Superposition principle for the simultaneous optimization of collective responses.

Abstract

We study the dynamics of sets of independent systems, all of which are coupled to the same time-dependent external force. Using optimal control theory, we compute the most efficient temporal pulse shape for this force that can maximize simultaneously the collective response of these systems. This response can be a weighted sum of all amplitudes at the final interaction time. Remarkably, it turns out that for certain systems this optimal force for the collective response can be related to the individual forces that would optimize each system separately. We illustrate this superposition principle for the simultaneous optimization of collective responses with numerical and also analytical solutions for sets of damped linear and nonlinear oscillators. We also apply this principle to predict the optimal temporal profile of a laser pulse that can maximize the final macroscopic polarization (total dipole moment) of a set of quantum mechanical two-level atoms.