Advances in Chemical Physics, Volume 148 E-Book

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Serie: Advances in Chemical Physics
• Sprache: Englisch

Seitenzahl: 735

Veröffentlichungsjahr: 2011

Beschreibung

The Advances in Chemical Physics series--the cutting edge of research in chemical physics The Advances in Chemical Physics series provides the chemical physics and physical chemistry fields with a forum for critical, authoritative evaluations of advances in every area of the discipline. Filled with cutting-edge research reported in a cohesive manner not found elsewhere in the literature, each volume of the Advances in Chemical Physics series offers contributions from internationally renowned chemists and serves as the perfect supplement to any advanced graduate class devoted to the study of chemical physics. This volume explores: * Control of Quantum Phenomena (Constantin Brif, Raj Chakrabarti, and Herschel Rabitz) * Crowded Charges in Ion Channels (Bob Eisenberg) * Colloidal Crystallization Between Two and Three Dimensions (H. Löwen, E.C. Oguz, L. Assoud, and R. Messina) * Statistical Mechanics of Liquids and Fluids in Curved Space (Gilles Tarjus, FranÇois Sausset, and Pascal Viot)

Leseprobe

Contents

Cover

Editorial Board

Title Page

Copyright

Contributors to Volume 148

Preface to The Series

Chapter 1: Control of Quantum Phenomena

I. Introduction

II. Controlled Quantum Dynamics

III. Controllability of Quantum Systems

IV. Quantum Optimal Control Theory

V. Adaptive Feedback Control in the Laboratory

VI. The Role of Theoretical Quantum Control Designs in the Laboratory

VII. Quantum Control Landscapes

VIII. Conclusions

Acknowledgments

References

Chapter 2: Crowded Charges in Ion Channels

I. Physical Chemistry and Life

II. Physical Chemistry and Biological Problems

III. Action Potential is a Cooperative Phenomenon

IV. Computation of the Action Potential

V. Reduced Models of Calcium and Sodium Channels

VI. Crowded Ions: Properties of the Model of Calcium Channels

VII. Balanced Forces and Structures in Crowded Systems

VIII. Inverse Methods and Selectivity Models

IX. Reduced Model of Transport Through a Channel

X. Reduced Model of a “Transport” Channel the Ryanodine Receptor

XI. Conclusions and Implications of the Crowded Charge Reduced Model

XII. Variational Approach

XIII. Outlooks: Unsolved Problems in Physical Chemistry

XIV. Conclusion

Appendix A: Models of Chemical Kinetics and the Law of Mass Action

Acknowledgments

References

Chapter 3: Colloidal Crystallization Between Two and Three Dimensions

I. Introduction

II. Freezing of Binary Systems in Two Spatial Dimensions

III. Exploring the Third Dimension: Buckling and Layering Transitions

IV. Conclusions

Acknowledgments

References

Chapter 4: Statistical Mechanics of Liquids and Fluids in Curved Space

I. Introduction

II. Theoretical Motivations and Physical Realizations

III. Thermodynamics and Boundary Effects

IV. Liquid-State Theory in Spaces of Constant Nonzero Curvature

V. Curvature Effect on the Thermodynamics and Structure of Simple Fluids

VI. Freezing, Jamming, and the Glass Transition

VII. Ground-State Properties, Order, and Defects

VIII. Conclusion

Appendix A: A Recap on Riemannian Manifolds

Appendix B: Periodic Boundary Conditions on the Hyperbolic Plane

References

Author Index

Subject Index

Color Plates

Editorial Board

Moungi G. Bawendi, Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA

Kurt Binder, Condensed Matter Theory Group, Institut für Physik, Johannes Gutenberg-Universität Mainz, Mainz, Germany

William T. Coffey, Department of Electronics and Electrical Engineering, Trinity College, University of Dublin, Dublin, Ireland

Karl F. Freed, Department of Chemistry, James Franck Institute, University of Chicago, Chicago, Illinois, USA

Daan Frenkel, Department of Chemistry, Trinity College, University of Cambridge, Cambridge, United Kingdom

Pierre Gaspard, Center for Nonlinear Phenomena and Complex Systems, Université Libre de Bruxelles, Brussels, Belgium

Martin Gruebele, School of Chemical Sciences and Beckman Institute, Director of Center for Biophysics and Computational Biology, University of Illinois at Urbana-Champaign, Urbana, Illinois, USA

Jean-Pierre Hansen, Department of Chemistry, University of Cambridge, Cambridge, United Kingdom

Gerhard Hummer, Chief, Theoretical Biophysics Section, NIDDK-National Institutes of Health, Bethesda, Maryland, USA

Ronnie Kosloff, Department of Physical Chemistry, Institute of Chemistry and Fritz Haber Center for Molecular Dynamics, The Hebrew University of Jerusalem, Israel

Ka Yee Lee, Department of Chemistry and The James Franck Institute, The University of Chicago, Chicago, Illinois, USA

Todd J. Martinez, Department of Chemistry, Stanford University, Stanford, California, USA

Shaul Mukamel, Department of Chemistry, University of California at Irvine, Irvine, California, USA

Jose Onuchic, Department of Physics, Co-Director Center for Theoretical Biological Physics, University of California at San Diego, La Jolla, California, USA

Steven Quake, Department of Physics, Stanford University, Stanford, California, USA

Mark Ratner, Department of Chemistry, Northwestern University, Evanston, Illinois, USA

David Reichmann, Department of Chemistry, Columbia University, New York, New York, USA

George Schatz, Department of Chemistry, Northwestern University, Evanston, Illinois, USA

Norbert Scherer, Department of Chemistry, James Franck Institute, University of Chicago, Chicago, Illinois, USA

Steven J. Sibener, Department of Chemistry, James Franck Institute, University of Chicago, Chicago, Illinois, USA

Andrei Tokmakoff, Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA

Donald G. Truhlar, Department of Chemistry, University of Minnesota, Minneapolis, Minnesota, USA

John C. Tully, Department of Chemistry, Yale University, New Haven, Connecticut, USA

Copyright © 2012 by John Wiley & Sons, Inc. All rights reserved

Published by John Wiley & Sons, Inc., Hoboken, New Jersey

Published simultaneously in Canada

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Library of Congress Catalog Number: 58-9935

ISBN: 978-1-118-12235-8

Contributors to Volume 148

L. Assoud, Institut für Theoretische Physik II: Weiche Materie, Heinrich-Heine-Universität Düsseldorf, D-40225 Düsseldorf, Germany

Constantin Brif, Department of Chemistry, Princeton University, Princeton, NJ 08544, USA

Raj Chakrabarti, School of Chemical Engineering, Purdue University, Forney Hall of Chemical Engineering, 480 Stadium Mall Drive, West Lafayette, IN 47907, USA

