J. Phys. Chem. 1993,97, 1211612121
12114
ARTICLES Optimal Control of Coherent Wave Functions: A Linearized Quantum Dynamical View Liyang Shen,? Shenghua Shi,#and kerschel Rabitz' Department of Chemistry, Princeton University, Princeton, New Jersey 08544 Received: April 13, 1993; In Final Form: September 22, 1993" We present an optimal control approach to study a linearized quantum dynamical system. Using quadratic forms for the physical objective of a target coherent wave function and for the physical penalties, we derive analytical expressions for the optimal fields. It is shown that the optimal fields can be decomposed into a finite number of monochromatic component fields. The numerical results suggest that the qualitative behavior of the optimal field is strongly dependent upon the cost functional weight factors. Moderate weight factors placed on the physical constraints can lead to excellent control of the desired target wave function. The effect of target time on the properties of the optimal field is also investigated. The numerical studies indicate that the optimal field is affected by the overall absolute phase of the target wave function, which provides a certain degree! of additional flexibility in choosing the optimal fields. In all the cases studied only negligible direct current components appear in the fields.
The linearized dynamical equation becomes
I . Introduction The rapid development of laser pulse shaping techniques1** has generated considerable interest in controlling molecular motion by using specially designed laser field~.~-l'Attempts to control molecular dynamics are motivated by the potential applications including the possibly high selectivity in chemical reactions.I3-l7 Nevertheless, for a given system to be controlled, it is often nontrivial to find the required laser field that drives the system to the desired state. Moreover, the feasibility of carrying out molecular control in practice may also be restricted by various physical constraints. Recently, considerabletheoretical advances have been made by employing optimal control theory to search for an optimal field that guides the system to achieve the target value while taking into account practical constraints.'* A number of studies have explored the existence of an optimal field for bounded19 and unbounded quantum systems,20and some of the conditions have been established. Given the fact that the interaction between a molecule and a laser field with low intensity (e.g., -0.1 MW/cm2) is generally weak, we showed in a previous paper12 (hereafter referred to as paper I) that control theory can be readily applied to the linearized Schradinger equation by taking a perturbation approach with respect to thecontrol field. Assuming that the molecule isinitially in an eigenstate, +(O), belonging to an eigenvalue, (YO, and that the relatively low intense control field acts among the N eigenstates, &, k = 1, 2, ..., N, of the field-free molecular Hamiltonian &, the wave function can then be described by first-order perturbation theory $(t)
E
e-'*d($(O)
+ x(')(t))
(1.1.1)
with N
(1.1.2)
7 Present address: Department of MacromolecularModeling, Bristol-Myers Squibb Pharmaceutical Research Institute, Princeton. NJ 08543. *Present address: Biosym Technologies, Inc., San Diego, CA 92121. Abstract published in Advonce ACS Absfrcrcfs.November 1, 1993.
