Control of quantum phenomena: Past, present, and future
aa r X i v : . [ qu a n t - ph ] M a y Control of quantum phenomena: Past, present, andfuture
Constantin Brif, Raj Chakrabarti ‡ and Herschel Rabitz Department of Chemistry, Princeton University, Princeton, New Jersey 08544E-mail: [email protected] , [email protected] and [email protected] Abstract.
Quantum control is concerned with active manipulation of physicaland chemical processes on the atomic and molecular scale. This work presents aperspective of progress in the field of control over quantum phenomena, tracing theevolution of theoretical concepts and experimental methods from early developmentsto the most recent advances. Among numerous theoretical insights and technologicalimprovements that produced the present state-of-the-art in quantum control, therehave been several breakthroughs of foremost importance. On the technology side,the current experimental successes would be impossible without the development ofintense femtosecond laser sources and pulse shapers. On the theory side, the twomost critical insights were (1) realizing that ultrafast atomic and molecular dynamicscan be controlled via manipulation of quantum interferences and (2) understandingthat optimally shaped ultrafast laser pulses are the most effective means for producingthe desired quantum interference patterns in the controlled system. Finally, thesetheoretical and experimental advances were brought together by the crucial concept ofadaptive feedback control, which is a laboratory procedure employing measurement-driven, closed-loop optimization to identify the best shapes of femtosecond laser controlpulses for steering quantum dynamics towards the desired objective. Optimizationin adaptive feedback control experiments is guided by a learning algorithm, withstochastic methods proving to be especially effective. Adaptive feedback controlof quantum phenomena has found numerous applications in many areas of thephysical and chemical sciences, and this paper reviews the extensive experiments.Other subjects discussed include quantum optimal control theory, quantum controllandscapes, the role of theoretical control designs in experimental realizations, andreal-time quantum feedback control. The paper concludes with a prospective of openresearch directions that are likely to attract significant attention in the future.
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New J. Phys. ‡ Present address: School of Chemical Engineering, Purdue University, Forney Hall of ChemicalEngineering, 480 Stadium Mall Drive, West Lafayette, Indiana 47907 ontrol of quantum phenomena: Past, present, and future
1. Introduction
For many decades, physicists and chemists have employed various spectroscopic methodsto carefully observe quantum systems on the atomic and molecular scale. The fascinatingfeature of quantum control is the ability to not just observe but actively manipulatethe course of physical and chemical processes, thereby providing hitherto unattainablemeans to explore quantum dynamics. This remarkable capability along with a multitudeof possible practical applications have attracted enormous attention to the field of controlover quantum phenomena. This area of research has experienced extensive developmentduring the last two decades and continues to grow rapidly. A notable feature of thisdevelopment is the fruitful interplay between theoretical and experimental advances.Various theoretical and experimental aspects of quantum control have beenreviewed in a number of articles and books [1–45]. This paper starts with a shortreview of historical developments as a basis for evaluating the current status of the fieldand forecasting future directions of research. We try to identify important trends, followtheir evolution from the past through the present, and cautiously project them into thefuture. This paper is not intended to be a complete review of quantum control, butrather a perspective and prospective on the field.In section 2, we discuss the historical evolution of relevant key ideas from the firstattempts to use monochromatic laser fields for selective excitation of molecular bonds,through the inception of the crucial concept of control via manipulation of quantuminterferences, and to the emergence of advanced contemporary methods that employspecially tailored ultrafast laser pulses to control quantum dynamics of a wide varietyof physical and chemical systems in a precise and effective manner. After this historicalsummary, we review in more detail the recent progress in the field, focusing on significanttheoretical concepts, experimental methods, and practical advances that have shaped thedevelopment of quantum control during the last decade. Section 3 is devoted to quantumoptimal control theory (QOCT), which is currently the leading theoretical approach foridentifying the structure of controls (e.g., the shape of laser pulses) that enable attainingthe quantum dynamical objective in the best possible way. We present the formalismof QOCT (i.e., the types of objective functionals used in various problems and methodsemployed to search for optimal controls), consider the issues of controllability andexistence of optimal control solutions, survey applications, and discuss the advantagesand limitations of this approach. In section 4, we review the theory of quantumcontrol landscapes, which provides a basis to analyze the complexity of finding optimalsolutions. Topics discussed in that section include the landscape topology (i.e., thecharacterization of critical points), optimality conditions for control solutions, Paretooptimality for multi-objective control, homotopy trajectory control methods, and thepractical implications of control landscape analysis. The important theoretical advancesin the field of quantum control have laid the foundation for the fascinating discoveriesoccurring in laboratories where closed-loop optimizations guided by learning algorithmsalter quantum dynamics of real physical and chemical systems in dramatic and often ontrol of quantum phenomena: Past, present, and future
2. Early developments of quantum control
The historical origins of quantum control lie in early attempts to use lasers formanipulation of chemical reactions, in particular, selective breaking of bonds inmolecules. Lasers, with their tight frequency control and high intensity, were consideredideal for the role of molecular-scale ‘scissors’ to precisely cut an identified bond, withoutdamage to others. In the 1960s, when the remarkable characteristics of lasers wereinitially realized, it was thought that transforming this dream into reality would berelatively simple. These hopes were based on intuitive, appealing logic. The procedureinvolved tuning the monochromatic laser radiation to the characteristic frequencyof a particular chemical bond in a molecule. It was suggested that the energy ofthe laser would naturally be absorbed in a selective way, causing excitation and,ultimately, breakage of the targeted bond. Numerous attempts were made in the 1970sto implement this idea [46–48]. However, it was soon realized that intramolecularvibrational redistribution of the deposited energy rapidly dissipates the initial localexcitation and thus generally prevents selective bond breaking [49–51]. This processeffectively increases the rovibrational temperature in the molecule in the same manneras incoherent heating does, often resulting in breakage of the weakest bond(s), which isusually not the target of interest.
Several important steps towards modern quantum control were made in the late 1980s.Brumer and Shapiro [52–55] identified the role of quantum interference in optical controlof molecular systems. They proposed to use two monochromatic laser beams withcommensurate frequencies and tunable intensities and phases for creating quantum ontrol of quantum phenomena: Past, present, and future
In the 1980s, Tannor, Kosloff, and Rice [76, 77] proposed a method for selectivelycontrolling intramolecular reactions by using two successive femtosecond laser pulseswith a tunable time delay between them. The first laser pulse (the “pump”) generatesa vibrational wave packet on an electronically excited potential-energy surface of themolecule. After the initial excitation, the wave packet evolves freely until the secondlaser pulse (the “dump”) transfers some of the population back to the ground potential-energy surface into the desired reaction channel. Reaction selectivity is achieved byusing the time delay between the two laser pulses to control the location at whichthe excited wave packet is dumped to the ground potential-energy surface [7, 11]. Forexample, it may be possible to use this method to move the ground-state wave-functionbeyond a barrier obstructing the target reaction channel. In some cases, the secondpulse transfers the population to an electronic state other than the ground state (e.g., ontrol of quantum phenomena: Past, present, and future
In the late 1980s, Bergmann and collaborators [91–94] demonstrated a very efficientadiabatic method for population transfer between discrete quantum states in atoms ormolecules. In this 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 lowerlevels via the intermediate upper level. Interestingly, the pulse sequence employed inthe STIRAP method is counter-intuitive, i.e., the Stokes laser pulse that couples theintermediate and final states precedes (but overlaps) the pump laser pulse that couplesthe initial and intermediate states. The laser electric fields should be sufficiently strongto generate many cycles of Rabi oscillations. The laser-induced coherence between thequantum states is controlled by tuning the time delay, so that the transient populationin the intermediate state remains almost zero, thus avoiding losses by radiative decay.Detailed reviews of STIRAP and related adiabatic passage techniques can be foundin [10, 16]. While the efficiency of the STIRAP method, under appropriate conditions,is very high, its applicability is restricted to control of population transfer between afew discrete states as arise in atoms and small (diatomic and triatomic) molecules. Inlarger polyatomic molecules, the very high density of levels generally prevents successfuladiabatic passage [10, 16].
Another two-pulse approach for control of population transfer between bound statesemploys Ramsey interference of optically excited wave packets [95, 96]. In this method, ontrol of quantum phenomena: Past, present, and future
Although the control approaches discussed in sections 2.1–2.4 were initially perceivedas quite different, it is now clear that on a fundamental level all of them employ themechanism of quantum interference induced by control laser fields. A common featureof these methods is that they generally attempt to manipulate the evolution of quantumsystems by controlling just one parameter: the phase difference between two laser fieldsin control via two-pathway quantum interference; the time delay between two laserpulses in pump-dump control, STIRAP, and WPI. While single-parameter control maybe relatively effective in some simple systems, more complex systems and applicationsrequire more flexible and capable control resources. The single-parameter controlschemes have been unified and generalized by the concept of control with speciallytailored ultrashort laser pulses. Rabitz and co-workers [111–113] and others [114, 115]suggested that it would be possible to steer the quantum evolution to a desired productchannel by specifically designing and tailoring the time-dependent electric field of thelaser pulse to the characteristics of the system. Specifically, QOCT may be used todesign laser pulse shapes which are best suited for achieving the desired goal [111–122].An optimally shaped laser pulse typically has a complex form, both temporally andspectrally. The phases and amplitudes of different frequency components are optimizedto excite an interference pattern amongst distinct quantum pathways, to best achieve thedesired dynamics. The first optimal fields for quantum control were computed by Shi,Woody, and Rabitz [111] who showed that the amplitudes of the interfering vibrationalmodes of a laser-driven molecule could add up constructively in a given bond. We willreview QOCT and its applications in more detail in section 3 (for earlier reviews ofQOCT, see [7, 11, 14, 38, 40]). ontrol of quantum phenomena: Past, present, and future Laser pulse-shaping technology rapidly developed during the early 1990s [4, 5, 12].However, the capabilities of pulse shaping were not fully exploited in quantum controluntil the first experimental demonstrations of adaptive feedback control (AFC) in1997–1998 [123, 124]. Initially, ultrashort laser pulses with time-varying photonfrequencies were used to tune just the linear chirp, which represents an increaseor decrease of the instantaneous frequency as a function of time under the pulseenvelope. § Linearly chirped femtosecond laser pulses were successfully applied forcontrol of various atomic and molecular processes, including control of vibrationalwave packets [125–131], control of population transfer between atomic states [132–134]and between molecular vibrational levels [135–137] via “ladder-climbing” processes,control of electronic excitations in molecules [138–142], selective excitation of vibrationalmodes in coherent anti-Stokes Raman scattering (CARS) [143], improvement of theresolution of CARS spectroscopy [144, 145], and control of photoelectron spectra [146]and transitions through multiple highly excited states [147] in strong-field ionizationof atoms. In particular, when the emission and absorption bands of a moleculestrongly overlap, pulses with negative and positive chirp excite vibrational modespredominately in the ground and excited electronic states, respectively [125, 129–131].Chirped pulses can be also used to control the localization of vibrational wave packets indiatomic molecules, with the negative and positive chirp increasing and decreasing thelocalization, respectively [126–128]. Based on this effect, pump pulses with negativechirp were used to enhance selectivity in pump-dump control of photodissociationreactions [128]. Recently, the localization effect of negatively chirped pulses was usedto protect vibrational wave packets against rotationally-induced decoherence [148]. Dueto their effectiveness in various applications, chirped laser pulses are widely used inquantum control. However, by the end of the 1990s, many experimenters realized thatmore sophisticated pulse shapes, beyond just linear chirp, provide a much more powerfuland flexible tool for control of quantum phenomena in complex physical and chemicalsystems. Femtosecond pulse-shaping technology is utilized to the fullest extent in AFCexperiments where laser pulses are optimally tailored to meet the needs of complexquantum dynamics objectives [13, 15, 18, 19, 22, 25–29, 31, 37, 41]. The enormous growthof this field during the last decade is reviewed in section 5.
Optimal control of quantum phenomena in atoms and molecules usually operates at laserintensities sufficient to be in the non-perturbative regime. Thus, controlled dynamicswill naturally utilize the dynamic Stark shift amongst other available physical processesin order to reach the target. In a recent quantum control development, Stolow and co-workers proposed and experimentally demonstrated manipulation of molecular processes § The instantaneous frequency ω ( t ) of a linearly chirped pulse with a carrier frequency ω is given attime t by ω ( t ) = ω + 2 bt , where b is the chirp parameter that can be negative or positive. ontrol of quantum phenomena: Past, present, and future One of the earliest examples of coherent control of quantum dynamics is manipulationof nuclear spin ensembles using radiofrequency (RF) fields [153]. The main applicationof nuclear magnetic resonance (NMR) control techniques is high-resolution spectroscopyof polyatomic molecules (e.g., protein structure determination) [154–157]. While controlof an isolated spin by a time-dependent magnetic field is a simple quantum problem,in reality, NMR spectroscopy of molecules containing tens or even hundreds of nucleiinvolves many complex issues such as the effect of interactions between the spins, thermalrelaxation, instrumental noise, and influence of the solvent. Therefore, modern NMRspectroscopy often employs thousands of precisely sequenced and phase-modulatedpulses. Among important NMR control techniques are composite pulses, refocusing,and pulse shaping. In particular, the use of shaped RF pulses in NMR makes it possibleto improve the frequency selectivity, suppress the solvent contribution, simplify high-resolution spectra, and reduce the size and duration of experiments [158]. In recent years,NMR became an important testbed for developing control methods for applications inquantum information sciences [159–163]. In order to perform fault-tolerant quantumcomputations, the system dynamics must be controlled with an unprecedented level ofprecision, which requires even more sophisticated designs of control pulses than in high-resolution spectroscopy. In particular, QOCT was recently applied to identify optimalsequences of RF pulses for operation of NMR quantum information processors [163,164].
