We consider the time-optimal control by magnetic fields of a spin 1/2 particle in a dissipative environment. This system is used as an illustrative example to show the role of singular extremals in the control of quantum systems. We analyze a simple case where the control law is explicitly determined. We experimentally implement the optimal control using techniques of nuclear magnetic resonance. To our knowledge, this is the first experimental demonstration of singular extremals in quantum systems with bounded control amplitudes.

International audienceWe show how to optimally control the creation of quantum superpositions in a bosonic Josephson junction within the two-site Bose-Hubbard-model framework. Both geometric and purely numerical optimal-control approaches are used, the former providing a generalization of the proposal of Micheli et al. [Phys. Rev. A 67, 013607 (2003)]. While this method is shown not to lead to significant improvements in terms of time of formation and fidelity of the superposition, a numerical optimal-control approach appears more promising, as it allows creation of an almost perfect superposition, within a time short compared to other existing protocols. We analyze the robustness of the optimal solution against atom-number variations. Finally, we discuss the extent to which these optimal solutions could be implemented with state-of-the-art technology