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- Herausgeber: John Wiley & Sons
- Kategorie: Fachliteratur
- Sprache: Englisch
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.
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Seitenzahl: 715
Veröffentlichungsjahr: 2014
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Table of Contents
Title Page
Copyright
About the Author
Preface
Chapter 1: Introduction
1.1 Quantum States
1.2 Quantum Systems Control Models
1.3 Structures of Quantum Control Systems
1.4 Control Tasks and Objectives
1.5 System Characteristics Analyses
1.6 Performance of Control Systems
1.7 Quantum Systems Control
1.8 Overview of the Book
References
Chapter 2: State Transfer and Analysis of Quantum Systems on the Bloch Sphere
2.1 Analysis of a Two-level Quantum System State
2.2 State Transfer of Quantum Systems on the Bloch Sphere
References
Chapter 3: Control Methods of Closed Quantum Systems
3.1 Improved Optimal Control Strategies Applied in Quantum Systems
3.2 Control Design of High-Dimensional Spin-1/2 Quantum Systems
3.3 Comparison of Time Optimal Control for Two-Level Quantum Systems
References
Chapter 4: Manipulation of Eigenstates—Based on Lyapunov Method
4.1 Principle of the Lyapunov Stability Theorem
4.2 Quantum Control Strategy Based on State Distance
4.3 Optimal Quantum Control Based on the Lyapunov Stability Theorem
4.4 Realization of the Quantum Hadamard Gate Based on the Lyapunov Method
References
Chapter 5: Population Control Based on the Lyapunov Method
5.1 Population Control of Equilibrium State
5.2 Generalized Control of Quantum Systems in the Frame of Vector Treatment
5.3 Population Control of Eigenstates
References
Chapter 6: Quantum General State Control Based on Lyapunov Method
6.1 Pure State Manipulation
6.2 Optimal Control Strategy of the Superposition State
6.3 Optimal Control of Mixed-State Quantum Systems
6.4 Arbitrary Pure State to a Mixed-State Manipulation
References
Chapter 7: Convergence Analysis Based on the Lyapunov Stability Theorem
7.1 Population Control of Quantum States Based on Invariant Subsets with the Diagonal Lyapunov Function
7.2 A Convergent Control Strategy of Quantum Systems
7.3 Path Programming Control Strategy of Quantum State Transfer
References
Chapter 8: Control Theory and Methods in Degenerate Cases
8.1 Implicit Lyapunov Control of Multi-Control Hamiltonian Systems Based on State Error
8.2 Quantum Lyapunov Control Based on the Average Value of an Imaginary Mechanical Quantity
8.3 Implicit Lyapunov Control for the Quantum Liouville Equation
References
Chapter 9: Manipulation Methods of the General State
9.1 Quantum System Schmidt Decomposition and its Geometric Analysis
9.2 Preparation of Entanglement States in a Two-Spin System
9.3 Purification of the Mixed State for Two-Dimensional Systems
References
Chapter 10: State Control of Open Quantum Systems
10.1 State Transfer of Open Quantum Systems with a Single Control Field
10.2 Purity and Coherence Compensation through the Interaction between Particles
Appendix 10.A Proof of Equation 10.59
References
Chapter 11: State Estimation, Measurement, and Control of Quantum Systems
11.1 State Estimation Methods in Quantum Systems
11.2 Entanglement Detection and Measurement of Quantum Systems
11.3 Decoherence Control Based on Weak Measurement
Appendix 11.A Proof of Normed Linear Space
References
Chapter 12: State Preservation of Open Quantum Systems
12.1 Coherence Preservation in a Λ-Type Three-Level Atom
12.2 Purity Preservation of Quantum Systems by a Resonant Field
12.3 Coherence Preservation in Markovian Open Quantum Systems
Appendix 12.A Derivation of
References
Chapter 13: State Manipulation in Decoherence-Free Subspace
13.1 State Transfer and Coherence Maintainance Based on DFS for a Four-Level Energy Open Quantum System
13.2 State Transfer Based on a Decoherence-Free Target State for a Λ-Type N-Level Atomic System
13.3 Control of Quantum States Based on the Lyapunov Method in Decoherence-Free Subspaces
References
Chapter 14: Dynamic Decoupling Quantum Control Methods
14.1 Phase Decoherence Suppression of an n-Level Atom in Ξ-Configuration with Bang-Bang Controls
14.2 Optimized Dynamical Decoupling in Ξ-Type n-Level Atom
14.3 An Optimized Dynamical Decoupling Strategy to Suppress Decoherence
References
Chapter 15: Trajectory Tracking of Quantum Systems
15.1 Orbit Tracking of Quantum States Based on the Lyapunov Method
15.2 Orbit Tracking Control of Quantum Systems
15.3 Adaptive Trajectory Tracking of Quantum Systems
15.4 Convergence of Orbit Tracking for Quantum Systems
References
Index
This edition first published 2014
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Library of Congress Cataloging-in-Publication Data
Cong, Shuang.
Control of quantum systems : theory and methods / Shuang Cong.
pages cm
Includes bibliographical references and index.
ISBN 978-1-118-60812-8 (hardback)
1. Quantum systems—Automatic control. 2. Control theory. I. Title.
TK7874.885.C66 2014
530.1201′1—dc23
2013037723
A catalogue record for this book is available from the British Library.
