Advanced research reference examining the closed and open quantum systems Control of Quantum Systems: Theory and Methods provides an insight into the modern approaches to control of quantum systems evolution, with a focus on both closed and open (dissipative) quantum systems. The topic is timely covering the newest research in the field, and presents and summarizes practical methods and addresses the more theoretical aspects of control, which are of high current interest, but which are not covered at this level in other text books. The quantum control theory and methods written in the book are the results of combination of macro-control theory and microscopic quantum system features. As the development of the nanotechnology progresses, the quantum control theory and methods proposed today are expected to be useful in real quantum systems within five years. The progress of the quantum control theory and methods will promote the progress and development of quantum information, quantum computing, and quantum communication. Equips readers with the potential theories and advanced methods to solve existing problems in quantum optics/information/computing, mesoscopic systems, spin systems, superconducting devices, nano-mechanical devices, precision metrology. Ideal for researchers, academics and engineers in quantum engineering, quantum computing, quantum information, quantum communication, quantum physics, and quantum chemistry, whose research interests are quantum systems control.

Control of Quantum Systems: Theory and Methods provides an insight into the system theory approaches to control of quantum systems evolution, with a focus on both closed and open quantum systems. The topic is timely, covering the newest research in the fi eld, and presents and summarizes practical methods and addresses the more theoretical aspects of control, which are of high current interest, but which are not covered at this level in other textbooks.

The book:

• Presents advanced research examining closed and open quantum systems
• Looks at stability analysis of quantum control systems with particular emphasis on the Lyapunov technique
• Equips readers with the potential theories and advanced methods needed to solve existing problems.

The quantum control theory and methods in this book are the result of a combination of macrocontrol theory and microscopic quantum system features. As the development of the nanotechnology progresses, the quantum control theory and methods proposed today are expected to be useful in real quantum systems within five years. Researchers, academics and engineers in quantum engineering, quantum computing, quantum information, quantum communication, quantum physics, and quantum chemistry whose research interests are quantum systems control will find this book useful. Graduate students whose interests are quantum system control will also find it valuable.

Preface xv

1 Introduction 1

1.1 Quantum States 2

1.2 Quantum Systems Control Models 3

1.2.1 Schrödinger Equation 4

1.2.2 Liouville Equation 4

1.2.3 Markovian Master Equations 5

1.2.4 Non-Markovian Master Equations 5

1.3 Structures of Quantum Control Systems 6

1.4 Control Tasks and Objectives 8

1.5 System Characteristics Analyses 9

1.5.1 Controllability 9

1.5.2 Reachability 9

1.5.3 Observability 10

1.5.4 Stability 10

1.5.5 Convergence 10

1.5.6 Robustness 10

1.6 Performance of Control Systems 11

1.6.1 Probability 11

1.6.2 Fidelity 11

1.6.3 Purity 12

1.7 Quantum Systems Control 13

1.7.1 Description of Control Problems 13

1.7.2 Quantum Control Theory and Methods 13

1.8 Overview of the Book 16

References 18

2 State Transfer and Analysis of Quantum Systems on the Bloch Sphere 21

2.1 Analysis of a Two-level Quantum System State 21

2.1.1 Pure State Expression on the Bloch Sphere 21

2.1.2 Mixed States in the Bloch Sphere 24

2.1.3 Control Trajectory on the Bloch Sphere 26

2.2 State Transfer of Quantum Systems on the Bloch Sphere 27

2.2.1 Control of a Single Spin-1/2 Particle 28

2.2.2 Situation with the Minimum t of Control Fields 30

2.2.3 Situation with a Fixed Time T 31

2.2.4 Numerical Simulations and Results Analyses 33

References 37

3 Control Methods of Closed Quantum Systems 39

3.1 Improved Optimal Control Strategies Applied in Quantum Systems 39

3.1.1 Optimal Control of Quantum Systems 40

3.1.2 Improved Quantum Optimal Control Method 42

3.1.3 Krotov-Based Method of Optimal Control 43

3.1.4 Numerical Simulation and Performance Analysis 45

3.2 Control Design of High-Dimensional Spin-1/2 Quantum Systems 48

3.2.1 Coherent Population Transfer Approaches 48

3.2.2 Relationships between the Hamiltonian of Spin-1/2 Quantum Systems under Control and the Sequence of Pulses 49

3.2.3 Design of the Control Sequence of Pulses 52

3.2.4 Simulation Experiments of Population Transfer 53

3.3 Comparison of Time Optimal Control for Two-Level Quantum Systems 57

3.3.1 Description of System Model 58

3.3.2 Geometric Control 59

3.3.3 Bang-Bang Control 61

3.3.4 Time Comparisons of Two Control Strategies 64

3.3.5 Numerical Simulation Experiments and Results Analyses 66

References 71

4 Manipulation of Eigenstates Based on Lyapunov Method 73

4.1 Principle of the Lyapunov Stability Theorem 74

4.2 Quantum Control Strategy Based on State Distance 75

4.2.1 Selection of the Lyapunov Function 76

4.2.2 Design of the Feedback Control Law 77

4.2.3 Analysis and Proof of the Stability 78

4.2.4 Application to a Spin-1/2 Particle System 80

4.3 Optimal Quantum Control Based on the Lyapunov Stability Theorem 81

4.3.1 Description of the System Model 82

4.3.2 Optimal Control Law Design and Property Analysis 84

4.3.3 Simulation Experiments and the Results Comparisons 86

4.4 Realization of the Quantum Hadamard Gate Based on the Lyapunov Method 88

4.4.1 Mathematical Model 89

4.4.2 Realization of the Quantum Hadamard Gate 90

4.4.3 Design of Control Fields 92

4.4.4 Numerical Simulations and Comparison Results Analyses 94

References 96

5 Population Control Based on the Lyapunov Method 99

5.1 Population Control of Equilibrium State 99

5.1.1 Preliminary Notions 99

5.1.2 Control Laws Design 100

5.1.3 Analysis of the Largest Invariant Set 101

5.1.4 Considerations on the Determination of P 104

5.1.5 Illustrative Example 105

5.1.6 Appendix: Proof of Theorem 5.1 107<...