Optimal control landscapes for quantum observables

THE JOURNAL OF CHEMICAL PHYSICS 124, 204107 共2006兲

Optimal control landscapes for quantum observables Herschel Rabitz and Michael Hsieha兲 Department of Chemistry, Princeton University, Princeton, New Jersey 08544

Carey Rosenthal Department of Chemistry, Drexel University, Philadelphia, Pennsylvania 19104

共Received 17 January 2006; accepted 30 March 2006; published online 25 May 2006兲 The optimal control of quantum systems provides the means to achieve the best outcome from redirecting dynamical behavior. Quantum systems for optimal control are characterized by an evolving density matrix and a Hermitian operator associated with the observable of interest. The optimal control landscape is the observable as a functional of the control field. The features of interest over this control landscape consist of the extremum values and their topological character. For controllable finite dimensional quantum systems with no constraints placed on the controls, it is shown that there is only a finite number of distinct values for the extrema, dependent on the spectral degeneracy of the initial and target density matrices. The consequences of these findings for the practical discovery of effective quantum controls in the laboratory is discussed. © 2006 American Institute of Physics. 关DOI: 10.1063/1.2198837兴 I. INTRODUCTION

The active control of quantum phenomena has been under intense study in recent years through theoretical simulations and in the laboratory. Typically, the quantum system without control evolves in an undesirable fashion. A quantum control field C共t兲 is introduced for the purpose of redirecting the dynamics with the hope of achieving a desired outcome. In all cases, it is natural to seek the best possible degree of control and this perspective leads to the use of optimal control techniques.1 Quantum optimal control theory 共OCT兲 and optimal control experiment 共OCE兲 methods based on openloop adaptive learning form the foundation of the subject. Many OCT calculations have successfully identified design controls for the optimal manipulation of a broad variety of phenomena.2–15 Most of these calculations have been concerned with maximizing the probability P j→i of making a transition from some initial state 兩j典 to a specified final state 兩i典 at a target time T. A limited number of studies have also considered more general circumstances of other observables as well as the system existing in a mixed initial state. Most of the OCT simulations have employed local algorithms to search for control fields C共t兲 attempting to find at least one field that maximizes the desired dynamical product; other simulations have utilized global genetic-type algorithms as feasibility studies for the laboratory experiments which employ these algorithms.30 A most striking general conclusion from the collective OCT simulations in recent years is the rather high quality, or even excellent quality, of the controlled dynamics achieved in virtually all cases. This behavior is especially surprising in cases where local search algorithms are employed, as the search should proceed to the nearest local extremum 共i.e., a maximum or minimum, as specified by the a兲

Electronic mail: [email protected]

0021-9606/2006/124共20兲/204107/6/$23.00

control optimization objective兲 of the optimal control cost function. The expectation is that local suboptimal solutions will exist, with most of them exhibiting poor quality control yields and the likely outcome would be trapping in one of these low quality solutions. Thus, the evident attractive behavior of OCT generally finding good quality control results presents a puzzle for explanation. In the laboratory, closed-loop learning techniques are being used to perform OCE with many successful studies emerging since the initial demonstration of the concept. While most of the OCT simulations have been carried out under ideal conditions, in the laboratory, the systems will be characterized by an initial temperature, fluctuations in the controls, noise in the observations, and the likely existence of various types of environmental decoherence. Despite these less than ideal circumstances, successful OCE has been achieved in broad classes of quantum phenomena. Although the absolute yields or cross sections in the present OCE studies are generally not known at this time, a fair conclusion is that it appears relatively easy to discover controls in the laboratory that can significantly, and even in some cases dramatically, increase the yield of a specified target objective.16–29 All the current laboratory OCE studies employ global genetic-type algorithms which appear to have an inherent robustness to various types of laboratory noise as well as the feature of possibly extricating the search from local suboptimal control solutions. The general conclusion evident from the collection of all the OCT and OCE studies is that finding successful optimal controls appears easier than one might expect. Regarding the latter point, typical optimal control calculations or experiments involve the manipulation of tens or even hundreds of control variables in the search for an optimal solution. Expressed as a generic search problem over the high dimensionality of these search spaces, along with the expected existence of mostly suboptimal control solutions, creates a