3. Quantum optimal control theory
The most comprehensive means for coherently controlling the evolution of a quantumsystem (e.g., a molecule) undergoing a complex dynamical process is through thecoordinated interaction between the system and the electromagnetic field whose spectral ontrol of quantum phenomena: Past, present, and future
Optimal control theory has an extensive history in traditional engineering applications[165, 166], but control of quantum phenomena imposes special features. Consider thetime evolution of a controlled quantum system,dd t U ( t ) = − i ~ [ H − ε ( t ) · µ ] U ( t ) , U (0) = I. (1)Here, U ( t ) is the unitary evolution operator of the system at time t , I is the identityoperator, H is the free Hamiltonian, µ is the dipole operator, and ε ( t ) is the controlfunction at time t . For the control function, we will use the notation ε ( · ) ∈ K (where K is the space of locally bounded, sufficiently smooth, square integrable functions oftime defined on some interval [0 , T ], with T fixed). Equation (1) adequately describesthe coherent quantum dynamics of a molecular system interacting with a laser electricfield in the dipole approximation or a spin system interacting with a time-dependentmagnetic field. We will consider finite-level quantum systems and denote the dimensionof the system’s Hilbert space as N . k For quantum systems undergoing environmentally-induced decoherence, there aremany dynamical models depending on the character of the system-environment coupling.If the system and environment are initially uncoupled, the evolution of the system’sreduced density matrix ρ from t = 0 to t is described by a completely positive, tracepreserving map: ρ ( t ) = Φ( ρ ) where ρ = ρ (0). This map (which is often called theKraus map) can be expressed using the operator-sum representation (OSR) [167]: ρ ( t ) = Φ( ρ ) = n X j =1 K j ( t ) ρ K † j ( t ) , (2)where K j are Kraus operators ( N × N complex matrices), which satisfy the condition P nj =1 K † j K j = I N , and I N denotes the identity operator on the Hilbert space ofdimension N . There exist infinitely many OSRs (with different sets of Kraus operators)for the same Kraus map. Various types of quantum master equations can be derivedfrom the Kraus OSR under additional assumptions [168, 169]. In particular, for the k If one is not concerned with the physically irrelevant global phase of the evolution operator, thecontrol problem can be restricted to the Hamiltonian represented by a traceless Hermitian matrix, and U ∈ SU( N ), where SU( N ) is the special unitary group. ontrol of quantum phenomena: Past, present, and future t ρ ( t ) = − i ~ [ H − ε ( t ) · µ, ρ ( t )] + N − X i =1 γ i (cid:20) L i ρ ( t ) L † i − L † i L i ρ ( t ) − ρ ( t ) L † i L i (cid:21) , (3)where γ i are non-negative constants and L i are Lindblad operators ( N × N complexmatrices) that represent the non-unitary effect of coupling to the environment. For aclosed system, (3) reduces to the von Neumann equation, ˙ ρ ( t ) = − (i / ~ )[ H − ε ( t ) · µ, ρ ( t )].For simplicity, we will formulate QOCT below using the Schr¨odinger equation (1)for the unitary evolution operator; analogous formulations using the von Neumannequation or the Lindblad master equation for the density matrix are available in theliterature [120, 171–174]. The general class of control objective functionals (cost functionals of the Bolza type)can be written as J [ U ( · ) , ε ( · )] = F ( U ( T )) + Z T G ( U ( t ) , ε ( t ))d t, (4)where F is a continuously differentiable function on U( N ), and G is a continuouslydifferentiable function on U( N ) × R . The optimal control problem may be stated as thesearch for J opt = max ε ( · ) J [ U ( · ) , ε ( · )] , (5)subject to the dynamical constraint (1). If only the term R T G ( U ( t ) , ε ( t ))d t 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 [166]. Three classes of problemscorresponding to different choices of F ( U ( T )) have received the most attention in thequantum 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 closeas possible to the target unitary transformation W . The Mayer-type cost functional inthis case can be generally expressed as F ( U ( T )) = 1 − k W − U ( T ) k , (6)where k · k is an appropriate normalized matrix norm, i.e., F ( U ( T )) is maximized whenthe distance between U ( T ) and W is minimized. This type of objective is common inquantum computing applications [175], where F ( U ( T )) is the fidelity of a quantumgate [176, 177]. One frequently used form of the objective functional F ( U ( T )) is ontrol of quantum phenomena: Past, present, and future k X k = (2 N ) − Tr( X † X )) [179–181]: F ( U ( T )) = 1 N ℜ Tr (cid:2) W † U ( T ) (cid:3) . (7)Other forms of the objective functional, which employ different matrix norms in (6),are possible as well [181–184]. For example, a modification of (7), F ( U ( T )) = N − (cid:12)(cid:12) Tr (cid:2) W † U ( T ) (cid:3)(cid:12)(cid:12) , which is independent of the global phase of U ( T ), can be used. Notethat F ( U ( T )) is independent of the initial state, as the quantum gate must producethe same unitary transformation for any input state of the qubit system [175].For state control, the goal is to transform the initial state ρ into a final state ρ ( T ) = U ( T ) ρ U † ( T ) that is as close as possible to the target state ρ f . The correspondingMayer-type cost functional is F ( U ( T )) = 1 − k U ( T ) ρ U † ( T ) − ρ f k , (8)where k · k is an appropriate normalized matrix norm (e.g., the Hilbert-Schmidt normcan be used) [185–188].For observable control, the goal is typically to maximize the expectation value of atarget quantum observable Θ (represented by a Hermitian operator). The correspondingMayer-type cost functional is [116, 189–192] F ( U ( T )) = Tr (cid:2) U ( T ) ρ U † ( T )Θ (cid:3) . (9)An important special case is state-transition control (also known as population transfercontrol), for which ρ = | ψ i ih ψ i | and Θ = | ψ f ih ψ f | , where | ψ i i and | ψ f i are eigenstatesof the free Hamiltonian H ; in this case, the objective functional (9) has the form F ( U ( T )) = P i → f = |h ψ f | U ( T ) | ψ i i| , which is the probability of transition (i.e., thepopulation transfer yield) between the energy levels of the quantum system [193, 194].In many chemical and physical applications of quantum control, absolute yields arenot known, and therefore maximizing the expectation value of an observable (e.g., thepopulation transfer yield) is a more appropriate laboratory goal than minimizing thedistance to a target expectation value.Also, in quantum control experiments (see section 5), measuring the expectationvalue of an observable is much easier than estimating the quantum state or evolutionoperator. Existing methods of quantum-state and evolution-operator estimation relyon tomographic techniques [195–202] which are extremely expensive in terms of thenumber of required measurements (e.g., in quantum computing applications, standardmethods of state and process tomography require numbers of measurements that scaleexponentially in the number of qubits [175, 202–206]). Therefore, virtually all quantumcontrol experiments so far have used observable control with objective functionals of theform (9). For example, in an AFC experiment [148], in which the goal was to maximizethe degree of coherence, the expectation value of an observable representing the degreeof quantum state localization was used as a coherence “surrogate,” instead of state ontrol of quantum phenomena: Past, present, and future F ( U ( T )) = n X k =1 α k Tr (cid:2) U ( T ) ρ U † ( T )Θ k (cid:3) , (10)which extends (9) to multiple quantum observables. Also, general methods of multi-objective optimization [220–222] have been recently applied to various quantum controlproblems [217–219].Another common goal in quantum control is to maximize a Lagrange-type costfunctional subject to a constraint on U ( T ) [223, 224]. For example, this type of controlproblem can be formulated as follows:max ε ( · ) Z T G ( ε ( t )) d t, subject to F ( U ( T )) = χ, (11)where F ( U ( T )) is the Mayer-type cost functional for evolution-operator, state, orobservable control (as described above), and χ is a constant that corresponds to thetarget value of F . Often, the goal is to minimize the total field fluence, in which case G ( ε ( t )) = − ε ( t ) is used. One of the fundamental issues of quantum control is to assess the system’s controllability.A quantum system is controllable in a set of configurations, S = { ζ } , if for any pair ofconfigurations ζ , ζ ∈ S there exists a time-dependent control ε ( · ) that can drive thesystem from the initial configuration ζ to the final configuration ζ in a finite time T .Here, the notion of configuration means either the state of the system ρ , the expectationvalue of an observable Tr( ρ Θ), the evolution operator U , or the Kraus map Φ, dependingon the specific control problem. Controllability of closed quantum systems with unitarydynamics has been well studied [225–244]. Controllability analysis was also extended toopen quantum systems [245–252].Controllability is determined by the equation of motion as well as properties ofthe Hamiltonian. For a closed quantum system with unitary dynamics (1), evolution-operator controllability implies that for any unitary operator W there exists a finitetime T and a control ε ( · ), such that W = U ( T ), where U ( T ) is the solution of (1).For an N -level closed system, a necessary and sufficient condition for evolution-operator ontrol of quantum phenomena: Past, present, and future G of the system (i.e., the Lie groupgenerated by the system’s Hamiltonian) be U( N ) (or SU( N ) for a traceless Hamiltonian)[235–237].Unitary evolution preserves the spectrum of the quantum state (i.e., the eigenvaluesof the density matrix). All density matrices that have the same eigenvalues form aset of unitarily equivalent states (e.g., the set of all pure states). Therefore, underunitary evolution, a quantum system can be state controllable only within a set ofunitarily equivalent states [235, 236]. Density-matrix controllability means that for anypair of unitarily equivalent density matrices ρ and ρ there exists a control ε ( · ) thatdrives ρ into ρ (in a finite time). It has been shown [235–237] that density-matrixcontrollability is equivalent to evolution-operator controllability. For specific classes ofdensity matrices, the requirements for controllability are weaker [229–231]. For example,pure-state controllability requires that the system’s dynamical Lie group G is transitiveon the sphere S N − . For infinite-level quantum systems evolving on non-compact Liegroups, such as those arising in quantum optics, the conditions for controllability aremore stringent [243, 244, 251, 253]. To solve for optimal controls that maximize an objective functional (of the typesdiscussed in section 3.2), it is convenient to define a functional ˜ J that explicitlyincorporates the dynamical constraint (1):˜ J [ U ( · ) , φ ( · ) , ε ( · )] = F ( U ( T )) + λ Z T G ( U ( t ) , ε ( t ))d t − ℜ Z T Tr (cid:26) φ † ( t ) (cid:20) d U ( t )d t + i ~ ( H − ε ( t ) · µ ) U ( t ) (cid:21)(cid:27) d t. (12)Here, λ is a scalar weight and an auxiliary operator φ ( t ) is a Lagrange multiplieremployed to enforce satisfaction of Eq. (1).Various modifications of the objective functional (12) are possible, for example,QOCT can be formulated for open systems with non-unitary dynamics [120, 171–174,254–258]. Modified objective functionals can also comprise additional spectral andfluence constraints on the control field [259,260], take into account nonlinear interactionswith the control field [261, 262], deal with time-dependent and time-averaged targets[256, 263–265], and include the final time as a free control parameter [266, 267]. It isalso possible to formulate QOCT with time minimization as a control goal (time optimalcontrol) [268–271]. As we mentioned earlier, QOCT can be also extended to incorporateoptimization of multiple objectives [215–219].A necessary condition for a solution of the optimization problem (5) subject to thedynamical constraint (1) is that the first-order functional derivatives of ˜ J with respect to U ( · ), φ ( · ), and ε ( · ) are equal to zero. Correspondingly, optimal controls can be obtainedby solving the resulting Euler-Lagrange equations. Equivalently, optimal controls can bederived through application of the Pontryagin maximum principle (PMP) [165,223,224]. ontrol of quantum phenomena: Past, present, and future ε ( · ). So-called Legendre conditions on theHessian, which depend on the type of cost, are also required for optimality [165, 166].An important issue is the existence of optimal control fields (i.e., maxima ofthe objective functional) for realistic situations that involve practical constraints onthe applied laser fields. It is important to distinguish between the existence of anoptimal control field and controllability; in the former case, a field is designed, subjectto particular constraints, that guides the evolution of the system towards a specifiedtarget until a maximum of the objective functional is reached, while in the lattercase, the exact coincidence between the attained evolution operator (or state) andthe target evolution operator (or state) is sought. The existence of optimal controlsfor quantum systems was analyzed in a number of works. Peirce et al. [112] provedthe existence of optimal solutions for state control in a spatially bounded quantumsystem that necessarily has spatially localized states and a discrete spectrum. Zhao andRice [272] extended this analysis to a system with both discrete and continuous statesand proved the existence of optimal controls over the evolution in the subspace of discretestates. Demiralp and Rabitz [185] showed that, in general, there is a denumerableinfinity of solutions to a particular class of well-posed quantum control problems; thesolutions can be ordered in quality according to the achieved optimal value of theobjective functional. The existence of multiple control solutions has important practicalconsequences, suggesting that there may be broad latitude in the laboratory, even understrict experimental restrictions, for finding successful controls for well-posed quantumobjectives. The existence and properties of critical points (including global extrema)of objective functionals for various types of quantum control problems were furtherexplored using the analysis of control landscapes [179–181, 189–194, 273] (see section 4).A number of optimization algorithms were adapted or specially developed foruse in QOCT, including the conjugate gradient search method [114], the Krotovmethod [177, 274, 275], monotonically convergent algorithms [276–282], non-iterativealgorithms [283], the gradient ascent pulse engineering (GRAPE) algorithm [164], ahybrid local/global algorithm [256], and homotopy-based methods [284–286]. Fasterconvergence of iterative QOCT algorithms was demonstrated using “mixing” strategies[287]. Also, the employment of propagation toolkits [288–290] greatly increases theefficiency of numerical optimizations and allows for fast combinatorial optimization [291].Detailed discussions of the QOCT formalism and algorithms are available in theliterature [11, 14, 38, 40]. In order to illustrate optimal control of molecules, we consider an instructive example.In one of the pioneering QOCT studies, Kosloff et al. [114] considered two electronicstates (ground and excited) of a model molecular system, with the wave function (in ontrol of quantum phenomena: Past, present, and future ¶ ψ ( r , t ) = h r | ψ ( t ) i = ψ e ( r , t ) ψ g ( r , t ) ! , (13)where ψ g and ψ e are the projections of the wave function on the ground and excited state,respectively. The time evolution of the wave function is determined by the Schr¨odingerequation: i ~ ∂∂t ψ e ( r , t ) ψ g ( r , t ) ! = H e ( r ) H ge ( r , t ) H † ge ( r , t ) H g ( r ) ! ψ e ( r , t ) ψ g ( r , t ) ! , (14)where H i ( r ) = p / (2 m ) + V i ( r ) ( i = g , e), p is the momentum operator, V g ( r ) and V e ( r ) are the adiabatic potential energy surfaces for the ground and excited state,respectively. The off-diagonal term H ge ( r , t ) represents the field-induced couplingbetween the molecular states: H ge ( r , t ) = − µ ge ( r ) ε ( t ) , (15)where µ ge ( r ) is the electric dipole operator and ε ( t ) is the time-dependent electric fieldof the control laser pulse applied to the molecule.The goal is to control a dissociation reaction in the presence of two distinct exitchannels on the ground potential energy surface. The corresponding objective functional(including the dynamical constraint) is given by˜ J = h ψ ( T ) | P | ψ ( T ) i − λ Z T ε ( t ) d t − ℜ Z T h χ ( t ) | (cid:18) ∂∂t + i ~ H (cid:19) | ψ ( t ) i d t, (16)The first term in (16) represents the main control goal, where P is the projectionoperator on the state corresponding to the target exit channel (i.e., the part of thewave function which is beyond the target saddle point on the ground-state surface andis characterized by the outgoing momentum); the second term is used to manage thefluence of the control field, with λ being a scalar weight factor; the third term includesan auxiliary state | χ ( t ) i that is a Lagrange multiplier employed to enforce satisfactionof the Schr¨odinger equation ( H is the 2 × J with respect to χ ( · ), ψ ( · ), and ε ( · ) are set to zero, producing the followingEuler-Lagrange equations:i ~ ∂∂t | ψ ( t ) i = H | ψ ( t ) i , | ψ (0) i = | ψ i , (17)i ~ ∂∂t | χ ( t ) i = H | χ ( t ) i , | χ ( T ) i = P | ψ ( T ) i , (18) ε ( t ) = − ~ λ ℑ {h χ g ( t ) | µ ge | ψ e ( t ) i + h χ e ( t ) | µ ge | ψ g ( t ) i} . (19) ¶ For the sake of notation consistency, the control problem is presented here slightly differently thanin the original work [114]. ontrol of quantum phenomena: Past, present, and future ε opt ( · ) that maximizesphotoinduced molecular dissociation into the target channel. Successful application ofQOCT to this model molecular system [114] demonstrated the benefits of optimallytailoring the time-dependent laser field to achieve the desired dynamic outcome. Originally, QOCT was developed to design optimal fields for manipulation of molecularsystems [111–122] and has been applied to a myriad of problems (e.g., rotational,vibrational, electronic, reactive, and other processes) [11, 14, 40]. Some recentapplications include, for example, control of molecular isomerization [292–295], controlof electron ring currents in chiral aromatic molecules [296], control of current flowpatterns through molecular wires [297], and control of heterogeneous electron transferfrom surface attached molecules into semiconductor band states [298]. Beyondmolecules, QOCT has been applied to various physical objectives including, forexample, control of electron states in semiconductor quantum structures [299–301],control of atom transport in optical lattices [302], control of Bose-Einstein condensatetransport in magnetic microtraps [303], control of a transition of ultracold atoms fromthe superfluid phase to a Mott insulator state [304], control of coherent populationtransfer in superconducting quantum interference devices [305], and control of thelocal electromagnetic response of nanostructured materials [306]. Recent interest hasrapidly grown in applications of QOCT to the field of quantum information sciences,including optimal protection of quantum systems against decoherence [188,257,307–317],optimal operation of quantum gates in closed systems [176, 177, 224, 253, 318–332] andin open systems (i.e., in the presence of decoherence) [184, 333–348], optimal generationof entanglement [266, 267, 347, 349–351], and optimal (i.e., maximum-rate) transfer ofquantum information [352]. In a recent experiment with trapped ion qubits, shapedpulses designed using QOCT were applied to enact single-qubit gates with enhancedrobustness to noise in the control field [353]. Optimal control methods were also appliedto the problem of storage and retrieval of photonic states in atomic media, includingboth theoretical optimization [354–356] and experimental tests [357–359].