0022-3654/93/2097- 12 114$04.00/0
= Hc(t) - c(t)d
(1.2.1)
c(t=O) = 0 where c(r) and d(t) are N-dimensional vectors, 0 is a N-dimensional zero vector, and H is a N X N matrix defined by [ ~ ( t ) ] k C k ( t ) , 1 Ik IN
[a],
f
dk = (4&$(0)),
1 Ik IN
(1.2.2) (1.2.3)
Here, ic is the electricdipole operator in the applied field direction. In paper I, weestablishedthenecessaryandsufficientconditions for a linearized quantum dynamical system to be completely controllable. The control field was obtained analytically and decomposed into N-monochromatic fields. We showed that both the conditions for controllability and the control field can be expressed explicitly in terms of the molecular transition frequencies (Ek - (YO) and the electric transition dipole moments dk, k = 1, 2, ..., N,as well as the target state. The purpose of this paper is to explore optimal control properties of a linearized quantum dynamical system under certain physical constraints. While in paper I we focused mainly on controllability and the unconstrained control field for a given system, we emphasize here optimal field design in the presence of physical constraints and their relative importance pitted against the physical objectives. For example, it is desirable to keep the laser energy fluence to a minimum since low-intensity pulse shaping is more feasible in the laboratory. The control field determined in paper I becomes the limit of the resultant optimal field with zero weight given to the physical constraints. In general, the choice of the constituent terms in the cost functional, including the weight for eachof the terms, criticallydetermines thebehavior and performance of an optimal field. Brumer and Shapiro have examined the control of relative yields of reaction products in molecular dissociation through a variety of means including, for example, a scheme with two weak p ~ 1 s e s . l ~One ~ pulse excites the molecule from the ground electronicstateto the excited state, while thesecondonedissociates
8 1993 American Chemical Society
Optimal Control of Coherent Wave Functions it to the ground-state continuum. The selectivity of the products is improved by adjusting the central frequency of the excitation pulse, the time delay between the two pulses, and the pulse intensities. In this paper, we consider the optimal control of a coherent state for a general linearized quantum system using a single pump field. We utilize optimal control theory to design the field to best achieve the control objective. As an application of the general theory developed in the present work, we examine the generation of the nuclear vibrational coherent state on the excited electronic state for the Liz molecule (cf. section 111). We demonstrate numerically that the target objectivecan be achieved nearly perfectly by the designed optimal field. As shown in paper I, for a linearized quantumdynamical system controllability is completely understood, and the control field can be expressed analytically. In general, an optimal field can be only sought numerically for a given system due to the inverse nature of an optimal control problem. However, as shown in section 11, in the present case, the optimal field also can be expressed analytically. Therefore, it is of interest to compare the optimal field with the control field of paper I to gain physical insight into the role of the physical constraints and the weight factors in determining the optimal field. We give an in-depth discussion on the general behavior of the optimal field and various effects on the optimal control properties. In paper I, the control field guided the system to exactly reach the target state. This may not be the case for the present optimal control problem. However, for given weight factors, optimal control theory provides a systematic algorithm to determine the optimal field that drives the system to best achieve the desired state. In section 111, we provide numerical examples to demonstrate the properties of optimal control of molecular coherent wave function generation. The results calculated for various weight factors of the constituent terms in the cost functional are then compared with those of control in paper 1. Insight is given about the role of the physical constraints and the corresponding weights in determining the behavior and performance of the optimal field. In paper I, we analytically showed that the control properties are not affected by the absolute constant phase of the target coherent wave functions. We demonstrate in section I11 that such a phase effect is significant for the optimal control system. Since observables are invariant to the overall phase, the phase may be chosen for convenience to produce optimal fields in better accordance with experimental conditions. Finally, the conclusions that can be derived from this work are summarized in section IV.