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puzzle as to why finding good solutions seems to be an easy task. In order to address the latter issue, this paper considers quantum control landscapes which consist of the target observable as a functional of the control C共t兲, and the landscape features of interest are its extrema. A preliminary examination of these landscapes considering the optimization of P j→i indicates behavior consistent with the optimal control findings.31 In the present work we investigate the landscape structure for the optimization of general quantum mechanical observables under the condition of the system evolving as a density matrix. The identification of the extremum values of the landscape is a classical mathematical problem studied by von Neumann32 that has reappeared in many guises.33–35 The landscape, as defined for signals corresponding to nonHermitian operators 共i.e., quadrature detection of complex magnetization in NMR兲, has also been considered.36,37 The present work aims to further explore this topic by analyzing the extremum structure from the perspective of optimizing physically motivated “action variables.” Section II will present the theoretical analysis for arbitrary quantum dynamic systems and Sec. III will discuss the physical consequences of the findings. A brief conclusion is given in Sec. IV. II. ANALYSIS OF OPTIMAL CONTROL LANDSCAPE EXTREMA

The quantum system represented by N states is assumed to undergo unitary evolution from an initial density matrix ␳共0兲. The evolution is described by the unitary matrix U共t兲 over the interval 0 艋 t 艋 T, where T is an arbitrary physically relevant target time. The system is assumed to be controllable at time T, implying that from U共0兲, any point on the unit evolutionary sphere may be reached by at least one control field C共t兲. The restriction to a finite number of states N is for the purpose of consistency with the present understanding of circumstances when controllability may be established. The observable matrix O† = O is associated with the physical objectives, and a laboratory observation corresponds to J = Tr关U␳U†O兴,

共1兲

where U ⬅ U共T兲 and ␳ ⬅ ␳共0兲. The matrices O and ␳ in general will be diagonal with respect to distinct sets of eigenvectors 兵兩␣i典其 and 兵兩␤ j典其, respectively, with their associated eigenvalues being 兵ei其 and 兵d j其 such that O兩␣i典 = ei兩␣i典,

共2兲

␳兩␤ j典 = d j兩␤ j典.

共3兲

It is convenient to evaluate the trace in Eq. 共1兲 with respect to basis 兵兩␣i典其 and express ␳ in terms of the basis 兵兩␤ j典其 to yield J = 兺 兩具␣i兩U兩␤ j典兩2eid j .

共4兲

ij

The matrix U = U关C共t兲兴 is a functional of the control field C共t兲. The general goal of quantum optimal control, whether

OCT or OCE, is to extremize 共i.e., maximize or minimize, depending on the nature of O兲 J with respect to C共t兲 corresponding to seeking the best roots of the relation

␦J ␦C共t兲

共5兲

= 0.

In this work we aim to identify the extrema satisfying Eq. 共5兲 and to elucidate the physical consequences of this analysis for practically identifying good quality optimal controls. A preliminary assessment of this matter has been performed in the special case where ␳ and O are pure-state projectors,31 and other work has emphasized the accessible upper and lower bounding values for J.34 To facilitate the present analysis, it is useful to employ the identity U = exp共iA兲 where A† = A and A = A关C共t兲兴. Introduction of the action matrix A provides a convenient and physically motivated set of variables, consisting of the matrix elements of A, to assess the issues presented above. In unitarily evolving quantum systems, the kinematic degrees of freedom in the A matrix describe in fully generic terms the time-ordered system dynamics. The use of such variables has the dual advantage of obviating any specific knowledge of the structure of the Hamiltonian, yet maintaining a qualitative connection between the Hamiltonian dynamics and the kinematic representation of the landscape. Combining Eqs. 共4兲 and 共5兲 along with this definition for U in terms of A produces

␦J ␦C共t兲

= 兺 兺 e id j pq

ij

⳵ ␦A pq 兩具␣i兩U兩␤ j典兩2 = 0. ⳵A pq ␦C共t兲

共6兲

Since the system is assumed to be controllable, then each element A pq关C共t兲兴 must be independently addressable, thereby implying that A pq关C共t兲兴 must be a unique functional of the control field C共t兲 共in keeping with the Hermiticity of the A兲. This behavior in turn means that the set of functions 兵␦A pq / ␦C共t兲其 must be linearly independent over the domain 0 艋 t 艋 T and Eq. 共6兲 can only generally be satisfied by requiring that ⌬qp = 兺 eidi ij

