The optimal control of the path to a specified 6nal 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 I.agrange-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 6nite di8'erences is used to reduce the in6nite-dimensional minimization problem to an approximate finite-dimensional problem. Numerical examples are provided for 6nal-state control of a diatomic molecule represented by a Morse potential. %ithia the context of this optimal control formulation, numerical results are given for the optimal pulsing strategy to demonstrate the feasibility of wavepacket control and 6nally to achieve a speci6ed dissociative wave packet at a given time. The optimal external optical Aelds 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.
The purpose of this brief note is to show that the set of states reachable from a given initial state for a finite dimensional quantum system is equal to the orbit under the Lie group generated by the Lie algebra generated by the internal and external
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