13736
J. Phys. Chem. 1995, 99, 13736-13747
Quantum Control of Molecular Dynamics: The Strong Response Regime Jeffrey L. Krause? Michael Messina, and Kent R. Wilson Department of Chemistry and Biochemistry, University of Califomia, San Diego, La Jolla, Califomia 92093-0339
YiJing Yan* Department of Chemistry, Hong Kong University of Science and Technology, Kowloon, Hong Kong Received: March 3, 1995; In Final Form: June 7, 1995@
We consider the quantum control of molecular dynamics in the strong response regime. The method used is based on a formulation in Liouville space and is valid for both the weak and strong response regimes. Several previous examples using this formulation in the weak response regime have shown that it is possible to overcome the natural tendency of wave packets to delocalize as they evolve on Bom-Oppenheimer potential surfaces. In this work we demonstrate that the weak field solutions for the optimal electric field can be surprisingly robust, in the sense that, by increasing their intensity without altering their form, they can be extended well beyond the first-order, perturbative limit to the regime in which significant population transfer occurs, while retaining a high degree of control. Further, since the weak field solutions are robust with respect to field strength, they can be used as initial guesses in a procedure to find locally optimal strong response solutions using an iterative search scheme. As a numerical application of the theory, we demonstrate the control of the vibrational dynamics of nuclear wave packets on an electronically excited state of 12.
I. Introduction Theory and experiment have now demonstrated that the quantum dynamics of atoms and molecules can be controlled with We have recently implemented a method to predict the electric field that best drives a sample of matter to a selected target in phase (positiodmomentum) space at a selected time.'4.28-30 We applied this method to control of electronic wave packets in the hydrogen and of vibrational wave packets in the iodine molecule and the sodium dimer.29*30An experiment based on our theoretical predictions has now been successfully completed.15$16-32 While the theory that forms the basis of our work was formulated in both the weak and strong response regimes, our theoretical and experimental applications to date have focused on the weak response case. By weak response, we refer to the regime in which the system responds to light with a simple power law dependence as the intensity of the light varies.2 This regime does not necessarily imply the use of weak electric fields. In fact, most spectroscopic experiments fall into this category. Typically, as the intensity increases, the signal responds linearly with the intensity over a fairly broad intensity range. However, second-harmonicgeneration, for example, can also be considered to be a weak response experiment, over the intensity range in which the conversion efficiency scales as the square of the intensity. In contrast, the response of a system to light in the strong response regime does not obey a simple functional form. Optical experiments in the saturation regime or experiments such as NMR, in which large population transfers can occur, are examples of the strong response regime. The advantage of the weak response regime is its simplicity. Perturbation theory is valid in this regime, and one's physical intuition is often correct. However, the strong response regime has the advantage that larger population transfers are possible, Permanent address: University of Florida, Quantum Theory Project, P.O. Box 118435, Gainesville, FL 326114435, * To whom correspondence should be addressed. Abstract published in Advance ACS Absrructs, August 1, 1995. @
OO22-3654/95/2099- 13736$09.00/0
and the additional complexity can allow richer dynamics to be incorporated into the control phenomenology. In the weak field limit of conventional laser spectroscopy, only a small percentage of the population can be controlled. In addition, as pointed out by Brumer and S h a p i r ~true , ~ ~control of asymptotic product distributions in the weak response regime with a single laser pulse is impossible. The system responds to the pulse simply as it would to a series of CW lasers with the same frequency spectrum. So, for example, in the weak response limit, any two laser pulses with the same frequency spectrum produce the same product distribution, regardless of how those frequencies are correlated in time. In the strong response regime this restriction no longer applies. For a theory of the quantum control of chemical reactions to be relevant to actually choosing products, the total asymptotic yield must be considered. However, photons are expensive, and so the number of photons required to produce a given outcome is certainly of relevance to the experimental design. In this work we illustrate in a unified fashion how the control of vibrational dynamics of evolving quantum nuclear wave packets varies in both the weak and strong response regimes. Further, we illustrate that the globally optimal electric field computed in the weak response limit can give excellent results when scaled into the strong response regime. The surprising robustness of the weak field solution with respect to the intensity of the electric field can be exploited by using it as a starting point for an iterative optimization scheme. We use the weak response solution as an initial guess in a primitive version of a gradient scheme similar to that used by Rabitz and c o - ~ o r k e r s ~ ~ and Tannor, Rice, and KosloffJ for obtaining the locally optimal field in the strong response regime. In Section 1I.A we review the theory of quantum control in the Liouville representation of quantum mechanics. In Section 1I.B we show how the formalism can be simplified in the weak response limit. Section I11 specializes the discussion to twostate systems in the rotating-wave approximation, and in Section IV the equations are simplified to the Schrodinger representation.
0 1995 American Chemical Society
J. Phys. Chem., Vol. 99, No. 37, 1995 13737
Quantum Control of Molecular Dynamics In Section V we define various types of targets and measures of success in reaching these targets. In Section VI we present example strong response computational results for control of vibrational dynamics on an excited state of I2 and compare the results to those obtained in the weak response limit. Finally, in Section VI1 we discuss our results and conclude.
11. Control Formalism in the Liouville Representation A. General Formulation. In this section we sketch a general formulation of quantum control in the Liouville representation. A more detailed derivation is presented in refs 28-30. As we will illustrate below, the Liouville formulation reduces to the conventional Schrodinger representation for a quantum system initially prepared in a pure state. The advantage of the Liouville representation is that mixed states, such as a thermal distribution of initial states, can be propagated and that environmental effects, such as a solvent or a thermal bath, can be incorporated easily into the dynamics. The Liouville representation also offers a convenient starting point for approximate c l a ~ s i c a land ’ ~ semiclassical3’ ~ ~ ~ ~ ~ ~ implementations of the theory. Consider a quantum molecular system evolving under the influence of the following Hamiltonian,
HT(?)= HM- DE(?)