ISBN: 978-1-118-60812-8
About the Author
Shuang Cong was born in Hefei, China. She obtained her Bachelor's degree from the Department of Automatic Control, Beijing University of Aeronautics and Astronautics, in 1982, and her PhD degree in systems engineering from the University of Rome “La Sapienza”, Italy, in 1995. She is currently Professor of Automation at the University of Science and Technology of China (USTC). Professor Cong has authored more than 340 research papers and invited papers published in academic journals at home and abroad or presented to international academic conferences. She has published one textbook in Chinese, entitled Neural Network Theory and Applications towards to the MATLAB Toolbox (third edition, 2009; the second edition was published in 2003 and the first edition in 1998), and five Chinese monographs: Neural Networks, Fuzzy System and the Application in Motion Control (2001), Vision-based Network Remote Control Systems (2013), Introduction to Quantum Mechanical Systems Control (2006), Applied Motion Control Technologies (first author, 2006), and Parallel Robots-Modeling, Control Optimization and Applications (first author, 2010). She has also edited an English book entitled Frontiers in Adaptive Control (2009). Control of Quantum Systems: Theory and Methods is her second monograph about quantum systems control.
Preface
Quantum control theory and methods are gaining increased attention and importance in the world. This area involves many fields of science, and there are many issues to be addressed so it is a continually developing research field. Until recently it has been difficult to find an integrated treatment of the relevant topics in one source, partly because of their rapidly changing nature. However, quantum control theory and methods are now sufficiently mature to warrant a reference book that extends the classical treatment of control theory and methods to the quantum domain. This book aims to provide a self-contained survey of these topics for graduate students and researchers in quantum engineering and quantum information sciences. The contents include the newest research achievements obtained in recent years. They are the result of a combination of macro-control theory and microscopic quantum system features.
The quantum control theory and methods in this book may have the potential to solve existing problems that cannot be solved by quantum physics, quantum chemistry, quantum computing, quantum communication, and quantum information. The progress of quantum control theory and methods will promote the progress and development of quantum information, quantum computing, and quantum communication. This book may open the door for researchers, academics, and engineers in relevant research fields to solve existing problems and provide them with new theories and methods for controling quantum systems.
My previous book on quantum system control, Introduction to Quantum Mechanical Systems Control (Scientific Press. 2006, ISBN 7-03-016474-1, in Chinese), covered the theoretical basis and modeling of quantum mechanical systems, the Lie group and Lie algebra and its applications, unitary evolution operator decomposition and its implementation, bilinear systems and their control, the controllability and reachability of quantum systems, feedback control of quantum systems, mixed and entangled states and their analysis, the geometric algebra of quantum systems, optimal control of quantum systems, quantum measurement, feedback coherent control of quantum systems, and the application of quantum systems. This book is my second book on quantum system control and is a research reference book.
There are many issues to be addressed in quantum information, quantum computing, and quantum communication. All of these issues are essentially quantum system control problems, and quantum control theory and methods are used to solve them. This book can be a research reference book for graduate students, researchers, academics, and engineers in quantum physics, chemistry, information, communication, electrical and mechanical engineering, applied mathematics, and computer science whose research interests involve quantum systems control. The only prerequisite is an introductory course in quantum mechanics at first-year graduate level, as typically taught in physics departments. One of the book's primary goals is to give the graduate student with a limited background in control theory, but a familiarity with quantum dynamical systems, the tools to engineer those systems. A second objective is to offer a convenient reference for active and experienced researchers in quantum engineering and quantum information theory. The first chapter introduces the basic concepts of quantum states and quantum control models, including Schrödinger equations, the Liouville equation, Markovian master equations, and non-Markovian master equations. For control systems we introduce structures of quantum control systems, control tasks and objectives, system characteristic analysis, performance of control systems, description of control problems, and quantum control theories and methods. However, this requires readers who are undergraduate students to have some knowledge of advanced mathematics and advanced algebra.
The book focuses on control theory and methods in quantum systems, divided into two parts: control theory and methods for closed and open quantum systems. The control theory and methods for closed quantum systems include geometric control, bang-bang control, improved optimal control strategy, and the Krotov-based method of optimal control. Because optimal quantum control is the most popular quantum method and it was introduced it in my earliert book, Introduction to Quantum Mechanical Systems Control, it has not been repeated in this book. However, throughout the book there are references to the concept of optimal control.
The Lyapunov-based quantum control method is a highlight of this book. Five chapters are used to introduce this control method, of which three cover the control method used for state-to-state transfer and population control, the transfer of states includes eigenstates, superposition states, and mixed states, and the convergence of this control method. A further chapter deals with the issues of degenerate cases. Other types of state manipulation, such as the preparation of entanglement states, the purification of mixed states, and Schmidt decomposition of quantum states, are also covered.
Five chapters cover the control methods of open quantum systems. For general cases, the control methods of state transfer with a single control field, purity and coherence compensation through the interaction between particles, and decoherence control based on weak measurement are introduced. One chapter investigates the state preservation of open quantum systems, which concerns the purity preservation of quantum systems through the resonant field and coherence preservation in Markovian open quantum systems. The decoherence-free subspace is a special control method for the control of open quantum systems, and this is covered in a separate chapter, as is the dynamic decoupling quantum control method, which is another important control method for open quantum systems
Finally, as with control in system engineering, systems control should be classified in two categories: state transfer (or state regulation) and trajectory tracking. Trajectory tracking of quantum systems is presented in the last chapter of this book.
This book required the cooperation of many people, including Dr Sen Kuang, Dr Yuesheng Lou, Dr Jie Yang, Dr Fangfang Meng, Ms Yuanyuan Zhang, Dr Fei Yang, Ms Jianxiu Liu, Mr Jie Wen, Mr Linping Chan, and Ms Yaping Zhu.
In writing this book I have benefited from interactions with many people. In particular, I would like to thank Professor Herschel Rabitz for his kind hospitality at the Princeton University for 5 months from January through June 2012, during which time I learnt many points of interest. My thanks also go to the Frick Chemistry Laboratory at Princeton University for its hospitality. Doctoral students and researchers at the laboratory have been very kind and I learnt a lot from discussions with them.