The Journal of Physical Chemistry, Vol. 97, No. 47, 1993 12115 linearized quantum dynamical system of eq 1.2, it can be shown that there is only a trivial solution for the optimal field. Unlike the control case, shown in paper I, the absolute phase of the desired wave function contained in the above cost functional will affect the resultant optimal field. However, this does not imply that an absolute phase is physically significant; the choice of the formulation here is guided by practical laboratory and computational considerations (Le., we will obtain a closed form solution for the optimal control field). There may exist distinct optimal fields resulting in the same state 4 but differing in overall phase. One may sampledifferent overall phases to seek fields that possibly have simpler structure. The cost functional in eq 11.1 provides much flexibility, allowing one to search a wide variety of optimal fields. An optimal field is achieved by minimizing the cost functional in eq 11.1. Introducing a Lagrange multiplier vector, A(r), we obtain an unconstrained functional
+ iHc(t) - it(t)d]
Q = Q - ReCA'(t)[$c(t)
dt (11.2)
The necessary condition for a minimum of Q is that the variation of Qvanishes with respect to the variations of c(t), A(t), and t(t), respectively. Applying this condition,one arrives at the following equations:
= HA(t) - iW,[c(t) - 51
(11.3.1)
and t(t)
1 = -Im[At(t)d]
(11.3.2)
B
along with eq 1.2. The optimal field can be obtained by solving the coupled eqs 1.2 and 11.3. Substituting eq 11.3.2 into eq 1.2 and then separating the real andimaginary partsofeqs 1.2 and 11.3.1, oneobtains the following equivalent equation:
"[' dt X(t) ] = R[$]
+
[LE]
(11.4.1)
where 0 is a 2N dimensional zero vector and
(11.4.2)
11. Optimal Control Algorithm
(11.4.3)
We first identify the physical objective and constraints to formulate the optimal control problem. As in paper I, a target wave function [ is the objective to be controlled at a target time T. In this paper, we consider two physical constraints. One constraint is to minimize the energy fluence of the electric field to make the experimental conditions more attractive. The other constraint is to require that the molecule achieve the desired wave function as quickly as possible. For computational simplicity, we employ quadratic forms for the cost functional, Q,
E= R=["
[;;I
(11.4.4)
- -bbT
]
(11.4.5)
W A with
61 dt + ;Brt2 dt(t (11.1) ) subject to eq 1.2. Here, W Tand W , are N X N diagonal weight matrices and j3 is a weight factor; they are used to balance the relative importance of the three terms. In conventionalquantum optimal control theory the absolute square of the projection of the wave function on the target state is used in the cost functional to measure the deviationof the molecular wave function from the desired one in order to avoid control of the physically irrelevant absolute phase.5 However, with such a cost functional, for the
(11.4.6) (11.4.7)
w=[?
O ,] w
(11.4.8)
12116 The Journal of Physical Chemistry, Vol. 97, No. 47, 1993
where 0 is a N X N zero matrix. The subscripts R and I are used to represent the real and imaginary parts of a vector, respectively. The boundary conditions for eq 11.4 are
(c(r=o) = o
(11.5.1) (11.5.2)
Equation 11.4.1 is an inhomogeneous linear equation, and the solution can be formally expressed as
Shen et al. and
As exp(At) decays with t for t 2 0, eqs 11.8.1 and 11.8.4 are numerically stable. Once the boundary values of zn(0) and Zb( Z) are determined, these two equations can be used to evaluate c(f) and X ( t ) , hence c ( t ) . Letting f = Tin eqs 11.8.1 and 11.8.2and r = 0 in eq 11.8.5, one obtains za(0) = (Ul)-'U2{A-' [I - exp(AT)](U-'),W,g
(11.6.1)
lTexp[R(f - s)]
[btg]ds (11.6.2)