An advantage of QOCT relative to the laboratory method of AFC (to be discussedin detail in section 5) is that the former can be used to optimize a well definedobjective functional of virtually any form, while the latter relies on information obtainedfrom measurements and thus is best suited to optimize expectation values of directlymeasurable observables. In numerical optimizations, there is practically no differencein effort between computing the expectation value of an observable, the density matrix, ontrol of quantum phenomena: Past, present, and future
4. Quantum control landscapes
An important practical goal of quantum control is the discovery of optimal solutionsfor manipulating quantum phenomena. Early studies [112,185,272] described conditionsunder which optimal solutions exist, but did not explore the complexity of finding them.Underlying the search for optimal controls is the landscape which specifies the physicalobjective as a function of the control variables. Analysis of quantum control landscapes ontrol of quantum phenomena: Past, present, and future
Properties of the search space associated with Mayer-type cost functionals play afundamental role in the ability to identify optimal controls. To characterize theseproperties, it is convenient to express the cost functional in a form where the dynamicalconstraints are implicitly satisfied. Consider a control problem with a fixed target time T for a closed quantum system with unitary evolution. Denote by V T : ε ( · ) U ( T )the endpoint map from the space of control functions to the space of unitary evolutionoperators, induced by the Schr¨odinger equation (1), so that U ( T ) = V T ( ε ( · )). A Mayer-type cost functional F ( U ( T )) itself describes a map F from the space of evolutionoperators to the space of real-valued costs. Thus the composition of these maps, J = F ◦ V T : K → R , is a map from the space of control functions to the spaceof real-valued costs. This map generates the functional J [ ε ( · )] = F ( V T ( ε ( · ))). We willrefer to the functional J [ ε ( · )] as the control landscape . The optimal control problemmay then be expressed as the unconstrained search for J opt = max ε ( · ) J [ ε ( · )] . (20)The topology of the control landscape (i.e., the character of its critical points, includinglocal and global extrema) determines whether local search algorithms will converge toglobally optimal solutions to the control problem [361]. Studies of quantum controllandscape topology are presently an active research area [39, 179–181, 189, 191–194, 233,362].The critical points (extrema) of the landscape are controls, at which the first-orderfunctional derivative of J [ ε ( · )] with respect to the control field is zero for all time, i.e., δJ [ ε ( · )] δε ( t ) = 0 , ∀ t ∈ [0 , T ] . (21)The critical manifold M of the control landscape is the collection of all critical points: M = { ε ( · ) | δJ/δε ( t ) = 0 , ∀ t ∈ [0 , T ] } . (22)A central concept in landscape topology is the classification of a critical point as regularor singular [363, 364]. Most generally, a critical point of J [ ε ( · )] is regular if the map V T is locally surjective in its vicinity, i.e., if for any local increment δU ( T ) in the ontrol of quantum phenomena: Past, present, and future δε ( · ) in the control function such that V T ( ε ( · ) + δε ( · )) = V T ( ε ( · )) + δU ( T ). This condition is equivalent to requiring thatthe elements µ ij ( t ) of the time-dependent dipole-operator matrix (in the Heisenbergpicture) form a set of N linearly independent functions of time [181]. In its turn, thiscondition is satisfied for all non-constant admissible controls if and only if the quantumsystem is evolution-operator controllable [181, 226]. Note that for landscapes of someparticular physical objectives the conditions for regularity of the critical points can beless stringent. For example, in the important special case of state-transition control, acritical point is regular if the matrix elements µ ij ( t ) contain a set of just 2 N − J [ ε ( · )] is singular if the map V T is not locally surjective in thepoint’s vicinity. Using the chain rule, one obtains: δJδε ( t ) = (cid:28) ∇ F ( U ( T )) , δU ( T ) δε ( t ) (cid:29) , (23)where ∇ F ( U ( T )) is the gradient of F at U ( T ), δU ( T ) /δε ( t ) is the first-order functionalderivative of U ( T ) with respect to the control field, and h A, B i = Tr( A † B ) is the Hilbert-Schmidt inner product. From (23), if a critical point of J is regular, ∇ F ( U ( T )) mustbe zero. A critical point is called kinematic if ∇ F ( U ( T )) = 0 and non-kinematic if ∇ F ( U ( T )) = 0. Thus, all regular critical points are kinematic. A singular critical pointmay be either kinematic or non-kinematic; in the latter case, δJ/δε ( t ) = 0 whereas ∇ F ( U ( T )) = 0 [364]. On quantum control landscapes, the measure of regular criticalpoints appears to be much greater than that of singular ones [364]. Therefore attentionhas been focused on the characterization of regular critical points, and several importantresults have been obtained [39]. Nevertheless, singular critical points on quantumcontrol landscapes have been recently studied theoretically [364] and demonstratedexperimentally [365].The condition for kinematic critical points, ∇ F ( U ( T )) = 0, can be cast in anexplicit form for various types of quantum control problems. For evolution-operatorcontrol with the objective functional J = F ( U ( T )) of (7), this condition becomes[179, 180] W † U ( T ) = U † ( T ) W, (24)i.e., W † U ( T ) is required to be a Hermitian operator. It was shown [179, 180] thatthis condition implies W † U ( T ) = Y † ( − I m ⊕ I N − m ) Y , where Y is an arbitrary unitarytransformation and m = 0 , , . . . , N . There are N + 1 distinct critical submanifoldslabeled by m , with corresponding critical values of J given by J m = 1 − (2 m/N ).The global optima corresponding to m = 0 and m = N (with J = 1 and J N = −
1, respectively) are isolated points, while local extrema corresponding to m = 1 , , . . . , N − N ). It can be shown that all regular local extrema are saddle-point regions [180]. ontrol of quantum phenomena: Past, present, and future J = F ( U ( T )) of (9), thecondition for a kinematic critical point becomes [189, 193, 233][ U ( T ) ρ U † ( T ) , Θ] = 0 , (25)i.e., the density matrix at the final time is required to commute with the targetobservable operator. This condition was studied in the context of optimization ofLagrange-type cost functionals with an endpoint constraint [362, 366, 367] as well asin the context of regular critical points for Mayer-type cost functionals [189, 233]. Let R and S denote unitary matrices that diagonalize ρ and Θ, respectively, and define˜ U ( T ) = S † U ( T ) R . The condition (25) that ρ ( T ) and Θ commute is equivalent to thecondition that the matrix ˜ U ( T ) is in the double coset M π of some permutation matrix P π [191]: ˜ U ( T ) ∈ M π = U( n ) P π U( m ) . (26)Here, U( n ) is the product group U( n ) × · · · × U( n r ), where U( n l ) corresponds tothe l th eigenvalue of ρ with n l -fold degeneracy, and U( m ) is the product groupU( m ) × · · · × U( m s ), where U( m l ) corresponds to the l th eigenvalue of Θ with m l -fold degeneracy. Thus, each critical submanifold M π corresponds to a particular choiceof the permutation π . All permutations on N indices form the symmetric group S N ,and the entire critical manifold M is given by M = S π ∈ S N M π . The structure of M depends on any degeneracies in the spectra of ρ and Θ. When both ρ and Θ arefully nondegenerate, then U( n ) = U( m ) = [U(1)] N , and M consists of N ! disjoint N -dimensional tori, labeled by the permutation matrices. The occurrence of degeneraciesin the spectra of ρ and Θ will merge two or more tori together, thereby reducing thenumber of disjoint critical submanifolds and increasing their dimensions [191]. Satisfaction of the condition (21) for a critical point is a necessary but not sufficientcondition for optimality of a control [166, 223]. For Mayer-type cost functionals, asufficient condition for optimality is negative semidefiniteness of the Hessian of J , whichis defined as H ( t, t ′ ) := δ Jδε ( t ′ ) δε ( t ) . (27)The characteristics of critical points (in particular, the presence or absence of localoptima) are important for the convergence properties of search algorithms [39]. Toclassify critical points as global maxima and minima, local maxima and minima, andsaddle points, one examines the second-order variation in J for an arbitrary controlvariation δε ( · ), which for Mayer-type functionals can be written as δ J = Q F ( δU ( T ) , δU ( T )) + h∇ F ( U ( T )) , δ U ( T ) i , (28)where δU ( T ) and δ U ( T ) are the first- and second-order variations, respectively, of U ( T ) caused by a control variation δε ( · ), and Q F is the Hessian quadratic form of ontrol of quantum phenomena: Past, present, and future F ( U ). Assuming that the critical point ε ( · ) is regular, one obtains: δ J = Q F ( δU ( T ) , δU ( T )) . (29)Explicit expressions for the Hessian and/or Hessian quadratic form were obtained forevolution-operator control [179–181] and observable control [190, 361].The optimality of regular critical points can be determined by inspecting thenumber of positive, negative and null eigenvalues of the Hessian (or, equivalently, thecoefficients of the Hessian quadratic form when written in a diagonal basis). An issueof special interest is to determine whether any of the regular critical points are localmaxima (frequently referred to as local traps due to their ability to halt searches guidedby gradient algorithms before reaching the global maximum). Detailed analyses forevolution-operator control and observable control reveal [39, 179–181, 190, 361] that allregular optima are global and the remainder of regular critical points (i.e., except forthe global maximum and global minimum) are saddles. This discovery means that nolocal traps exist in the control landscapes of controllable closed quantum systems. Thesame result was also obtained for observable-control landscapes of controllable openquantum systems with Kraus-map dynamics [273]. Due attention still needs to be givento consideration of singular critical points, although numerical evidence suggests thattheir effect on optimization is likely insignificant [364]. Many practical quantum control problems seek to optimize multiple, often competing,objectives. In such situations the usual notion of optimality is replaced by that ofPareto optimality. The
Pareto front of a multi-objective control problem is the setof all controls such that all other controls have a lower value for at least one of theobjectives [220–222]. The analysis of the Pareto front reveals the nature of conflictsand tradeoffs between different control objectives. The structure of the landscape formulti-observable control is of interest and follows directly from that of single-observablecontrol [217]. Of particular relevance to many chemical and physical applications is theproblem of simultaneous maximization of the expectation values of multiple observables.Such simultaneous maximization is possible if the intersection T k M (max) k (where M (max) k is the maximum submanifold for the k th observable) is nonempty and a point in theintersection can be reached under some control ε ( · ); in this regard, the dimension ofthe intersection manifold T k M (max) k has been analyzed [218]. It has been shown thatthe common QOCT technique of running many independent maximizations of a costfunctional like (10) (using different weight coefficients { α k } ) is incapable of samplingmany regions of the Pareto front [218]. Alternative methods for Pareto front samplingare discussed further below. ontrol of quantum phenomena: Past, present, and future The absence of local traps in landscapes for observable control and evolution-operator control with Mayer-type cost functionals has important implications for thedesign of optimization algorithms. Many practical applications require algorithmscapable of searching quantum control landscapes for optimal solutions that satisfyadditional criteria, such as minimization of the field fluence or maximization ofthe robustness to laser noise. So-called homotopy trajectory control algorithms (inparticular, diffeomorphic modulation under observable-response-preserving homotopy,or D-MORPH) [284–286] can follow paths to the global maximum of a Mayer-typecost functional, exploiting the trap-free nature of the control landscape, while locallyoptimizing auxiliary costs. The essential prerequisite for successful use of thesealgorithms is the existence of a connected path between the initial and target controls.Homotopy trajectory control is closely related to the notion of a level set which is definedas the collection of controls that all produce the same value of the cost functional J .Theoretical analysis [39, 285, 286] predicts that for controllable quantum systems eachlevel set is a continuous manifold. A homotopy trajectory algorithm is able to move onsuch a manifold exploring different control solutions that result in the same value of thecost functional, but may differ in other properties (e.g., the field fluence or robustness).A version of the D-MORPH algorithm was also developed for evolution-operator controlof closed quantum systems; it was able to identify optimal controls generating a targetunitary transformation up to machine precision [324].Homotopy trajectory algorithms are also very useful for exploring quantum controllandscapes for multiple objectives. For example, in order to track paths in the space ofexpectation values of multiple observables while locally minimizing a Lagrange-type cost,multi-observable trajectory control algorithms were developed [217]. Such algorithmsare generally applicable to the treatment of multi-objective quantum control problems(Pareto quantum optimal control) [218]. They can traverse the Pareto front to identifyadmissible tradeoffs in optimization of multiple control objectives (e.g., maximizationof multiple observable expectation values). This method can continuously sample thePareto front during the course of one optimization run [218] and thus can be moreefficient than the use of standard QOCT with cost functionals of the form (10). Also,the D-MORPH algorithm was recently extended to handle optimal control problemsinvolving multiple quantum systems and multiple objectives [219]. The absence of local traps in control landscapes of controllable quantum systems hasvery important implications for the feasibility of AFC experiments (see section 5).The relationship between the quantum control landscape structure and optimizationcomplexity of algorithms used in AFC has been the subject of recent theoreticalanalyses [39,368–370]. Results of these studies support the vast empirical evidence [361]indicating that the favorable landscape topology strongly correlates with fast mean ontrol of quantum phenomena: Past, present, and future
Significant efforts have been recently devoted to experimental observation of quantumcontrol landscapes, aiming both at testing the predictions of the theoretical analysis andat obtaining a better understanding of control mechanisms. Roslund et al. [372] observedquantum control level sets for maximization of non-resonant two-photon absorption in amolecule and second harmonic generation (SHG) in a nonlinear crystal and found themto be continuous manifolds (closed surfaces) in the control landscape. A diverse familyof control mechanisms was encountered, as each of the multiple control fields forminga level set preserves the observable value by exciting a distinct pattern of constructiveand destructive quantum interferences.Wollenhaupt, Baumert, and co-workers [373, 374] used parameterized pulse shapesto reduce the dimensionality of the optimization problem (maximization of theAutler-Townes contrast in strong-field ionization of potassium atoms) and observedthe corresponding two-dimensional quantum control landscape. In order to betterunderstand the performance of AFC, the evolution of different optimization procedureswas visualized by means of trajectories on the surface of the measured control landscape.Marquetand et al. [375] observed a two-dimensional quantum control landscape (formaximization of the retinal photoisomerization yield in bacteriorhodopsin) and used itto elucidate the properties of molecular wave-packet evolution on an excited potentialenergy surface.The theoretical analysis of control landscape topology has been carried out withno constraints placed on the controls (see section 4.1). A main conclusion fromthese studies is the inherent lack of local traps on quantum control landscapes undernormal circumstances. Recently, Roslund and Rabitz [376] experimentally demonstratedthe trap-free monotonic character of control landscapes for optimization of frequencyunfiltered and filtered SHG. For unfiltered SHG, the landscape was randomly sampledand interpolation of the data was found to be devoid of traps up to the level of data noise.In the case of narrow-band-filtered SHG, trajectories taken on the landscape revealedthe absence of traps, although a rich local structure was observed on the landscape inthis case. Despite the inherent trap-free nature of the landscapes, significant constraintson the controls can distort and/or isolate portions of the erstwhile trap-free landscapeto produce apparent (i.e., false) traps [376]. Such artificial structure arising from theforced sampling of the landscape has been seen in some experimental studies [373–375], ontrol of quantum phenomena: Past, present, and future
