside singularities a pseudo-Riemannian metric in dimension ve. Complete quadratures are given for this subcase by means of Weierstra elliptic functions. Preliminary computations of cut and conjugate loci are also provided for a two-dimensional restriction using .
Mathematics Subject Classification: Primary: 49K15, 81Q05.