I would like to thank John Wiley & Sons for its assistance and for agreeing to publish this book and the National Science Foundation for financial support during the research work.
Shuang Cong June 29, 2013 University of Science and Technology of China, Hefei, P. R. China
Chapter 1
Introduction
The theory of quantum mechanics was one of the major discoveries of the history of science in the twentieth century. It is very important to study the properties of quantum mechanical systems and their control. As quantum technologies have matured, a lot of practical applications of quantum control have been realized in quantum optics, cavity quantum electro-dynamics (QED), atomic spin ensembles, ion trapping, and Bose–Einstein condensation, and so on, which means that the manipulation of quantum phenomena is a rapidly growing research field. The improvements in nanotechnology and its manufacture process as well as increasing interests in new applications of quantum effects, including quantum information process, mean the control of quantum phenomena is becoming a growing concern all over the world in areas such as quantum computation, quantum chemistry, nano-material, and quantum physics. In the past three decades, researchers have been trying to expand the control theories that are obtained from the macroscopic world to the microscopic world, and this has gradually become a new system control theory in interdisciplinary fields: quantum control theory. The methods and technologies of quantum control have become one of the leading research areas in the world. The main topics of quantum control theory are, from the control system perspective, to investigate how to manipulate a system state trajectory and its evolution. For this purpose, quantum control theory is used to design an external realizable control law to achieve a desired control goal by combining control theory and the characteristics of quantum systems. Developing a special control theory and methods for quantum systems has been a challenging task. This task requires interdisciplinary researchers with interest in the development and applications of novel quantum control methodologies to fundamental physical, chemical, and biological problems from the quantum physics, chemistry, quantum information, mathematical and computer sciences, and control engineering communities. On the other hand, the field of quantum information involves the complex task of designing and effectively manipulating multi-qubit systems. However, this problem is beset by significant difficulties, such as the corruption of quantum information caused by decoherence. Finding solutions to the problem of decoherence, resulting from the unavoidable interaction of a quantum system with its environment, is one of the most critical challenges impeding practical realization of a quantum information processor (QIP). Current strategies for decoherence management are being developed by researchers from three distinct communities within quantum information science (QIS), namely, dynamical decoupling (DD), optimal control (OC), and quantum error correction (QEC). All of the problems to be solved in these areas are in fact control problems, which should be solved by means of control theory and methods. The aim of a control theory is to find a method of transforming a system by means of controlling action in order to achieve its prescribed behavior. A control theory can be used effectively only when it is executed in the whole process of control systems design because control theory is one part of the whole process of control systems design.
The whole process of control system design and implementation, which can also be called control system engineering, in the order in which it is done is (i) modeling, including identification, estimation, and filter; (ii) system synthesis, including controllability, observability, and/or reachability; (iii) control laws design; (iv) control systems analysis, including stability and/or convergence; (v) the numerical simulation of the control system; and (vi) actual system experimental implementation. A designer could do every part of the process if necessary, but because it is a huge control engineering process in fact it is better for one person to study only one or two parts of the design. The problems that exist in each part of the process may be solved by several available theories, methods, or tools, so in practice no one person can do all the control system design and implementation. In most cases the focus is on the study of control methods for a system in which the model of a system to be controlled does not need to be built because it is given. The controllability does not be studied because it is known that the system to be controlled is controllable. If it is not the case, controllability analysis has to be done. No-one can design a control law for an uncontrollable system. In other words, no control method can be used to achieve the desired behavior for an uncontrollable system. Not doing some work in a control system design does not mean that it is not important, but that it is known or the requirement has been satisfied. This book is mainly concerned with steps (iii) to (v) of the process of control system design. Because the quantum control system concerns interdisciplinary knowledge, let us start with quantum states.
1.1 Quantum States
A quantum system can be completely described by its state vector in a complex vector space with an inner product known as Hilbert space. is a unit vector in the system's state space and is called the wave function. In physics, bra-ket notation is often used to denote such vectors. The notation is represented by a single vector known as a ket, while is a bra. This notation is known as Dirac representation in a complex Hilbert space H. A quantum state is also called as a qubit. The wave function represents a pure state. This “state” in quantum mechanics is different from that in classical systems. For a classical system, the state usually describes some real physical properties such as the position or the momentum, which are generally observable. However, a quantum state cannot be directly observed and also does not directly correspond to the physical quantity of the quantum system. Since the global phase of a quantum state has no observable physical effect, we often say that the vectors and , in which and , describe the same physical state. For example, in quantum information theory the information is coded by a two-level (two-state) quantum system and the state of a qubit can be written as
1.1
where and . Then and correspond to the states 0 and 1 for a classical bit.
A quantum system can be closed or open according to whether or not the system is isolated from the external environment. The closed quantum system is under conditions of absolute zero temperature or does not interact with the external environment, and its state evolution is unitary. However, quantum systems usually cannot meet these ideal conditions in practical quantum information processing and quantum computing, and have interactions with the external environment, and are therefore treated as open quantum systems. In practical applications, the quantum systems to be controlled are usually not simple closed systems. They may be quantum ensembles or open quantum systems and their states cannot be written in the form of unit vectors . In this case, it is necessary to introduce the density operator or density matrix to describe quantum states of quantum ensembles or open quantum systems. A density operator is positive and has a trace equal to one. Suppose that a quantum system is in an ensemble of pure states; that is, in a mixture of a number of pure states with respective probabilities . The density matrix for the system is defined as
1.2
where and . Here, the operation refers to the conjugate transpose. For a pure state , there is and . If the state of a quantum system satisfies , we call the quantum state a mixed state.
