Optimally controlled quantum molecular dynamics: A perturbation formulation and the existence of multiple solutions

Metin Demiralp, Herschel Albert Rabitz

Research output: Contribution to journalArticle

91 Scopus citations

Abstract

This work considers optimal control of quantum-mechanical systems within the framework of perturbation theory with respect to the controlling optical electric field. The control problem is expressed in terms of a cost functional including the physical objective, the penalties, and constraints. The resultant nonlinear variational equations are linearized by considering the lowest-order term in an expansion in powers of the optical-field strength. The optical field is found to satisfy a linear integral equation, and the solution may be expressed in terms of a generalized eigenvalue problem associated with the corresponding kernel. A full determination of the field is specified through the solution to the integral equation and the roots of an accompanying linearized spectral equation for a characteristic multiplier parameter. Each discrete value of the latter parameter corresponds to a particular solution to the variational equations. As a result, it is argued that under very general conditions there will be a denumerably infinite number of solutions to well-posed quantum-mechanical optimal-control problems.

Original languageEnglish (US)
Pages (from-to)809-816
Number of pages8
JournalPhysical Review A
Volume47
Issue number2
DOIs
StatePublished - Jan 1 1993

All Science Journal Classification (ASJC) codes

  • Atomic and Molecular Physics, and Optics

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