Control landscape for ultrafast manipulation by a qubit

In this work we study extrema of objective functionals for ultrafast manipulation by a qubit. Traps are extrema of the objective functionals which are optimal for manipulation by quantum systems only locally, not globally. Prior work has devoted a large amount of effort to the analysis of traps for quantum systems controlled by laser pulses which are long enough, and, for example, manipulation by a qubit with long control pulses was shown to be trap-free. Ultrafast femtosecond and attosecond control has now become widely applicable, which makes the analysis of traps on the ultrafast time scale a necessity. We complete such analysis for a qubit and show that ultrafast state transfer in a qubit remains trap-free for a wide range of the initial and final states of the qubit. We prove that for this range the probability of transition between the initial and the final states has a saddle but no traps. (paper)