Bob Eisenberg, Department of Molecular Biophysics and Physiology, Rush University, Chicago, IL 60305, USA; Mathematics and Computer Science Division, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, IL 60439, USA

H. Löwen, Institut für Theoretische Physik II: Weiche Materie, Heinrich-Heine-Universität Düsseldorf, D-40225 Düsseldorf, Germany

R. Messina, Institut für Theoretische Physik II: Weiche Materie, Heinrich-Heine-Universität Düsseldorf, D-40225 Düsseldorf, Germany

E. C. Ouz, Institut für Theoretische Physik II: Weiche Materie, Heinrich-Heine-Universität Düsseldorf, D-40225 Düsseldorf, Germany

Herschel Rabitz, Department of Chemistry, Princeton University, Princeton, NJ 08544, USA

François Sausset, Department of Physics, Technion, Haifa 32000, Israel

Gilles Tarjus, Laboratoire de Physique Théorique de la Matière Condensée, Université Pierre et Marie Curie-Paris 6, UMR CNRS 7600, 4 place Jussieu, 75252 Paris Cedex 05, France

Pascal Viot, Laboratoire de Physique Théorique de la Matière Condensée, Université Pierre et Marie Curie-Paris 6, UMR CNRS 7600, 4 place Jussieu, 75252 Paris Cedex 05, France

Preface to The Series

Advances in science often involve initial development of individual specialized fields of study within traditional disciplines, followed by broadening and overlapping, or even merging, of those specialized fields, leading to a blurring of the lines between traditional disciplines. The pace of that blurring has accelerated in the last few decades, and much of the important and exciting research carried out today seeks to synthesize elements from different fields of knowledge. Examples of such research areas include biophysics and studies of nanostructured materials. As the study of the forces that govern the structure and dynamics of molecular systems, chemical physics encompasses these and many other emerging research directions. Unfortunately, the flood of scientific literature has been accompanied by losses in the shared vocabulary and approaches of the traditional disciplines, and there is much pressure from scientific journals to be ever more concise in the descriptions of studies, to the point that much valuable experience, if recorded at all, is hidden in supplements and dissipated with time. These trends in science and publishing make this series, Advances in Chemical Physics, a much needed resource.

The Advances in Chemical Physics is devoted to helping the reader obtain general information about a wide variety of topics in chemical physics, a field that we interpret very broadly. Our intent is to have experts present comprehensive analyses of subjects of interest and to encourage the expression of individual points of view. We hope that this approach to the presentation of an overview of a subject will both stimulate new research and serve as a personalized learning text for beginners in a field.

Stuart A. Rice Aaron R. Dinner

Chapter 1

Control of Quantum Phenomena

Constantin Brif,1 Raj Chakrabarti,2 and Herschel Rabitz1

1Department of Chemistry, Princeton University, Princeton, NJ 08544, USA

2School of Chemical Engineering, Purdue University, West Lafayette, IN 47907, USA

I. Introduction

During the past two decades, considerable effort has been devoted to the control of physical and chemical phenomena at the atomic and molecular scale governed by the laws of quantum mechanics [1–11]. Quantum control offers the ability to not just observe but also actively manipulate the course of processes on this scale, thereby providing hitherto unattainable means to explore quantum dynamics and opening the way for a multitude of practical applications [8, 9, 12–23]. Owing to the tremendous growth of research in this area, it would be impossible to cover in one chapter all the advances. Therefore, this chapter is not intended to be a complete review of quantum control, but rather will focus on a number of important topics, including controllability of quantum systems, quantum optimal control theory (QOCT), adaptive feedback control (AFC) of quantum phenomena in the laboratory, and the theory of quantum control landscapes (QCLs).

The origins of quantum control can be traced back to early attempts at the use of lasers for the selective breaking of bonds in molecules. The concept was based on the application of monochromatic laser radiation tuned to the particular vibrational frequency that would excite and, ultimately, break the targeted chemical bond. However, numerous attempts to implement this strategy [24–26] were largely unsuccessful due to intramolecular vibrational redistribution of the deposited energy that rapidly dissipated the initial local excitation and thus generally prevented selective bond breaking [27–29].

Important advances toward selective quantum control of chemical and physical processes were made in the late 1980s, when the role of quantum interference in optical control of molecular systems was identified [30–40]. In particular, Brumer and Shapiro [30–33] proposed to use two monochromatic laser beams with commensurate frequencies and tunable intensities and phases for creating quantum interference between two reaction pathways. In this approach, control over branching ratios of molecular reactions, in principle, can be achieved by tuning the phase difference between the two laser fields [41–43]. The method of coherent control via two-pathway quantum interference was experimentally demonstrated in a number of applications in atomic, molecular, and semiconductor systems [44–59]. Although the practical effectiveness of this method is limited (by a number of factors including the problem of identifying two particular pathways among a dense set of other transitions, the difficulty of matching excitation rates along the two pathways, and undesirable phase and amplitude locking of the two laser fields in optically dense media [60]), the concept of control via two-pathway quantum interference has played an important role in the historical development of the field [1, 6, 61, 62].

Another important step toward selective control of intramolecular reactions was made by Tannor, Kosloff, and Rice [34, 35], who proposed the method of pump–dump control, based on the use of two successive femtosecond laser pulses with a tunable time delay between them. In this approach, a vibrational wave packet generated by the first laser pulse (the “pump”) evolves on the potential energy surface (PES) of an excited electronic state of the molecule until the second laser pulse (the “dump”) transfers the population back to the ground-state PES into the desired product channel. Product selectivity can be achieved by using the time delay between the two pulses to control the location at which the wave packet is dumped to the ground-state PES [3, 5]. The pump–dump control method was applied in a number of experiments [63–67]. A useful feature of pump–dump control experiments is the possibility of qualitatively interpreting the control mechanism by developing a simple and intuitive picture of the system dynamics in the time domain. Moreover, the pump–dump scheme also can be used as a time-resolved spectroscopy technique to explore transient molecular states and thus obtain information about the molecular dynamics at various stages of a reaction [68–75]. While the employment of transform-limited laser pulses in the pump–dump method can be satisfactory for some applications [3, 5, 14], the effectiveness of this technique as a practical control tool can be immensely increased by optimally shaping one or both the pulses.

