Measuring the distance from saddle points and driving to locate them over quantum control landscapes

Optimal control of quantum phenomena involves the introduction of a cost functional J to characterize the degree of achieving a physical objective by a chosen shaped electromagnetic field. The cost functional dependence upon the control forms a control landscape. Two theoretically important canonical cases are the landscapes associated with seeking to achieve either a physical observable or a unitary transformation. Upon satisfaction of particular assumptions, both landscapes are analytically known to be trap-free, yet possess saddle points at precise suboptimal J values. The presence of saddles on the landscapes can influence the effort needed to find an optimal field. As a foundation to future algorithm development and analyzes, we define metrics that identify the 'distance' from a given saddle based on the sufficient and necessary conditions for the existence of the saddles. Algorithms are introduced utilizing the metrics to find a control such that the dynamics arrive at a targeted saddle. The saddle distance metric and saddle-seeking methodology is tested numerically in several model systems.