⳵ 兩具␣i兩U兩␤ j典兩2 ⳵A pq

冋

= 兺 eid j 具␣i兩U兩␤ j典* ij

+ 具␣i兩U兩␤ j典

⳵ 具␣i兩U兩␤ j典 ⳵A pq

册

⳵ 具␣i兩U兩␤ j典* = 0, ⳵A pq

共7兲

for all p and q. In transforming from Eq. 共6兲 to Eq. 共7兲, the system-specific and generally highly complex mapping between the control field C共t兲 and A was subsumed into a simple statement of linear independence for the functions 兵␦A pq / ␦C共t兲其. This step reduces the analysis to a problem of kinematics in Eq. 共7兲. The kinematic nature of Eq. 共7兲 is evident through the use of the identity U = exp共iA兲. Thus, we may analyze the general nature of the extrema in Eq. 共5兲 by exploring the solutions to Eq. 共7兲 without specific resort to solving Schrödinger’s equation for any particular application.

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Optimal control landscapes

The analysis of Eq. 共7兲 requires evaluation of the derivatives 共⳵ / ⳵A pq兲具␣i兩U兩␤ j典, which may be facilitated by defining the unitary matrix V共s兲 = exp共iAs兲 that satisfies the equation i

⳵ V共s兲 = − AV共s兲, ⳵s

共8兲

V共0兲 = 1.

* ⬘ = ␴rs ⌬rs F共␭r,␭s兲 + ␴srF*共␭r,␭s兲 = 0,

for all r and s. First consider the case of ␭r ⫽ ␭s in Eq. 共11兲, and evaluating Eq. 共14兲 yields

冋

具␥r兩O exp共iA兲␳兩␥s典* exp

By differentiating this equation with respect to A pq, we may solve for 共⳵ / ⳵A pq兲具␣i兩U兩␤ j典 upon noting that V共1兲 = U, to produce

⳵ 具␣i兩U兩␤ j典 = i具␣i兩 ⳵A pq ⫻

冕

1

ds exp共iA共1 − s兲兲

0

⳵A exp共iAs兲兩␤ j典. ⳵A pq

共9兲

The expression in Eq. 共9兲 may be reduced to a practical form by utilizing the eigenvectors 兵兩␥ᐉ典其 of A, A兩␥ᐉ典 = ␭ᐉ兩␥ᐉ典, where 兵␭ᐉ其 are the associated eigenvalues of A. In addition, we have the expression ⳵A / ⳵A pq = 兺mn兩m典共⳵Amn / ⳵A pq兲具n兩 = 兩p典具q兩, where the set of states 兵兩k典其 utilized here corresponds to the representation chosen for the matrix A. Combining all of these statements reduces Eq. 共9兲 to the form

⳵ 具␣i兩exp共iA兲兩␤ j典 = 兺 具␣i兩␥ᐉ典具␥ᐉ兩p典具q兩␥ᐉ⬘典 ⳵A pq ᐉᐉ ⬘

⫻具␥ᐉ⬘兩␤ j典F共␭ᐉ,␭ᐉ⬘兲,

冦

i exp共i␭ᐉ兲

where F共␭ᐉ,␭ᐉ⬘兲 = exp共i␭ᐉ兲 − exp共i␭ᐉ⬘兲 ␭ ᐉ − ␭ ᐉ⬘

共10兲 if ␭ᐉ = ␭ᐉ⬘ if ␭ᐉ ⫽ ␭ᐉ⬘ .

冧

共11兲

The first case in Eq. 共11兲 is also understood to include the situation when ␭ᐉ = ␭ᐉ⬘ + 2␲m, where m is an integer, while the second case excludes this circumstance. In Eq. 共7兲 we also need the derivative 共⳵ / ⳵A pq兲具␣i兩U兩␤ j典* = 共⳵ / ⳵A pq兲 ⫻具␤ j兩exp共−iA兲兩␣i典, which may be readily expressed in a form like that of Eqs. 共10兲 and 共11兲. These derivative relations with respect to A pq may be directly substituted into Eq. 共7兲, but the resultant expression is difficult to analyze. Without any loss of generality, this analysis can be facilitated by making a unitary transformation of ⌬qp,

⬘ = 兺 具␥s兩q典⌬qp具p兩␥r典 ⌬rs pq

= 兺 eid j关具␣i兩U兩␤ j典*具␣i兩␥r典具␥s兩␤ j典F共␭r,␭s兲 + 具␣i兩U兩␤ j典 ij

⫻具␤ j兩␥r典具␥s兩␣i典F*共␭r,␭s兲兴 = 0,

共12兲

for all r and s. Equation 共12兲 can be simplified by defining the matrix ␴ whose elements are given by