5. Adaptive feedback control in the laboratory
There are important differences between quantum control theory and its experimentalimplementation. Control solutions obtained in theoretical studies strongly depend onthe employed model Hamiltonian. However, for real systems controlled in the laboratory,the Hamiltonians usually are not known well (except for the simplest cases), andthe Hamiltonians for the system-environment coupling are known to an even lesserdegree. An additional difficulty is the computational complexity of accurately solvingthe optimal control equations for realistic polyatomic molecules. Another importantdifference between control theory and experiment arises from the difficulty of reliablyimplementing theoretical control designs in the laboratory, due to instrumental noiseand other limitations. As a result, optimal theoretical control designs generally willnot be optimal in the laboratory. Notwithstanding these comments, control simulationscontinue to be very valuable, and they even set forth the logic leading to practicallaboratory control as explained below.A crucial step towards selective laser control of physical and chemical phenomenaon the quantum scale was the introduction of AFC (also referred to as closed-looplaboratory control or learning control). AFC was proposed and theoretically groundedby Judson and Rabitz in their paper “Teaching lasers to control molecules” in 1992 [377].In AFC, a loop is closed in the laboratory, with results of measurements on the quantumsystem used to evaluate the success of the applied control and to refine it, until thecontrol objective is reached as best as possible. At each cycle of the loop, the externalcontrol (e.g., a shaped laser pulse) is applied to the system (e.g., an ensemble ofmolecules). The signal (e.g., the yield of a particular reaction product or populationin a target state) is detected and fed back to the learning algorithm (e.g., a geneticalgorithm). The algorithm evaluates each control based on its measured outcome withrespect to a predefined control goal, and searches through the space of available controlsto move towards an optimal solution.While AFC can be simulated on the computer [377–390], the important advantageof this approach lies in its ability to be directly implemented in the laboratory. Mostimportantly, the optimization is performed in the laboratory with the actual system, andthus is independent of any model. As a result, the AFC method works remarkably wellfor systems even of high complexity, including, for example, large polyatomic moleculesin the liquid phase, for which only very rough models are available. Second, there isno need to measure the laser field in AFC, because any systematic characterizationof the control “knobs” (such as pulse shaper parameters) is sufficient. This set ofcontrol “knobs” determined by the experimental apparatus defines the parameter spacesearched by the learning algorithm for an optimal laser shape. This procedure naturallyincorporates any laboratory constraints on the control laser fields. Third, optimalcontrols identified in AFC are characterized by a natural degree of robustness to ontrol of quantum phenomena: Past, present, and future
The majority of current AFC experiments employ shaped ultrafast laser pulses. Insuch experiments, one usually starts with a random or nearly random selection of trialshaped pulses of length ∼ − s or less. The pulses are shaped by modulating thephases and/or amplitudes of the spatially resolved spectral components, for example,by means of a liquid crystal modulator (LCM), acousto-optic modulator (AOM), ora micromechanical mirror array (MMA). The experiments employ fully automatedcomputer control of the pulse shapes guided by a learning algorithm. The shaped laserpulses produced by this method can be viewed as “photonic reagents,” which interactwith matter at the atomic or molecular scale to facilitate desired controlled outcomesof various physical and chemical phenomena.Significant femtosecond pulse-shaping capabilities were already available in theearly 1990s, with the development of a programmable multi-element liquid-crystal phasemodulator that operated on a millisecond time scale [397]. Devices with two LCMs madepossible simultaneous and independent phase and amplitude modulation of spectralcomponents [398, 399]. Similar capabilities are also available with AOM-based pulseshapers [400]. These and other developments have been reviewed [4, 5, 12, 27]. Duringthe last decade, physical and chemical applications of AFC motivated further advancesin femtosecond pulse-shaping technology, including arbitrary amplitude and phasemodulation in an acousto-optic programmable dispersive filter [401], enhanced resolutionof LCMs [402, 403], compact and robust pulse-shaping [404], pulse-shape modulation atnanosecond time scales using an electro-optical gallium arsenide array with controlledwaveguides [405], and spectral line-by-line pulse shaping [406, 407]. The developmentof polarization pulse shaping [408, 409] brought an additional dimension to control ofquantum phenomena, which is particularly important in some applications (e.g., forincreasing the yield of multiphoton ionization in molecules); recent improvements in thisarea also include full control of the spectral polarization of ultrashort laser pulses [410],simultaneous phase, amplitude, and polarization shaping [411–415], and a simplifiedultrafast polarization shaper using a birefringent prism [416]. Most recently, the ontrol of quantum phenomena: Past, present, and future The AFC approach can be used to produce optical fields with prescribed properties,which, in turn, can be applied to control physical and chemical phenomena (e.g., inatoms, molecules, and semiconductor structures). In particular, some of the earliestAFC experiments aimed at the maximal compression of femtosecond laser pulses[421–426]. In these experiments, the light produced through SHG of the shaped pulsein a thin nonlinear crystal served as the feedback signal. The SHG yield is directlyproportional to the intensity of the incident light pulse, and, for pulses with a fixedenergy, the most intense pulse is the shortest one. AFC-optimized compression ofbroadband laser pulses was also demonstrated using a feedback signal derived fromtwo-photon absorption in semiconductors [427]. Application of AFC makes it possibleto generate maximally compressed laser pulses in a simple and effective way, withoutrequiring knowledge of the input pulse’s shape. Such adaptive pulse compressors(with AOM-based pulse shapers and SHG-based feedback signal) are now employed asbuilt-in components in some commercially available femtosecond amplification systems.However, since the amplification process is nearly linear, the full-scale application ofAFC is usually not necessary, as pulses can be compressed in a single feedback step usingspectral interferometry. The resulting transform-limited pulses can be used as a startingpoint for the study and control of various photophysical and photochemical processes(e.g., they can be used to excite and track localized fine-structure and Rydberg wavepackets in atoms and vibrational wave packets in molecules). In many AFC experiments,transform-limited pulses are used as a reference, to separate off the intensity dependencewhich is ubiquitous in nonlinear processes.Another optical application of AFC is optimal amplification of chirped femtosecondlaser pulses [428, 429]. The AFC method is used to minimize the higher-order phasedispersion that is inherent in the amplification process. Furthermore, AFC was appliedto generate almost arbitrary target temporal shapes starting with uncharacterized inputpulses [430, 431]. These experiments used a cross-correlation measurement [430] andelectric field characterization via frequency-resolved optical gating (FROG) [431] ofthe output pulses as the feedback signal. As polarization shaping technology forfemtosecond laser pulses developed [408, 409], AFC was used to generate pulses withtarget polarization profiles [432, 433]. One experiment [432] used the SHG feedbacksignal to compensate for material dispersion and time-dependent modulation of thepolarization state. Another experiment [433] employed a sophisticated feedback signalbased on dual-channel spectral interferometry to generate shaped femtosecond pulseswhose ellipticity increased at a constant rate. In a further development, a recentexperiment [434] used polarization-shaped laser pulses and AFC to manipulate the ontrol of quantum phenomena: Past, present, and future
Among important physical applications of AFC is coherent manipulation of soft X-raysproduced via high-harmonic generation. In a pioneering experiment, Murnane, Kapteyn,and co-workers [435] used shaped ultrashort, intense laser pulses (with 6-8 opticalcycles) for AFC of high-harmonic generation in atomic gases. Their results demonstratethat optimally shaped laser pulses identified by the learning algorithm can improvethe efficiency of X-ray generation by an order of magnitude, manipulate the spectralcharacteristics of the emitted radiation, and “channel” the interaction between nonlinearprocesses of different orders. All these effects result from complex interferences betweenthe quantum amplitudes of the atomic states, created by the external laser field. Thelearning algorithm guides the pulse shaper to tailor the laser field to produce the optimalinterference pattern. Several consequent AFC experiments [436–438] explored variousaspects of optimal high-harmonic generation in atomic gases, including the analysisof the control mechanism via a comparison of experimental data with predictions oftheoretical models. Further experimental studies used AFC for optimal spatial controlof high-harmonic generation in hollow fibers [439,440], optimal control of the brilliance ofhigh-harmonic generation in gas jet and capillary setups [441], and optimal control of thespectral shape of coherent soft X-rays [442]. The latter work [442] has been a precursorto a more recent development, in which spectrally shaped femtosecond X-ray fields werethemselves used to adaptively control photofragmentation yields of SF [443]. Advancesin optimal control of high-harmonic generation (including related AFC experiments)have been recently reviewed [35, 41].Beyond the physical interest in achieving control over high-harmonic generation,these experiments also demonstrated a dramatic degree of inherent robustness to laser-field noise in strongly non-linear control. This behavior can be understood in termsof an extensive null space of the Hessian at the top of the control landscape, implyinga very gentle slope near the global maximum [190, 194]. This characteristic of thequantum control landscape makes it possible to tolerate much of the laser noise whilemaintaining a high control yield. Such robustness is expected to be a key attractivefeature of observable control across virtually all quantum phenomena. Control of bound-to-bound multiphoton transitions in atoms with optimally shapedfemtosecond laser pulses provides a vivid illustration of the control mechanism basedon multi-pathway quantum interference. Non-resonant multiphoton transitions involvemany routes through a continuum of virtual levels. The interference pattern excitedby the multiple frequency components of the control pulse can enhance or diminishthe total transition probability. The interference effect depends on the spectral phasedistribution of the laser pulse. A number of experiments [444–446] used AFC to ontrol of quantum phenomena: Past, present, and future dark pulses that do notexcite the atom at all due to destructive quantum interference. On the other hand,AFC was able to find shaped pulses that induce transitions as effectively as transform-limited pulses, even though their peak intensities are much lower. Due to the relativesimplicity of the atomic systems studied, it was possible to compare the results of theAFC experiments with theoretical predictions and verify the control mechanism basedon quantum interference of multiple laser-driven transition amplitudes. In a relatedexperiment [447], AFC was helpful for demonstrating that transform-limited pulses arenot optimal for inducing resonant multiphoton transitions. It was shown that optimallyshaped pulses enhance resonant multiphoton transitions significantly beyond the levelachieved by maximizing the pulse’s peak intensity. A recent experiment [448] considerednon-resonant multiphoton absorption in atomic sodium in the strong-field limit. Itwas demonstrated that in this regime the stimulated emission induced by the dynamicStark shift becomes important, which makes transform-limited pulses not optimal forstrong-field non-resonant multiphoton transitions. AFC was used to discover strong-field shaped laser pulses that optimally counteract the dynamic Stark shift-inducedstimulated emission and thus maximize the absorption probability.A more complex problem in atomic physics is control of multiphoton ionization. Inone experiment [449], AFC was applied to optimize multiphoton ionization of atomiccalcium by shaped femtosecond laser pulses. The feedback signals were measured usingion and electron spectroscopy, and the optimization results were used to elucidate theintermediate resonances involved in the photoionization process. Another experiment[450] studied photoionization of potassium atoms controlled by phase-locked pairs ofintense femtosecond laser pulses. Measurements of the Autler-Townes doublet in thephotoelectron spectra enabled analysis of the induced transient processes. The AFCexperimental results were helpful for exploring the control mechanism based on theselective population of dressed states.
In one of the first applications of AFC, in 1999, Bucksbaum and co-workers [451]manipulated the shape of an atomic radial wave function (a so-called Rydberg wavepacket). Non-stationary Rydberg wave packets were created by irradiating cesium atomswith shaped ultrafast laser pulses. A variation of the quantum holography method [452]was used to measure the atomic radial wave function generated by the laser pulse. Inorder to reconstruct the wave function, the amplitude of each energy eigenstate in thetotal wave packet was measured independently via state selective field ionization. Thedistance between the measured and target wave packet provided the feedback signal.In the weak-field limit, a simple linear relationship exists between the amplitudes ofthe energy eigenstates in the wave packet decomposition and the amplitudes of thecorresponding spectral components of the control laser field. Based on this relationship, ontrol of quantum phenomena: Past, present, and future
29a simple gradient-type algorithm was employed to adjust the spectral phase distributionof the control field. AFC equipped with this algorithm was able to change the shapeof the Rydberg wave packet to match the target within two iterations of the feedbackcontrol loop. If the wave packet is created in the strong-field regime, then a moresophisticated learning algorithm is generally required to implement AFC.
