Optimal control of quantum-mechanical systems: Existence, numerical approximation, and applications

Anthony P. Peirce, Mohammed A. Dahleh, Herschel Albert Rabitz

Research output: Contribution to journalArticle

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Abstract

The optimal control of the path to a specified final state of a quantum-mechanical system is investigated. The problem is formulated as a minimization problem over appropriate function spaces, and the well-posedness of this problem is is established by proving the existence of an optimal solution. A Lagrange-multiplier technique is used to reduce the problem to an equivalent optimization problem and to derive necessary conditions for a minimum. These necessary conditions form the basis for a gradient iterative procedure to search for a minimum. A numerical scheme based on finite differences is used to reduce the infinite-dimensional minimization problem to an approximate finite-dimensional problem. Numerical examples are provided for final-state control of a diatomic molecule represented by a Morse potential. Within the context of this optimal control formulation, numerical results are given for the optimal pulsing strategy to demonstrate the feasibility of wave-packet control and finally to achieve a specified dissociative wave packet at a given time. The optimal external optical fields generally have a high degree of structure, including an early time period of wave-packet phase adjustment followed by a period of extensive energy deposition to achieve the imposed objective. Constraints on the form of the molecular dipole (e.g., a linear dipole) are shown to limit the accessibility (i.e., controllability) of certain types of molecular wave-packet objectives. The nontrivial structure of the optimal pulse strategies emphasizes the ultimate usefulness of an optimal-control approach to the steering of quantum systems to desired objectives.

Original languageEnglish (US)
Pages (from-to)4950-4964
Number of pages15
JournalPhysical Review A
Volume37
Issue number12
DOIs
StatePublished - Jan 1 1988

All Science Journal Classification (ASJC) codes

  • Atomic and Molecular Physics, and Optics

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