␴sr = 兺 eid j具␥s兩␣i典具␣i兩exp共iA兲兩␤ j典具␤ j兩␥r典 ij

= 具␥s兩O exp共iA兲␳兩␥r典, thereby producing

共13兲

共14兲

i共␭r + ␭s兲 2

冋

− 具␥s兩O exp共iA兲␳兩␥r典exp

册

册

− i共␭r + ␭s兲 = 0, 2

共15兲

where the relation F共␭r , ␭s兲 = 兩F共␭r , ␭s兲兩exp关i共␭r + ␭s兲 / 2兴 was utilized, recognizing that 兩F共␭r , ␭s兲兩 ⫽ 0 under the stated condition. Equation 共15兲 may be rearranged with the definition ⍀ = exp共iA / 2兲 to have the following form: ˜ 兩␥ 典 − 具␥ 兩O ˜ ␳兩␥ 典 = 0, 具␥s兩˜␳O r s ˜ r

共16兲

where ˜ = ⍀†O⍀, O

共17a兲

˜␳ = ⍀␳⍀† .

共17b兲

The opposite circumstance in Eq. 共11兲 for the case of ␭r = ␭s also leads to the satisfaction of Eq. 共16兲. Since Eq. 共16兲 ˜兴 is valid for all r and s values, we may conclude that 关˜␳ , O = 0 is a necessary criterion for Eq. 共12兲 to be valid, implying ˜ share a common set of eigenvectors. These that ˜␳ and O eigenvectors may be identified by suitably pre- and postmultiplying Eqs. 共17a兲 and 共17b兲 by ⍀ and ⍀† and noting that ⍀2 = U to yield ˜ ⍀† = O, ⍀O

共18a兲

⍀†˜␳⍀ = U␳U† .

共18b兲

˜ and ˜␳ commute, they share a common set of eigenSince O vectors 兵兩␣典其, and in this basis, ⍀ diagonalizes both operators. Equation 共18b兲 shows that U = exp共iA兲 diagonalizes ␳ in the 兵兩␣典其 basis as well. Since ␳ was already diagonal in the 兵兩␤典其 basis, U must necessarily be a permutation matrix. Let e and d denote vectors containing the eigenvalues of ␳ and O. Substituting an arbitrary permutation matrix P for U in Eq. 共4兲, the cost function at the extrema is J = 兺 ei Pijd j = eT · P · d.

共19兲

ij

Since there are at most N! distinct permutations on N objects, the maximum number of distinct landscape extremum values in N!. Three limiting cases commonly arise in Eq. 共19兲. First, when e has only one nonzero element whose value we may assign as ei = 1 associated with the observable operator O = 兩␣i典具␣i兩, for a particular state 兩␣i典, then there will be N distinct values for J corresponding to J j = d j, j = 1 , 2 , 3 , . . . , N. The second case concerns the opposite circumstance where ␳共0兲 = 兩␤ j典具␤ j兩 for a particular state 兩␤ j典 with d j = 1 and again J takes on the N values Ji = ei, i = 1 , 2 , 3 , . . . , N. Finally, a third limiting case corresponds to only a single element of both e and d being nonzero, thereby reducing the optimization problem to maximization of the

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transition probability P j→i = 兩Uij兩2. In this case, only a single optimal value J = 1 is consistent with satisfying Eq. 共5兲 共as well as the trivial case of J = 0兲. This latter, perhaps surprising behavior of there being no suboptimal extrema for maximizing P j→i, was examined carefully with a different analysis in another work.31 It should be emphasized that in this situation, a unique critical value J = 1 does not imply a unique critical solution, as there may be multiple dynamical mechanisms by which full population transfer is achieved. The structure of the result in Eq. 共19兲 was also observed earlier in an effort to identify the upper and lower bounds for J. The focus here is on the physical consequences of the result in Eq. 共19兲 for the practical achievement of laboratory optimal control of quantum systems.