(1)
In this equation, HM is independent of time and describes all of the internal degrees of freedom of the material system. The material system is coupled to the field, E(?), by the dipole operator, D. We assume that the system begins in equilibrium at time to. The system density matrix at time z is then given by
e(z>= 4%)@(to)
position, momentum, and time. One way to do this is via the application of a second light pulse that opens a “window” in phase space and time during which the system can be excited to a higher-lying excited state38or back to the ground state.39 Photon emission or ionization40 from the higher-lying excited state or stimulated emission or transient absorption from the excited state back to the ground state gives an experimental observable that can be used to infer the degree of control that has been achieved. Another possible probe is ultrafast diffraction, using X - r a y ~ ~ or I - ~electron^.^^,^-^^ ~ For simplicity, we set no bias with respect to any particular form of the control field E(?) and simply specify that the total incident field energy must be finite and non-zero. In this case, the control functional J(tf) can be written as
(4) where A is a Lagrange multiplier that constrains the total energy of the field. Other constraints or penalty terms could also be added that, for example, govem various expectation values or that force the system to follow a prescribed path through phase space.4-8A9-5 1 To find the optimal fields that lead to an extrema in the yield, we perform a functional variation of J with respect to E(?) and set the first-order variation equal to zero, 0 = &I =( dA(tf) ?,1 ) Jtfddt to &(z) E(Z)
(5)
Here, dA(tf), the first-order variation in the target yield with respect to the control field, is28
dA(tf) = JI:’dt
&(t)
K(t;t,)
(6)
(2) with K(z;tf),the control kernel, given by
where @z,to) is the Green function or Liouville space propagator, which depends on the electric field E ( t ) through the total Hamiltonian HT in eq 1. The goal of quantum control is to find the field that best drives the system to a desired target at a specified time.* Formally we can represent any specific target by an operator, A, and the goal is to find the electric field that maximizes the target yield, or the expectation value,
A(?,) = Tr[&(t,)l
i K(z;tf) = E Tr[&z;t,)@~(z)] = h
In this equation, e(t)is the density matrix at time z, as given by eq 2, while A(t;tf) is defined as
(3)
at the target time tf, nsubject to various constraints. In this work, the target operator A is chosen to be a projection operator onto a minimum uncegainty Gaussian distribution in phase space, but in principle A can define any outcome that is physically observable. In this work, we concentrate on targets defined by a projection onto a desired distribution in phase space on a specified potential energy surface at a specified time. That is, we choose a mean position and momentum about which we wish the system to be localized. For example, in a diatomic molecule, we choose a mean bond length and a mean relative momentum for the two atoms. To be more explicit, we choose a phase space distribution, that is a minimum uncertainty Gaussian. In contrast to the control of asymptotic final states, the system, in this case, has maximum overlap with the target only at the target time. Such targets might be useful intermediate steps in the efficient quantum control of chemical reactions by preparing or “preconditioning” the system for subsequent interactions with light or matter, for example a second light pulse that locks in the final asymptotic state distributions. Detection of the controlled system requires a probe that is sensitive to the probability to find the system as a function of
Tr[&z;t,)De(t) - e(t)D&z;tf)] (7)
A(z;t,) = d&(tf,t)= 4z,t,)A
(8)
and can be considered as the target propagated backward in time from the target time tf to time t. LZJ is the dipol? commutator, defined by its action on an arbitrary observable 0 as52
@6=06 - 60
(9)
where, as previously, D is the dipole operator. Setting 6J(tf) = 0 leads to the following equation for the locally optimal fields,28 E(t)
= K(z;t,)/A
(10)
Note that since the control kemel depends implicitly on the control field, eq 10 must be solved iteratively. We can see that the operational content of eq 10 is quite simple. We guess an initial control field d0)(z)and then calculate the control kemel associated with this field as a forward propagation of the density operator to time z and then a backward propagation of the target from tf to z. Then the trace is performed over all space. The resultant control kemel gives a new field, &)(z), that is the initial
13738 J. Phys. Chem., Vol. 99, No. 37, 1995 field for a subsequent iteration. This procedure is repeated until convergence is attained. The direct iterative solution of eq 10 for the optimal control field is ill-conditioned. We have found that this is true even with a very good initial guess for the field and that upon each iteration the control objective was not guaranteed to be optimized. As other workers have demonstrated, however, more stable methods for searching for the local minima do exi~t.~~' For example, the first derivative of the functional J(tf) with respect to E ( Z ) can be used as the input to a gradient ~ e a r c h ,where ~ . ~ ~the functional derivative of J(tf) is given by
In this case, the optimal field is determined as
and a is a small constant whose value is determined empirically to enhance the convergence. Other, more powerful methods for finding the optimal control fields, such as conjugate gradient methods and the Krotov m e t h ~ d , ~have ~,~ also ~ .been ~ used. We have found that the primitive scheme outlined in eq 12 is adequate when we have a good initial guess for the field. B. Weak Response Regime. The formulas in the preceding section are valid for both the strong response and the weak response regimes. However, in the weak response or perturbative regime they can be considerably simplified by expanding the target in a perturbation series about the control field. As shown previously, the lowest-order, nontrivial and nonvanishing term in this series is the second-order term, AQ). Thus, eq 3 can be rewritten as28
= Tr[&'2'(t,)] where is the density matrix to second order in the control field and
Krause et al.
