Assessing the structure of classical molecular optimal control landscapes

We consider laser-driven optimal control landscape of a molecule from a classical mechanical perspective. The goal of optimal control in the present work is to steer the molecule from an initial state to a target state, denoted by two distinct points in phase space. Thus, a particular control objective is given as the difference between the final achieved phase space point and the target. The corresponding control landscape is defined as the latter control objective as a functional of the control field. While previous examination of the landscape critical points (i.e., a suboptimal point on the landscape where there is a zero gradient) has shown that the landscape topology is generally trap-free, the structure of the landscape away from these critical points is not well understood. We explore the landscape structure by examining an underlying metric defined as the ratio R of the gradient-based optimization path length of the control field evolution to the Euclidean distance between a given initial control field and the resultant optimal control field, where the latter field corresponds to a point at the top of the landscape. We analyze the path length-to-distance ratio R analytically for a linear forced harmonic oscillator and numerically for a nonlinear forced Morse oscillator. For the linear forced harmonic oscillator, we find that R⩽2 and reaches its minimum value of 1 (i.e., corresponding to “a straight shot” through control space) in the large target time limit, as well as at special finite target times. The ratio R is similarly small for Morse oscillator simulations when following a steepest-ascent path to the top of the landscape, implying that the landscape is quite smooth and devoid of gnarled features. This conclusion is exemplified for a path discovered with R≃1.0 where simply following the initial gradient direction takes the climb very close to the top of the landscape. These findings are consistent with a variety of previous like simulations examining R in quantum control scenarios.