Exploring the level sets of quantum control landscapes

A quantum control landscape is defined by the value of a physical observable as a functional of the time-dependent control field E (t) for a given quantum-mechanical system. Level sets through this landscape are prescribed by a particular value of the target observable at the final dynamical time T, regardless of the intervening dynamics. We present a technique for exploring a landscape level set, where a scalar variable s is introduced to characterize trajectories along these level sets. The control fields E (s,t) accomplishing this exploration (i.e., that produce the same value of the target observable for a given system) are determined by solving a differential equation over s in conjunction with the time-dependent Schrödinger equation. There is full freedom to traverse a level set, and a particular trajectory is realized by making an a priori choice for a continuous function f (s,t) that appears in the differential equation for the control field. The continuous function f (s,t) can assume an arbitrary form, and thus a level set generally contains a family of controls, where each control takes the quantum system to the same final target value, but produces a distinct control mechanism. In addition, although the observable value remains invariant over the level set, other dynamical properties (e.g., the degree of robustness to control noise) are not specifically preserved and can vary greatly. Examples are presented to illustrate the continuous nature of level-set controls and their associated induced dynamical features, including continuously morphing mechanisms for population control in model quantum systems.