Substituting eq 17 into eq 4 and performing the functional variation with respect to ~(t), we obtain
Setting 6J = 0 leads to the following equation for the optimal fields in the weak response limit,28
Equation 19 can be discretized onto a numerical grid and solved as an eigenequation in which the eigenvectors E ( Z ) are the optimal fields and the eigenvalues are the target yields with' respect to unit incident field energy,
The eigenvector associated with the largest eigenvalue is therefore the globally optimal field in the weak response limit. Thus we see that, in the weak response regime, a specially constructed second-order response function gives us directly (Le. without iteration) the globally optimal field for a given target, while the strong response regime requires an iterative solution to determine a locally optimal field. 111. Control Formalism for Two-Surface Systems
In this section we apply the general formulation derived in the previous section to the control of the molecular dynamics of systems involving two electronic potential energy surfaces. Further, for computational efficiency, we implement the rotatingwave approximation (RWA) in both the strong and the weak response regimes. We start with the electronically adiabatic description for the material, where the material Hamiltonian can be written as
= f ' d t S'dt' M(t, - t,t-r') E(Z) ~ ( r ' )(14) fo
fo
and the dipole operator as
In this equation, M, the second-order material response function, is given by
Notice in eq 15 that the Green function & is a function only of a single time argument (or time interval), rather than the double time argument of the Green functions in eq 8. These single-time-argumentGreen functions correspond to the material Hamiltonian HMin the absence of the extemal field in eq 1 and not to the full, time-dependent Hamiltonian, HT. Therefore, the material response function M,the control kemel in the weak response regime, depends only on the material properties and the target but not, as in the strong response regime, explicitly on the field. This allows us to find a direct, noniterative method of solution for the optimal fields. By introducing a symmetrized version of the M function as
d(t,t') = k?(t',z) we can recast eq 14 as28,29
M(f,-t,t-t'),
for t 1 t' (16)
The notations lg) and le) refer to the electronic degrees of freedom on the ground and excited states, respectively. The above equations are equivalent to the following matrix forms in the two-state electronic basis of lg) and le),
In these equations, Hg and He are the adiabatic Hamiltonians which govem the material nuclear dynamics on the ground and excited electronic states in the absence of extemal fields, p is the electronic transition dipole moment which couples the two states to the extemal electric field, and 000 is the purely electronic transition frequency, which is assumed to be large compared with the characteristic frequencies of the nuclear dynamics govemed by the Hamiltonians Hg and He. We will consider the situation in which the system is initially a steady state on the ground electronic surface, Ig),
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J. Phys. Chem., Vol. 99, No. 37, 1995 13739
and the target is on the excited electronic state, le),
and
For simplicity, we consider control fields that resonantly excite the material system. Therefore, we can represent the control field c(t) in terms of a single optical reference frequency Q, and a slowly varying complex amplitude E(t),
Note that D- = (D+)" and K- = K$ In eq 33, @ I ( t ) and Al(t;tf)are expressed in the interaction picture, so their dynamics are governed by the following interaction Hamiltonian that is obtained by substitutingeq 31 into eq 29 and invoking the RWA,
c(t) = E(t)e-iRr'
+ E*(t)eiRr'
(26)
This representation is consistent with the separation of time scales for electronic and nuclear dynamics. The reference frequency Q, can be chosen as any frequency within the material absorption band, such that
AS2, = S2, - om*Cwm
(27)
A. Strong Response Regime. For computational purposes, it is convenient to remove the highly oscillatory factor due to the purely electronic contribution WOO. To do this, we consider an interaction picture via the rotating frame defined by the unitary operator,
n! this picture, the density matrix @(r)and the target operator A(z;tf) in the contrcl kemel (eq 7),are transformed to @ 1 ( t ) = Uf(z)@(z)U(t) and Al(z;tf)= Uf(z)A(t;tf)U(t), and their dynamics are governed by eqs 2 and 8, respectively, with the Hamiltonian HT replaced by the following interaction Hamiltonian,
Here, the molecular adiabatic Hamiltonian remains unchanged (that is, UtHoU = Ho), while the electric dipole operator assumes the form
Using the form of c(t) in eq 26, the electric dipole interaction, Dl(t) c(t), can be written as
In this paper we consider only one-photon resonant control processes, in which the second term on the right-hand side (rhs) of eq 31 can be neglected, since it contributes only to higherorder resonant processes. In the interaction representation, the strong field control kemel is of the same form as eq 7, with all quantities on the rhs being replaced by their counterparts in the interaction picture. Substituting eq 30 into eq 8, we obtain the following expression:
+
K(t;t,) = K+(z;rf)e-iwWr K-(z;tf)eiwWt with
(32)
Substituting eqs 32 and 26 into eq 10 and neglecting the highly oscillatory terms (RWA), we obtain the control equation for EW,
E(z) = K+(z;t,)eiARrr/A
(36)
This equation is one of the main results of this paper. It is valid when there is a separation of time scales between the electronic and nuclear dynamics in a two-surface molecular system. The molecular dynamics involved in the control equation for the complex slowly varying field amplitude E ( t ) (eq 36) is governed by the RWA Hamiltonian in eq 35, which can be numerically solved more efficiently than the control equation for the real field in eq 10. B. Weak Response Regime. Here we consider how the RWA affects the control equations in the weak response regime for a system consisting of two electronic surfaces. The Hamiltonian for this case is given by eqs 21 and 23. We assume, for simplicity, that the target is specified on the excited state and that the wave function on the ground state is uncontrolled. A target of this form is given by eq 25. Substituting eqs 21,23, and 25 into eq 15 leads to the following expression for the material response function,
Here, Me is the excited state material response function and C.C. denotes the complex conjugate of the first term on the righthand side of the equation. The excited state material response function is given by
In eq 38, &e and Ggare matrix elements of the molecular Green function. They are Liouville space oper!tors and are defined by their action on an arbitrary operator 0 as
@mn,m,n, is the dipole operator in the Liouville representation and is defined as
Once again, we consider control fields that resonantly excite the system, and represent the electric field in terms of a slowly varying complex amplitude E(t) and a reference frequency Q, as in eq 26. Substituting eqs 26 and 37 into eq 14 or 17 and invoking the RWA leads to the following expression for the yield,
13740 J. Phys. Chem., Vol. 99, No. 37, 1995
A'*'(tf) = s t f d rf'dr'i@(z,r') to
to
Krause et al.
E*(r) E(r')
(41)
Here, @ is the Hermite symmetrized material response function, defined as
i@(r,r')
[i@(r',r)l* Me(tf-t,t-z')e
the notation
eq 49 can be recast in the following form
iAQr(r-r'). 9
for t Ir' (42) which is related to the real symmetrized @ function of eq 16 via
~~(t,= r 'i@(r,d)e-i*r(r-r'' >
+
C.C.