The first AFC experiment was reported in 1997 by Wilson and co-workers [123].Femtosecond laser pulses shaped by a computer-controlled AOM were used to excitean electronic transition in molecules (laser dye IR125 in methanol solution). Themeasured fluorescence served as the feedback signal in AFC to optimize the populationtransfer from the ground to first excited molecular electronic state. Both excitationefficiency (the ratio of the excited state population to the laser energy) and effectiveness(the total excited-state population) were optimized. Similar AFC experiments werelater performed with different molecules in the liquid phase: laser dyes LDS750 inacetonitrile and ethanol solutions [453], DCM in methanol solution [454], rhodamine 101in methanol solution [455], coumarin 515 in ethanol solution [456], coumarin 6 in a rangeof non-polar solvents (linear and cyclic alkanes) [457], the charge-transfer coordinationcomplex [Ru(dpb) ](PF ) (where dpb is 4,4 ′ -diphenyl-2,2 ′ -bipyridine) in methanolsolution [458] and acetonitrile solution [459–461], a donor-acceptor macromolecule (aphenylene ethynylene dendrimer tethered to perylene) in dichloromethane solution [462],and perylene in chloroform solution [463]. Two-photon electronic excitations in flavinmononucleotide in aqueous solution were controlled using multi-objective optimization(a genetic algorithm was employed to simultaneously maximize the fluorescence intensityand the ratio of fluorescence and SHG intensities) [464]. Molecular electronic excitationswere also optimized in AFC experiments in the solid state, with a crystal of α -perylene[463, 465].Since these AFC experiments are performed in the condensed phase, an importantissue is the degree of coherence of the controlled dynamics. This issue has been recentlyexplored in a series of AFC experiments [457], in which the level of attained controlwas investigated by systematically varying properties of the environment. Specifically,AFC was applied to optimize the stimulated emission from coumarin 6 (a laser dyemolecule) dissolved in cyclohexane, and the recorded optimal pulse shape (characterizedby a significantly nonlinear negative chirp) was used with several other solvents (linearand cyclic alkanes). In these experiments, the molecule was excited in the linearabsorption regime in order to exclude the trivial intensity dependence characteristic ofmultiphoton processes. The results revealed an inverse correlation between the obtaineddegree of control (as measured by the enhancement of stimulated emission relative tothat achieved by excitation with the transform-limited pulse) and the viscosity of thesolvent. This study indicates that the control mechanism involves a coherent process(i.e., based on quantum interference of coherent pathways) and that environmentally- ontrol of quantum phenomena: Past, present, and future ∼ A long-standing goal of photochemistry is selective control of molecular fragmentation.During the last decade, AFC employing shaped femtosecond laser pulses has beenapplied to achieve significant successes towards meeting this goal [18, 26, 37]. Selectivequantum control of photodissociation reactions in molecules using AFC was firstdemonstrated by Gerber and co-workers in 1998 [124]. They studied photodissociation ofthe organometallic complex CpFe(CO) Cl (where Cp = C H ) that contains particulartypes of iron-ligand bonds and exhibits different fragmentation channels upon excitationwith shaped femtosecond laser pulses. The branching ratio [CpFe(CO)Cl] + /[FeCl] + wasmaximized and minimized in AFC experiments employing an evolutionary algorithm.The experiment was performed in a molecular beam, and the feedback signal wasobtained from measurements of the ionized photofragments in a time-of-flight massspectrometer. Using AFC, it was possible to change the branching ratio between 5:1and 1:1.The success of the AFC experiment described above triggered an ongoing wave ofresearch activity in this area. In particular, Gerber’s group explored various aspectsof AFC of photodissociation and photoionization reactions in molecules. In one AFCexperiment [467], the relative yields of photodissociation and photoionization of ironpentacarbonyl, Fe(CO) , were controlled in the gas phase, using femtosecond laserpulses with carrier wavelengths at 800 nm and 400 nm (the latter produced via SHGof the former). The AFC-based optimization (both maximization and minimization)of the branching ratio [Fe(CO) ] + /Fe + demonstrated that the control mechanism isnot simply intensity-dependent, but rather employs the spectral phase distributionof the shaped laser pulse to steer the dynamics of the excited molecular vibrationalwave packet towards the target reaction channel. Another gas-phase AFC experiment[468] also analyzed the relative importance of intensity-dependent and coherent effectsin control of photochemical reactions that involve nonlinear (multiphoton) opticalexcitations. The control goals were the direct photoionization of CpFe(CO) Cl (i.e.,maximization of the [CpFe(CO) Cl] + yield) and selective photofragmentation (i.e., ontrol of quantum phenomena: Past, present, and future + /[FeCl] + ). For each pulse shapeduring the AFC-based optimization, the target reaction yield and SHG efficiency(which is directly proportional to the pulse intensity) were recorded. In the caseof direct ionization control, a clear correlation between the [CpFe(CO) Cl] + yieldand SHG efficiency was observed, which implies that the photoionization controlmechanism is mainly intensity-dependent. However, for fragmentation control, nocorrelation between the [CpFe(CO)Cl] + /[FeCl] + ratio and SHG efficiency was found.Moreover, for different pulses with the same SHG intensity, a large range of different[CpFe(CO)Cl] + /[FeCl] + values was obtained, depending on the specific pulse shape.These results indicate that while photofragmentation involves multiphoton excitation,it is not regulated by the pulse intensity alone. Rather, the control mechanism fora particular photofragmentation reaction requires a specially tailored laser pulse toguide the complex wave packet dynamics towards the desired outcome. Results ofa similar AFC experiment [469] that analyzed photofragmentation of CH ClBr inthe gas phase (including maximization and minimization of the [CH Br] + /[CH Cl] + ratio) also indicate that the control mechanism involves manipulation of the wavepacket dynamics on neutral dissociative surfaces rather than purely intensity-dependenteffects. Experiments that demonstrated AFC of photofragmentation in the moleculesCpFe(CO) Cl and CpFe(CO) Br [470] will be discussed later in the context of optimaldynamic discrimination of similar quantum systems.In 2001, Levis and co-workers [471] used AFC with shaped, strong-field laser pulsesto demonstrate selective cleavage and rearrangement of chemical bonds in polyatomicorganic molecules (in the gas phase), including (CH ) CO (acetone), CH COCF (trifluoroacetone), and C H COCH (acetophenone). Control over the formation ofCH CO from (CH ) CO, CF or CH from CH COCF , and C H CH (toluene) fromC H COCH was achieved with high selectivity. The use of strong laser fields (withintensities of about 10 W/cm ) helps to effectively increase the available bandwidth,as transitions to excited molecular states are facilitated by the dynamic Stark shift.This effect opens up many reaction pathways which are inaccessible in the weak-field(perturbative) regime due to resonant spectral restrictions [18]. While theoreticaltreatment of the complex strong-field molecular dynamics is extremely difficult, thiscomplexity in no way affects employment of AFC in the laboratory, where the moleculesolves its own Schr¨odinger equation on a femtosecond time scale. By operating at ahigh-duty control cycle, a learning algorithm is typically able to identify optimal laserpulses in a matter of minutes.Significant attention has been devoted to the analysis of quantum dynamicalprocesses involved in molecular photofragmentation control achieved in gas-phase AFCexperiments with shaped femtosecond laser pulses. W¨oste and co-workers [472–474]studied mechanisms of photofragmentation control for CpMn(CO) optimizing thebranching ratios [CpMn(CO)] + /[CpMn(CO) ] + and [CpMn(CO) ] + /[CpMn(CO) ] + .Weinacht and co-workers [475–478] analyzed mechanisms underlying control ofphotofragmentation in a series of AFC experiments with similar molecules: CH COCF ontrol of quantum phenomena: Past, present, and future COCCl (trichloroacetone), and CH COCD (tri-deuteratedacetone). The yield of the [CX ] + fragment and the [CX ] + /[CH ] + ratio (whereX is F, Cl, and D for trifluoroacetone, trichloroacetone, and tri-deuterated acetone,respectively) were optimized in these AFC experiments using intense shaped laserpulses. AFC was also used to optimize the branching ratios Br + /[CH Br] + and[CH I] + /[CH Br] + in photofragmentation of CH BrI (bromoiodomethane) [478,479]. Ina number of works [472,474–476], AFC experiments were supplemented by theoretical abinitio quantum calculations to help clarify photofragmentation control mechanisms. Inseveral other studies [477–479], a change in the basis of the control variables madeit possible to reduce the dimension of the search space and thus elucidate controlmechanisms of selective molecular photofragmentation. In another work [480], pump-probe spectroscopy was utilized to explore the control mechanism of photofragmentationof CHBr COCF (1,1-3,3,3 dibromo-trifluoroacetone) in AFC experiments with intenseshaped laser pulses. In particular, optimization of the [CF ] + /[CHBr ] + ratio revealeda charge-transfer-based control mechanism.Selective control of molecular fragmentation in the gas phase was demonstratedin several other AFC experiments with shaped femtosecond laser pulses. W¨oste andco-workers [481–484] controlled photoionization and photofragmentation dynamics ofalkali clusters. Jones and co-workers [485] optimized the S + N /S + M ratios for various N and M values in strong-field photofragmentation of S molecules. It was found thatoptimally shaped pulses dramatically outperform the transform-limited pulses. Wells et al. [486] controlled the vibrational population distribution in the transient CO to manipulate the branching ratio of the CO and C + + O + products. Hill andco-workers [487] controlled the amplitude of the bending vibrational mode in highlyionized CO (during strong-field Coulomb explosion) to enhance the symmetric six-electron fragmentation channel, CO → O + C + O . They constrained the searchspace by expressing the spectral phase of the laser pulse as a Taylor series, in order toelucidate the controlled photodissociation dynamics. Laarmann et al. [488,489] achievedselective cleavage of strong backbone bonds in amino acid complexes (in particular,a peptide bond in Ac-Phe-NHMe and Ac-Ala-NHMe), while keeping weaker bondsintact. Based on these results, they suggested the possibility of employing AFC withoptimally tailored laser pulses as an analytical tool in mass spectrometry of complexpolyatomic systems (with potential applicability, e.g., to protein sequencing of largebiopolymers). In a recent AFC experiment, Levis and co-workers [490] used intenselaser pulses to manipulate branching ratios of various photofragmentation productsof dimethyl methylphosphonate (DMMP), a simulant for the nerve agent sarin. Theoptimization in this experiment was performed in the presence of a high background ofa hydrocarbon and water in the extraction region of a time-of-flight mass spectrometer.The ability to achieve highly selective control under these conditions demonstrates thatAFC may provide the means to identify complex airborne molecules. As mentionedin section 5.3, photofragmentation of SF was controlled (including optimization ofthe ratio [SF ] + /[SF ] + ) in an AFC experiment [443] that used spectrally shaped ontrol of quantum phenomena: Past, present, and future The use of polarization-shaped femtosecond laser pulses can significantly enhance thelevel of control over multiphoton ionization in molecules. In 2004, Brixner et al. [494]demonstrated that a suitably polarization-shaped laser pulse increased the photoion-ization yield in K beyond that obtained with an optimally shaped linearly polarizedlaser pulse. This effect is explained by the existence of different multiphoton ionizationpathways in the molecule involving dipole transitions which are preferably excited bydifferent polarization directions of the laser field. Suzuki et al. [495] applied AFC withpolarization-shaped laser pulses to multiphoton ionization of I molecules and optimizedthe production of oddly charged (I +2 and I ) and evenly charged (I ) molecular ions.Weber et al. [496] performed AFC experiments with polarization-shaped laser pulses tooptimize the photoionization yield in NaK molecules. Free optimization of the pulsephase, amplitude, and polarization resulted in a higher ionization yield than parame-terized optimization with a train of two pulses.W¨oste and co-workers [497–502] investigated AFC of multiphoton ionization inK and NaK using femtosecond laser pulses with phase and amplitude modulation,but without polarization shaping. In particular, good agreement between the optimal ontrol of quantum phenomena: Past, present, and future was sufficient to remove unnecessary pulsecomponents and thus expose the participating vibronic transitions. For ionization ofNaK, strong pressure was applied and multi-objective optimization was performed. Theresultant Pareto-optimal curve revealed the correlation of the two conflicting objectivesof maximizing the ionization yield versus cleaning the control pulse. The optimalionization pathway depends on the CPC strength, which helps to identify the importantelectronic transitions to particular vibrational states. These results demonstrate thatthe spectra of optimal pulses obtained with CPC contain important information aboutthe control mechanism. In another series of AFC experiments [501], the controlmechanism of multiphoton ionization in NaK was analyzed by systematically reducingthe complexity of the search space. The spectral phase function of the control pulse wasexpressed as a truncated Fourier series, whose parameters were examined with respectto the ionization yield and the obtained optimal field. By progressively reducing thenumber of phase modulation parameters, it was possible to generate optimized pulsesthat allowed for a simple mechanistic interpretation of the controlled dynamics. In anearlier study, Leone and co-workers [503] applied AFC to optimize the weak-field pump-probe photoionization signal in Li and used first-order time-dependent perturbationtheory to investigate the dynamics of a rotational wave packet excited by the pumppulse and explain the corresponding control mechanism.W¨oste and co-workers [498, 504–506] also demonstrated that AFC is capable ofachieving isotope-selective ionization of diatomic molecules such as K and NaK. Theyshowed that optimally tailored control pulses can increase the divergence between thedynamics of excited vibrational wave packets in distinct isotopomers (these studies willbe discussed in more detail in section 5.12 below). The controlled alignment of molecules has attracted considerable attention as it canprovide a well defined sample for subsequent additional control experiments. Athigh laser intensities ( ∼ –10 W/cm ), dynamical variations of the molecularpolarization can have a significant effect on alignment. By shaping the temporalprofile of such an intense femtosecond laser pulse, it is possible to achieve controlover molecular alignment. Quantum dynamics of laser-induced molecular alignmentis amenable to theoretical treatment and optimization [216, 507–511], and can besuccessfully controlled using simple ultrafast laser pulses [512–516]. AFC provides a ontrol of quantum phenomena: Past, present, and future [517, 518] and CO [519]. Shaped femtosecond laser pulses were successfully used to enhance resolution andimprove detection in several areas of nonlinear spectroscopy and microscopy [44]. Ofparticular interest are experiments that employ AFC to identify optimal pulse shapes.One area of nonlinear molecular spectroscopy is control of vibrational modes viastimulated Raman scattering (SRS). In the gas phase, a number of AFC experiments[520–522] manipulated molecular vibrations excited via SRS by intense ultrafast laserpulses in the impulsive regime (i.e., when the duration of the control laser pulse is shorterthan the vibrational period). In one of these experiments [520], control of the vibrationaldynamics of K was achieved via impulsive SRS in a degenerate four-wave-mixing opticalsetup. Different parameterizations of shaped femtosecond laser pulses in the frequencyand time domains were employed to decipher the physical mechanism responsible for theachieved control. In other gas-phase experiments, mode suppression and enhancement insulfur hexafluoride [521], mode-selective excitation in carbon dioxide [521], and creationof shaped multimode vibrational wave packets with overtone and combination modeexcitation in CCl [522] were demonstrated using impulsive SRS at room temperatureand high pressures. In the liquid phase, AFC was applied to control relative intensitiesof the peaks in the Raman spectrum, corresponding to the symmetric and antisymmetricC–H stretch modes of methanol [523–526]. The modes were excited via SRS in the non-impulsive regime (i.e., the duration of the control laser pulse exceeded the vibrationalperiod). The control pulse was shaped and the forward scattered Raman spectrum wasmeasured to obtain the feedback signal, with the goal of achieving selective control ofthe vibrational modes. However, it was argued [527, 528] that in non-impulsive SRSthe relative peak heights in the Raman spectrum do not reflect the relative populationsof the vibrational modes and that control of the spectral features demonstrated in theexperiments [523–526] does not involve quantum interference of vibrational excitations,but rather is based on classical nonlinear optical effects.Another important area of nonlinear molecular spectroscopy is control of molecularvibrational modes via CARS. Materny and co-workers [529–534], Zhang et al. [535] andvon Vacano et al. [536] used AFC in a CARS setup to optimally control vibrationaldynamics in complex molecules. The Stokes pulse was shaped and the feedbacksignal was derived from the intensities observed in the CARS spectrum. In an AFCexperiment with polymers (fluorobenzene sulfonate diacetylenes), selective excitation ofone vibrational ground-state mode and suppression of all other modes was achieved,and the decay times of different modes were modified [529]. In liquid-phase AFCexperiments, selective enhancement or suppression of one or more vibrational modeswas demonstrated for toluene [530, 531], benzene [535], and β -carotene in hexane ontrol of quantum phenomena: Past, present, and future et al. [536] employed the method of single-beamCARS spectroscopy with shaping of broadband pulses from a photonic crystal fiber.Each broadband pulse provides numerous pairs of pump and Stokes frequencies, and thespectral phase of the pulse was optimized with AFC to produce the desired interferencepattern of the molecular vibrational modes. The optimally compressed and shapedpulses enabled the unambiguous assignment of the participating vibrational modes oftoluene between 500 and 1000 cm − in a blue-shifted CARS signal.Control of molecular vibrational dynamics is possible not only through Raman-typeprocesses, but also directly in the infra-red (IR) regime. Zanni and co-workers [537]demonstrated selective control of vibrational excitations on the ground electronic stateof W(CO) , using shaped femtosecond mid-IR (5.2 µ m, 1923 cm − ) pulses. The spectralphase distribution of the pulse was optimized using AFC to achieve selective populationof the excited vibrational levels of the T u CO-stretching mode. Systematic truncationof optimal pulses was employed to analyze the control mechanism. In a related AFCexperiment [538], polarization-shaped mid-IR pulses were used to selectively controlvibrational excitations of the two carbonyl stretching modes in Mn(CO) Br.