A composite quantum system assumed to be made up of two subsystems A and B is defined on a Hilbert space , which is the tensor product of the Hilbert spaces and . For the composite quantum system, its state can be described by the tensor product of the states of its subsystems: . Consider any bipartite pure state . If it can be written as a tensor product of pure states and ,
1.3
we call it a separable state; otherwise, we call it an entangled state. Quantum entanglement is a uniquely quantum mechanical phenomenon that plays a key role in many interesting applications of quantum communication and quantum computation.
When performing a particular measurement on a quantum state, the result is usually described by a probability distribution, and the distribution is completely determined by the quantum state and the observable describing the measurement. These probability distributions are necessary for both mixed states and pure states.
1.2 Quantum Systems Control Models
There are some different descriptions of a system model to be controlled. If the system to be controlled is a closed quantum system, its model is generally described by the Schrödinger or quantum Liouville equations, both of which are bilinear models. Bilinear models are widely used to describe closed quantum control systems such as molecular systems in physical chemistry and spin systems in nuclear magnetic resonance (NMR). For example, consider a spin-1/2 system in a constant magnetic field along the z-axis and controlled by magnetic fields along the x-axis and y-axis.
1.2.1 Schrödinger Equation
The Schrödinger equation, describing states of quantum particles, has analytical solutions that determine precisely how the state changes with time. The state of a closed quantum system evolves according to the Schrödinger equation
1.4
where is the free Hamiltonian of the system and a Hermitian operator on H, and is the reduced Planck's constant. For convenience, we usually assume = 1. For simplicity, we consider finite dimensional quantum systems, which are appropriate approximations in many practical situations.
The control of the system may be realized by a set of control functions coupled to the system via time-independent interaction Hamiltonians . Then the total Hamiltonian determines the controlled evolution
1.5
Equation 1.5 is a bilinear quantum system control model.
1.2.2 Liouville Equation
If we use the density matrix to describe the state of a closed quantum system, the evolution equation of the density matrix can be described by the quantum Liouville equation
1.6
where is the commutation operator.
A control system with density matrix as its state has the control system model
1.7
where is the variable to be controlled, is the free (or internal) Hamiltonian, and is the control (or external) Hamiltonian. Usually we can assume that and are all independent of time; is an external control field, which is a real value.
Generally, the evolution of a Hamiltonian system is unitary in a closed quantum system. Unitary evolution preserves the spectrum of the quantum state, that is, the eigenvalues of the density matrix. All density matrices that have the same eigenvalues form a set of unitarily equivalent states, for example the set of all pure states. The control problem involving pure states is always expected to be described by the wave function and its Schrödinger equation in Hilbert space. Equation 1.7 can also be used to control a mixed state. In practice, the system equation chosen depends on the problem to be solved. Compared to the Liouville equation, the Schrödinger equation in which the wave function is a variable is simple. However, the wave function can be used only in the pure states systems and not in the systems of mixed states. There is no such a limitation for the Liouville equation with density matrix as its variable. When pure states are manipulated, Equation 1.5 is equivalent to the expression of density operator as . But Equation 1.7 is valid for mixed states manipulations. It is to be noted that although we can regard the model in Equation 1.5 as a particular case of Equation 1.7, the case in Equation 1.5 for pure states always gives more straightforward results and provides some inspiring ideas for studying Equation 1.7.
1.2.3 Markovian Master Equations
In many practical applications, the quantum systems to be controlled are open quantum systems. In fact, this is the case for most quantum control systems since such systems unavoidably interact with their external environments, including control inputs and measurement devices. For an open quantum system, a quantum master equation with the density matrix is suitable for describing the characteristics of the state. One of the simplest cases is when a Markovian approximation can be applied where a short environmental correlation time is supposed and memory effects may be neglected. For an N-dimensional open quantum system with Markovian dynamics, the state can be described by the following Markovian master equation:
1.8
where the generator of the semigroup represents a super-operator. The explicit form of this matrix can be derived using rigorous master equation formalism. The first term of Equation 1.8 describes the standard dynamics and the last term accounts for the gain and the damping mechanism, which has the form of the Liouville super-operator and can be written in the Lindblad form (Dacies, 1976):
1.9
where and form a collection of generalized atomic creation and annihilation operators characteristic for a particular problem.
The Lindblad form of the master equation guarantees that the interaction with the damping reservoir preserves the positivity of the density operator.
1.2.4 Non-Markovian Master Equations
In the case of weak coupling, assuming the form of the interaction Hamiltonians between the system and the environment is bilinear, the two-level reduced system model described by the non-Markovian time-convolution-less master equation can be written as follows:
1.10
1.11
where is the total Hamiltonian, and are the system and control Hamiltonian, respectively, is the transition frequency of the two-level system, and is the modulation by the time-dependent external control field. The control Hamiltonians can be described by (), where , , are the Pauli matrices and are the rising and lowering operators, respectively. describes the interaction between the system and the environment. In the Ohmic environment, the analytic expression for the dissipation coefficient appearing in Equation 1.11 is
1.12
and the diffusion coefficient is (Maniscalco et al., 2004):
1.13
where is the coupling constant, , , ( is the environment temperature), is the high-frequency cutoff, , and is the Gauss hypergeometric function (Gradshtein and Ryzhik, 1994).
Under conditions of high temperature, one has
1.14
From Equations 1.12 and 1.14, at high temperature both and hold. In such a case, the diffusion coefficient plays a dominant role in the non-unitary dynamics of the system. The essential difference between Markovian systems and non-Markovian systems is the existence of the environment memory effect. If the decay rate is defined as , then the difference is distinguished by the sign of , that is, when , the system mainly presents Markovian characteristics; when , non-Markovian characteristics are predominant. In the case of high temperature, one can easily get since . Note that at medium and low temperatures the approximation conditions in the Gauss hypergeometric function used to derive Equation 1.14 are not available, and can no longer be negligible, so is related to both and .