(43)
Substituting eqs 26 and 43 into the general weak field control equation (eq lo), we obtain the eigenequation for the optimal slowly varying field E(t) within the weak response regime and the RWA,28.29
IV. Control Formalism in the Schrijdinger Representation In the last two sections, we presented the general Liouville space control formulations for both the strong and weak response regimes. These formulations can be implemented by a variety of numerical approaches, such as quantum, classical, and semiclassical dynamics. In this section, we specialize to the case of a two-surface molecular system, as defined in section I11 and express the control kemel K+ (eq 33) in the strong response regime and the response function Me (eq 38) in the weak response regime in the Schrodinger representation. The Schrodinger wave function description is valid when the complete Hamiltonian of the system is known. A. Strong Response Regime. For simplicity, we first assume that the initially stationary material system is in a pure state, vg(to), such as an eigenstate on the ground electronic surface. In this case, the system density matrix remains as a pure state within the Hilbert space dynamics. We have
The numerical implementation of eq 51 is canied out as follows: it begins with the Hilbert space propagation forward from q(ro)to Virf) using the coupled Hamiltonian HI(t) of eq 35; x(tf;tf) = AIv(tf)) is then evaluated, followed by the backward propagation to obtain X(z;rf) (eq 50) for all z with to 5 t 5 tf. Finally the transition dipole overlaps between and W are calculated to obtain the desired control kernel (eq 51). If the system is initially in thermal equilibrium (a mixed state), eq 45 should be replaced by a linear combination of the occupied states weighted by proper Boltzmann factors. The resulting control kemel should be replaced by the same linear combination of the individual contributions from the initially occupied states. Equations 50 and 51 have been derived previously by several workers for a two-surface system in the Schrodinger representation, using a somewhat different f o r m a l i ~ m . ~ . ' ' , ~In~the ,~~-~~ original derivation, the Schrodinger equation is included explicitly in the control functional J , and the wave function % is introduced as a dynamical Lagrange multiplier. In principle, this procedure is only justified within the exact Schrodinger dynamics. In our formulation the equations of motion are not included explicitly as dynamical constraints. Although the final formulations are identical, our approach allows, in principle, a hierarchy of approximate methods that can be employed when exact solutions are difficult or impossible to obtain due to the complexity of the problem. Another advantage of our approach is that we are able to obtain a variety of control formulations in terms of the Schrodinger wave function dynamics. Let us consider the case when the target is a pure state = IN44 and
x
with where
In this case, we obtain the following control kemel: In eq 46, G(t,b)is the Schrodinger Hilbert space Green function, defined by the RWA Hamiltonian HI of eq 35 for t 2 t' as (47)
where
The numerical implementation of eq 54 begins with the backward propagation of the target wave function, from the target time &) to the initial time $(to); at this moment, v(tO;tf) = e(to)#(to;tf)is evaluated, followed by the forward Hilbert space propagation (eq 5 3 , and finally the transition dipole overlaps are evaluated to obtain the desired control kemel. This approach is particularly useful when the system is initially in a mixed state. Using the approach of eq 51 involves multiple forward-backward Hilbert space propagations for each individual thermally occupied level, and the final control kemel is
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J. Phys. Chem., Vol. 99, No. 37, 1995 13741
the Boltzmann-weighted linear combination of the individual contributions. Equation 54,in this case, however, requires only a single backward-forward propagation to obtain the control kernel for a given field. There is no need to perform the Hilbert space propagation for each individual Boltzmann thermally occupied pure state. Now we consider the case where both the initial system and the final target are pure states. This is the general situation in the theoretical study of optimal control, since, within the Hilbert space dynamics, a pure state will remain a pure state. It is clear that in this case we can use either eq 51 or 54 for evaluating the control kernel. Moreover, we can easily obtain a third approach to the Hilbert space dynamical computation of control of the kernel, given by
However, we also note that the control objective, such as the target yield A(tf) (in eq 3), or the control functional J(tf) in eq 4 may need to be modified for different control targets. In the following we consider several types of control targets and discuss whether the target yield A(tf) of eq 3 reaches its global maximum, if the system density matrix at !he target time is coincident with the target operator, e(tf) = A. Let us :tart with a pure state control target in which the target operator A can be defined as a projection operator:
In this case, we have Trfi’ = T r f i = 1
i
K+(t;tf) = ~[C(4g(t;t,)IrUIlYe(z))- ~*(1Vg(z>Ir~I4e(z;tf))I (56) with
(64)
We can further express the density matrix at time tf by its own diagonal representation
c = ( w g ~ ~ o ~ 1 4 g ~ ~ o ; ~ f ~ > (57)
Q(tf) = Cc,iqiCOXqjCff)I
(65)
1
Here, W(z), given by the forward propagation in eq 46,is the system wave function at time z, while $(z), given by eq 53, is the target wave function backward propagated from ff to time z. In this approach, the forward q propagation and the backward 4 propagation are performed independently. This is to be contrasted with eq 51 or 54,where the forward and the backward propagations must be carried out in proper order (forward fiist and backward later in eq 5 1 and the reversed order in eq 54)and the wave function for later dynamics depends on the previous propagation. Therefore, the accumulated numerical error is expected to be smaller in the implementation of eq 56, compared with those in eqs 51 and 54. B. Weak Response Regime. In the weak response limit, the equations in the Schrodinger representation can be considerably simplified. Consider the excited state control target, as shown in eq 25, and assume that the system is in a pure state. The target yield A”(tf) (eq 13) in the weak response limit then reduces to
= Tr[fi&’(rf)] = (q~’(ff)IfiIq~l’(tf)) (58) Here, the time-evolved wave packet on the excited surface, y(’)(t), can be written in first-order perturbation theory as a convolution of the field with the time-evolved wave packet in the absence of the field. Using the RWA Hamiltonian in eq 35 and omitting the trivial phase factor, we obtain
qkl)(t)= l rto d t e-iARr‘E(t)q~(t-z)
(59)
with Tr
= &(Vi(tf)lq,(ff)) = i
A(tf) = Ccj(4IVi(t,))(qj(ff)I4) 5 i
~cj(4I4)(qj(ff)Iqi(ff)) = 1 (67) i
This is the Schwartz inequality, and the equal sign holds when 4 = Vi(tf) = q(tf) and is the upper bound for the control target yield in the pure state case. In a molecular system with two electronic surfaces as in section 111, the Hilbert space wave vector 14) has the form
14) =
where, for a pure (or projection) target state, we have = I4mX4n I
(70)
(60)
(61)
The material response function for excited state control, Me of eq 38, then reduces to a particularly simple form,29 Me(tf-z,r-z’) (Ill~> V. Some Comment on Targets and Control Yields
[
which corresponds to the form of target operator
fimn
Here, we assumed that the system is initially in an eigenstate on the ground electronic state, Iqg(to)) = lv), with q v g ( t o )= ) Hg14= % I 4
(66)
A diagonal representation of e(ff)(eq 65) can always be found, since e(@) is a Hermitian operator. The target yield A(tf) in eq 3 now reduces to
where
I,&~)) = ti- Ie-i(He-~Nfi. IulVg(~0))l
=1 I
Equation 69 means that control can be specified not only for the population and shape of the wave packet on each electronic surface but also for the electronic phase between the wave functions on the two surfaces. The most general target in a two-surface system can be defined via eq 69 without imposing the pure state relation of eq 70. For example, we can choose the control target as
(62)
As we mentioned previously, the target operator f can in principle specify any outcome that is a function of observables.