Equation 11.6 cannot be used directly to evaluate c ( t ) and X(t), since the boundary values of c ( T ) and X(0) are not explicitly given. This difficulty can be overcome below by making use of the properties of R. The analysis has a structure parallel to that of an earlier work on the mathematically analogous problem of the optimal control of coupled harmonic oscillator^.^ In this paper, we consider the case in which R can be diagonalized. However, the following procedure to obtain the optimal field is general and can be applied to the case in which R can only be block-diagonalized in Jordan form. In the Appendix, it is shown that the eigenvalues of the 4N X 4N matrix R occur in groups of four: f a k , f a k * if ak is complex, or f ( a k ) I , *(a& if a k is real, k = 1, 2, ..., N . Let
-
exp(AT)zb(T)) (11.9.1) while letting t = Tin eq 11.8.5 and making use of eqs 11.5.2and 11.9.1,the other boundary value can be identified as zb(T) = A-'{(w&l - U,)[exp(AT)(U,)-'U,A-'[exp(AT) I](U-'),
+ A-I[exp(AT) - I](U-'),]W,
- W & (11.9.2)
where
A = U, - W,U2
+ (W,Ul
- U,) exp(AT)(Ul)-'U2 exp(A2') (11.9.3)
With the boundary values now determined, an analytical expression for the optimal field can be obtained. Notice that the matrix U is formed by the eigenvectorsof R as defined by eq 11.7. By making use of the properties of the eigenvectors presented in the Appendix as well as UU-l = I, after much algebra, it can be shown that the optimal field can be decomposed into N-monochromatic fields N
(11.10.1) be the matrix that diagonalizes R
[^0
]
(11.7.1) -A where A is a 2N X 2N diagonal matrix of elements consisting of eigenvalues with nonpositiue real parts defined by U-'RU=
where (Nkk
=
k
(or (ak)'if
k-N
(or
(Yk is real), for 1 5 k 5 N if ak-Nis real), for ( N 1) 5 k 5 2 N
+
(11.7.2) By eq 11.7,one can transform eq 11.6 into
= exp(Ar)z,(O) + A-'[exp(At) - I](U-1)2Wfg zb(t) = exp(-At)zb(o) A-' [exp(-At) - I] (U-1)4Wf$ z,(t)
(11.8.1) (11.8.2)
and z,(t) = exp[-A(T-t)]za(7')
zb(r)
+
A-'(exp[-A(T- t)] - I](U-')2Wrg = exp[A(T-t)lzb(T) A-'(exp[A(T-r)J - IJ(U-'),W,g
(11.8.3) (11.8.4)
where
(11.8.5)
with
The Journal of Physical Chemistry, Vol. 97, No. 47,1993 12117
Optimal Control of Coherent Wave Functions
TABLE I: Potential Parametersz1of Lif (cm-1) we (cm-1) re (A) T~(cm-'1 AIZ:
X'Z;
9352.5 8516.9
255.47 351.39
3.108 2.675
14068 0
(le
(A-1)
0.603 0.869
The reduced mass is 3.509 amu. constraints, the component field frequencies will coincide with the molecular transition frequencies. The optimal field is strongly influenced by the properties of the matrix R. In the Appendix, we show that the characteristic polynomial of R can be expressed as
A(a) =
(11.1 1)
(11.10.14)
Equation 11.10 states that the optimal control field is a linear combinationof N monochromaticcomponent electric fields. Each of the component fields consists of three pulses. For the kth component, the first two pulses have the same frequency, equal to the absolute value of the imaginary part of the eigenvalue LYk, but with different initial phases and amplitudes. One pulse decays exponentially as exp[(&t], and the other is modulated by an exponentially growing function exp[(CYk)R(T - t ) ] . The corresponding amplitude factors (&)k and (&)kin general are unequal, and their relative magnitudes play an important role in determining the overall shape of the optimal field envelope. Ths third pulse is a constant. The presence of such a direct current (DC) field component is of considerable practical concern. In the numerical examplesstudied in section 111, the DC amplitudes are very small. It is instructive to compare the optimal field with the control field derived in paper I. Both fields are composed of N monochromatic component fields. However, the form of the optimal field components is different from that of the control field components found in paper I, each of which only has a single pulse with a constant amplitude, although the present results should reduce to that of paper I when W, = 0 and j3 = 0. Physically, the component fields of the optimal field are no longer solely determined by the molecular properties and the target state but are also dependent upon the imposed physical constraints. The required frequencies, phases, and amplitudes of the component fields vary with different emphasis on the relative importance of the target objective and the physical constraints. For instance, as will be illustrated in the numericalresults later, when we heavily weight the target objective and emphasize less the physical
