Formulation and Numerical Solution of Quantum Control Problems

This book provides an introduction to representative nonrelativistic quantum control problems and their theoretical analysis and solution via modern computational techniques. The quantum theory framework is based on the Schrodinger picture, and the optimization theory, which focuses on functional spaces, is based on the Lagrange formalism. The computational techniques represent recent developments that have resulted from combining modern numerical techniques for quantum evolutionary equations with sophisticated optimization schemes. Both finite and infinite-dimensional models are discussed, including the three-level Lambda system arising in quantum optics, multispin systems in NMR, a charged particle in a well potential, Bose-Einstein condensates, multiparticle spin systems, and multiparticle models in the time-dependent density functional framework.

This self-contained book covers the formulation, analysis, and numerical solution of quantum control problems and bridges scientific computing, optimal control and exact controllability, optimization with differential models, and the sciences and engineering that require quantum control methods.

Audience: This book is intended for mathematicians working on ODE/PDE control and optimization problems and the numerical analysis of differential equations; physicists; chemists; and engineers who focus on quantum control problems. It is suitable for advanced courses on ODE/PDE quantum control problems and provides extensively elaborated problems that help the reader develop insight into the main ideas and techniques of quantum control problems. It is also suitable for advanced graduate students and scientists of mathematics, the natural sciences, and engineering.

Contents: Chapter 1: Introduction; Chapter 2: Quantum mechanics and the Schrödinger equation; Chapter 3: Optimal control theory for quantum systems; Chapter 4: Controllability of quantum systems; Chapter 5: Discretization schemes; Chapter 6: Numerical optimization methods; Chapter 7: Application to quantum control problems; Appendix.

Alfio Borzi is professor and chair of scientific computing at the Institute for Mathematics of the University of Wurzburg, Germany. He has contributed considerably to the development of computational methodologies for simulation and optimization with PDEs, with a focus on multilevel methods and on the modeling and numerical solution of quantum and stochastic optimal control problems.

Gabriele Ciaramella is currently a postdoctoral researcher in the Numerical Analysis group of the University of Geneva, Switzerland. He has contributed to the development of methodologies for nonsmooth optimization and control of finite- and infinite-dimensional quantum systems and to the analysis of domain-decomposition methods.

Martin Sprengel is currently a doctoral student in the Scientific Computing group of the Institute for Mathematics of the University of Wurzburg, Germany. He works on the formulation and solution of optimal control problems in the framework of time-dependent density functional theory.