In this case, the functional forms of phase space wave packets on both surfaces, as well as their relative populations,
N , = Tr A,,,
with n = g or e
(72)
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13742 J. Phys. Chem., Vol. 99, No. 37, I995
+
and N, Ne = 1, are to be controlled. However, the electronic coherence is totally free. Such a control target in general represents a mixed state, since Tr = Tr[A;, INi N: I1. We can show that the target yield (eq 3), in this case, satisfies the following inequality:
A:
+
+
Here, the Tr denotes the trace over the nuclear degrees of freedom and
P, = Tr Qnn(rf), with n = g or e
(74)
+
and P, P, = 1. The equal sign in eq 73Polds oqly in the case when both the nuclear control targets A,, and A,, are in pure states and they are proportional to egg(tf) and Qee(tf), respectively. Even in such an ideal pure nuclear state control case, the target yield has the upper bound A,,(tf) = Ne, in which Pg = 0 and P, = 1, if N, < Ne, or Ama,(tf)= N,, in which P, = 1 and P, = 0, if N, > Ne. In other words, the target yield defined by eq 73 is not sufficient to characterize the control objective, and additional constraint terms must be included in the control fulnctional J(tf)(cf eq 4). The target A1 (eq 71) defines the simultaneous control of both the relative populations as well as the specific functional forms of the evolving nuclear wave packet in phase space on both surfaces. Less demanding yet still physically motivated objectives can also be chosen. In particular, we can choose a target that does not attempt to control the functional form of the ground electronic state wave packet but only its relative population. In this case, we define the target as
(75) and rewrite A(tf) as
Again, the yield defined by eq 76 is not sufficient to characterize the control objective, and additional constraint(s) would be required. The specific case that we will consider is the control of nuclear vibrational dynamics on one of the two electronic BornOppenheimer states which are radiatively coupled by the control field. For ground state control, we can define the ground state projection target by eqs 63 and 68 with 16,) = 0. However, in the weak response regime, where the controlled system density matrix depends quadratically on the extemal field, we can only control a Raman coherent hole, involving at least one initially occupied rovibrational level in the ground electronic state. In order to compare control in both strong and weak response regimes, y e will focus on the excited state 5ontrol target by choosing A with the same form as eq 25 with Aee = I@e)($el, or
The control target of eq 77, in the strong response regime, defines only the functional form on the excited surface, unless the constraint that Tr[~,,(tf)]remains constant is further included in the control functional J . To determine the degree to which an optimal field reaches the objective defined by the target operator, we define an achievement a(tf)as29
Here, ee(t)= @,,(t) is the density matrix for the electronic excited state at time t. The achievement in eq 78 is constructed to range from 0 (no control) to 1 (perfect control). The achievement can also be contructed in Schrodinger space by expressing the density matrix at the target time in the Schrodinger representation and performing the trace in eq 78. Some further discussion of the achievement function at the target time, a(tf), is merited here. In particular, we would like to clearly enunciate the relation between the measure of achievement, a(tf), and the optimal yield per energy, A given by eq 20, in the weak response regime. For an optimal weak control field, c(t), we have the relation a(tf) = {AZ/[Tr A, Tr ~ ~ ( t f ) ] }where ” ~ , Z = Jdt c 2 ( t )is the incident field intensity. In the weak response regime, the c(t) giving the largest value of the target yield, Max[A], does not necessarily correspond to the largest value of the achievement at the target time, Max[a(tf)]. It is not too surprising that A is proportional to the maximum yield A@) in the weak response regime, which is only approximately, but not exactly, coincident with the maximum value of the achievement function at the target time, a(rf). We have found, in many cases, that the globally optimal control field associated with the maximum eigenvalue Max[A] (cf. eq 19) also gives a value of a(tf) close to its maximum of ~nity.*~,~O The achievement alone is not a completely satisfactory measure of the degree of control attained by a given laser field, because it does not take into account the amount of population created by the field nor the number of photons required to excite that population. To overcome this deficiency, we define a quantum efficiency, y , as the amount of population that overlaps the target, normalized by the intensity of the laser field, where y = oA(tf)/Z
This equation is defined with only single-photon interactions in mind. The parameter u is problem-dependent and is related to the response of the system to light in the perturbative limit. In the scenario considered in this work, we know from firstorder, time-dependent perturbation theory that the population excited by the laser scales linearly with the laser intensity. We define u as the initial (weak response) slope of a plot of the population produced on the excited state versus the intensity of the electric field, @e
o = lim1-0
This is a special pure state case, described by eqs 63 and 68 with I&) = 0. This target, in the weak response regime, attempts to control only the functional form of the evolving nuclear wave packet on the excited electronic state, independent of both the functional form of the wave packet on the ground state and the relative populations on the two states. In the strong response regime, both the population and the functional form of the wave packet in principle can be cpntrolled. However, the target yield is of the form A(tf) = Tr[Aeeeee(tf)]in this case.
