Control of quantum systems : theory and methods

Select an Action

Place Hold(s)
Add to My Lists
Email
Print

Title:

Personal Author:

Cong, Shuang

Publication Information:

Singapore : John Wiley & Sons Inc., 2014

Physical Description:

xi, 425 pages : illustrations ; 25 cm.

ISBN:

9781118608128

Abstract:

"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"--provided by publisher

Subject Term:

Quantum systems -- Automatic control

Control theory

Available:*

Library	Item Barcode	Call Number	Material Type	Item Category 1	Status
Searching... PSZ JB	30000010338237	TK7874.885 C66 2014	Open Access Book	Book	Searching... Unknown

On Order

Summary

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.

Author Notes

Shuang Cong University of Science and Technology of China

About the Author	p. xiii
Preface	p. xv
1 Introduction	p. 1
1.1 Quantum States	p. 2
1.2 Quantum Systems Control Models	p. 3
1.2.1 Schrodinger Equation	p. 4
1.2.1 Liouville Equation	p. 4
1.2.1 Markovian Master Equations	p. 5
1.2.2 Non-Markovian Master Equations	p. 5
1.3 Structures of Quantum Control Systems	p. 6
1.4 Control Tasks and Objectives	p. 8
1.5 System Characteristics Analyses	p. 9
1.5.1 Controllability	p. 9
1.5.2 Reachability	p. 9
1.5.3 Observability	p. 10
1.5.4 Stability	p. 10
1.5.1 Convergence	p. 10
1.5.6 Robustness	p. 10
1.6.1 Performance of Control Systems	p. 11
1.6.1 Probability	p. 11
1.6.1 Fidelity	p. 11
1.6.3 Purity	p. 12
1.7 Quantum Systems Control	p. 13
1.7.1 Description of Control Problems	p. 13
1.7.2 Quantum Control Theory and Methods	p. 13
1.8 Overview of the Book	p. 16
References	p. 18
2 State Transfer and Analysis of Quantum Systems on the Bloch Sphere	p. 21
2.1 Analysis of a Two-level Quantum System State	p. 21
2.1.1 Pure State Expression on the Bloch Sphere	p. 21
2.1.2 Mixed States in the Bloch Sphere	p. 24
2.1.3 Control Trajectory on the Block Sphere	p. 26
2.2 State Transfer of Quantum Systems on the Bloch Sphere	p. 27
2.2.1 Control of a Single Spin-1/2 Particle	p. 28
2.2.2 Situation with the Minimum ¿t of Control Fields	p. 30
2.2.3 Situation with a Fixed Time T	p. 31
2.2.4 Numerical Simulations and Results Analyses	p. 33
References	p. 37
3 Control Methods of Closed Quantum Systems	p. 39
3.1 Improved Optimal Control Strategies Applied in Quantum Systems	p. 39
3.1.1 Optimal Control of Quantum Systems	p. 40
3.1.2 Improved Quantum Optimal Control Method	p. 42
3.1.3 Krotov-Based Method of Optimal Control	p. 43
3.1.4 Numerical Simulation and Performance Analysis	p. 45
3.2 Control Design of High-Dimensional Spin-1/2 Quantum Systems	p. 48
3.2.1 Coherent Population Transfer Approaches	p. 48
5.2.1 Relationships between the Hamiltonian of Spin-1/2 Quantum Systems under Control and the Sequence of Pulses	p. 49
3.2.1 Design of the Control Sequence of Pulses	p. 52
3.2.2 Simulation Experiments of Population Transfer	p. 53
3.3 Comparison of Time Optimal Control for Two-Level Quantum Systems	p. 57
5.3.1 Description of System Model	p. 58
3.3.1 Geometric Control	p. 59
3.3.2 Bang-Bang Control	p. 61
3.3.3 Time Comparisons of Two Control Strategies	p. 64
3.3.4 Numerical Simulation Experiments and Results Analyses	p. 66
References	p. 71
4 Manipulation of Eigenstates - Based on Lyapunov Method	p. 73
4.1 Principle of the Lyapunov Stability Theorem	p. 74
4.2 Quantum Control Strategy Based on State Distance	p. 75