One can immediately deduce the following basic properties of the matrix R. (1) The eigenvalues LYk, k = 1, 2, ..., 4N, of R depend on the weights W, and 3/ only through the ratios (Wf)j,/j3, j = 1, 2, ..., N. Therefore, the shape parameters Re(ak) and the frequencies Im(ak) of the component optimal fields are determined by these ratios. (2) If the molecular states defined by eq 1.2 are not completely controllable Le., either some of the eigenstates of the field-free Hamiltonian involved in the superposition to form the coherent wave function are degenerate or there exist zero electric dipole transition moments from the initial state to the Neigenstates (see paper I for details), then the matrix R cannot be fully diagonalized. (3) If the molecular states defined by eq 1.2 are completely controllable,and also (W,)j, = 0 with /3 being finite, then R cannot be fully diagonalized, and it has two-fold degeneracy with eigenvalues *iE$. (4) If the molecular states defined by eq 1.2 are completely controllable, and also W, # 0 with j3 being finite, then R does not have pure imaginary eigenvalues. ( 5 ) If the molecular states defined by eq 1.2 are completely controllable,and also j3- -, then R cannot be fully diagonalized, and it has two-fold degeneracy with eigenvalues ME$,j = 1,2, ..., N. 111. Applications
In this section, we apply the theory developed in section I1 to the optimal control of molecular coherent wave functions on an excited potential energy surface. To facilitate the comparison of the results between controlpresented in paper I and optimalcontrol here, we once again consider the creation of specified vibrational wave function profiles on the excited potential energy curve AIZ: for the Liz molecule. As in paper I, we assume that the molecule is initially in the ground vibrational level of the ground electronic state X'Z: (Le., a0 = 174.79 cm-I). Morse functions are used to describe the potential curves X'Z: and A'Z: with the parametersz1 summarized in Table I. The electronic dipole transition moments for X'Z: A'Z: as a function of the nuclear distance are taken from ref 22. The first seven vibrational eigenfunctionsof A'Z:, &(r), k = 0,1, ...,6,are chosen to form the basis set to create the superposition of molecular coherent wave functions. We showed in paper I that this system is completely controllable. To simplify the calculation, in this paper we set the weight matrices to be constant, i.e., WT= UTIand W, = w,I, where I is a unit matrix and UT and w, are weight factors. Physically, only the relative ratios of the weights in the cost functional are important in determining an optimal field. In the following examples, we fix the weight factor for the target wave function control to be U T = lo4 and adjust the other weight factors in accordance with different physical emphasis. The optimal field
-
Shen et al.
12118 The Journal of Physical Chemistry, Vol. 97, No. 47, 1993
is determined by the overall behavior of the cost functional, not by the individual constituent terms acting alone. For the sake of comparison with the results obtained from the previous control study in paper I, we select the target wave function as
for ~ ( 1 to ) achieve. Furthermore, the target time is chosen to be T = 0.2 ps, unless stated otherwise. The fundamental issue we are concerned with is whether a controllable target coherent wave function, prescribed under the necessary and sufficient conditions given by paper I, can also be successfully controlled by an optimal field when other competing physical criteria are present. By choosing @ = lo3 and w1 = 2 X 10-9, we mainly emphasize the control of the target wave function whilegiving a reasonable emphasis to the energy fluence. The calculated optimal field as a function of time and the achieved probability density as a function of the nuclear distance are depicted in parts a and b of Figure 1, respectively. Compared with the target wave function displayed in Figure 1b as a dotted line, which is indistinguishable from the solid line, it is seen that the desired target wave function can be achieved almost perfectly. Comparing the optimal field in Figure la here with the control field shown in Figure 2 of paper I for the same target wave function, we find that these two fields are almost identical. This finding is confirmed by the values of the parameters of the seven components of the optimal field presented in Table 11. It is seen that the field component frequencies are completely determined by the molecular transition frequenciesdue to the heavy emphasis on the target objective. In Figure IC, we display the evolution of the wave function probability density under the optimal field. It is seen that the molecular excitation propagates through various profiles before finally