1.3 Structures of Quantum Control Systems
Systems, in one sense, are devices that take input and produce an output. A system can be thought to operate on the input to produce the output. The output is related to the input by a relationship known as the system response. The system response usually can be modeled by a mathematical relationship between the system input and the system output. A control system is a device, or a set of devices, that manages, commands, directs, or regulates the behavior of other devices or systems.
Control systems are broadly classified as either closed-loop or open-loop control systems. From the system control point of view, whether or not a system is feedback (or closed-loop) control depends on the expression of its control law: it is feedback when control law is a function of the output variable of the system. An open-loop control system is controlled directly and runs only in pre-arranged ways, or only by an input signal. Open-loop control systems do not make use of feedback. The benefit of an open-loop system is often the low cost associated with running the processes. In this case, there is no need for feedback to be taken into consideration. The drawbacks of the open-loop system are less accuracy of control and less robustness of the disturbance and uncertainty because no measurement of the system output is used to alter the control. The open-loop control has been very successful in the control of some simple quantum systems. However, it has had some difficulties in more complex quantum control tasks such as suppressing the decoherence and dealing with the disturbances in quantum systems. A natural solution to this problem is to explore closed-loop control strategies.
In closed-loop control the system is self-adjusting. Data do not flow one way. They may pass back from a specific device to the start of the control system, telling it to adjust itself accordingly. Feedback loops take the system output into consideration, which enables the system to adjust its performance to meet a desired output response. Figure 1.1 is a basic feedback control system structure in which the output value of the system y is used to help prepare the next output value. In this way, one can create a system that correct error e. Figure 1.1 shows a feedback loop with the value 1. We call this a unity feedback control system.
Figure 1.1 Basic feedback control system structure
A feedback control has many advantages: (i) it can increase the robustness of the system; (ii) it can increase the stability of the control system; (iii) it can automatically implement the control; and (iv) it can increase the performance of a control system. It is not an exaggeration to say that the outstanding achievements of control theory in engineering, including aeronautics and astronautics, in the last 50 years have become possible owing to the development of efficient feedback design methods.
As we know, feedback is an effective strategy in classical control theory and the aim of feedback is to compensate for the effects of unpredictable disturbances on a system under control or to make automatic control possible when the initial state of the system is unknown. In classical control, many results have shown that feedback control is superior to open-loop control. In feedback control, it is usually necessary to obtain information about the state of the system through measurement. However, the measurements of a quantum system will unavoidably destroy the state of the measured quantum system, which makes the situation more complex when applying feedback to quantum systems. In spite of this difficulty, important progress has been made and feedback has been used to improve the control performance for squeezed states (Chen and Elliott, 1990; Misawa and Kobayashi, 2000), quantum entanglement (Malkmus et al., 2005), and quantum state reduction (Muller et al., 1990; Assion et al., 1996) in many areas such as quantum optics (Herek, Materny, and Zewail, 1994), superconducting quantum systems (Assion et al., 1996), Bose–Einstein condensate (Cerullo et al., 1996), and nanomechanical systems (Bardeen et al., 1998).
In quantum feedback control, the two main approaches to information acquisition are projective measurement and continuous weak measurement. The system to be controlled is a quantum system. However, the controller may be quantum, classical, or a quantum/classical hybrid. Several paradigms of quantum feedback have been proposed, such as Markovian quantum feedback (Brixner, Damrauer, and Gerber, 2001), Bayesian quantum feedback (Dupont et al., 1995), and coherent quantum feedback (Brown and Rabitz, 2002). In Markovian quantum feedback, any time delay is ignored and a memory-less controller is assumed, that is, the measurement record is immediately fed back into the system to alter the system dynamics and may then be forgotten (Knutsen et al., 2004). Hence, the equation describing the resulting evolution is a Markovian master equation. In Bayesian quantum feedback, the process is divided into two steps involving state estimation and feedback control. The best estimates of the dynamical variables are obtained continuously from the measurement record and feed back to control the system dynamics (Dupont et al., 1995). In coherent quantum feedback, the feedback controller itself is a quantum system and it processes quantum information. Feedback control is typically a closed-loop control system. In fact, a feedback control system with the output state obtained from mathematical models can also be designed. The premise of state feedback based on mathematical models is that the state of the system rightly evolves according to the mathematical model, which requires a closed quantum system due to the requirement of consistency between states from the model and from the real system.
1.4 Control Tasks and Objectives
In contrast to the states controlled in macroscopic systems, the states in quantum systems and their applications involve the manipulation and tracking of some particular states, such as active control in the molecular dynamics of the chemical reaction to selectively obtain the resultant, using the laser intensity and phase to manipulate the system's population, which transfers the molecular system from a initial base state to an excited state with high probability. In quantum computation, population transfer to the target state by radiation-induced excitation of atoms and molecules is in fact a kind of computing operation, and population transfer is used to prepare initial pure states. Fast parallel computing in a quantum system depends on its particular coherence, which is the computation of the superposition states. In quantum application of secret communication, the most important quantum states used are entangled states. The preparation, manipulation, and preservation of quantum states are the aims of using quantum control methods.
Generally, the objective in a control system, according to Figure 1.1, is to make some output, say y, behave in a desired way by manipulating some input, say u. The simplest objective might be to keep y small (or close to some equilibrium point), a regulation or manipulation problem, or to keep y − r small for r, a reference or command signal, in some set, the trajectory tracking problem. From the control system perspective, the control tasks and objectives of a quantum system can be itemized as follows:
The first four tasks can be for both closed and open quantum systems, while the fifth is the control goal specifically for open quantum systems.