4.2.1 Selection of the Lyapunov Function	p. 76
4.2.2 Design of the Feedback Control Law	p. 77
4.2.5 Analysis and Proof of the Stability	p. 78
4.2.4 Application to a Spin-1/2 Particle System	p. 80
4.3 Optimal Quantum Control Based on the Lyapunov Stability Theorem	p. 81
4.3.1 Description of the System Model	p. 82
4.3.2 Optimal Control Law Design and Property Analysis	p. 84
4.3.3 Simulation Experiments and the Results Comparisons	p. 86
4.4 Realization of the Quantum Hadamard Gate Based on the Lyapunov Method	p. 88
4.4.1 Mathematical Model	p. 89
4.4.2 Realization of the Quantum Hadamard Gate	p. 90
4.4.3 Design of Control Fields	p. 92
4.4.4 Numerical Simulations and Comparison Results Analyses	p. 94
References	p. 96
5 Population Control Based on the Lyapunov Method	p. 99
5.1 Population Control of Equilibrium Slate	p. 99
5.1.1 Preliminary Notions	p. 99
5.1.2 Control Laws Design	p. 100
5.1.3 Analysis of the Largest Invariant Set	p. 101
5.1.4 Considerations on the Determination of P	p. 104
5.1.5 Illustrative Example	p. 105
5.1.6 Appendix: Proof of Theorem 5.1	p. 107
5.2 Generalized Control of Quantum Systems in the Frame of Vector Treatment	p. 110
5.2.1 Design of Control Law	p. 110
5.2.2 Convergence Analysis	p. 113
5.2.3 Numerical Simulation on a Spin-1/2 System	p. 114
5.3 Population Control of Eigenstates	p. 117
5.3.1 System Model and Control Laws	p. 117
5.3.2 Largest Invariant Set of Control Systems	p. 118
5.3.3 Analysis of the Eigenstate Control	p. 118
5.3.4 Simulation Experiments	p. 119
References	p. 123
6 Quantum General State Control Based on Lyapunov Method	p. 125
6.1 Pure State Manipulation	p. 125
6.1.1 Design of Control Law and Discussion	p. 125
6.1.2 Control System Simulations and Results Analyses	p. 129
6.2 Optimal Control Strategy of the Superposition State	p. 131
6.2.1 Preliminary Knowledge	p. 132
6.2.2 Control Law Design	p. 133
6.2.3 Numerical Simulations	p. 134
6.3 Optimal Control of Mixed-State Quantum Systems	p. 135
6.3.1 Model of the System to be Controlled	p. 136
6.3.2 Control Law Design	p. 137
6.3.3 Numerical Simulations and Results Analyses	p. 142
6.4 Arbitrary Pure State to a Mixed-State Manipulation	p. 145
6.4.1 Transfer from an Arbitrary Pure State to an Eigenstate	p. 146
6.4.2 Transfer from an Eigenstate to a Mixed State by Interaction Control	p. 147
6.4.3 Control Design for a Mixed-State Transfer	p. 149
6.4.4 Numerical Simulation Experiments	p. 151
References	p. 154
7 Convergence Analysis Based on the Lyapunov Stability Theorem	p. 155
7.1 Population Control of Quantum States Based on Invariant Subsets with the Diagonal Lyapunov Function	p. 155
7.1.1 System Model and Control Design	p. 155
7.1.2 Correspondence between any Target Eigenstate and the Value of the Lyapunov Function	p. 156
7.1.3 Invariant Set of Control Systems	p. 157
7.1.4 Numerical Simulations	p. 161
7.1.5 Summary and Discussion	p. 164
7.2 A Convergent Control Strategy of Quantum Systems	p. 165
7.2.1 Problem Description	p. 165
7.2.2 Construction Method of the Observable Operator	p. 166
7.2.3 Proof of Convergence	p. 168
7.2.4 Route Extension Strategy	p. 173
7.2.5 Numerical Simulations	p. 174
7.3 Path Programming Control Strategy of Quantum State Transfer	p. 176
7.3.1 Control Law Design Based on the Lyapunov Method in the Interaction Picture	p. 177
7.3.2 Transition Path Programming Control Strategy	p. 178
7.3.3 Numerical Simulations and Results Analyses	p. 182