reaching the target profile. Since the cost functional does not significantly weight the first integral term in eq 11.1, it isnot surprisingthat thecontrolled wave functionreaches its target shape while passing through profiles of considerably different from before the target time. To investigate the effect of the weight factors on the performance of the optimal field, we select @ = 106 and 0 1 = 2 X 10-3. In this case, we give much more emphasis to the energy fluence and achieving the target as soon as possible. The resultant optimal field is displayed in Figure 2a which shows that its overall qualitativestructure is similar to theoptimal field shown in Figure la. This can be explained as follows. First, if the ratio cot/@ is sufficiently small, e.g., 5 10-9, then the frequenciesof the resultant N component optimal fields will be very close to the N molecular transition frequencies. Moreover, as indicated by Table 111, the shape factors ( ( Y ~ ) Rof the component fields are still quite small, and the amplitude factors ( B J k and (B& of the decaying and growing exponential functions are similar in magnitude for each k value. Furthermore, the DC component fields are very small and can thus be neglected. Figure 2b showsthat the performance of the optimal field in Figure 2a is somewhat different from the one shown in Figure 1. The deviation of the controlled coherent wave function from the desired shape results from a heavy weighting of the energy fluence as well as the path to the target. Next, we lower the weight factor for the energy fluence to @ = lo3 while keeping the integral weight factor the same, i.e. 0 1 = 2 X 10-3, and thus the ratio o f / @becomes much larger. The calculated optimal field is displayed in Figure 3. The coherent wave function can be controlled to very high precision like the case shown in Figure lb, and the behavior of the optimal field is significantly different from those shown in Figures la and 2a. The envelope of the optimal field exponentially increases as the target time is approached. This can be understood by analyzing the component fields whose parameters are listed in Table IV. Now the component frequencies (a& are no longer equal to the molecular transition frequencies due to the large ratio wt/@. Moreover, the shape factors ( ( Y ~ ) R are now much larger. In
0.10
0.05
0.00
0.20
0.15
0.3 I
3
v
x
d .-
v)
5
0.2
a
x
-.-5 P
P
2
0.1
a
0.0 4.0
4.5
5.0
5.5
6.0
6.5
7.0
7.5
0.0
r (a.u.) n
co I
-2
0.40
.O
o.20
t
'r
Figure 1. Optimal control results for the target coherent vibrational wave function on the excited potential energy curve A'Z; for Liz. The target wave function is cp(r) = (10-3/~)(e'r/4&,(r) + &(r)), where )&#t denotes the kth vibrational eigenstate of the potential A'Z:. The same target state is employed in the remaining examples. The weight factors here are = 104, of = 2 X 1W9,and B = 103,and the target time is T = 0.2 ps: (a) theoptimal field; (b) the target wave function probability density profile (dotted line) and the achieved profile (solid line) under the optimal field at the target time; the two curves are indistinguishable on the scale of the graph; (c) the evolution profile of wave function probability density under the optimal field.
addition, the exponentially rising amplitudes (&)&are about one order of magnitude larger than the exponentially damped amplitudes ( B a ) k . Furthermore, from Table IV, one observes that the optimal field consists mostly of the two component fields
The Journal of Physical Chemistry, Vol. 97, No. 47, 1993 12119
Optimal Control of Coherent Wave Functions
TABLE III: Parameters of the Components of the Optimal Field' Shown in Figure 2a
0
14020.5 14272.5 14521.0 14766.0 15 007.5 15245.5 14480.0
1 2 3 4 5 6
-1.043 -1.618 -1.864 -1.832 -1.625 -1.339 -1.043
0.108 0.020 0.065 0.024 0.003 0.009 0.002
0.130 0.019 0.087 0.023 0.002 0.009 0.002
0.12 -0.0 0.30 -0.0 -0.0 -0.0 -0.0
0.355 0.366 1.259 1.264 1.220 1.160 2.514 2.551 2.625 -0.579 2.753 2.805 2.008 1.180
a See footnote a of Table I1 for comments on the symbols. Table I1 also has the transition frequencies Efk for comparison with (a& here. bx10-6.cxlo-~o.
0.05
0.00
0.10
0.15
0.20
-
t (PS)
1.0
a
1
d
0.3 I
SI
w
4
g
x
t:
8
-1.0
0.2
Q
x
t:
d
n 2 a
i
-2.0 I 0.00
2
0.1
i 0.05
0.15
0.10
t
0.20
(PS)
Figure 3. Optimal control field for the same target wave function shown l = lo3,and T = 0.2 ps. This in Figure 1, with U T = 104,w1 = 2 X lt3f,
0.0 4.0
, 4.5
5.0
5.5
6.0
6.5
7.0
7.5
8.0
r (a.u.)