An important application of AFC with shaped femtosecond laser pulses is in multiphotonexcited fluorescence (MPEF) microscopy. For a given pulse energy, the transform-limited pulse has the maximum peak intensity, which helps to increase the fluorescencesignal intensity, but unfortunately also increases the rate of photobleaching of themolecules (which is especially undesirable with samples of live cells). Use of optimallyshaped pulses instead of a transform-limited pulse can reduce the bleaching rate,enhance spatial resolution, and increase contrast in biological fluorescence imaging.In a series of AFC experiments with shaped laser pulses, Midorikawa and co-workers[539–542] optimally controlled MPEF microscopy in different fluorescent biomolecules.Attenuation of photobleaching by a factor of four (without decreasing the fluorescencesignal intensity) was demonstrated in two-photon excitation fluorescence (TPEF) froma green fluorescent protein [539]. Another AFC experiment [540] achieved selectivecontrol of two-photon and three-photon fluorescence in a mixture of two biosamples.The use of optimally tailored pulses helped to minimize the harmful three-photon ontrol of quantum phenomena: Past, present, and future
Discrimination of similar systems is important for many practical problems in scienceand engineering. In particular, selective identification of target molecules in a mixtureof structurally and spectroscopically similar compounds is a challenge in such areasas selective excitation of multiple fluorescent proteins in microscopy of live samples,targeted component excitation in solid-state arrays, and selective transformation ofchemically similar molecules. Theoretical studies [219, 543–546] indicate that quantumsystems differing even very slightly in structure may be distinguished by means of theirdynamics when acted upon by a suitably tailored ultrafast control field. Such optimaldynamic discrimination (ODD) can in principle achieve dramatic levels of control, andhence also provides a valuable test of the fundamental selectivity limits of quantumcontrol despite noise and constrained laser resources. AFC provides a very effectivelaboratory means for practical implementation of ODD.In 2001, Gerber and co-workers [547] experimentally demonstrated selective multi-photon excitation of two complex molecules, a laser dye DCM and [Ru(dpb) ](PF ) ,in methanol solution. The goal was to electronically excite DCM while simultaneouslysuppressing electronic excitation of [Ru(dpb) ] . While these two molecules are elec-tronically and structurally distinct, the DCM/[Ru(dpb) ] emission ratio is practicallyunaffected by variations in single control parameters such as wavelength, intensity, andlinear chirp. Nevertheless, selective excitation was successfully achieved using AFC withshaped femtosecond laser pulses. The DCM/[Ru(dpb) ] emission ratio was used asthe feedback signal, and the evolutionary algorithm identified optimally shaped con-trol pulses that improved the signal by approximately 50%. These results obtained inthe presence of complex solvent/solute interactions aroused significant interest to ODD.As mentioned in section 5.7, Gerber and co-workers [470] demonstrated AFC of pho-toproduct branching ratios in CpFe(CO) Cl and CpFe(CO) Br. Despite the chemicalsimilarity of these two molecules, AFC was sensitive enough to detect differences due tothe electronic metal-halogen bonding properties. This finding suggests the possibility ofperforming ODD of individual compounds in mixtures of chemically similar molecules.Other examples of ODD include molecule-specific manipulation of CARS spectra from ontrol of quantum phenomena: Past, present, and future
38a mixture of benzene and chloroform [533], selective excitation of multiple fluorophoresin TPEF microscopy [541,542], and quantitative differentiation of dyes with overlappingone-photon spectra [548].A recent experimental demonstration of ODD by Roth et al. [466] achieved dis-tinguishing excitations of two nearly identical flavin molecules in aqueous phase. Theabsorption spectra for flavin mononucleotide (FMN) and riboflavin (RBF) are practi-cally indistinguishable throughout the entire visible and far UV. This implementation ofODD used a shaped UV pulse centered at 400 nm and a time-delayed unshaped IR pulsecentered at 800 nm. The first pulse creates a coherent vibrational wave packet on an ex-cited electronic state, and the second pulse disrupts the wave packet motion and resultsin additional excitation to a higher state and consequential depletion of the recordedfluorescence signal. The effect of slight differences in the vibronic structure of the twomolecules upon the dynamics of the excited wave packets is amplified by tailoring thespectral phase of the UV pulse. Since further excitation produced by the second pulsedepends on the precise structure, position, and coherence of the tailored wave packet,it is possible to dynamically interrogate the two statically nearly identical systems andthereby produce a discriminating difference in their respective depleted fluorescence sig-nals. In contrast, if the UV pulse is transform-limited, then the fluorescence depletionsignals from the flavins are indistinguishable. UV pulse shapes that optimally discrimi-nate between FMN and RBF were discovered using AFC. The optimized depletion ratio D FMN /D RBF could be changed by ∼ ± ∼ ∼
10 nm of IR bandwidth, significant selectivity wasachieved with optimal UV pulses working in concert with the time-delayed unshapedIR pulse. Although the static spectra appear nearly identical, subtle differences inthe molecular structure are nonetheless made profound during the tailored evolution ofwave packets generated by optimal controls. System complexity (e.g., high vibrationalstate density, thermal population, solvent-induced line broadening) effectively enhancesthe ODD capabilities of the control field and compensates for the limited bandwidthconstraint, thus making dramatic levels of control possible even in the weak-field limit.Another example of ODD is isotope-selective ionization of molecules demonstratedin a number of AFC experiments by W¨oste and co-workers [498,504–506]. In particular,in an illustrative study [505, 506], they applied shaped femtosecond laser pulses to the , K and , K isotopomers and used AFC to maximize and minimize the isotope ionratio R = I ( , K ) /I ( , K ). K molecules can be ionized in a three-photon processat relatively low pulse energies within the available wavelength range (810–833 nm).Differences between the evolutions of vibrational wave packets on an excited electronicstate in the two isotopomers can be amplified by optimized control fields. Operation inthis fashion made it possible to achieve significant selectivity of isotope ionization, witha variation by a factor of R max /R min ∼
140 between the maximal and minimal valuesof the isotope ion ratio. Information about the dynamics of the controlled vibrationalwave packets was extracted from the optimal pulse shapes to help reveal ionization ontrol of quantum phenomena: Past, present, and future
Applications of quantum control to increasingly complex molecular systems havebeen considered. In particular, Motzkus and co-workers used AFC with shapedfemtosecond laser pulses to control and analyze the energy flow pathways in the light-harvesting antenna complex LH2 of
Rhodopseudomonas acidophila (a photosyntheticpurple bacterium) [549, 550] and β -carotene [551]. They demonstrated that by shapingthe spectral phase distribution of the control pulse, it is possible to manipulate thebranching ratio of energy transfer between intra- and inter-molecular channels in thedonor-acceptor system of the LH2 complex [549]. Analysis of the transient absorptiondata was used to decipher the control mechanism and identify the molecular statesparticipating in energy transfer within LH2 [550] and β -carotene [551]. These resultsindicate that coherent quantum control is possible even in very complex molecularsystems in a condensed-phase environment. AFC has been applied to quantum control of inter-molecular photoinduced electrontransfer. Yartsev and co-workers [552] reported an AFC experiment that maximizedthe yield of ultrafast electron injection from the sensitizer to TiO nanocrystals in thecore part of a dye-sensitized solar cell. The electron transfer process was monitoredusing the transient absorption signal. The impulsive structure of the optimal laser pulsewas observed to correlate with the coherent nuclear motion of the photoexcited dye.The pulse shape and the transient absorption kinetics were explained by an impulsivestimulated (anti-Stokes) Raman scattering process, followed by electronic excitation. The control of molecular structure transformations is a coveted goal in chemistry. Inparticular, control of cis-trans isomerization has attracted much attention due to theimportance of this process in chemistry and biology (e.g., it is a primary step ofvision). AFC of cis-trans photoisomerization in cyanines (in the liquid phase) withshaped femtosecond laser pulses was first reported by Gerber and co-workers [553]. Thisexperiment demonstrated that by using optimally shaped laser pulses it is possible toenhance or reduce isomerization efficiencies. The mechanism underlying isomerizationcontrol in this experiment was discussed in a number of theoretical works [554–556].In particular, Hoki and Brumer [554] suggested that isomerization control involvesan incoherent pump-dump process and that the role of quantum coherence effectsin the evolution of the excited vibrational wave packet is negligible due to strongenvironmentally-induced decoherence. On the other hand, Hunt and Robb [555] andImprota and Santoro [556] used a more sophisticated model and argued that control of ontrol of quantum phenomena: Past, present, and future trans versus cis product is affected by the presence of an extended conical intersectionseam on the potential-energy surface, and that photoisomerization can therefore becoherently controlled by tuning the distribution of momentum components in thephotoexcited vibrational wave packet. The validity of this coherent control mechanismwas corroborated in a further AFC experiment by Yartsev and co-workers [557]. Theydemonstrated that optimally shaped laser pulses can be used to modify the momentumcomposition of the photoexcited wave packet and thus achieve significant control of theabsolute yield of isomerization (i.e., the photoisomer concentration versus the laser pulseenergy). The coherent character of liquid-phase control of cis-trans photoisomerizationin cyanines was further studied in another AFC experiment by Yartsev and co-workers[558]. They used a control scheme with an unshaped pump pulse and a time-delayedshaped dump pulse (an unshaped probe pulse was also applied to measure the effectof control). By using the optimally shaped dump pulse, they achieved control ofphotoisomerization closer to the decisive points of the reaction. This approach made itpossible to explore the effect of the wave-packet’s momentum composition at differenttime scales and assign the dynamics to distinct parts of the excited-state potential.AFC of the retinal molecule in bacteriorhodopsin (from the all- trans to the 13- cis state) was demonstrated by Miller and co-workers [559]. This experiment employed bothphase and amplitude modulation of femtosecond laser pulses and operated in the weak-field regime (with pulse energies of 16–17 nJ). By using optimally shaped pulses, it waspossible to enhance or suppress the quantity of molecules in the 13- cis state by about20%, relative to the yield observed using a transform-limited pulse with the same energy.They further explored the mechanism of coherent control of retinal photoisomerization inbacteriorhodopsin using time- and frequency-resolved pump-probe measurements [560].Experimental data together with a theoretical analysis suggest that the isomerizationyield depends on the coherent evolution of the photoexcited vibrational wave packeton an excited-state potential-energy surface in the presence of a conical intersection.According to this analysis, control of retinal photoisomerization is dominated byamplitude modulation of the spectral components of the excitation pulse. Gerber andco-workers [561] also demonstrated AFC of retinal isomerization in bacteriorhodopsin.In this experiment, they pioneered the control scheme with unshaped-pump and time-delayed shaped-dump femtosecond laser pulses. As mentioned above, the use of theoptimally shaped dump pulse makes it possible to control the molecule in a regionof the potential-energy surface where the decisive reaction step occurs. Moreover, bychanging the time delay between the pulses, it is possible to obtain information on thewave packet evolution.The role of quantum coherence effects in control of retinal isomerization inbacteriorhodopsin is still not fully clear, as a recent experiment by Bucksbaum and co-workers [562] found no dependence of the isomerization yield on the control pulse shape ontrol of quantum phenomena: Past, present, and future cis isomer is maximized by a transform-limitedpulse, which could indicate that the yield depends only on the pulse intensity, withquantum coherence not playing a significant role. These findings (especially for lowerintensities) apparently contradict the optimization results obtained by Miller and co-workers [559]. It is possible that these discrepancies could be explained by differencesin experimental setups. In particular, Bucksbaum and co-workers [562] used only phasemodulation of the control pulse, whereas Miller and co-workers [559, 560] argued thatcontrol is mainly achieved by amplitude modulation. Additional experimental andtheoretical work will be needed to fully explore the mechanisms underlying condensed-phase control of photoisomerization in complex molecular systems and clarify the role ofquantum coherence in the controlled dynamics. For example, recent theoretical studies[563, 564] suggest that coherent control of photoisomerization and other branchingreactions in an excited state may be affected by and, moreover, take advantageof environmentally-induced relaxation effects. Such cooperation between coherentcontrol and environmentally-induced decoherence may be important in various quantumphenomena [565] and hence its optimal exploitation deserve further investigation.Other examples of structural transformations in complex molecules include ringopening in cyclohexadiene along with isomerization as well as cyclization reactions in cis -stilbene. Carroll et al. [566, 567] demonstrated AFC of the photoinduced ring-openingreaction of 1,3-cyclohexadiene (CHD) to form 1,3,5-cis-hexatriene (Z-HT). The feedbacksignal was obtained from measurements of the UV absorption spectrum. The learningalgorithm was able to identify optimal pulse shapes that increased the formation of Z-HT by a factor of two. For a different control objective, the AFC optimization producedpulse shapes that decreased solvent fragmentation while leaving the formation of Z-HTessentially unaffected. Kotur et al. [568] used AFC with shaped ultrafast laser pulsesin the deep UV to control the ring opening reaction of CHD to form 1,3,5-hexatriene(HT). The feedback signal was obtained from measurements of fragmentation productsfollowing strong-field ionization with a time-delayed IR laser pulse. The learningalgorithm discovered shaped UV pulses that increased the HT yield by ∼
37% relative toan unshaped (nearly transform-limited) pulse of the same energy. Greenfield et al. [569]demonstrated AFC of the photoisomerization and cyclization reactions in cis -stilbenedissolved in n -hexane. This experiment employed phase-modulated 266 nm femtosecondpulses to maximize or minimize the yields of cis - to trans -stilbene isomerization as well as cis -stilbene to 4 a ,4 b -dihydrophenanthrene cyclization. The yields of both isomerizationand cyclization were minimized by transform-limited pulses that enhanced competingmultiphoton processes, while the yields were maximized by complex pulse shapes thathelped to avoid multiphoton effects. ontrol of quantum phenomena: Past, present, and future Fullerenes are a class of molecules of considerable interest in many areas of science.Laarmann et al. [570] employed AFC-optimized femtosecond laser pulses to coherentlyexcite large-amplitude oscillations in C fullerene. The structure of the optimal pulsesin combination with complementary two-color pump-probe data and time-dependentdensity functional theory calculations provided information on the underlying controlmechanism. It was found that the strong laser field excites many electrons in C , andthe nuclear motion is excited, in turn, due to coupling of the electron cloud to a radiallysymmetric breathing mode. Despite the complexity of this multi-particle system withvarious electronic and nuclear degrees of freedom, the optimal control fields generatedessentially one-dimensional oscillatory motion for up to six cycles with an amplitude of ∼ Quantum control has found applications beyond atomic and molecular phenomena.In particular, it is possible to use optimal control methods to manipulate variousprocesses in semiconductors. Kunde et al. [571,572] demonstrated AFC of semiconductornonlinearities using phase-modulated femtosecond laser pulses, with the purpose ofcreating an ultrafast all-optical switch. The feedback signal was obtained by measuringthe differential transmission (DT) in a control-probe setup. Optimizations wereperformed on the spectrally integrated DT as well as DT in narrow spectral windows.The learning algorithm was able to identify optimal pulse shapes that enhanced ultrafastsemiconductor nonlinearities by almost a factor of four. Chung and Weiner [573]used AFC with phase-modulated femtosecond laser pulses to coherently control two-photon-induced photocurrents in two different semiconductor diodes. Because of theirspectrally distinct two-photon absorption responses, the diodes generated noticeablydifferent photocurrent yields depending on the pulse shape. An evolutionary algorithmguided the AFC experiment to discover pulse shapes that maximize or minimize thephotocurrent yield ratio for the two diodes.