There also exist other control methods employing pairs of time-delayed laser pulses; however, in contrast to pump–dump control, the goal of these methods is primarily to transfer population between discrete quantum states in atoms and molecules. In one approach, known as stimulated Raman adiabatic passage (STIRAP), two time-delayed laser pulses (typically, of nanosecond duration) are applied to a three-level Λ-type configuration to achieve complete population transfer between the two lower levels via the intermediate upper level [76–83]. The laser-induced coherence between the quantum states is controlled by tuning the time delay, in order to keep the transient population in the intermediate state almost at zero (thus avoiding losses by radiative decay). The applicability of the STIRAP method is limited to control of population transfer between a few discrete states as arise in atoms and small diatomic and triatomic molecules; in larger polyatomic molecules, adiabatic passage is generally prevented by the very high density of levels [82, 83]. In another method, referred to as wave packet interferometry (WPI) [20], population transfer between bound states in atoms, molecules, and quantum dots is controlled by employing quantum interference of coherent wave packets excited by two laser pulses with a tunable time delay between them [84–96].

The possibility of significantly improving control capabilities by using specially shaped ultrafast laser pulses to produce desired quantum interference patterns in the controlled system was proposed in the late 1980s [36–40]. Independently, significant advances have been made in the technology of femtosecond laser pulse shaping [97, 98, 99]. In first applications of pulse shaping to optical control of quantum phenomena, only the linear chirp1 was tuned. Linearly chirped ultrashort laser pulses were employed for control of various processes in atoms and molecules [100–122]. In particular, when applied to molecules with overlapping emission and absorption bands, pulses with negative and positive chirp excite vibrational modes predominately in the ground and excited electronic states, respectively (this effect was utilized for state-selective control of vibrational wave packets [100–106]). Another useful property is the ability of negatively and positively chirped pulses to increase and decrease, respectively, the localization of optically excited vibrational wave packets in diatomic molecules [101–103](the localization effect of negatively chirped pulses was employed to enhance selectivity in pump–dump control of photodissociation reactions [103] and protect vibrational wave packets against rotationally induced decoherence [123]).

The control approaches discussed above share the same fundamental mechanism based on quantum interference induced by laser fields in the controlled system. However, another common feature of these methods is the reliance on just one control parameter (e.g., the phase difference between two monochromatic laser fields, the time delay between two laser pulses, or the linear chirp rate), which is only a nascent step toward the full exploitation of control field resources for controlling quantum phenomena. While single-parameter control may be effective in some simple circumstances, more flexible and capable control resources are required for more complex systems and applications. The concept of control with specially tailored ultrafast laser pulses has unified and generalized the single-parameter control schemes. Rabitz and coworkers [36–38] and others [39, 40] proposed to steer quantum evolution toward a desired target by specifically designing and tailoring the time-dependent electric field of the laser pulse to the characteristics of the system and the control objective. In particular, QOCT [36–40,124–129] has emerged as the leading theoretical tool to design and explore laser pulse shapes that are best suited for achieving the desired goal (see, for example, Refs. [3, 5, 11, 130, 131] for earlier reviews). An optimally shaped laser pulse typically has a complex form, in both time and frequency domains, in keeping with the rich dynamical capabilities of strongly coupled multiparticle quantum systems. In this fashion, the phases and amplitudes of the available spectral components of the field are optimized to excite an interference pattern among multiple quantum pathways to best achieve the control objective.

The fruitful synergistic influence of theoretical and experimental advances has played a central role in the development of the quantum control field. A combination of the conceptual insights described above and breakthroughs in ultrafast laser technology ultimately resulted in the establishment of the AFC laboratory procedure proposed by Judson and Rabitz [132]. AFC has proved to be the most effective and flexible practical tool for the vast majority of quantum control applications [4, 7–9, 12, 13, 19, 133–138]. In AFC experiments, femtosecond pulse-shaping technology is utilized to the fullest extent, guided by measurement-driven, closed-loop optimization to identify laser pulses that are optimally tailored to meet the needs of complex quantum dynamical objectives. Optimization in AFC uses a learning algorithm, with stochastic methods (e.g., genetic algorithms and evolutionary strategies) being especially effective due to their inherent robustness to noise and to operational uncertainties [4, 139, 140]. A sizeable body of experimental research has demonstrated the capability of AFC to manipulate the dynamics of a broad variety of quantum systems and explore the underlying physical mechanisms [8, 9]. Owing to modeling limitations, QOCT-based control designs are typically less effective in practice than solutions optimized via AFC directly in the laboratory for the actual quantum system. However, theoretical studies employing QOCT and other similar methods are extremely valuable for analyzing the feasibility of controlling new classes of quantum phenomena and providing a basic understanding of controlled quantum dynamics [141].

Theoretical research in the field of quantum control involves the exploration of several fundamental issues with important practical implications. One such issue is controllability that addresses the question whether a control field, in principle, exists that can drive the quantum system to the target goal [10, 134, 142, 143]. A related but distinct issue is concerned with the existence and multiplicity of optimal controls, that is, solutions that maximize the chosen objective functional [144, 145]. More generally, the objective as a function of the control variables forms the QCL [146, 147], and exploration of its entire structure is of fundamental importance. The study of QCL topology builds on controllability results. In turn, the characterization of the critical points2 of the QCL forms a foundation for analyzing the complexity of finding optimal control solutions [148–151]. This theoretical analysis has direct practical implications as its results provide a basis to determine the ease of finding effective and robust controls in the laboratory [148] and can help identify the most suitable optimization algorithms for various applications of quantum control [152, 153].

This chapter is organized as follows. Controlled quantum dynamics is described in Section II, including a review of the basic notions of quantum theory needed for the remaining presentation. Controllability of closed and open quantum systems is discussed in Section III. QOCT is presented in Section IV, including the mathematical formalism of objective functionals and methods employed to find optimal control solutions, as well as a survey of applications. In section V, numerous laboratory implementations of AFC of quantum phenomena are reviewed. Section VI discusses the relationship of QOCT and AFC, along with the role of theoretical control designs in experimental realizations. Section VII is devoted to the theory of QCLs, including such topics as the characterization of critical points that determine the QCL topology, conditions of control optimality, Pareto optimality for multiobjective control, methods of homotopy trajectory control, and the practical implications of QCL analysis. A summary of this chapter is presented in Section VIII.

II. Controlled Quantum Dynamics

In this section, we present basic results from quantum theory that are needed to describe the dynamics of closed and open quantum systems under the influence of external time-dependent controls. Coherent control of quantum phenomena involves the application of classical fields (e.g., laser pulses) to quantum systems (e.g., atoms, molecules, quantum dots, etc.). Consider first a coherently controlled closed quantum system (i.e., a system isolated from the environment during the control process) whose evolution is governed by the time-dependent Hamiltonian of the form

(1)

Here, H0 is the free Hamiltonian of the system and Hc(t) is the control Hamiltonian (at time t) that represents the interaction of the system with the external field. In the mathematically oriented literature, the control Hamiltonian is usually formally written as Hc(t)=∑mcm(t)Hm, where {cm(t)} are real-valued control functions at time t and {Hm} are Hermitian operators through which the controls couple to the system. In physical and chemical applications, the control Hamiltonian is often given by

(2)

where μ is the dipole operator and ε(t) is the control field at time t. The Hamiltonian of the form given in Eq. (2) adequately describes the interaction of an atomic or molecular system with a laser electric field in the dipole approximation or the interaction of a spin system with a time-dependent magnetic field. However, other forms for Hc(t) can arise including those nonlinear in ε(t). For the control fields, we will use the notation and , where is the space of locally bounded, sufficiently smooth, square integrable functions of time defined on some interval [0, T], with T being the target time for achieving the desired control outcome.