III. QUANTUM OPTIMAL CONTROL LANDSCAPES

The main conclusion of the analysis in the previous section is that for finite dimensional controllable quantum systems without constraints placed on the controls, there are only a limited number of distinct values for the target cost function J corresponding to solutions of Eq. 共5兲. At first sight, this result may appear counterintuitive, as controllability implies that J can take on any dynamically accessible value, and simple intuition might suggest that J could have an arbitrary number of local extremum values throughout the control space. Although J can take on any accessible value, only a discrete set of distinct values of J satisfy Eq. 共5兲. This behavior is most encouraging for the practical achievement of quantum control. Except for the upper and lower bounding values for these extrema, Eqs. 共5兲 and 共19兲 do not reveal whether any particular extremum value corresponds to a maximum, a minimum, or possibly a saddle point. An eigenvalue analysis of the Hessian ␦2J / ␦C共t兲␦C共t⬘兲 at each extremum would be needed to establish the detailed structure around each maximum, minimum, or saddle point. This matter is addressed with topological methods in a separate, forthcoming work.38 The initial density matrix ␳共0兲 may generally be represented as a mixture over the eigenstates 兵兩␸ᐉ典其 of the system control-free Hamiltonian H0 such that ␳共0兲 = 兺ᐉᐉ⬘兩␸ᐉ典␳ᐉᐉ⬘共0兲 ⫻具␸ᐉ⬘兩. This matrix, if not already diagonal, may be diagonalized by the basis 兵兩␤i典其 as indicated in Eq. 共3兲. The presence of nonzero off-diagonal elements ␳ᐉᐉ⬘, ᐉ ⫽ ᐉ⬘, implies that an initial coherence exists with respect to the basis 兵兩␸ᐉ典其. It is evident that the number of distinct values for J in Eq. 共19兲 is directly related to the number of nonzero eigenvalues n␳ of ␳共0兲 residing in the vector d, regardless of the nature of the coherence built into ␳共0兲. An analogous conclusion also applies to considering the observable operator O with the number of distinct, nonzero eigenvalues nO in the vector e again controlling the number of values for J. For the case in which there are n␳ and nO nondegenerate eigenvalues of ␳ and O, an analytical expression for the number ⌳ of the distinct values of J can be obtained in terms of n␳ and nO. The general case in which there may be arbitrary degeneracies in the eigenvalues of ␳ and O is treated in a separate work.39 It can be seen from Eq. 共19兲 that ⌳ is

equal to the number of distinct elements of the set 兵e · P · d, with P being a permutation matrix on N objects其 composed of N! such distinct matrices. We first examine the case where nO ⬎ n␳. The minimum and maximum numbers of zeros in the first nO positions of P · d are qmin = nO − n␳ and qmax = min兵N − n␳ , nO其, respectively. Without loss of generality, we can assume that the nO nonzero elements of e are all located in the first nO positions in e. The number of nonzero elements of d existing in the first nO positions in P · d is nO − q, where q = number of zeros in the first nO positions in P · d. We wish to determine the number of ways nO − q nonzero elements can be permuted into n␳ positions. For this, we need to account for ordering because we are counting the number of distinct values that can be obtained by scrambling the elements of d and then taking the dot product e · P共d兲. Therefore, the number of ways is given by the permutation term n␳! / 共n␳ − nO + q兲!. Next, we wish to determine the number of ways n␳ positions can be distributed into nO positions. We may neglect ordering because this task just calls for counting the number of ways to configure the positions into which the nonzero elements of d are sorted into n␳ positions. Therefore, the number of ways is given by the combinatorial term nO! / 共nO − q兲!q!. For a given value of q, the product of the permutation and the combinatorial term is equal to the number of distinct values of e · P共d兲. By summing over all possible values of q, we obtain qmax

⌳=

n O! n ␳! . 共n␳ − nO + q兲! 共nO − q兲!q! min

兺 q=q

共20兲

For the case of nO ⬍ n␳, let q⬘ = number of zeros in the first n␳ positions in P共e兲. Then qmin ⬘ = n␳ − nO, qmax ⬘ = min兵N − nO , n␳其, and qmax ⬘

⌳=

兺

q⬘=qmin ⬘

n ␳! n O! . 共nO − n␳ + q⬘兲! 共n␳ − q⬘兲!q⬘!