(79)
dz
Here, P , Tr ee(tf)is the population that has been promoted to the excited state at the target time. We find that there is a low-intensity regime where the population that has been promoted to the excited state scales linearly with the laser intensity.
VI. Numerical Results As an example of the implementation of the formalism presented in the previous sections, consider the scenario depicted
J. Phys. Chem., Vol. 99, No. 37, 1995 13743
Quantum Control of Molecular Dynamics
al
1-
c
m
G
0.8
C
0.6
a 0
C
.-0m
-
c.
Initial State
0.4
n 0.2 3
0
a
R (pm) Figure 1. Schematic diagram of the 12 molecular reflectron. The molecule begins on the ground X electronic state and is excited by the field e(?) to the excited B electronic state. The target, on the B state, is a minimum uncertainty Gaussian distribution with a negative momentum; that is, the two atoms are approaching one another.
-
19.51
8
.
1
,
.
v
,
8
,
I
,
. . 1
.
r
, I
Time (fs) Figure 2. Globally optimal field in the weak response limit for the molecular reflectron in Figure 1 . The field is depicted in a Wigner
time-frequency representation. in Figure 1 for the control of the vibrational dynamics of the iodine molecule. The molecule begins in its ground X state and is excited by an electric field ~ ( to t )the electronically excited B state. For simplicity, we assume that the molecule begins in its ground vibrational level and ignore the effects of molecular rotation. The objective is to find the electric field ~ ( tthat ) drives the system to maximum overlap with a target distribution in phase space at the target time. The target in this case is given by eq 77; that is, the target is a pure state, and we do not attempt to control the relative populations. In particular, we choose as the target a minimum uncertainty Gaussian distribution in phase space,
1.o
1.5
Intensity (W/cm*xlo'* Figure 3. Population created on the B state of 12 by E&), the amplitude-scaled globally optimal weak field, as a function of the intensity. The inset shows a magnification of the curve at low intensity, in the weak response regime, where it is linear.
self-interference normally present in this representation. We use a Gaussian window function of the form W(z) = exp ( - z?/(). Here zCis a correlation time that is chosen to be on the order of the pulse width. As can be seen in Figure 2, the globally optimal field is simple and smooth, and its dominant characteristic is a sizable negative chirp, visible as the overall negative slope of the contours in the figure. The achievement, a in eq 78, for the globally optimal field is 0.97, indicating that the wave packet excited by the field has a nearly perfect overlap with the target at the target time. We emphasize that we did not enforce a priori that the optimal field have the simple chirped form in Figure 2. The functional form of the field in the allotted interval is totally unconstrained by the formalism. We now investigate the robustness of the effect of the weak response globally optimal field with respect to intensity. To do this we scale the amplitude of the globally optimal weak field so that its maximum intensity can be tuned from the weak response through the strong response regime. We call this the amplitude-scaled weak field: ew(t). The dipole moment p is assumed to be independent of bond length, with a magnitude of 1.O D, the approximate value of the X B transition dipole moment in the Frank-Condon regions6' Using C W ( ~ ) , we solve for the exact dynamics of the coupled-surface system in eq 46. The populations on the ground and excited surfaces are obtained as
-
and where wqqis the variance of the wave packet in position, X is the expectation value of the position of the wave packet, and p is the expectation value of the momentum of the wave packet. For the case illustrated in Figure 1, X = 372 pm and jj is chosen such that p2/2m = 403 cm-' (0.05 eV), where m is the reduced mass of 12, p 0 (that is, the momenta of the atoms are directed toward the center of mass), and wqq= W 2 m o . The target time tf, the maximum time interval within which ~ ( tmay ) operate, is chosen to be 550 fs. We have termed this case the molecular r e f l e c t r ~ nbecause ,~~ the wave packet must reflect from the outer portion of the potential to reach the target region with a negative momentum. The globally optimal field, in the weak response limit, for the reflectron is shown in Figure 2, in a Wigner representation Fw(z,o), where
Fw(t,w) = 2 Re h w d r e*(t
+ z/2)e(t - z/2)e-'"rW(t)
(82)
W(z) is a window function included for clarity to remove the
+
and the normalization is chosen such that P,(t) P,(t) = 1. P&) and Pe(t) are the populations that are actually produced with the light field and should be distinguished from N g and Ne, which are target parameters (see eq 7 2 ) . Note that in this model the only coupling between the two electronic surfaces is the radiative coupling due to pew(& so that after the field is over, the populations on the ground and excited surfaces are constant for all time. In Figure 3 we show the final population on the excited surface as a function of the peak intensity. At low intensity, as expected, the excited state population scales linearly with the intensity. This is clearly shown by the inset in the figure. At higher intensities, not shown, the population begins to saturate. This is a signature of the onset of the strong response regime. As is evident in the figure, at the intensities examined, up to 99% of the population of the ground electronic state can be
13744 J. Phys. Chem., Vol. 99, No. 37, 1995
intensity (W/cm2x10'2 Figure 4. Achievement a (eq 77) at the target time, rf, as a function of intensity, for the iteratively improved field, E [ ( ? ) (solid squares), the Gaussian constrained field CG(Z) (open squares), and the amplitudescaled globally optimal weak field E W ( ? ) (solid circles).