References	p. 186
8 Control Theory and Methods in Degenerate Cases	p. 187
8.1 Implicit Lyapunov Control of Multi-Control Hamiltonian Systems Based on State Error	p. 187
8.1.1 Control Design	p. 188
5.1.2 Convergence Proof	p. 192
8.1.3 Relation between Two Lyapunov Functions	p. 193
8.1.4 Numerical Simulation and Result Analysis	p. 193
8.2 Quantum Lyapunov Control Based on the Average Value of an Imaginary Mechanical Quantity	p. 195
8.2.1 Control Law Design and Convergence Proof	p. 195
8.2.2 Numerical Simulation and Result Analysis	p. 199
8.3 Implicit Lyapunov Control for the Quantum Liouville Equation	p. 200
8.3.1 Description of Problem	p. 201
8.3.2 Derivation of Control Laws	p. 202
8.3.3 Convergence Analysis	p. 205
8.3.4 Numerical Simulations	p. 209
References	p. 211
9 Manipulation Methods of the General State	p. 213
9.1 Quantum System Schmidt Decomposition and its Geometric Analysis	p. 213
9.1.1 Schmidt Decomposition of Quantum States	p. 214
9.1.2 Definition of Entanglement Degree Based on the Schmidt Decomposition	p. 215
9.1.3 Application of the Schmidt Decomposition	p. 216
9.2 Preparation of Entanglement States in a Two-Spin System	p. 220
9.2.1 Construction of the Two-Spin Systems Model in the Interaction Picture	p. 220
9.2.2 Design of the Control Field Based on the Lyapunov Method	p. 223
9.2.3 Proof of Convergence for the Bell States	p. 226
9.2.4 Numerical Simulations	p. 227
9.3 Purification of the Mixed State for Two-Dimensional Systems	p. 230
9.3.1 Purification by Means of a Probe	p. 230
9.3.2 Purification by interaction Control	p. 232
9.3.3 Numerical Experiments and Results Comparisons	p. 233
9.3.4 Discussion	p. 234
References	p. 235
10 State Control of Open Quantum Systems	p. 237
10.1 State Transfer of Open Quantum Systems with a Single Control Field	p. 237
10.1.1 Dynamical Model of Open Quantum Systems	p. 237
10.1.2 Derivation of Optimal Control Law	p. 238
10.1.3 Control System Design	p. 241
10.1.4 Numerical Simulations and Results Analyses	p. 242
10.2 Purity and Coherence Compensation through the Interaction between Particles	p. 246
10.2.1 Method of Compensation for Purity and Coherence	p. 247
10.2.2 Analysis of System Evolution	p. 250
10.2.3 Numerical Simulations	p. 253
10.2.4 Discussion	p. 255
Appendix 10.A Proof of Equation 10.59	p. 257
References	p. 258
11 State Estimation. Measurement, and Control of Quantum Systems	p. 261
11.1 Suite Estimation Methods in Quantum Systems	p. 261
11.1.1 Background of State Estimation of Quantum Systems	p. 262
11.1.2 Quantum State Estimation Methods Based on the Measurement of Identical Copies	p. 262
11.1.3 Quantum State Reconstruction Methods Based on System Theory	p. 267
11.2 Entanglement Detection and Measurement of Quantum Systems	p. 268
11.2.1 Entanglement States	p. 269
11.2.2 Entanglement Witnesses	p. 271
11.2.3 Entanglement Measures	p. 273
11.2.4 Non-linear Separability Criteria	p. 277
11.3 Decohercnce Control Based on Weak Measurement	p. 278
11.3.1 Construction of a Weak Measurement Operator	p. 279
11.3.2 Applicability of Weak Measurement	p. 280
11.3.3 Effects on States	p. 282
Appendix 1 LA Proof of Normed Linear Space (A, \|\| · \|\|)	p. 286
References	p. 287
12 Stale Preservation of Open Quantum Systems	p. 291
12.1 Coherence Preservation in a ¿-Type Three-Level Atom	p. 291
12.1.1 Models and Objectives	p. 292
12.1.2 Design of Control Field	p. 294