共21兲

If nO = n␳, either Eq. 共20兲 or 共21兲 is valid, with qmin = 0. Consider the special case where ␳共0兲 = 兩␤ j典具␤ j兩 and O = 兩␣i典具␣i兩 such that d and e have only a single nonzero eigenvalue corresponding to the goal of maximizing the transition probability P j→i = 兩Uij兩2. This case has the attractive feature of all optimal control solutions strictly having the extremum value of J = 1 or the trivial solution J = 0, as noted earlier corresponding to ⌳ = 2. In OCT calculations seeking control field design, this case has been a common goal for optimization. A general finding from many such published studies is that excellent quality controls appear easy to identify. The reason for the evidently puzzling success of the many OCT design studies using local search algorithms to produce very high product yields for P j→i = 兩Uij兩2 is now clear, as any initial trial guess for the control will always be next to a perfect control solution. The fact that J takes on maximum values J ⬍ 1 in actual OCT calculations is due to the constraints in one form or another always being placed on the controls. These constraints could have many forms, including the specification of a particular target time T, the inclusion of an additional cost for minimizing the control field fluence 兰T0 C共t兲2dt, and possibly specific constraints on the form of

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Optimal control landscapes

FIG. 1. Enumeration of landscape critical values as a function of the number of distinct eigenvalues of the system density operator ␳ and target observable operator ␪.

the control field. All these constraints can reduce the value of J and in especially unfavorable cases introduce new suboptimal local extrema. Notwithstanding the special case of a cleanly prepared single initial state 兩␤ j典 and a single target state 兩␣i典 when seeking to maximize P j→i, the prospect of the number of extrema ⌳共n␳ , nO兲 growing very rapidly with increasing n␳ and nO shown in Fig. 1 deserves special consideration for the practical achievement of effective quantum control. This matter will be examined from several perspectives below. In the laboratory, a common OCE goal is to maximize the probability of making a transition P j→i = 兩Uij兩2. But, in practice, the system will be at a finite temperature producing a distribution of states to be averaged over in the initial density matrix and the final goal may also involve a summation over product channels40 corresponding to the general structure in Eq. 共1兲. At first sight, this circumstance would appear to produce a dynamical space of very high dimension N which in turn would lead to a very large number ⌳ of extrema with all but one corresponding to suboptimal values of J. However, in very recent experimental and computational studies, it has been found that the number of “significant” dimensions, defined as the dominant components of a principal component analysis, can be very low.41 These observations may be rationalized heuristically from the perspective of energetics. Commonly, the control goal is to induce a “high energy” electronic or vibrational transition, regardless of the rotational degrees of freedom and possibly other lowfrequency vibrational degrees of freedom which all may be collectively denoted as “low energy” background states. Under these circumstances as a simple model, we may consider the free Hamiltonian H0 to be approximately broken into two HE LE HE separate parts H0 ⯝ HLE 0 + H0 , where H0 and H0 , respectively, depend on the low 共high兲 physical variable 共e.g., coordinates and momenta兲 portions of the system. Assuming that the corresponding dynamics associated with HLE 0 is effectively uncoupled from the physically interesting dynamics driven by the control C共t兲, then 具␣i兩exp共iA兲兩␤ j典 of HHE 0 HE LE HE → 具␣ᐉ 兩exp共iALE兲兩␤ᐉLE⬘ 典具␣HE k 兩exp共iA 兲兩␤k⬘ 典. Here we have followed the approximate separation of the Hamiltonian to accordingly separate A into ALE and AHE, and the states have LE HE HE the factorized form 兩␣i典 → 兩␣LE ᐉ 典兩␣k 典 and 兩␤ j典 → 兩␤ᐉ⬘ 典兩␤k⬘ 典. HE LE HE LE Taking the observable operator as O = 兺ᐉ⬘兩␣k* ␣ᐉ⬘ 典具␣k* ␣ᐉ⬘ 兩 for some particular high energy product channel state