Figure 2. Optimal control results under UT = lo4,w1 = 2 X lt3, @= IO6, and T = 0.2 ps: (a) the optimal field; (b) the target wave function
probability density profilegiven by the dotted line and the achieved result under the optimal field indicated by the solid line.
TABLE 11: Parameters of the Components of the Optimal Field' Shown in Figure la and the Transition Frequencies E'& from the Ground Vibrational State of the Potential X'Z; to the First Seven Vibrational Eigenstates of the Potential A'Z: E'k
(cm-I)
(a31
3 14 766.0 4 15007.5 5 15 245.5 6 15 480.0
(cm-I)
14 766.0 15007.5 15 245.5 15 480.0
(ps-I)
(Balkb
-0.058 -0.051 -0.042 -0.033
(au)
(Bb)kb
@c)k
0.020 0.001 0.008 0.002
(au)
(&)k
(eb)k
(au) (rad) (rad) 0 14020.5 14020.5 -0.033 0.112 0.112 -0.0 0.359 0.359 1 14272.5 14 272.5 -0.051 0.018 0.018 -0.0 1.290 1.291 2 14521.0 14521.0 -0.059 0.064 0.065 -0.0 1.189 1.187
k
0.020 0.001 0.008 0.002
-0.0 -0.0 -0.0 -0.0
2.525 2.693 2.764 1.703
2.525 2.716 2.765 1.677
( a r ) field ~ frequencies; ( a k ) R , shape parameters; (Ea)&and (&)a amplitude factors; (ea)& and (ob)&, phases; (Ec)&,constant DC pulse amplitudes. X 10-6.
at the frequencies 14 638.9 and 14 676.9 cm-1. In the present case, the DC terms are small, but perhaps not negligible. Compared with the previous results in Figures l a and 2a, the intensity of the optimal field is increased due to a lower fluence weight. The evolution of the probability density (not shown here) is different from that of Figure IC. In the present case, the molecular excitation starts near 0.1 ps and then gradually approaches the target profile. The behavior of the optimal fields in the above examples is very typical and can be categorized by the ratio ut/& The first case is with a relatively small ratio, and the optimal field has a
field gives rise to the same excellent control results as the one shown in Figure lb.
TABLE IV: Parameters of the Component Fields of the Optimal Field' Shown in Figure 3a (ad1
k
(cm-9
0 1 2 3 4 5 6
14043.5 14 330.1 14638.9 14676.9 14954.7 15 212.9 15455.8
(DS-~) (au)
-19.67 -29.28 -39.54 -61.46 -31.20 -25.41 -20.84
0.032 0.039 0.102 0.232 0.024 0.018 0.008
(Bb)&b
(au)
0.950 0.783 4.624 6.978 0.914 0.359 0.172
(&)ab
(h)k
(ob),
(au)
(rad)
(rad)
0.002 -0.006 -0.036 0.094 -0,008 -0.003 -0.001
-1.058 -0.446 -1.117 1.665 -2.822 -2.575 -1.682
-0.126 -0.72 -1.498 0.302 -0.258 2.327 -0.920
a See footnote a of Table I1 for comments on the symbols. Table I1 also has the transition frequencies E'& for comparison with (a&)lhere.
bX10-6.
slowly varying envelope, e.g., Figures 1a and 2a; the second case is with a larger ratio, and the optimal field has a sharp envelope variation, e.g., Figure 3. However, the performance of the optimal fields upon the molecule is dependent not only upon the ratio but also on the absolute values of the weight factors, as demonstrated by the examples in Figures 1 and 2. In general, large values of the weight factors for the nontarget terms will result in a loss of control to some extent, as shown in Figure 2. Next, we consider the effect of the target time Ton the optimal control properties. A longer target time will allow for a lower field intensity to excite the wave function if the fluence penalty is significant. Equation 1I.lOshows that the intensity of an optimal field approaches a finite value as T goes to infinity. We have found through numerical calculations that if a target time is too short, e.g.,