The Hilbert space of a quantum system is spanned by the eigenstates of the free Hamiltonian H0. Let be the space of trace class operators on . For example, for an N-level quantum system, is the space of complex vectors of length N and is the space of N×N complex matrices. The set of admissible states of a quantum system, which are represented by density matrices on the Hilbert space , is denoted as . Any density matrix is a positive operator of trace one, that is, ρ≥0 and Tr(ρ)=1 (thus, ). The density matrix of a pure state satisfies Tr(ρ2)=1 and can be expressed as ρ=|ψψ|, where |ψ is a normalized complex vector in . Any quantum state that is not pure can be represented as a statistical mixture of pure states and therefore is called mixed; the density matrix of a mixed state satisfies Tr(ρ2)<1.

The time evolution of a closed quantum system from t=0 to t is given in the Schrödinger picture by

(3)

where ρ(t) is the density matrix of the system at time t, ρ0ρ(0) is the initial state, and U(t) is the system's unitary evolution operator (for an N-level quantum system, U(t) is an N×N unitary matrix). If the state is initially pure: ρ0=|ψ0ψ0|, it will always remain pure under unitary evolution: ρ(t)=|ψ(t)ψ(t)|, where

(4)

The evolution operator satisfies the Schrödinger equation:

(5)

where I is the identity operator on . The corresponding evolution equation for the density matrix (called the von Neumann equation) is

(6)

and the Schrödinger equation for a pure state is

(7)

In practice, it is often necessary to take into account interactions between quantum systems (e.g., between molecules in a liquid or between electron spins and nuclear spins in a semiconductor material). If for a particular problem only one of the interacting subsystems is of interest (now referred to as the system), all other subsystems that surround it are collectively referred to as the environment. A quantum system coupled to an environment is called open [154]. A molecule in a solution or an atom coupled to the vacuum electromagnetic field are examples of open quantum systems. The interaction with the environment typically results in a process of decoherence, in which a coherent superposition state of an open quantum system is transformed into a statistical mixture (decoherence is often accompanied by dissipation of the initial excitation, although in some situations pure dephasing is possible; see Refs. [154, 155] for details). Generally, all quantum systems are open; however, whether environmentally induced processes are important depends on their rate relative to the rate of the coherent evolution. From a practical perspective, the importance of decoherence also depends on the control objective. In chemical applications such as control over the yield of a reaction product, environmental effects may play a significant role in the liquid phase where relaxation processes happen on the timescale of the order of one picosecond [8, 9, 156, 157]. On the other hand, decoherence induced by collisions in the gas phase can often be neglected (at least at low pressures), as the time between collisions is much longer than the characteristic period of vibronic dynamics controlled by femtosecond laser pulses [8]. In contrast, in quantum information processing, an unprecedented level of control accuracy is required to minimize quantum gate errors, and therefore decoherence needs to be taken into account for practically all physical realizations [158].

The state of an open quantum system is described by the reduced density matrix ρTrenv(ρtot), where ρtot represents the state of the system and environment taken together, and Trenv denotes the trace over the environment degrees of freedom. There are many models of open-system dynamics depending on the type of environment and character of the system–environment coupling [154, 155]. If the system and environment are initially uncoupled,, then the evolution of the system's reduced density matrix from t=0 to t is described by a completely positive, trace-preserving map Φt:

(8)

where ρ0ρ(0). A linear map is called completely positive if the map (where Il is the identity map in ) is positive for any . A map Φ is called trace preserving if Tr(Φρ)=Tr(ρ) for any . The map (8) can be defined for any time t≥0, and the entire time evolution of the open quantum system is described by a one-parameter family {Φt|t≥0} of dynamical maps (where Φ0 is the identity map I).

Any completely positive, trace-preserving map has the Kraus operator-sum representation (OSR) [159, 160, 161]:

(9)

where {Kj} are the Kraus operators (N×N complex matrices for an N-level quantum system). The trace preservation is ensured by the condition

(10)

Here, is the number of Kraus operators. We will refer to completely positive, trace-preserving maps simply as Kraus maps. Unitary transformations form a particular subset of Kraus maps corresponding to n=1. There exist infinitely many OSRs (with different sets of Kraus operators) for the same Kraus map. For any Kraus map for an N-level quantum system, there always exists an OSR with a set of n≤N2 Kraus operators [161].

While the Kraus-map description of open-system dynamics is very general, it is often not the most convenient one for numerical calculations. With additional assumptions, various types of quantum master equations can be derived from the Kraus OSR [154, 155]. In particular, for a Markovian environment (i.e., when the memory time for the environment is zero), the set of Kraus maps {Φt|t≥0} is a semigroup, which means that any two maps in the set satisfy the condition [154]

(11)

In this situation, the dynamics of an open quantum system is described by the quantum master equation of the form [154]

(12)

where the linear map (also referred to as the Liouville superoperator) is the generator of the dynamical semigroup, that is, . The evolution equation of the form (12) is often referred to as the Liouville–von Neumann equation. For an N-level quantum system coupled to a Markovian environment, the most general form of the map can be constructed, resulting in the quantum master equation of the Lindblad type [154, 155, 162]:

(13)

Here, {γi} are nonnegative constants and {Li} are the Lindblad operators (N×N complex matrices) that form (together with the identity operator) an orthonormal operator basis on . By convention, {Li} are traceless. The first term in Eq. (13) represents the unitary part of the dynamics governed by the Hamiltonian H and the second (Lindblad) term represents the nonunitary effect of coupling to the environment. The constants {γi} are given in terms of certain correlation functions of the environment and play the role of relaxation rates for different decay modes of the open system [154].

III. Controllability of Quantum Systems

Before considering the design of a control or a control experiment, a basic issue to address is whether in principle a control exists to meet the desired objective. Assessing the system's controllability is an important issue from both fundamental and practical perspectives. A quantum system is called controllable in a set of configurations, , if for any pair of configurations and there exists a control that can drive the system from the initial configuration ζ1 to the final configuration ζ2 in a finite time T.3 Here, possible types of the configuration ζ include the system's state ρ, the expectation value Tr(ρΘ) of an observable (a Hermitian operator) Θ, the evolution operator U, and the Kraus map Φ, with the particular choice depending on the specific control problem.