1.5 System Characteristics Analyses
1.5.1 Controllability
Controllability is a major issue in system analysis before a control strategy is applied, and is used to judge whether it is possible to control or stabilize the system. The controllability relates to the possibility of controlling a particular system by using an appropriate control signal. If the system is not controllable, then no control signal will be able to manipulate its state. The controllability of quantum systems is a fundamental theoretical notion in quantum control as well as having practical importance because of its close connection with the universality of quantum computation and the possibility of attaining atomic- or molecular-scale transformations. We can more precisely define the concept of controllability: A state is controllable at time if for some finite time there exists an input that transfers the state from to the origin at time . In this book, we assume that the systems to be controlled are all controllable.
Generally, the results on the controllability of quantum systems only show if the system is controllable, but do not provide constructive methods to design a control law for manipulating a quantum system from an initial state to a predetermined target state. If a quantum system is controllable, the design of a proper control law to realize desired control goals is another important task, and one that this book will consider. Hence, it is desirable to develop useful methods to design such a control law, and these will be provided in this book.
1.5.2 Reachability
The reachability is also an important essential characteristic of a system. We can write the definition of reachability more precisely: A state is called reachable at time if for some finite initial time there exists an input that transfers the state from the origin at to . A system is reachable at time if every state in the state space is reachable at time . Similar to the controllability, if a system is reachable then one can design a control law to manipulate its state to reach the target state.
The difference between the controllability and the reachability is that the former refers to the possibility of transferring a system state to the origin, while the latter refers to the possibility of transferring a system state from an initial state to the target state. So in a quantum control system, the controllability in fact refers to the reachability.
1.5.3 Observability
The term observability describes whether or not the internal state variables of the system can be externally measured. If a state is not observable, the controller will not be able to determine the behavior of an unobservable state and hence cannot use it to stabilize the system directly. If a system is said to be observable at time with the system in state , it is possible to determine this state from the observation of the output over a finite time interval.
Controllability (reachability) and observability play an important role in the design of control systems in state space. In fact, the conditions of controllability and observability may govern the existence of a complete solution to the control system design problem. The solution to this problem may not exist if the system considered is not controllable. Although most physical systems are controllable and observable, the corresponding mathematical model may not possess the properties of controllability and observability. In this case it is necessary to know the conditions for a system to be controllable and observable.
1.5.4 Stability
Stability is a very important concept in system control theory, that is, under what conditions will a system become unstable? If it is unstable, how should we stabilize the system? Whether or not a system is stable is a property of the system itself and does not depend on the input or driving function of the system.
Stability is also one of the hardest function properties to prove. There are several different criteria for system stability, but the most common requirement is that the system must produce a finite output when it is subjected to a finite input.
1.5.5 Convergence
Convergence refers to the notion that some functions and sequences approach a limit under certain conditions.
There is a difference between classical control theory and quantum control theory: probability is the control objective of quantum systems, which means the control goal is not achievable completely for a stable control system with non-zero error. For this reason, a convergent control with zero error is required in quantum control systems. In summary, a mature quantum control theory is able to not only manipulate various states of particular needs, but also to design a convergent control law.
1.5.6 Robustness
A control system must always have some robustness. A robust control system is one in which the performance does not change much if the actual system is slightly different from the mathematical model used for the synthesis and controller design. This specification is important: no real physical system truly behaves like the series of differential equations used to represent it mathematically. Typically, a simpler mathematical model is chosen in order to simplify the calculation because the true system dynamics can be so complicated that a complete model is impossible.
1.6 Performance of Control Systems
1.6.1 Probability
A quantum system can be in two possible states, for example the polarization of a photon. When the polarization is measured, it could be horizontal, labeled as state , or vertical, labeled state . Until its polarization is measured, the photon can be in a superposition of both these states, so its wave function would be written:
The probability amplitudes of states and are and , respectively. When the photon's polarization is measured, the system is horizontally polarized with probability and vertically polarized with probability . Therefore, a photon with the wave function , whose polarization is measured, would have a probability of to be horizontally polarized and a probability of 2/3 to be vertically polarized. The measurement must give either or , so the total probability of measuring and must be 1. This leads to a constraint that ; more generally, the sum of the squared moduli of the probability amplitudes of all the possible states is equal to 1. The wave function that fulfills this constraint is called the normalized wave function.
The probability amplitude interprets the physical meaning of the wave function. In quantum mechanics, a probability amplitude is a complex number whose modulus squared represents a probability or probability density. For example, if the probability amplitude of a quantum state is , the probability of measuring that state is . The values taken by a normalized wave function at each point x are probability amplitudes, since gives the probability density at position x.
1.6.2 Fidelity
In quantum information theory, the fidelity is a measure of the “closeness” of two quantum states. It is not a metric on the space of density matrices, but it can be used to define the Bures metric on this space.
1.6.3 Purity
In quantum mechanics, and especially quantum information theory, the purity of a state is a scalar defined as
where is the density matrix of the state.
For the closed quantum system, the purity of a pure state is always equal to 1, that is, , while the purity of a mixed state is less than 1, that is, .
The purity is trivially related to the linear entropy of a state by
1.7 Quantum Systems Control
1.7.1 Description of Control Problems
The greatest advantage of a control law designed according to system control theory is that one can design a better or OC law and determine its parameters by the control theory instead of by tentative experiments. The control law derived from theory will lead an experiment to its desired results. In this sense, the design of the control law is the task of finding the best parameters according to some control theory, which results in many challenges for quantum control engineering:
The parameterized control field in a general form is , where , , and are parameters to be optimized according to the control theory. The most typical problem in a quantum system is to design a set of control functions , , to steer the system from its initial state to a desired target one. For unitary evolution in a Hamiltonian system, the frequency spectrum of is time-independent or . That is to say, to make sure the target state is achievable, the same frequency spectrum (or entropy) is used for both and . The system is controllable for its density matrix. If the entropies of and are different from each other, one can minimize the distance index to accomplish the control task.
