Approximate Controllability Via Adiabatic Techniques for the Three-Inputs Controlled Schrödinger Equation

We consider a system described by a controlled bilinear Schrödinger equation with three external inputs. We provide a constructive method to approximately steer the system from a given energy level to a superposition of energy levels corresponding to a given probability distribution. The method is based on adiabatic techniques and works if the spectrum of the Hamiltonian, as a function of the control parameters, admits conical intersections of eigenvalues. We provide sharp estimates of the relation between the error and the controllability time, and we show how to improve these estimates by selecting special control paths. As a by-product of our results we show that conical intersections are stable with respect to general perturbations of the Hamiltonian, and we also provide some results on the regularity of the eigenfamilies along paths locally around conical intersections.