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Introduction to Quantum Control and Dynamics




ISBN 9781584888840
Published August 3, 2007 by Chapman and Hall/CRC
360 Pages - 49 B/W Illustrations

 
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Book Description

The introduction of control theory in quantum mechanics has created a rich, new interdisciplinary scientific field, which is producing novel insight into important theoretical questions at the heart of quantum physics. Exploring this emerging subject, Introduction to Quantum Control and Dynamics presents the mathematical concepts and fundamental physics behind the analysis and control of quantum dynamics, emphasizing the application of Lie algebra and Lie group theory.

After introducing the basics of quantum mechanics, the book derives a class of models for quantum control systems from fundamental physics. It examines the controllability and observability of quantum systems and the related problem of quantum state determination and measurement. The author also uses Lie group decompositions as tools to analyze dynamics and to design control algorithms. In addition, he describes various other control methods and discusses topics in quantum information theory that include entanglement and entanglement dynamics. The final chapter covers the implementation of quantum control and dynamics in several fields.

Armed with the basics of quantum control and dynamics, readers will invariably use this interdisciplinary knowledge in their mathematical, physics, and engineering work.

Table of Contents

QUANTUM MECHANICS
States and Operators
Observables and Measurement
Dynamics of Quantum Systems

MODELING OF QUANTUM CONTROL SYSTEMS: EXAMPLES
Quantum Theory of Interaction of Particles and Fields
Approximations and Modeling: Molecular Systems
Spin Dynamics and Control
Mathematical Structure of Quantum Control Systems

CONTROLLABILITY
Lie Algebras and Lie Groups
Controllability Test: The Dynamical Lie Algebra
Notions of Controllability for the State
Pure State Controllability
Equivalent State Controllability
Equality of Orbits

OBSERVABILITY AND STATE DETERMINATION
Quantum State Tomography
Observability
Observability and Methods for State Reconstruction

LIE GROUP DECOMPOSITIONS AND CONTROL
Decompositions of SU(2) and Control of Two Level Systems
Decomposition in Planar Rotations
Cartan Decompositions
Levi Decomposition
Examples of Application of Decompositions to Control

OPTIMAL CONTROL OF QUANTUM SYSTEMS
Formulation of the Optimal Control Problem
The Necessary Conditions of Optimality
Example: Optimal Control of a Two Level Quantum System
Time Optimal Control of Quantum Systems
Numerical Methods for Optimal Control of Quantum Systems

MORE TOOLS FOR QUANTUM CONTROL
Selective Population Transfer via Frequency Tuning
Time Dependent Perturbation Theory
Adiabatic Control
STIRAP
Lyapunov Control of Quantum Systems

ANALYSIS OF QUANTUM EVOLUTIONS: ENTANGLEMENT, ENTANGLEMENT MEASURES, AND DYNAMICS
Entanglement of Quantum Systems
Dynamics of Entanglement
Local Equivalence of States

APPLICATIONS OF QUANTUM CONTROL AND DYNAMICS
Nuclear Magnetic Resonance Experiments
Molecular Systems Control
Atomic Systems Control: Implementations of Quantum Information Processing with Ion Traps

APPENDIX A: POSITIVE AND COMPLETELY POSITIVE MAPS, QUANTUM OPERATIONS, AND GENERALIZED MEASUREMENT THEORY
Positive and Completely Positive Maps
Quantum Operations and Operator Sum Representation
Generalized Measurement Theory

APPENDIX B: LAGRANGIAN AND HAMILTONIAN FORMALISM IN CLASSICAL ELECTRODYNAMICS
Lagrangian Mechanics
Extension of Lagrangian Mechanics to Systems with Infinite Degrees of Freedom
Lagrangian and Hamiltonian Mechanics for a System of Interacting
Particles and Field

APPENDIX C: CARTAN SEMISIMPLICITY CRITERION AND CALCULATION OF THE LEVI DECOMPOSITION
The Adjoint Representation
Cartan Semisimplicity Criterion
Quotient Lie Algebras
Calculation of the Levi Subalgebra in the Levi Decomposition
Algorithm for the Levi Decomposition

APPENDIX D: PROOF OF THE CONTROLLABILITY TEST OF THEOREM 3.2.1

APPENDIX E: THE BAKER-CAMPBELL-HAUSDORFF FORMULA AND SOME EXPONENTIAL FORMULAS

APPENDIX F: PROOF OF THEOREM 6.2.1

REFERENCES

INDEX

Notes and Exercises appear at the end of every chapter.

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Reviews

"Overall, this book is a fantastic success in terms of organization, clarity, and strictness . . . I will actively recommend this book to my colleagues and students."

– Jiangbin Gong, Department of Physics and Centre of Computational Science and Engineering, National University of Singapore, in International Journal of Quantum Information, 2008, Vol. 6, No. 2

"...the book by D. D'Alessandro provides a consistent and well-written introductory to quantum dynamics and quantum control that can be recommended for students and teachers, and also for general readers of a rather broader circle with some background in quantum mechanics."                                                                                                                            Eugene Kryachko, Zentralblatt MATH, 2008, 1139.