Geometric analysis of minimum time trajectories for a two-level quantum system

We consider the problem of controlling in minimum time a two-level quantum system which can be subject to a drift. The control is assumed to be bounded in magnitude, and to affect two or three independent generators of the dynamics. We describe the time optimal trajectories in SU(2), the Lie group of possible evolutions for the system, by means of a particularly simple parametrization of the group. A key ingredient of our analysis is the introduction of the optimal front line. This tool allows us to fully characterize the time-evolution of the reachable sets, and to derive the worst-case operators and the corresponding times. The analysis is performed in any regime: controlled dynamics stronger, of the same magnitude or weaker than the drift term, and gives a method to synthesize quantum logic operations on a two-level system in minimum time.