兩␣kHE * 典 and summed over all low energy states leads to the cost functional for maximization reducing to HE 2 HE J = 兺k⬘dkHE兩具␣kHE 典兩 , where the sum over all the 兩exp共iA 兲兩 ␤ * k⬘ ⬘ final low energy states and the average over the initial low energy states were analytically eliminated. The yield for going to a particular final high energy state 兩␣kHE * 典 produces a cost function J which may still have a number of distinct extrema dictated by the number of populated initial high energy stats n␳HE. But, we should have n␳HE Ⰶ N, and at room temperature it will likely be found that n␳HE ⬃ 1. Thus, when large numbers of physically uninteresting low energy background states exist, the effective controlled dynamics of interest again may have only one or a few distinct control extremum values for J. This behavior is consistent with what appears to be occurring in many of the laboratory OCE studies of control over molecular dynamics, including dissociative phenomena, where modest search efforts can find significant enhancements in the product yield. From another physical perspective, an effective reduction in the number of distinct extrema values for J will arise when degeneracies occur in the eigenvectors d and e of ␳ and O, respectively. For example, this will occur when the initial and final manifolds of states contain high degrees of near degeneracy due to the presence of low energy accessible states. Under these conditions, the number of distinct values ⌳ of the cost functional will be dictated by n␳HE, as the degenerate low energy states are summed away within each high energy manifold. From a mathematical perspective, the effect of eigenvalue degeneracy on the enumeration of critical values for the landscape is explored in closer detail in Ref. 39. Considering the discussion in the last paragraph regarding the expected landscape behavior of many OCE studies, these results suggest that local search algorithms 共e.g., simplex algorithms兲 may be effective in the laboratory. However, robust global search procedures provided by genetictype algorithms still may be the best way to perform OCE searches for effective controls. In the laboratory, the search algorithm must operate in the presence of control field fluctuations,42 and stochastic-based algorithms may have better performance in this environment. Search algorithms tailored to the special nature of quantum optimal control may ultimately be the most effective. Beyond issues of robustness, in the laboratory additional factors can arise including the presence of environmental decoherence and noise in the observations. Control field fluctuations and environmental interactions can both contribute to a reduction in the degree of coherence in the dynamics. IV. CONCLUSION

This paper explores the nature of quantum control landscapes by drawing on the fact that the difficult task of unraveling the dynamical mapping C共t兲 → U may be put aside under the assumption of system controllability. The general structure of control landscapes then becomes amenable to a thorough analysis of the system kinematics through the introduction of U = exp共iA兲, leading to an identification of the number of distinct control extrema and their values. The possibly complex dependence of the dynamics on the subtle

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204107-6

structure of C共t兲 will enter again in any particular OCT or OCE study attempting to home in one of the cost function extrema. Nevertheless, the landscape is a universal underlying structure upon which the search would occur. The discussion in Sec. III indicated that various issues and circumstances need to be factored in when considering the OCT or OCE discovery of effective quantum controls. A most important conclusion from the analysis is that realistic control over many aspects of quantum system dynamics should produce a relatively modest number of distinct extrema values for the target cost functions. Approaching this ideal behavior in the laboratory is a matter of engineering better and higher quality flexible controls as well as cleaner environments for the quantum systems subjected to manipulation. This work provides the basis to generally expect that high quality control over quantum dynamics phenomena should be attained in the laboratory and under broad circumstances. Armed with these conclusions, an interesting goal ahead will be the through laboratory exploration of wide classes of optimally controlled quantum dynamics phenomena for fundamental reasons and possible practical applications. S. Shi and H. Rabitz, Chem. Phys. 139, 185 共1989兲. C. Meier and M. Heitz, J. Chem. Phys. 123, 044504 共2005兲. 3 I. Serban, J. Werschnik, and E. Gross, Phys. Rev. A 71, 053810 共2005兲. 4 M. Sukharev and T. Seideman, Phys. Rev. Lett. 93, 093004 共2004兲. 5 C. M. Tesch and R. de Vivie-Riedle, J. Chem. Phys. 121, 12158 共2004兲. 6 E. Luc-Koenig, R. Kosloff, F. Masnou-Seeuws, and M. Vatasescu, Phys. Rev. A 70, 033414 共2004兲. 7 D. Babikov, J. Chem. Phys. 121, 7577 共2004兲. 8 B. Lan, I. Vrabel, and W. Jakubetz, J. Chem. Phys. 121, 10401 共2004兲. 9 J. Palao and R. Kosloff, Phys. Rev. A 68, 062308 共2003兲. 10 D. Geppert, A. Hoffman, and R. de Vivic-Riedle, J. Chem. Phys. 119, 5901 共2003兲. 11 T. Hornung, S. Gordienko, R. de Vivie-Riedle, and B. Verhaar, Phys. Rev. A 66, 043607 共2002兲. 12 J. Palao and R. Kosloff, Phys. Rev. Lett. 89, 188301 共2002兲. 13 T. Mancal, U. Kleinekathofer, and V. May, J. Chem. Phys. 117, 636 共2002兲. 14 T. Hornung, M. Motzkus, and R. de Vivie-Riedle, J. Chem. Phys. 115, 3105 共2001兲. 1 2

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