First, consider the well-studied issue of controllability of closed quantum systems with unitary dynamics [142, 143, 163–179]. A closed quantum system is called kinematically controllable in a set of states if for any pair of states and , there exists a unitary operator U, such that ρ2Uρ1U†. Any two quantum states that belong to the same kinematically controllable set are called kinematically equivalent. As unitary evolution preserves the spectrum of a density matrix, two states ρ1 and ρ2 of a closed quantum system are kinematically equivalent if and only if they have the same eigenvalues [171, 172]. Therefore, all quantum states that belong to the same kinematically controllable set have the same density-matrix eigenvalues (and, correspondingly, the same von Neumann entropy and purity [158]). For example, all pure states belong to the same kinematically controllable set. However, any pure state is not kinematically equivalent to any mixed state. For a closed quantum system, all states on the system's Hilbert space are separated into disconnected sets of kinematically equivalent states.

It is also possible to consider controllability in the dynamic picture. Assume that the Hamiltonian H(t) that governs the dynamics of a closed quantum system through the Schrödinger equation (5) is a function of a set of time-dependent controls: H(t)=H(c1(t), . . ., ck(t)). A closed quantum system is called dynamically controllable in a set of states if for any pair of states and there exist a finite time T and a set of controls {c1(·), . . ., ck(·)}, such that the solution U(T) of the Schrödinger equation (5) transforms ρ1 into ρ2: ρ2U(T)ρ1U†(T). Since a closed system is controllable only within a set of kinematically equivalent states, a dynamically controllable set of states is always a subset of the corresponding kinematically controllable set . If the dynamically controllable set of pure states coincides with its kinematically controllable counterpart (i.e., the set of all pure states), the closed quantum system is called pure-state controllable. If for any pair of kinematically equivalent states ρ1 and ρ2 there exists a set of controls that drives ρ1 into ρ2 in finite time (i.e., if all dynamically controllable sets of states coincide with their kinematically controllable counterparts), the closed system is called density-matrix controllable.

It is also possible to consider controllability of closed quantum systems in the set of unitary evolution operators. A closed quantum system is called evolution-operator controllable if for any unitary operator W there exists a finite time T and a set of controls {c1(·), . . ., ck(·)}, such that WU(T), where U(T) is the solution of the Schrödinger equation (5) with H(t)=H(c1(t), . . ., ck(t)). For an N-level closed quantum system, a necessary and sufficient condition for evolution-operator controllability is [143, 171–173] that the dynamical Lie group of the system (i.e., the Lie group generated by the system's Hamiltonian) be U(N) [or SU(N) for a traceless Hamiltonian]. It was also shown [171–173] that density-matrix controllability is equivalent to evolution-operator controllability. For specific classes of states, the requirements for controllability are weaker [165, 166, 167]. For example, pure-state controllability requires that the system's dynamical Lie group is transitive on the sphere . For infinite-level quantum systems evolving on noncompact Lie groups, such as those arising in quantum optics, the conditions for controllability are more stringent [179, 180].

Controllability analysis was also extended to open quantum systems [181–188]. An open quantum system with Kraus-map evolution of the form (9) is called kinematically controllable in a set of states if for any pair of states and there exists a Kraus map Φ, such that ρ2=Φρ1. It was recently proved [186] that for a finite-level open quantum system with Kraus-map evolution, kinematic state controllability is complete, that is, the system is kinematically controllable in the set of all states ρ on the Hilbert space. Moreover, for any target state ρf on the Hilbert space of a finite-level open system, there exists a Kraus map Φ such that Φρρf for all states ρ on [186].

The issue of dynamic state controllability of open quantum systems is yet to be fully explored. To analyze this problem, first it is necessary to specify dynamic capabilities, that is, the set of available controls. While unitary evolution of a closed quantum system can be induced only by coherent controls, Kraus-map evolution of an open system can involve both coherent and incoherent controls (the former act only through the Hamiltonian part of the dynamics, while the latter include interactions with other quantum systems and measurements). Let be the set of all available finite-time controls, which may include coherent electromagnetic fields, an environment with a tunable distribution function [189], coupling to an auxiliary system [190–197], measurements [181–203], and so on. Each particular set of controls, , induces the corresponding time evolution of the system through the Kraus map Φξ,t that transforms an initial state ρ0 into the state ρ(t)=Φξ,tρ0 at time t. An open quantum system with Kraus-map evolution is called dynamically controllable in the set of states if for any pair of states and , there exists a set of controls and a finite time T, such that the resulting Kraus map Φξ,T transforms ρ1 into ρ2: ρ2=Φξ,Tρ1. An open quantum system is called Kraus-map controllable if the set of all available controls can generate any Kraus map Φ from the identity map I. It follows from these definitions and the result on kinematic state controllability described above that Kraus-map controllability is sufficient for an open quantum system to be dynamically controllable in the set of all states ρ on the Hilbert space [186]. In general, complete dynamic state controllability is a weaker property than Kraus-map controllability, since the former can be achieved even if the set of available controls generates only a particular subset of all possible Kraus maps [186].

In principle, there exist various methods to engineer arbitrary finite-time Kraus-map evolutions of open quantum systems. One method relies on the ability to coherently control both the system of interest and an ancilla (an auxiliary quantum system that serves as an effective environment). If a quantum system with Hilbert space dimension Ns is coupled to an ancilla with Hilbert space dimension Na, such that (1) and (2) the ancilla is initially prepared in a pure state, then evolution-operator controllability of the system and ancilla taken together is sufficient for Kraus-map controllability (and thus for complete dynamic state controllability) of the system [186]. Another method of Kraus-map engineering employs a combination of a measurement and a coherent control action in a feedback setup. Specifically, the ability to perform a single simple measurement on a quantum system, together with the ability to apply coherent control conditioned upon the measurement outcome, allows enacting an arbitrary finite-time Kraus-map evolution and thereby suffices for complete dynamic state controllability [181].

Analyses of controllability in closed and open quantum systems have important implications for the theory of QCLs, which in turn is the basis for understanding the optimization complexity of quantum control simulations and experiments. Specifically, controllability is one of the underlying assumptions required to establish the absence of regular critical points that are local traps in QCLs for several important types of quantum control objectives. This fundamental property of the QCL topology and its practical implications will be discussed in detail in Section VII.

IV. Quantum Optimal Control Theory

In the majority of physical and chemical applications, the most effective way to coherently control complex dynamical processes in quantum systems is via the coordinated interaction between the system and the electromagnetic field whose temporal profile may be continuously altered throughout the control period. For a specified control objective, and with restrictions imposed by many possible constraints, the time-dependent field required to manipulate the system in a desired way can be designed using QOCT [11, 130, 131]. This general formulation encompasses both weak and strong field limits and incorporates as special cases the single-parameter methods such as control via two-pathway quantum interference and pump–dump control.