Manipulation of quantum interference requires that the system under control remainscoherent, avoiding (or at least postponing) the randomization induced by couplingto an uncontrolled environment. Therefore, the ability to manage environmentally-induced decoherence would bring substantial advantages to control of many physical andchemical phenomena. In particular, decoherence is a fundamental obstacle to quantuminformation processing [175], and therefore the ability to protect quantum informationsystems against decoherence is indispensable.The possibility of using AFC for optimal suppression of decoherence was firstproposed by Brif et al. [574], and numerical simulations in a model system were ontrol of quantum phenomena: Past, present, and future at400 ◦ C was irradiated by a shaped femtosecond laser pulse, inducing a vibrational wavepacket in the lowest excited electronic state A Σ +u of the molecules. This wave packetundergoes dephasing (a form of decoherence that does not involve dissipation of energy).Dephasing is caused by coupling of the vibrational mode to the thermalized rotationalquasi-bath. The amplitude of quantum beats in the fluorescence signal (measured ata chosen delay time after the excitation pulse) served as the feedback signal. Thisamplitude provides an estimate of the degree of wave packet localization in the phasespace and therefore can be used as a coherence surrogate. The optimal pulse identifiedby AFC increased the quantum-beat visibility from zero to more than four times thenoise level and prolonged the coherence lifetime by a factor of ∼ ∼ Be + ions (cooled to a temperatureof ∼ ∼
124 GHz transition in a microwave setupsimilar to optical ones. A sequence of microwave π pulses used for qubit control inthis laboratory configuration is technologically quite different from shaped femtosecondoptical laser pulses typically employed in molecular control experiments; however, thefundamental concept of AFC is still fully applicable. The feedback signal was obtainedfrom fluorescence detection on a cycling transition (with decoherence-induced errorsmanifested as non-zero fluorescence). The Nelder-Mead simplex method was utilizedto search for optimal pulse positions in a fixed-length sequence of n pulses. Optimalpulse sequences discovered in the AFC experiment, without a priori knowledge of the ontrol of quantum phenomena: Past, present, and future The learning algorithm is an important component of laboratory AFC. The majority ofAFC experiments employ stochastic algorithms such as evolutionary strategies [595] andgenetic algorithms [596]. Historically, genetic algorithms were characterized by the useof recombination, while evolutionary strategies primarily relied on mutation; however,modern algorithms guiding AFC experiments and simulations typically incorporate bothtypes of genetic operations and are variably called genetic algorithms or evolutionaryalgorithms. These algorithms are very well suited to laboratory optimizations as theynaturally match the discrete structure of control “knobs” (e.g., the pixels of a pulseshaper) and are robust to noise. Moreover, robustness to noise in AFC experimentscan be enhanced by incorporating the signal-to-noise ratio into the control objectivefunctional [378, 597]. Various aspects of evolutionary algorithms and their applicationto AFC of quantum phenomena were assessed [598,599]. Evolutionary algorithms can bealso used in multi-objective optimization [600,601], and the application of this approachto quantum control problems was studied theoretically [216–218, 602] and demonstratedin AFC experiments [464, 500].Other types of stochastic algorithms include, for example, simulated annealing [603]and ant colony optimization [604, 605]. Simulated annealing was utilized in someAFC experiments [430, 542], and it seems best suited to situations where just a fewexperimental parameters are optimized [606, 607]. Ant colony optimization recently hasbeen used in an AFC simulation [390], but it is yet to be tested in the laboratory.As mentioned in section 4, the absence of local traps in control landscapes forcontrollable quantum systems has important practical consequences for the optimizationcomplexity of AFC experiments. In particular, deterministic search algorithms can beused to reach a globally optimal solution. Deterministic algorithms (in particular, thedownhill simplex method) were successfully implemented in several AFC experiments[431, 569, 594]. Recently, Roslund and Rabitz [371] demonstrated the efficiency of agradient algorithm in laboratory AFC of quantum phenomena. They implementeda robust statistical method for obtaining the gradient on a general quantum controllandscape in the presence of noise. The experimentally measured gradient wasutilized to climb along steepest-ascent trajectories on the landscapes of three quantumcontrol problems: spectrally filtered SHG, integrated SHG, and excitation of atomicrubidium. The optimization with the gradient algorithm was very efficient, as itrequired approximately three times fewer experiments than needed by a standard geneticalgorithm in these cases. High algorithmic efficiency is especially important for AFC oflaser-driven processes in live biological samples, as damage (e.g., due to photobleaching)can be reduced by decreasing the number of trial laser pulses. Still, evolutionary or otherstochastic algorithms may be preferable over deterministic algorithms in many AFC ontrol of quantum phenomena: Past, present, and future
6. The role of theoretical quantum control designs in the laboratory
A very significant portion of theoretical research in the area of quantum control isdevoted to model-based computations which employ QOCT (or other similar methods)to design optimal control fields for various physical and chemical problems. Suchcomputations are widespread in theoretical studies of molecular applications of quantumcontrol and are becoming increasingly popular in considering control of quantuminformation systems, including optimal operation of quantum gates and optimalgeneration of entanglement (see section 3.6). Notwithstanding these extensive QOCT-based control field designs, experiments seeking optimal control of molecular processesoverwhelmingly employ AFC methods as described in section 5. Such experiments inmost cases work remarkably well with random initial trials, and thus exhibit no evidentneed to operate or possibly start with theoretical control designs. This raises importantquestions about the practical usefulness of open-loop control and role of theoreticalmethods such as QOCT in control experiments [23]. In considering this matter it isimportant to keep in mind that the AFC procedure grew out of observations fromQOCT simulations and associated analyses.
As discussed in section 3.7, the practical laboratory relevance of theoretical designsdepends on the complexity of the system of interest, with simpler cases yieldingtheoretical models closer to reality. For example, in numerous NMR experimentsemploying RF fields to manipulate nuclear spins [153–157], including NMR realizationsof quantum gates [159–162], theoretically designed sequences of pulses (some of whichwere developed using QOCT [163, 164]) can function quite well. The model of acollection of spins (with empirical coupling and decay constants) interacting withclassical fields is often sufficiently accurate for NMR-based applications, allowing forsuccessful employment of open-loop control. A QOCT-based design was also successfullyapplied experimentally to enhance robustness of single-qubit gate operations in a systemof trapped ions [353].At the other extreme of system complexity are electronic and vibrational processesin polyatomic molecules whose dynamics cannot be accurately modelled at the presenttime. An objective assessment is that models used for polyatomic molecules in controlcomputations are currently too simplified and computational techniques inadequatefor the true levels of complexity, resulting in theoretical designs that are not directlyapplicable to control experiments which work with real systems. There are severalaspects of laser control of molecules, which make the difference between theoretical ontrol of quantum phenomena: Past, present, and future ab initio quantitative theoretical account of laser-driven molecular dynamicsis presently not feasible, unless the studies are limited to cases of very simple moleculesand weak fields. Third, it is difficult to calibrate the laser and pulse-shaping apparatus toreliably reproduce theoretical control designs in the laboratory. In many cases, directlyusing AFC optimization is much easier and much more effective, than calibrating thelaser and pulse-shaper for generation of theoretically computed control fields with therequired accuracy.These considerations lead to the conclusion that open-loop control experimentsemploying theoretical designs may be useful for some systems and impractical for others,depending on how well the system is known and which computational capabilities areavailable, consequently determining how accurately the controlled dynamics can bemodelled. Thus, the boundary between the systems for which modelling is sufficientlyreliable and the systems for which it is not, depends on available Hamiltonian data,numerical algorithms and computational power. Of course, with time, better modellingwill become available for more complex systems, although the exponential increase ofthe Hilbert-space dimension with the system complexity is a fundamental feature ofquantum mechanics, which significantly hinders the effectiveness of numerical controldesigns for practical laboratory implementation.Consider, for example, quantum information processing systems which are typicallymodelled as collections of two-level particles (qubits) with controlled interactionsbetween them [175].
Prima facie , such a system appears to be quite simple, sothat controls for all desired transformations can be theoretically designed (e.g., usingQOCT). However, the difficulty of accurately modelling the actual environmentalnoise is significant even for simple few-qubit systems. As was recently demonstratedwith trapped-ion qubits, dynamical decoupling pulse sequences obtained via AFCsignificantly outperformed the best available theoretical designs [594]. Moreover, asthe Hilbert-space dimension increases exponentially with the number of qubits, theunwanted effects of uncontrolled couplings between the qubits in multi-particle systems ontrol of quantum phenomena: Past, present, and future
In molecules, the interactions between the atoms are inherently strong in order to holdthe atoms together. Therefore, in the foreseeable future, for optimal manipulationof electronic and vibrational processes in molecules with four and more atoms, AFCwill continue to be much more effective than employing theoretical control designs.Notwithstanding this assessment, theoretical control studies should continue to havehigh significance; however, for complex systems the value of theoretical studies is notin generating specific control designs for immediate laboratory use. Control solutionsobtained via theoretical model-based computations (in particular, those employingQOCT) should play an important role by advancing the general understanding of thecharacter of controlled dynamics and control mechanisms. One practically importantissue is that while each cycle of a typical AFC experiment is very fast (from microsecondsto milliseconds) and cheap, the initial setup of the experiment is usually quite difficultand expensive, since advanced methods of pulse shaping and control-yield measurementneed to be incorporated together as well as adjusted to the particular nature of a physicalor chemical system. Therefore, theory can be especially important in exploring thefeasibility of various control outcomes for quantum dynamics of a system of interest; evensemiquantitative modelling may be successful for such purposes in many applications.Theoretical control simulations can provide important guidance for the selection of theexperimental configuration, thereby helping to make AFC a more effective practicaltool. Additional such high value utilizations of theory and simulations can be expectedin the future.