It can be shown that is stationary under the condition that and are commutable, viz. . Thus, quantum state control for most target states is a problem of the transfer. For a non-stationary target, it is a trajectory tracking problem: find a function to make a trajectory of with initial state track the target trajectory of . In the latter case, the problem may need to distinguish trajectory tracking. Both orbit tracking and functional tracking exist in quantum systems. It is considered that itself is a trajectory and the target state evolves according to its own orbit. The orbit is expressed as the globe phase of the quantum state, which does not influence the states' amplitude and does not need to be considered. In both orbit and functional tracking, the trajectory of is a time-dependent function and can be decided by the control value.
1.7.2 Quantum Control Theory and Methods
Quantum control theory is a rapidly developing research area. Controlling quantum phenomena has been an implicit goal in much quantum physics and chemistry research since the establishment of quantum mechanics (Warren, Rabitz, and Dahleh, 1993; Chu, 2002). One of the main goals in quantum control theory is to establish a firm theoretical footing and develop a series of systematic methods for the active manipulation and control of quantum systems (Mabuchi and Khaneja, 2005). This goal is non-trivial since microscopic quantum systems have many unique characteristics (e.g., entanglement and coherence) which do not occur in classical mechanical systems and the dynamics of quantum systems must be described by quantum theory. In recent years, the development of the general principles of quantum control theory has been recognized as an essential requirement for the future application of quantum technologies (Dowling and Milburn, 2003).
Similar to macroscopic control systems, we can divide quantum control systems into two classes: state transfer and trajectory tracking. The former can be subdivided into control of pure states and general states. Many kinds of control methods have been proposed for every control problem. Each control method has its own characteristics and a range of suitable applications. Among the different control methods there are certain differences in the amount of computation, the performance of the solution and the degree of difficulty of the realization. A complete control process of quantum systems involves choosing the proper system model to be controlled and designing a control strategy to reach the desired control goal, which requires knowledge of various mathematical models and related control theories.
Generally, the premise to design a controller for a system is the system's controllability, otherwise the desired output state cannot be achieved by any control input. The controllability of the controlled system must therefore be studied before designing a controller. The controllability conditions themselves only provide the criterion to determine whether or not a system is controllable and do not take a part in the controller design. The basic condition required so that the controller designed by a control theory can be used is stability of the whole control system. If the controlled system is not only stable but also convergent, the error between the reference and the controlled system will be zero. The stability can only guarantee an allowable error range, not the convergence. However, a convergent system must be stable. So far controllability in quantum control theory has received much attention and great progress has been made, especially for finite-dimensional closed quantum systems.
It is worth noting that automatic control systems have been widely used in physical experiments for a long time. Between the late 1980s and the early 1990s, ultrafast lasers, so-called femtosecond lasers, appeared along with the rapid development of laser industry. This new generation of lasers has the ability to generate pulses with durations of a few femtoseconds (1 fs = 10e−15 seconds) and even less. The duration of such a pulse is comparable with the period of a molecule's natural oscillation, therefore a femtosecond laser can, in principle, be used as a means of controlling a single molecule or atom. A consequence of such an application is the possibility of realizing the chemist's dream of changing the natural course of chemical reactions. In addition, control is an important part of many recent nanoscale applications: nanomotors, nanowires, nano chips, nano robots, and so on. Using the apparatus of modern control theory, new horizons in studying the interactions of atoms and molecules may bring new ways and possible limits for the intervention in intimate processes of the micro-world.
Many control methods and technologies for systems have been proposed. The easiest one is pulses dynamics, different pulse areas of which in resonant single-photo transition or resonant double-photos Raman transition are used to build coherent states. This can realize the complete population overturn of a two-level system but comes with the difficulties of accurate control of pulse intensity and duration time. Based on adiabatic theory, some manipulation technologies have been proposed and realized, such as Chirped Adiabatic Passage (CHIRAP) and Stimulated Raman Adiabatic Passage (STIRAP). In both of these the adjustment of pulse parameters such as frequency, amplitude envelope, and so on is extremely slow to make the transition go along adiabatic defined dressed states. The strategy is expected to achieve complete population transfer of the target state. In view of the facts mentioned above, it is very difficult to apply these theories to more general needs such as the control of a superposition state or a mixed state. Furthermore, control law is based on the feedback of measure, where measurement is described by measure operators. In general, the reliability of measure operators is a serious problem that feedback control has to face.
Among the control methods based on control theories, quantum OC based on optimal control theory (OCT) is the most successful, and turns the problem of state manipulation into the one of global optimization. In the OC approach, the quantum control problem can be formulated as a problem of seeking a set of admissible controls satisfying the system dynamic equations and simultaneously minimizing a cost functional. The cost functional may be different according to the practical requirements of the quantum control problems, such as minimizing the control time and the control energy, the error between the initial state and target state, or a combination of these requirements. Many useful tools in traditional OC, such as the variational method, the Pontryagin minimum principle, and convergent iterative algorithms, can be adapted to quantum systems and applied to search for OCs. OC techniques have been widely applied to control quantum phenomena in physical chemistry and NMR experiments. In such a control method, time-independent performance functions are more flexible and suitable for diverse optimal problems, but the main weakness of these is that the optimal problem is a two-point boundary value problem, and the evolution of system dynamical equations and their states depend on the initial control field. One has to guess an initial value to start the design process of the control field and to optimize by continuous iterations. This requires a large amount of computing, which limits its application to quantum physics, where a rapid response is needed.