A. Control Objective Functionals

The formulation of a quantum control problem necessarily includes the definition of a quantitative control objective. Consider a coherently controlled N-level closed quantum system, with the Hamiltonian

(14)

and the unitary evolution operator that obeys the Schrödinger equation (5). In QOCT, the control objective for such a system can be a functional of the set of evolution operators , where T is the target time, as well as of the control fields ε(·). The general class of control objective functionals (also referred to as cost functionals) can be written as

(15)

where F is a continuously differentiable function on U(N), and G is a continuously differentiable function on . Usually, the first term in (15) represents the main physical goal, while the second term is used to incorporate various constraints on the dynamics and control fields. The optimal control problem may be stated as the search for

(16)

subject to the dynamical constraint (5). For the sake of consistency, throughout this chapter we will consider only maximization of cost functionals; any control problem can be easily reformulated from minimization to maximization and vice versa by changing the sign of the functional. The cost functional of the form (15) is said to be of the Bolza type. If only the term is present, the cost functional is said to be of the Lagrange type, whereas if only the term F(U(T)) is present, the functional is said to be of the Mayer type [204]. Three classes of problems corresponding to different choices of F(U(T)) have received the most attention in the quantum control community to date: (i) evolution-operator control, (ii) state control, and (iii) observable control.

For evolution-operator control, the goal is to generate U(T) such that it is as close as possible to the target unitary transformation W. The Mayer-type cost functional in this case can be generally expressed as

(17)

where ||·|| is an appropriate normalized matrix norm; that is, F1(U(T)) is maximized when the distance between U(T) and W is minimized. This type of objective is common in quantum computing applications [158], where F1(U(T)) is the fidelity of a quantum gate [205, 206]. One frequently used form of the objective functional F1(U(T)) is obtained utilizing the squared Hilbert–Schmidt norm [207] in (17) with an appropriate normalization (i.e., ||X||=(2N)−1Tr(X†X)) [208–210]:

(18)

Other forms of the objective functional, which employ different matrix norms in (17), are possible as well [210–213]. For example, a modification of (18)

(19)

which is independent of the global phase of U(T), can be used. Note that F1(U(T)) is independent of the initial state, as the quantum gate must produce the same unitary transformation for any input state of the qubit system [158].

For state control, the goal is to transform the initial state ρ0 into a final state ρ(T)=U(T)ρ0U†(T) that is as close as possible to the target state ρf. The corresponding Mayer-type cost functional is

(20)

where ||·|| is an appropriate normalized matrix norm (e.g., the Hilbert–Schmidt norm can be used) [145, 214–216].

For observable control, the goal is typically to maximize the expectation value of a target quantum observable Θ (represented by a Hermitian operator). The corresponding Mayer-type cost functional is [124, 217–220]

(21)

An important special case is state transition control (also known as population transfer control), for which ρ0=|ψiψi| and Θ=|ψfψf|, where |ψi and |ψf are eigenstates of the free Hamiltonian H0. In this case, the objective functional (21) has the form

(22)

which is the probability of transition (i.e., the population transfer yield) between the energy levels of the quantum system [146, 221]. In many chemical and physical applications of quantum control, absolute yields are not known, and therefore maximizing the expectation value of an observable (e.g., the population transfer yield) is a more appropriate laboratory goal than minimizing the distance to a target expectation value.

Also, in quantum control experiments (see Section V), measuring the expectation value of an observable is much easier than estimating the quantum state or evolution operator. Existing methods of quantum state and evolution-operator estimation rely on tomographic techniques [222–229] that are extremely expensive in terms of the number of required measurements (e.g., in quantum computing applications, standard methods of state and process tomography require numbers of measurements that scale exponentially in the number of qubits [158, 229–233]). Therefore, virtually all quantum control experiments so far have used observable control with objective functionals of the form (21) or (22). For example, in an AFC experiment [123], in which the goal was to maximize the degree of coherence, the expectation value of an observable representing the degree of quantum state localization was used as a coherence “surrogate,” instead of state purity or von Neumann entropy, which are nonlinear functions of the density matrix and hence would require state estimation. Nevertheless, future laboratory applications of quantum control, in particular in the field of quantum information sciences, will require evolution-operator control and state control, with the use of objective functionals of the types (17) and (20), respectively, together with novel state and process estimation methods [232–241].

Recently, attention has turned to problems requiring simultaneous maximization of several control objectives [242–245]. In the framework of QOCT, these optimization problems are sometimes handled through the use of a weighted-sum objective functional, such as [243,244]

(23)

which extends (21) to multiple quantum observables. Also, general methods of multiobjective optimization [246–248] have been recently applied to various quantum control problems [244, 245].

Another common goal in quantum control is to maximize a Lagrange-type cost functional subject to a constraint on U(T) [249, 250]. For example, this type of control problem can be formulated as follows:

(24)

where F(U(T)) is the Mayer-type cost functional for evolution-operator, state, or observable control (as described above), and f0 is a constant that corresponds to the target value of F. Often, the goal is to minimize the total field fluence, in which case G(ε(t))=−(1/2)ε2(t) is used.

B. Searching for Optimal Controls

To identify optimal controls that maximize an objective functional J (of the types discussed in Section IV.A), it is convenient to define a functional that explicitly incorporates the dynamical constraint. For example, many QOCT studies [11, 130, 131] considered pure-state evolution of a closed quantum system, for which the dynamical constraint is satisfaction of Eq. (7). The corresponding objective functional (e.g., for observable control) often is taken to have the form

(25)

Here, the first term represents the main control goal of maximizing the expectation value of the target observable Θ at the final time T; the second term is used to restrict the fluence and shape of the control field, with α(t) being a weight function; the third term includes an auxiliary state |χ(t) that is a Lagrange multiplier employed to enforce satisfaction of the Schrödinger equation for the pure state [Eq. (7)], and H(t) is the Hamiltonian (14) that includes the time-dependent control term. More generally, satisfaction of the Schrödinger equation for the evolution operator of a closed quantum system [Eq. (5)] can be used as the dynamical constraint for different types of objectives, including evolution-operator, state, and observable control. The corresponding general form of the objective functional is

(26)

Here, an auxiliary operator V(t) is a Lagrange multiplier employed to enforce satisfaction of Eq. (5), and, for the sake of simplicity, we assumed that G depends only on the control field.