The open-loop control procedure is not limited to the use of optimal theoretical designsgenerated via QOCT and similar methods. Moreover, optimality in not always requiredin quantum control. In the conceptually allied field of synthetic chemistry, progress hasoften been achieved via intuition-guided trials leading to a gradual increase of reactionyields. Following this venerable tradition, some recent open-loop control experimentsseek improvement by employing ultrafast shaped laser pulses with so-called “rational”or “judicious” control designs obtained using a combination of intuition and argumentsbased on some knowledge of system properties (e.g., spectral information or symmetry).This approach is popular in nonlinear spectroscopy and microscopy [44, 609–622] as wellas some other atomic [623–631], molecular [512–516, 632, 633], and solid-state [634–637] ontrol of quantum phenomena: Past, present, and future
7. Concepts and applications of real-time feedback control
Feedback is very important in classical engineering where it is routinely used for controlof complex systems in the presence of uncertainties. In quantum control, two importantapproaches based on the concept of feedback have been introduced for similar reasons.One is AFC, which was extensively discussed in section 5. A fundamental characteristicof AFC is that in each control cycle a fresh quantum ensemble is used (either a newsample is prepared or the system is reset to its initial state before each run), whichmakes measurement back action irrelevant. The other approach is real-time feedbackcontrol (RTFC) [391–396, 638], in which the same quantum system is followed in realtime around the feedback loop, and the measurement (or interaction with a quantum“controller”) has a significant effect on the system’s evolution.There are two distinct types of RTFC, which differ by the nature of the controller.In one approach to RTFC, measurements are employed to probe the quantum system,and the gathered information is processed classically off-line in real time to assess thebest, next control action [391–396]. The evolution of the controlled quantum system isgoverned by two effects: coherent (unitary) action exerted by the classical controller + and incoherent (non-unitary) back action exerted by the measurements. A generalizeddescription of measurement-based RTFC employs quantum filtering theory [639–641].Recently, another type of RTFC — referred to as coherent feedback control ∗ — hasdrawn much attention [638, 642–648]. In this approach, no measurements with aclassical output signal are performed; instead, an ancillary quantum system serves asthe controller. The controller influences the evolution of the system of interest via adirect interaction between them. Additionally, external classical forces also can be usedto act upon the system, the controller, or both. The system of interest together withthe controller are characterized by the entirely quantum nature of the information flow— coherence is not destroyed by measurements, which is the source of the name givento this type of control [638,642]. Coherent feedback control can be viewed as a quantumanalog of Watt’s flyball governor — an automatic self-regulating quantum machine [649].It was recently shown [650] that the evolution of a quantum system undergoing any + The free evolution of the system can be included while the off-line modelling is performed. ∗ We will also use the term coherent RTFC. The choice of terminology is standard in the field andshould not be confused with the notion of coherent control employed in AFC and QOCT. ontrol of quantum phenomena: Past, present, and future
8. Future directions of quantum control
Common sense dictates that the future is notoriously difficult to predict, but it is alsothe nature of science to try and anticipate new directions that will expand currentknowledge. The evident paths followed in the development of the quantum control fieldduring the last two decades provide a basis for projection, with due caution, on how ontrol of quantum phenomena: Past, present, and future
Except for the special situation of measurement-based RTFC (where measurement backaction is a distinctively non-classical feature), one may naively conclude that thereare no fundamental differences between designing controls for quantum and classicalsystems. The distinctions seem to lie in solving classical equations of motion in onecase and the Schr¨odinger equation in the other, but otherwise the method of finding theoptimal control solution is basically the same. However, from a practical perspective,the difference between solving classical and quantum equations of motion is fundamentaldue to the exponential increase of the Hilbert space dimension characteristic of quantumsystems. This is the reason why simulating controlled quantum dynamics of multi-particle systems is so difficult.A qualitative breakthrough in open-loop control of complex quantum systems wouldbe possible, if a way could be found to replace the laborious calculations of quantumdynamics with “black-box” models that essentially capture the main features of theprocesses leading to a particular control objective (e.g., breaking of a specific molecularbond). The goal is to perform a modest number of simulations and use the informationto generate an input-output map from the applied control field to the resultant changein the control objective. In this fashion, the input-output map aims to capture therelationship between the control and the system’s reaction to it. This approach iscommonly used in classical control problems in many areas of engineering; however, atthe present time, we do not know how to effectively determine these input-output mapsfor complex quantum systems, such as molecules.An example of a method proposed for identifying nonlinear input-output mapsfor quantum control studies is the high-dimensional model representation (HDMR)technique [696–700]. The total number of points in the search space for a quantumcontrol optimization problem (and for many other problems in science and technology)grows exponentially with the number of input variables (this situation is sometimescalled the “curse of dimensionality”). In HDMR, the input-output map is characterizedby a hierarchy of contributions from the input variables acting independently, in pairs,triples, etc. For many important problems, with an appropriate choice of the variables,only low order input variable cooperativity is significant. This property can be usedto dramatically reduce the effort required to explore the map. Approaches such asHDMR are designed for systems with a large number of input variables with the aim oflearning the input-output map using a number of simulations that grows relatively slowly(e.g., polynomially) with the number of input variables [700]. Specifically, the use ofnonlinear functional HDMR-type maps for quantum control problems was discussed byGeremia et al. [701]. Such input-output maps would be of value as well when generated ontrol of quantum phenomena: Past, present, and future
The introduction of quantum control landscapes in the last few years is an importanttheoretical advance in the field. The nature of the control landscape topology has directimplications for the ease of finding effective controls in the laboratory. The analysisof the control landscape topology and other structural features can provide the basisfor investigating the complexity of optimizing different types of control objectives. Inturn, this understanding can help identify the most suitable optimization algorithms forvarious theoretical and experimental applications of quantum control (see sections 4.4,4.5, and 5.19). In addition, the landscape analysis may be extended to the study ofquantum control problems involving simultaneous optimization of multiple objectives(see sections 4.3 and 4.4). This research area is still rapidly developing with muchremaining for investigation. In particular, an open issue that deserves significantattention is the effect of field constraints (e.g., due to limited laboratory resources)upon the accessible regions of quantum control landscapes.There are several additional research directions for which the analysis of the controllandscape features may provide important insights. One ubiquitous problem with wide-ranging implications is evaluation of the robustness of control solutions to noise andimperfections, which depends on the degree of flatness of the control landscape aroundan optimal solution. Another interesting issue is related to a phenomenon discovered forobservable control of an open quantum system prepared in a mixed state and coupledto a thermal environment [273]. Specifically, the range of the control landscape (i.e.,the difference between the maximum and minimum expectation values of the targetobservable) decreases when the temperature of the environment raises. Therefore,an important application of control landscape analysis would be determination of thefundamental thermodynamic limits on the control yield for open quantum systems. ontrol of quantum phenomena: Past, present, and future As discussed in section 5, AFC has proved to have broad practical success as a meansfor achieving optimal control of quantum phenomena in the laboratory. Particularlyimpressive is the breadth of applications, ranging from optical systems, to atoms, tosemiconductor structures, to biologically relevant photochemical processes in complexmolecules, etc. One clear trend is the extension of AFC applications towards themanipulation of increasingly more complex systems and phenomena. Along this avenue,implementation of AFC could bring significant benefits to such areas as near and evenremote detection of chemical compounds (first steps in this direction have been recentlymade [490, 706]), optimal control of molecular electronics devices, and optimal controlof photochemical phenomena in live biological samples (including nonlinear microscopyand ODD, as indicated by several recent experiments [539–542]).We can also envision increasing use of AFC for optimal quantum control ofphotophysical phenomena. One important area is coherent manipulation of quantumprocesses in solid-state systems, especially in semiconductor quantum structures[707, 708]. Another potential application is optimal storage and retrieval of photonicstates in atomic-vapor and solid-state quantum memories [358, 359, 709–714]. The AFCmethodology may be also applicable to physical problems where, instead of laser pulses,other means (e.g., voltages applied to an array of electrodes) are used to implementthe control. Examples could include optimal control of coherent electron transport insemiconductors by means of adaptively shaped electrostatic potentials [715], coherentcontrol of charge qubits in superconducting quantum devices by gate voltages [716,717],and coherent control of photonic qubits in integrated optical circuits via the thermo-optic effect [718]. Several types of quantum systems (e.g., flux qubits in superconductingquantum devices, hyperfine-level qubits in trapped neutral atoms and ions, electronspins of donor atoms in silicon, etc.) can be controlled by pulses of microwave radiation(e.g., AFC-optimized dynamical decoupling [594] of trapped-ion qubits by a sequenceof microwave π pulses was discussed in section 5.18). Many possible applicationsof AFC could have significant implications for the progress in the field of quantuminformation sciences. A new domain of quantum control involves manipulation ofrelativistic quantum dynamics with extremely intense laser fields [719] for acceleratingparticles and even intervening in nuclear processes in analogy with what is happeningin atomic-scale control. It is reasonable to forecast that AFC methods could becomeuseful for optimal control of such laser-driven high-energy phenomena.Despite significant advances achieved in the field of quantum control during thelast decade, the experimental capabilities are limited by currently available laserresources. It is likely that existing practical limitations, in particular, the relativelynarrow bandwidth of femtosecond lasers, restrict the achievable yields in some AFCexperiments. One might expect that many new applications would open up, if reliablesources of coherent laser radiation with a much wider bandwidth became available. Suchresources could make possible the simultaneous manipulation of rotational, vibrational, ontrol of quantum phenomena: Past, present, and future Despite the significant technological difficulties on the path to routine practicalapplication of RTFC, the potential benefits are alluring (see section 7). An interestingquestion is whether AFC, whose practical utility has already been well established,can be used to aid in the implementation of RTFC (measurement-based, coherent, orboth). Due to the apparent technological differences between AFC and RTFC, thusfar they have been considered as separate branches of quantum control. However,it has been recently shown [650] that AFC and RTFC share a common fundamentallandscape topology characterized by the absence of local traps (i.e., all sub-optimalextrema are saddles provided that the controllability condition is satisfied). Since thecontrol landscape topology strongly influences the optimization complexity, this findingmay have immediate practical importance. Moreover, the unification of the seeminglydifferent AFC and RTFC approaches at a fundamental level suggests the possibility ofdeveloping new laboratory realizations that combine these currently distinct techniquesof quantum feedback control in a synergistic way. For example, some form of AFCmight be used to optimize the design or construction of quantum controllers employedin coherent RTFC. Development of “hybrid” quantum control schemes incorporatingboth AFC and RTFC (in particular, for control and stabilization of quantum computingsystems) could provide significantly enhanced flexibility in the laboratory.
In addition to the manipulation of quantum dynamics via application of optimalexternal fields, there is the prospect of performing material control through alterationof the internal system properties (i.e., the spatial structure or matrix elements of thesystem Hamiltonian), with the aim of identifying optimal materials and system designs.Analogous to the circumstance of a particular quantum system acted upon by a familyof homologous external control fields, we can consider the controlled response of a familyof homologous quantum systems to a particular field. In the former case, a control levelset consists of all homologous control fields that produce the same expectation value ofthe target observable when applied to a particular quantum system. This level set canbe explored, for example, by homotopy trajectory methods (e.g., D-MORPH), in orderto identify control solutions with desired properties (see section 4.4). In the latter caseof material control, a level set consists of all dynamically homologous quantum systemsthat produce the same expectation value of the target observable when controlled bya particular field. For example, each quantum system may be specified by a point in ontrol of quantum phenomena: Past, present, and future
54a hypercube whose edges are labeled by Hamiltonian matrix elements. A variationof the D-MORPH method can be used to explore a system level set by continuouslywarping the corresponding Hamiltonian [720]. At this juncture little is known abouteither homologous control fields or homologous quantum systems. Exploration of thesetopics could reveal the systematic aspects of control over quantum phenomena.Morphing through Hamiltonian structure in the laboratory can be physicallyrealized in many different ways, with broad and yet largely unexplored possibilities.For example, the properties of light-sensitive materials could be varied using familiesof structurally similar chemical compounds, characteristics of doped semiconductorscan be varied by changing the concentration of dopant atoms and the depth ofimplanting, etc. Material control is potentially applicable to a wide set of problemsin various areas of science and technology. Possible applications include, for example,development of photodetectors with higher efficiency and faster response time, molecularswitches with increased sensitivity and durability, quantum computing systems withenhanced immunity to environmentally-induced decoherence and improved robustnessto instrumental noise, etc. Exploiting the accessible variations in Hamiltonian structureas a means for achieving optimal quantum control is a potentially important area offuture research, including exploration of the associated control landscapes, developmentof adaptive and open-loop techniques, investigation of effective methods of combiningmaterial and electromagnetic control, and adaptation of the theoretical concepts tovarious practical applications.
The general goal of science is to understand nature, including the structure of physicalsystems and characteristics of the system dynamics, while the goal of engineering isto design and implement a system that will function in a prescribed manner in thebest possible way. Optimal quantum control draws together science and engineering toincorporate both goals: (1) to understand the dynamical behavior of quantum systemsand the mechanisms by which these processes can be managed, and (2) to require optimalfunctional performance through the achievement of prescribed control objectives in thebest possible way.An important feature evident in the prior development of quantum control is theimpact of progress in one aspect of the subject on advancing another. We expectthat this trend will continue in the future, as a better understanding of the underlyingphysical processes would aid in choosing better control tools and thereby achievinga higher degree of performance. In turn, the ability to steer system evolution in anoptimal fashion should facilitate the acquisition of knowledge about the underlyingcontrol mechanisms and other properties of the system. For example, in many AFCexperiments, the characteristics of the resultant optimal control fields were used (oftenin combination with additional measurements and/or simulations) to decipher physicalmechanisms responsible for the achieved control [459–463,465,467,468,472,474–480,501, ontrol of quantum phenomena: Past, present, and future ontrol of quantum phenomena: Past, present, and future
9. Concluding remarks
It would be impossible to cover in a paper of any reasonable length all of the advancesthat have been made in the last two decades in the field of quantum control. Some areasthat did not receive full attention here were considered in more detail in other reviewarticles and books (in particular, those cited in section 1), to which we refer the interestedreader. For example, thematic reviews are available on control via two-pathway quantuminterference [1,8,21,24], pump-dump control [7,11], control via STIRAP [10,16], controlvia WPI [42], the formalism and applications of QOCT [14, 36, 38, 40], controllabilityof quantum systems [36], the formalism of quantum control landscape theory [39],femtosecond pulse-shaping technology [4, 5, 12, 27], femtosecond laser control of X-raygeneration [35, 41], quantum control experiments with “rational” control designs [30],quantum control applications in nonlinear spectroscopy and microscopy [34, 44], andcontrol of quantum dynamics on the attosecond time scale [45]. While we tried toprovide a comprehensive account of laboratory AFC of quantum phenomena, moredetailed discussions of some important AFC experiments are available in earlier reviews[15, 18, 19, 25, 26, 31, 34, 35, 37, 41]. New papers, often containing significant theoreticaland experimental results in quantum control, appear now almost on a daily basis.In this paper, our goal was to give a perspective and prospective on the fieldhighlighting the evolution of important trends in quantum control. A look into the pasttogether with a review of current, cutting-edge research were used to cautiously forecasttopics of future interest. We also attempted to emphasize the synergistic connectionbetween the theoretical and experimental advances, which has been immensely beneficialfor the development of the field. We believe that sustaining this productive interplaybetween theory and experiment will be critical for future progress. This paper aimed toprovide the basis to better understand which aspects of theoretical research are havinga high impact on laboratory control of quantum phenomena. At the same time, acomplementary goal of this work was to point out the experimental aspects of quantumcontrol that have special significance and relation to theory. Although the scope ofexperimental and theoretical research in quantum control is vast, we hope that thiswork provides a valuable bridge for the community involved as well as for those outsidewho are interested in understanding the reasons for the fervor in the field.
Acknowledgments
This work was supported by DOE, NSF, ARO, and Lockheed Martin.
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