Flatness Based Trajectory Generation of Quantum Systems

A two-states quantum system with one control is proved to be flat. This provides a simple procedure to design smooth open-loop controls that steer in finite time from one eigen-state to the other one. A three-states quantum system with one control is not flat in general. Following the Rabi oscillations used by physicists to control stimulated transition, we associate to this system an averaged control system where the number of controls is increased and where flatness-based motion planning techniques can be used. This allows to steer directly from one eigen-state to any other one without using an additional intermediate eigen-state. In certain energy configurations our method is a noticeable improvement of the standard Rabi transitions strategy and leads to ” active tunnelling control”, as illustrated by simulations. This method can be extended without major difficulties to higher dimensional cases.