D’Alessandro D. Introduction to Quantum Control and Dynamics

Boca Raton: Taylor & Francis Group, LLC. – 2008. –
339. In many experimental setups, an electromagnetic field interacts with a system, such as an atom, a nucleus or an electron, whose dynamics follows the laws of quantum mechanics. While it is appropriate to treat the latter system as a quantum mechanical one, the electromagnetic field can often be treated as a classical field, giving predictions that agree with macroscopic observation. This is the so-called semiclassical approximation. This book is an introduction to the analysis and control of quantum dynamics which emphasizes the application of Lie algebra and Lie group theory. It was developed from lecture notes written during a course given at Iowa State University in the spring semester of 2004. This course was intended for advanced undergraduate and beginning graduate students in mathematics, physics and engineering. The book is organized as follows. Chapter 1 introduces the basics of quantum mechanics, with an emphasis on dynamics and a quantum information theory point of view. In Chapter 2, from fundamental physics, a class of models for quantum control systems is derived. In Chapter 3, we study the controllability of quantum systems and, in the process, we introduce many concepts of Lie algebra and Lie group theory as well as Lie transformation groups. Chapter 4 deals with observability of quantum systems and the related problem of quantum state determination and measurement. Chapter 5 introduces Lie group decompositions as a tool to analyze dynamics and design control algorithms. In Chapters 6 and 7 we describe more methods for control. Chapter 8 presents some topics in quantum information theory. Chapter 9 discusses physical set-ups of implementations and applications of quantum control and dynamicsQuantum Mechanics
Modeling of Quantum Control Systems; Examples
Controllability
Observability and State Determination
Lie Group Decompositions and Control
Optimal Control of Quantum Systems
More Tools for Quantum Control
Analysis of Quantum Evolutions; Entanglement, Entanglement Measures and Dynamics
Applications of Quantum Control and Dynamics
Positive and Completely Positive Maps, Quantum Operations and Generalized Measurement Theory
Lagrangian and Hamiltonian Formalism in Classical Electrodynamics
Cartan Semisimplicity Criterion and Calculation of the Levi Decomposition
Proof of the Controllability Test of Theorem 3.2.1
The Baker-Campbell-Hausdorff Formula and Some Exponential Formulas
Proof of Theorem 6.2.1