Lyapunov-based control methods are powerful tools for feedback controller design in classical control theory (Dong and Petersen, 2010). In quantum control, the acquisition of feedback information through measurements usually destroys the state being measured, which makes it difficult to directly apply Lyapunov approaches to quantum feedback controller design. However, one may first complete the feedback control design by simulation on a computer, which will give a sequence of controls, then apply the control sequence to the quantum system to be controlled in an open-loop form (Mirrahimi, Rouchon, and Turinici, 2005; Kuang and Cong, 2008). This is a feedback design and open-loop control strategy at the current level of technology. This strategy is especially useful for some difficult quantum control tasks (Altafini, 2007). The most important aspects of Lyapunov-based control design are the construction of the Lyapunov function, the determination of the control law, and the analysis of the convergence.
The advantage of the Lyapunov-based method is that control law can be obtained directly from the Lyapunov indirect stability theorem without iterations from the solution of partial differential equations, which makes it possible to realize rapid quantum control. The main shortcoming of the Lyapunov-based method is that it is usually only a stable control method, not a convergent control method. A given target state requires the control system to be convergent to guarantee that the system achieves the desired target state. To achieve this goal the convergence of the Lyapunov-based method has been studied intensely in recent years.
Bang-bang control and geometry control are alternative control methods often used in quantum systems. Geometry control is suitable for lower level systems, especially when combined with the Bloch sphere, which can help the physical meaning of the quantum state in a two-level system to be understood. The trajectory of geometry control of a pure state in two-level systems is just the trajectory on a Bloch sphere, while the trajectory of the mixed state is the points inside the Bloch sphere. The bang-bang control in control theory is a type of simple switch control, which corresponds to the pulse control in quantum systems. Because of its simple realization, bang-bang control was used earlier in quantum systems. In open quantum systems several special control theories and methods have been developed, such as the decoherence-free subspace (DFS) method and the dynamic decoupling quantum control method.
All of the quantum control theories and methods mentioned above are covered in this book.
There are important differences between quantum control theory and its experimental implementation. Control solutions obtained from theoretical studies strongly depend on the employed model Hamiltonian. However, for real systems controlled in the laboratory, the Hamiltonians usually are not well known (except for the simplest cases) and the Hamiltonians for the system-environment coupling are known to an even lesser degree. An additional difficulty is the computational complexity of accurately solving the OC equations for realistic polyatomic molecules. Another important difference between control theory and experiment arises from the difficulty of reliably implementing theoretical control designs in the laboratory because of instrumental noise and other limitations. As a result, optimal theoretical control designs generally will not be optimal in the laboratory. Notwithstanding these comments, control simulations continue to be very valuable and they even set forth the logic leading to practical laboratory control.
1.8 Overview of the Book
This book presents the latest developments and achievements in control theories and methods in both closed and open quantum systems. The contents are suitable for both active researchers and non-experts who wish to enter the field. Each chapter discusses how to use and design the particular control method or theory, which types of systems can use it, and what information can be learned from the control system. Some possible further developments and extensions of the methods that may be expected in the near future are also mentioned. The book places an emphasis on ideas and concepts, with a fair to moderate amount of mathematical rigor. The simulation experiments and results display all essential information about the quantum state under the control methods used and will greatly enhance the reader's understanding of quantum mechanics and the effectiveness of control theories.
The book can be divided into two parts: control theory and methods for closed and open quantum systems. Chapter 1 is provides an introduction to the subject. Chapters 2–9 cover control methods for closed quantum systems, and Chapters 10–14 cover control methods for open quantum systems. Chapter 15 discusses the trajectory tracking of quantum systems. In the control theory and methods for the closed quantum systems section, Chapters 4–8 concentrate on quantum control theory based on the Lyapunov method.
The book is organized as follows.
We begin the basic analysis of system states in a simple two-level quantum system on the Bloch sphere in Chapter 2. We also introduce the state transfer of quantum systems on the Bloch sphere by means of geometric control. The Bloch sphere is a suitable tool used to present a qubit because it gives us an intuitive vision to understand the physical meanings of quantum bits or variables. We first present the descriptions of pure states, superposition states, and mixes states. Then we propose the control methods of a single spin-1/2 particle in a Bloch sphere in which we focus on the situations of a minimum control field and a fixed time T, respectively.
Chapter 3 is about the general control methods of closed quantum systems. Two improved OC strategies applied to quantum systems are presented first. After that the design of the control sequence of pulses for a high-dimensional spin-1/2 quantum system is used to prepare the entangled state. Chapter 3 also provides a comparison study between geometric control and bang-bang control, which are the two earliest control methods used in quantum systems.
Chapters 4–8 introduce quantum control theory based on the Lyapunov method. The first three chapters are the Lyapunov methods that are used in state transfer between diverse quantum states. They are the manipulation of eigenstates, population control, and general state control, respectively. Chapter 7 covers the convergence analysis of the Lyapunov method. The Lyapunov method is discussed in so many chapters because this method is an increasingly interesting method used in quantum control systems. It will be applied in quantum systems as it is in control engineering applications. From the convergence conditions obtained in Chapter 7, it can be seen that the convergence conditions are so strong that most practical systems are not able to satisfy them. In order to relax the convergence conditions to comply with actual cases, we specifically introduce control theory and method in degenerate cases in Chapter 8.
Chapter 4 introduces the eigenstates transfer control law design process for the selected Lyapunov function based on the state distance according to the Lyapunov stability theorem. We also present an optimal quantum control based on the Lyapunov stability theorem, and the realization of the quantum Hadamard gate based on the Lyapunov method.
There are three parts in Chapter 5: the population control of equilibrium states, the generalized control of quantum systems in the frame of vector treatment, and the population control of eigenstate and its simulation.
Chapter 6 covers the general state control method in quantum systems, including superposition state manipulation, pure state control strategy, and the OC of the mixed state, and from any pure state to mixed state manipulation.