QOCT can also be formulated for open systems with nonunitary dynamics [128, 251–258]. For example, for a quantum system coupled to a Markovian environment, the Liouville–von Neumann equation (12) must be satisfied. The corresponding objective functional (e.g., for observable control) has the form

(27)

Here, is the Liouville superoperator (the generator of the dynamical semigroup), and an auxiliary density matrix σ(t) is a Lagrange multiplier employed to enforce satisfaction of Eq. (12). Extensions of QOCT to non-Markovian open-system dynamics were also considered [256– 260].

Various modifications of the objective functionals (25), (26), and (27) are possible. For example, modified objective functionals can comprise additional spectral and fluence constraints on the control field [261, 262], take into account nonlinear interactions with the control field [263, 264], deal with time-dependent and time-averaged targets [258, 265, 266, 267], and include the final time as a free control parameter [268, 269]. It is also possible to formulate QOCT with time minimization as a control goal (time optimal control) [270–272]. As mentioned earlier, QOCT can also be extended to incorporate optimization of multiple objectives [242–245].

A necessary condition for a solution of the optimization problem (16) subject to the dynamical constraint (5) is that the first-order functional derivatives of the objective functional of Eq. (26) with respect to V(·), U(·), and ε(·) are equal to zero. The resulting set of Euler–Lagrange equations is given by

(28)

(29)

(30)

where ∇F(U(T)) is the gradient of F at U(T) in U(N). Critical points of the objective functional, which include optimal controls, can be obtained by solving this set of equations (various algorithms employed for numerical solution are discussed below). In the special case of the objective functional of Eq. (25), setting the first-order functional derivatives of with respect to χ(·), ψ(·), and ε(·) to zero results in the following Euler–Lagrange equations:

(31)

(32)

(33)

Analogously, in the special case of the objective functional of Eq. (27), setting the first-order functional derivatives of with respect to σ(·), ρ(·), and ε(·) to zero results in the following Euler–Lagrange equations:

(34)

(35)

(36)

An equivalent method for deriving optimal control equations is based on applying the Pontryagin maximum principle (PMP) [249, 250, 273]. For a bilinear control system of the form (5) evolving on the unitary group, the PMP function (also referred to as the PMP Hamiltonian) is defined as

(37)

where A, BTr(A†B) is the Hilbert–Schmidt inner product. According to the PMP, all solutions to the optimization problem (16) satisfy equations

(38)

with the boundary conditions U(0)=I, V(T)=∇F(U(T)). It is easy to see that for the PMP function of the form (37), the conditions (38) produce Eqs. (28)–(30). Satisfaction of the first-order condition or, equivalently, is a necessary but not sufficient condition for optimality of a control ε(·). So-called Legendre conditions on the Hessian ∂2H/∂ε(t')∂ε(t) are also required for optimality [204, 273]. The optimality criteria are discussed in Section VII.B.

1. Existence of Optimal Controls

An important issue is the existence of optimal control fields (i.e., maxima of the objective functional) for realistic situations that involve practical constraints on the applied laser fields. It is important to distinguish between the existence of an optimal control field and controllability; in the former case, a field is designed, subject to particular constraints, that guides the evolution of the system toward a specified target until a maximum of the objective functional is reached, while in the latter case, the exact coincidence between the attained evolution operator (or state) and the target evolution operator (or state) is sought. The existence of optimal controls for quantum systems was analyzed in a number of studies. Peirce et al. [37] proved the existence of optimal solutions for state control in a spatially bounded quantum system that necessarily has spatially localized states and a discrete spectrum. Zhao and Rice [144] extended this analysis to a system with both discrete and continuous states and proved the existence of optimal controls over the evolution in the subspace of discrete states. Demiralp and Rabitz [145] showed that, in general, there is a denumerable infinity of solutions to a particular class of well-posed quantum control problems; the solutions can be ordered in quality according to the achieved optimal value of the objective functional. The existence of multiple control solutions has important practical consequences, suggesting that there may be broad latitude in the laboratory, even under strict experimental restrictions, for finding successful controls for well-posed quantum objectives. The existence and properties of critical points (including global optima) of objective functionals for various types of quantum control problems were further explored using the analysis of QCLs [146,208–210, 217–221, 274](see Section VII).

2. Algorithms Employed in QOCT Computations

A number of optimization algorithms have been adapted or specially developed for use in QOCT, including the conjugate gradient search method [39], the Krotov method [206, 275, 276], monotonically convergent algorithms [277–282], noniterative algorithms [283], the gradient ascent pulse engineering (GRAPE) algorithm [284], a hybrid local/global algorithm [258], and homotopy-based methods [285–287]. Faster convergence of iterative QOCT algorithms was demonstrated using “mixing” strategies [288]. Also, the employment of propagation toolkits [289–291] can greatly increase the efficiency of numerical optimizations. Detailed discussions of the QOCT formalism and algorithms are available in the literature [5, 11, 130, 131].

C. Applications of QOCT

Originally, QOCT was developed to design optimal fields for manipulation of molecular systems [36–40, 124–129] and has been applied to a myriad of problems (e.g., rotational, vibrational, electronic, reactive, and other processes) [5,11,130]. Some recent applications include, for example, control of molecular isomerization [292–295], control of electron ring currents in chiral aromatic molecules [296], and control of heterogeneous electron transfer from surface attached molecules into semiconductor band states [297]. Beyond molecules, QOCT has been applied to various physical objectives including, for example, control of electron states in semiconductor quantum structures [298–300], control of atom transport in optical lattices [301], control of Bose–Einstein condensate transport in magnetic microtraps [302], control of a transition of ultracold atoms from the superfluid phase to a Mott insulator state [303], control of coherent population transfer in superconducting quantum interference devices [304], and control of the local electromagnetic response of nanostructured materials [305].

In the context of open-system dynamics in the presence of coupling to a Markovian environment, QOCT applications include control of unimolecular dissociation reactions in the collisional regime [128], laser cooling of molecular internal degrees of freedom coupled to a bath [252, 253, 306], control of vibrational wave packets in a model molecular system coupled to an oscillator bath at finite temperatures in the weak-field (perturbative) regime [251, 307], creation of a specified vibronic state, population inversion, wave packet shaping in the presence of dissipation in the strong-field regime [255], control of ultrafast electron transfer in donor–acceptor systems where the reaction coordinate is coupled to a reservoir of other coordinates [308], control of photodesorption of NO molecules from a metal surface in the presence of strong dissipation [309], control of excitation of intramolecular and molecule-surface vibrational modes of CO molecules adsorbed on a metal surface in the presence of dissipation to baths of substrate electrons and phonons [310, 258], and control of current flow patterns through molecular wires coupled to leads [311]. Also, QOCT was actively applied to the problem of protection of open quantum systems against environmentally induced decoherence [216, 257, 312–321].

