Control of Photochemical Branching: Novel Procedures for Finding Optimal Pulses and Global Upper Bounds

Abstract

Given the great progress in femtosecond laser technology — with amplitude, frequency and phase control of the electric field — the systematic design of optical pulse sequences to control microscopic atomic and molecular motion has become an inexorable topic of study. The problem may be formulated variationally as the search for the optimal pulse sequence (i.e. complex electric field amplitude, ε(t)) which prepares a wavepacket such as to maximize the expectation value of an appropriate projection operator at some final time. For control of photofragmentation, for instance, the projection operator is equal to 1 for a particular chemical arrangement channel and 0 elsewhere. The optimization must be performed subject to the constraint that the wavepacket satisfy the time dependent Schrodinger equation in the presence of the field, leading to a mathematical structure of the type found in Optimal Control Theory (OCT). The first part of the paper provides the physical motivation for the definition of the objective functional. Then, the original objective, subject to the constraint of the TDSE, is transformed into an unconstrained, or modified objective and partitioned in a novel way following the work of Krotov. The partitioned form for the objective functional leads to a great deal of insight into the problem, as well as to a variety of remarkable new results. It is first used to introduce and contrast the necessary and sufficient conditions for optimal pulses. The partitioned objective is then used to derive a novel and remarkably efficient iterative algorithm to find optimal pulses. The new algorithm, which can be said to derive from the sufficient conditions of optimality, explicitly includes both linear and quadratic dependence of the objective on the field and the wavefunction, and as such is able to take macrosteps in the field at every iteration. In contrast, the usual gradient type methods include only the linear dependence of the objective on the field and the wavefunction; as a result they require a line search at each iteration and converge very slowly. Numerically, the new method several times more efficient than gradient type methods for the systems we have studied. In the last part of the paper the sufficient conditions for global optimality are used to obtain the first known upper bound on control of chemical reactions, subject to a penalty on the field energy. Although the upper bound is less than one (one represents 100% yield in the desired chemical arrangement channel) the discrepancy vanishes for zero penalty, consistent with 100% control of the channel from which the wavepacket exits.

Keywords

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Alfred P. Sloan Foundation Fellow 1991–93