transferred to the excited state by simply scaling the amplitude of the globally optimal weak field into the strong response regime. At higher intensities, not shown, the population on the excited state starts to decrease (with a concomitant increase of the population on the ground state). At intensities above about 2 x 10l2 W/cm2, Rabi flopping (that is, rapid transfer of population between the ground and excited states) begins to occur. We choose not to investigate this regime here because the validity of the two-state model is questionable, as additional effects, such as ionization and multiphoton processes, should be included in the model. The ability to excite a nontrivial fraction of the ground state population to the excited state does not prove that the vibrational dynamics on the excited state are still being controlled. To investigate this, we calculate the achievement a as a function of intensity and present the results in Figure 4, in which the line connecting the circles represents the achievement values for the amplitude-scaled weak field, c ~ ( t ) As . can be seen in the figure, even at an intensity of 1 x 10l2W/cm2, an intensity at which 95% of the population is transferred to the excited state, the achievement remains surprisingly high, 0.73, compared to 0.97, the achievement for the globally optimal field in the weak response limit. This is a strong indication that the weak response solution is quite robust with respect to intensity and that it should serve as an excellent initial guess for calculations in the strong response regime. As expected, as seen in Figure 4,the achievement does eventually drop to 0.53 as the intensity increases to 1.8 x 10l2 W/cm2. We should point out that intensities on the order of 10l2 W/cm2 are quite modest with current laser systems (which can reach on the order of 1020 W/cm2) and that such intensity can be achieved with conventional focused femtosecond sources. To analyze how the form of the optimal field changes as we increase the intensity, we next use the weak response globally optimal field E&) as the initial guess for the primitive gradient search in eq 12. As a way of isolating the effect of the intensity on the control, the total integrated intensity jrle(t)12,is set to a constant during the iterative process. We use the target operator given by eq 77 and calculate the achievement a. This procedure gives us a lower bound to the achievement attainable at a given intensity, starting with the form of the weak response globally optimal weak field. (Allowing the relative populations to change results in a somewhat higher achievement.) We also find that "adiabatically" continuing the optimal field into the strong response regime, that is, using the locally optimal field computed at the slightly lower intensity in the previous step for the initial guess, rather than the form of the weak response globally optimal field, leads to somewhat faster convergence but does not significantly alter the achievement or the form of the locally
Krause et al. optimal strong field. We refer to the iteratively improved, locally optimal field as cI(t). Figure 4 (solid line connecting solid squares) shows the achievement a as a function of intensity, for ~ ( tderived ) from an initial field consisting of the globally optimal field in the weak response limit with amplitude scaled such that the initial peak intensity of the field is 1 x lo9 W/cm2. Using that field as an initial guess, we find that typically about ten iterations are required to converge to q ( r ) . Also shown in Figure 4, as discussed above, is the achievement for cw(t), also as a function of the peak intensity. As can be seen in the figure, the achievement can be dramatically improved by using the iterative procedure. As the intensity increases, the form of e[(t)becomes more complicated. Figure 5 shows the Wigner representations of the iteratively improved optimal fields, cI(t), at moderate (6.2 x 10" W/cm2) and high (1.8 x 1OI2 W/cm2) intensities, in panels a and b, respectively. By 1.8 x 10l2 W/cm2, q(t) has departed considerably from the simple, smooth form. Once again, we emphasize that the optimal fields shown in Figure 5 are locally optimal fields, and we cannot prove that the achievements obtained with them are the maximum possible, Le. the globally optimal, at a given intensity. However, as seen in Figure 4, their achievement is quite high, and operationally, for an experiment, they should be more than adequate. It is instructive to compare the actual wave packets created at the target time via excitation by c ~ ( tto) the wave packets created via excitation with c ~ ( t ) Figure . 6 shows the Wigner representations of the phase space wave packets created at the target time, at an intensity of 1.8 x 10l2 W/cm2. The wave packet created by cW(t), shown in panel c of Figure 6, is considerably delocalized, and its mean momentum is -6.75 x kg m / s , while the mean momentum of the target is -3.97 x kg m/s. The combination of spatial delocalization and the shift in momentum leads to a rather small achievement of 0.53 for this field. In contrast, the phase space wave packet created by excitation with c ~ ( t )shown , in panel a of Figure 6, is somewhat more localized in the target region, and its mean kg m/s, a value rather momentum is at about -4.8 x closer to the mean momentum of the target. (Note that all these momenta are relatively small compared to the average momenta nearer the bottom of the well.) The achievement for this field is 0.78. Thus we see that iteratively optimizing the control field tends to create a phase space wave packet that is better localized spatially, with a momentum closer to the target value, than the wave packet created by cW(t). As discussed previously, the achievement alone is not a completely satisfactory measurement of the degree of control attained by a given laser field, because it does not consider the amount of population created by the field nor the number of photons required to excite this population. We now calculate the value of the quantum efficiency, y(tf), defined in eq 79. In order to calculate y(tf) we must first calculate the value of u (see eq 79) for this system. As discussed previously, u can be d e t e d n e d as the inverse of the slope of the line in the inset of Figure 3 (eq 80). In this case the value of 0 is 3.82 x 10l2 cm2/W. Figure 7 shows a plot of quantum efficiency, y(tf) from eq 79, as a function of the laser intensity, for both the amplitudescaled weak field cW(t) (solid circles) and the iteratively improved field cI(t) (solid squares). Both curves in the figure approach zero efficiency asymptotically as saturation occurs. The iteratively optimized control fields always have a greater efficiency than the scaled globally optimal fields, because they have higher achievements. Comparing the efficiencies in Figure 7 gives a better idea of the relative cost, in terms of population excited per photon, than comparing the achievements in Figure
J. Phys. Chem., Vol. 99, No. 37, 1995 13745
Quantum Control of Molecular Dynamics
19.5
n
n
fr
19.0
T
-6 E
u
18.5
c
g 18.0 E! W
l7m50
LL
100
200
300
400
500
Time(fs) n
n
19.5
n
w N
0
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'ZX
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5 Time(fs)
0
5
L
320
360
400
440
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* 0
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2 18.0 U
....,.........,.........,....,...LE
l7q50
LL
100 200 300 400 Time(fs)
500
320
360
400
440
19.5
n
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Figure 6. Wigner transform of the coordinate space wave packets created by (a) E&), the iteratively improved locally optimal field, (b) e&), the Gaussian constrained field, and (c) E W ( ~ ) the , amplitude-scaled globally optimal weak field. The dotted line in each panel showns the Wigner transform of the target. The intensity in all cases is 1.8 x IO1? W/cm*. The internuclear distance is given as R.
z 19.0
r
-
'E u >I
18.5
0
c
2 18.0 W E l7m50
100
200 300 400 Time (fs)
500
Figure 5. Wigner transform of (a) cl(f), the iteratively improved field, at a light intensity of 6.2 x 10" W/cm*, (b) E @ ) at a light intensity of 1.8 x loi2Wlcm2, (c) E&), the Gaussian constrained field, at a light intensity of 6.2 x 10" Wlcm*, and (d) E&) at a light intensity of 1.8 x 10l2 W/cm2.