12.1.3 Analysis of Singularities Issues	p. 297
12.1.4 Numerical Simulations	p. 299
12.2 Purity Preservation of Quantum Systems by a Resonant Field	p. 301
12.2.1 Problem Description	p. 302
12.2.2 Purity Property Preservation	p. 303
12.2.3 Discussion	p. 306
12.3 Coherence Preservation in Markovian Open Quantum Systems	p. 307
12.3.1 Problem Formulation	p. 308
12.3.2 Design of Control Variables	p. 311
12.3.3 Numerical Simulations	p. 313
12.3.4 Discussion	p. 315
Appendix 12 A Derivation of H C	p. 316
References	p. 317
13 State Manipulation in Decoherence-Free Subspace	p. 321
13.1 State Transfer and Coherence Maintenance Based on DFS for a Four-Level Energy Open Quantum System	p. 321
13.1.1 Construction of DFS and the Desired Target State	p. 322
13.1.2 Design of the Lyapunov-Based Control Law for State Transfer	p. 325
13.1.3 Numerical Simulations	p. 326
13.2 State Transfer Based on a Decoherence-Free Target State for a A-Type N-Level Atomic System	p. 328
13.2.1 Construction of the Decoherence-Free Target State	p. 328
13.2.2 Design of the Lyapunov-Based Control Law for State Transfer	p. 331
13.2.3 Numerical Simulations and Results Analyses	p. 332
13.3 Control of Quantum States Based on the Lyapunov Method in Decoherence-Free Subspaces	p. 336
13.3.1 Problem Description	p. 336
13.3.2 Control Design in the Interaction Picture	p. 338
13.3.3 Construction of P and Convergence Analysis	p. 339
13.3.4 Numerical Simulation Examples and Discussion	p. 345
References	p. 348
14 Dynamic Decoupling Quantum Control Methods	p. 351
14.1 Phase Decoherence Suppression of an n-Level Atom in ¿;-Configuration with Bang-Bang Controls	p. 351
14.1.1 Dynamical Decoupling Mechanism	p. 352
14.1.2 Design of the Bang-Bang Operations Group in Phase Decoherence	p. 355
14.1.3 Examples of Design	p. 357
14.2 Optimized Dynamical Decoupling in ¿-Type n-Level Atom	p. 360
14.2.1 Periodic Dynamical Decoupling	p. 361
14.2.2 Uhrig Dynamical Decoupling	p. 361
14.2.3 Behaviors of Quantum Coherence under Various Dynamical Decoupling Schemes	p. 362
14.2.4 Examples	p. 365
14.2.5 Discussion	p. 366
14.3 An Optimized Dynamical Decoupling Strategy to Suppress Decoherence	p. 366
14.3.1 Universal Dynamical Decoupling for a Quint	p. 367
14.3.2 An Optimized Dynamical Decoupling Scheme	p. 369
14.3.3 Simulation and Comparison	p. 369
14.3.4 Discussion	p. 375
References	p. 378
15 Trajectory Tracking of Quantum Systems	p. 381
15.1 Orbit Tracking of Quantum States Based on the Lyapunov Method	p. 382
15.1.1 Description of the System Model	p. 382
15.1.2 Design of Control Law	p. 384
15.1.3 Numerical Simulation Experiments and Results Analysis	p. 385
15.2 Orbit Tracking Control of Quantum Systems	p. 389
15.2.1 System Model and Control Law Design	p. 390
15.2.1 Numerical Simulation Experiments	p. 391
15.3 Adaptive Trajectory Tracking of Quantum Systems	p. 394
15.3.1 Description of the System Model	p. 396
15.3.2 Control System Design and Characteristic Analysis	p. 398
15.3.3 Numerical Simulation and Result Analysis	p. 400
15.4 Convergence of Orbit Tracking for Quantum Systems	p. 402
15.4.1 Description of the Control System Model	p. 403
15.4.2 Control Law Derivation	p. 404
15.4.3 Convergence Analysis	p. 404
15.4.4 Applications and Experimental Results Analyses	p. 411
References	p. 416
Index	p. 419

Available:*

On Order

Summary

Summary

Author Notes

Table of Contents