Recently, there has been rapidly growing interest in applications of QOCT to the field of quantum information sciences. As mentioned above, one of the important problems in this field is optimal protection of quantum systems against decoherence. Applications of QOCT to quantum information processing also include optimal operation of quantum gates in closed systems [180, 205, 206, 250, 322–335] and in open systems (i.e., in the presence of decoherence) [213, 336–351] and optimal generation of entanglement [268, 269, 350–354]. One particular area where QOCT methods have proved to be especially useful is design of optimal sequences of radiofrequency (RF) pulses for operation of quantum gates in systems of coupled nuclear spins in a nuclear magnetic resonance (NMR) setup [284, 355, 356]. In a recent experiment with trapped-ion qubits, shaped pulses designed using QOCT were applied to enact single-qubit gates with enhanced robustness to noise in the control field [357]. Optimal control methods were also applied to the problem of storage and retrieval of photonic states in atomic media, including both theoretical optimization [358–360] and experimental tests [361–363].

D. Advantages and Limitations of QOCT

An advantage of QOCT relative to the laboratory execution of AFC (to be discussed in detail in Section V) is that the former can be used to optimize a well-defined objective functional of virtually any form, while the latter relies on information obtained from measurements and thus is best suited to optimize expectation values of directly measurable observables. In numerical optimizations, there is practically no difference in effort between computing the expectation value of an observable, the density matrix, or the evolution operator. In the laboratory, however, it is much more difficult to estimate a quantum state or, even more so, the evolution operator than to measure the expectation value of an observable. Moreover, state estimation error increases rapidly with the Hilbert space dimension [225, 364]. The very large number of measurements required for accurate quantum state/process tomography [229–232] renders (at least presently) the use of adaptive laboratory methods for state/evolution-operator control rather impractical.

QOCT is often used to explore new quantum phenomena in relatively simple models to gain physical insights. The realization of quantum control is ultimately performed in the laboratory, and in this context QOCT fits into what is called open-loop control. Generally, in open-loop control, a theoretical control design (e.g., obtained by using QOCT or another theoretical method) is implemented in the laboratory with the actual system. Unfortunately, there are not many problems for which theoretical control designs are directly applicable in the laboratory. QOCT is most useful when detailed knowledge of the system's Hamiltonian is available. Moreover, for open quantum systems, it is essential to know the details of the system–environment interaction. Therefore, the practical applicability of QOCT in the context of open-loop control is limited to very simple systems, that is, mostly to cases when a small number of degrees of freedom can be controlled separately from the remainder of the system. This may be possible when the controlled subsystem has characteristic frequencies well separated from those of other transitions and evolves on a timescale that is very different from that of the rest of the larger system. A well-known example of such a separately controllable subsystem is nuclear spins in a molecule, which can be very well controlled using RF fields without disturbing rotational, vibrational, and electronic degrees of freedom. Another example is a subset of several discrete levels in an atom or diatomic molecule, the transitions between which frequently can be controlled in a very precise way without any significant leakage of population to other states. However, for a majority of interesting physical and chemical phenomena, controlled systems are too complex and too strongly coupled to other degrees of freedom. For such complex systems, the accuracy of control designs obtained using model-based QOCT is usually inadequate, and hence laboratory AFC is generally the preferred strategy. In these situations, QOCT may be more useful for feasibility analysis and exploration of control mechanisms, as basic features of the controlled dynamics possibly can be identified in many cases even using relatively rough models.

V. Adaptive Feedback Control in the Laboratory

Many important aspects of quantum control experiments are not fully reflected in theoretical analyses. In particular, control solutions obtained in theoretical studies strongly depend on the employed model Hamiltonian. However, for real systems controlled in the laboratory, the Hamiltonians usually are not known well (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 optimal control 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 owing to 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 as explained below.

A crucial step toward selective laser control of physical and chemical phenomena on the quantum scale was the introduction of AFC (also referred to as closed-loop laboratory control or learning control). AFC was proposed and theoretically grounded by Judson and Rabitz in 1992 [132]. In AFC, a loop is closed in the laboratory (see Fig. 1.1), with results of measurements on the quantum system used to evaluate the success of the applied control and to refine it, until the control objective is reached as best as possible. At each cycle of the loop, the external control (e.g., a shaped laser pulse) is applied to the system (e.g., an ensemble of molecules). The signal (e.g., the yield of a particular reaction product or population in a target state) is detected and fed back to the learning algorithm (e.g., a genetic algorithm). The algorithm evaluates each control based on its measured outcome with respect to a predefined control goal and searches through the space of available controls to move toward an optimal solution.

Figure 1. A schematic depiction of a closed-loop process employed in adaptive feedback control (AFC) of quantum phenomena. The input to the loop is an initial control guess which can be a design estimate or even a random field in many cases. A current design of the laser control field is created with a pulse shaper and then applied to the sample. The outcome of the control action is measured, and the results are fed back to a learning algorithm. The algorithm searches through the space of possible controls and suggests an improved field design. Excursions around the loop are repeated until a satisfactory value of the control objective is achieved. Adapted from Ref.[4]. (See the color version of this figure in Color Plates section.)

While AFC can be simulated on the computer [132, 365–378], the important advantage of this approach lies in its ability to be directly implemented in the laboratory. Most important, the optimization is performed in the laboratory with the actual system and thus is independent of any model. As a result, the AFC method works remarkably well for systems even of high complexity, including, for example, large polyatomic molecules in the liquid phase, for which only very rough models are available. Second, there is no need to measure the laser field in AFC because any systematic characterization of the control “knobs” (such as pulse shaper parameters) is sufficient. This set of control “knobs” determined by the experimental apparatus defines the parameter space searched by the learning algorithm for an optimal laser shape. This procedure naturally incorporates any laboratory constraints on the control laser fields. Third, optimal controls identified in AFC are characterized by a natural degree of robustness to instrumental noise, since nonrobust solutions will be rejected by the algorithm. Fourth, in AFC, it is possible to operate at a high-duty cycle of hundreds or even thousands of experiments per second by exploiting (i) the conceptual advantage of the evolving quantum system solving its own Schrödinger equation in the fastest possible fashion and (ii) the technological advantage of high repetition rate pulsed laser systems under full automation. Fifth, in AFC, a new quantum ensemble (e.g., a new molecular sample) is used in each cycle of the loop that completely avoids the issue of back action exerted by the measurement process on a quantum system. Thus, AFC is technologically distinct from measurement-based real-time feedback control [379, 380, 381, 382, 383, 384], in which the same quantum system is manipulated until the final target objective is reached and for which measurement back action is an important effect that needs to be taken into account.

The majority of current AFC experiments employ shaped ultrafast laser pulses. In such experiments, one usually starts with a random or nearly random selection of trial shaped pulses of length 10−13