4. Figure 7 does not support the viewpoint that strong fields
are necessarily the route to greater quantum efficiency. But, it should be remembered that there is no guarantee that the globally optimal strong field solution has been found. The falloff in quantum efficiency to reach the excited state shown in Figure 3 is the major cause of the falloff of efficiency with intensity.
Intensity (W/cm2 X I O ' ~ Figure 7. Efficiency y(tf) (eq 79) as a function of intensity for
~l(t),
the iteratively improved field, E G ( ~ ) the , Gaussian constrained field, and cw(t), the amplitude-scaled weak field.
Allowing the control field to have an unrestricted functional form in the optimization procedure is certainly the most general
13746 J. Phys. Chem., Vol. 99, No. 37, 1995
Krause et al.
TABLE 1: Parameters for the Optimized Constrained Gaussian Field for the Reflectron (See Eqs 85 and 86), which Maximizes the Value of the Achievement Function a optimal value parameter
r
optimal value EW 391 cm-'
wC
235 fs 18 580 cm-I
CI
- 1136 fs2
co
+ wep
I = 6.2 x 10" W/cm2
364 cm-I 253 fs 18 450 cm-' - 1303 fs2
EG
I = 1.8 x 10l2 W/cm2 347 cm-I 254 fs 18 480 cm-I - 1029 fs2
approach from the viewpoint of theory but might not be the most useful from the viewpoint of experiment. Can the optimal fields be chosen from a smaller subset of experimentally convenient functional forms29and still retain high achievement and efficiency? That is, the Wigner plots of the iteratively improved optimal fields shown in Figure 5 (specifically, panels 5a and 5b) all have a nearly Gaussian profile, with a negative linear chirp. From an experimental viewpoint, Gaussian pulses with linear chirps are fairly easy to approximately synthesize, so it is useful to attempt to optimize the control fields with the restriction. For ease of discussion we define the Fourier transform of 6&) as Q(o),which obeys the following functional form.
optimized field ~ ( t ) ) Finally . in panel b of Figure 6 we show the Wigner plot of the wave packet created by excitation with the optimized Gaussian constrained field at an intensity of 1.8 x 10l2 W/cm2. These results are encouraging, because they indicate that light fields that can be synthesized easily in the laboratory can in fact yield high achievements in the strong response regime.
VII. Discussion and Conclusions We have demonstrated by an example that the form of the globally optimal field found in the weak response regime may be effectively used to control the vibrational dynamics of wave packets well into the strong response regime. In this I2 calculation, for example at 0.5 x loT2 W/cm2, 85% of the total population can be transferred to the excited electronic state with an achievement of 0.88, just by scaling the amplitude of the globally optimal weak field into the strong response regime. This is an important result because it shows that the vibrational dynamics of a nontrivial proportion of the population of a quantum molecular system can be controlled using a relatively simple and robust light field. The optimal light fields computed in the weak response regime for this system, and used above, are generally found to be of a simple form that can be sufficiently closely synthesized in the laboratory, as has been demonstrated experimentally for the weak response regime. We also show that the achievement obtained using the amplitudescaled weak response globally optimal field can be improved in the strong response regime via an iterative solution of the control equations (cf. eqs 11 and 12). The value of the achievement function can thus be increased by as much as 30% over that obtained with the amplitude-scaled weak response globally optimal field. The fields obtained by this iterative search are generally more complicated than the weak response fields (Figure 5) and may not be as easily synthesized in the laboratory. To overcome this obstacle, we have further demonstrated that an experimentally convenient chirped Gaussian functional form for the optimal field can also lead to significant improvement in the achievements over that for the amplitude-scaled weak response globally optimal field. We feel that optimization within a simple class of experimentally accesible fields29 might be the best practical way to achieve high yields along with effective control. 15916,32
The phase function p(w) is expanded in a Taylor series as
+
+
q ( ~=)qo co(u - oc)
1 / 2 ~ 1( ~w
,)~
(86)
Here, po is the arbitrary mean phase, ocis the excess carrier frequency, co is the group delay coefficient, r is related to the fwhm of the pulse in frequency space, and CI is responsible for the linear chirp. We note that the true carrier frequency is wc weg. EO is the peak field amplitude. We will always optimize a field for a particular value of the peak intensity, so the amplitude of the field in eq 85 is not a variable parameter. Thus, the only parameters that need to be optimized are wc, CO, r, and cl, As an initial guess for the parameters, we use the best Gaussian fit to the globally optimal weak field, as listed in the first column of Table 1. The goal, then, is to maximize the achievement a with respect to the parameters in eq 85. To do this, we use Powells' method of multidimensional minimization.62 Figure 5c shows the Wigner representation of c G ( f ) with the parameters found via the optimization procedure described above, at a moderate intensity of 6.2 x 10" W/cm2. The achievement, a, obtained with this field is 0.88, compared to a value of 0.92 using the unrestricted optimization. At a higher intensity, 1.8 x 10l2W/cm2, we find a value for the achievement with the optimized CG(O) of 0.75, compared to 0.79 for the field obtained with the unrestricted method. The optimized Gaussian constrained field for this intensity is shown in Figure 5d. The optimal parameters for both fields are listed in the second and third columns of Table 1. The parameters in the second column are for an intensity of 6.2 x 10' I W/cm2, and the parameters in the third column are for an intensity of 1.8 x 10l2 W/cm2. The achievement for C G ( W ) as a function of intensity is indicated by the open squares in Figure 4,for several values of the intensity (note that these points are very close to the solid square points for the interatively optimized field ~ [ ( t ) )We . also show the values of the quantum efficiency y for the field E G ( ~ in ) Figure 7 (open squares) for several values of the intensity (note that these points are close to the solid square points for the iteratively
+
References and Notes (1) (2) (3) (4)
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