J. Phys. Chem. 1995, 99, 13731- 13735
13731
Unified Formulation for Control and Inversion of Molecular Dynamics Zi-Min Lu and Herschel Rabitz" Department of Chemistry, Princeton University, Princeton, New Jersey 08544-1009 Received: February 28, 1995; In Final Form: May 11, I995@
In this paper we present a unified approach for the control and inversion of molecular dynamics. The concept of molecular tracking ties the subjects to a common formulation. For molecular control the time-dependent track of an observable operator is imposed a priori and the control field is determined to meet the track. For extraction of the potential and/or dipole function the time-dependent track is observed laboratory data. A common three-step algorithm is presented to treat both seemingly unrelated problems. The essential features of the algorithm are presented in the paper.
1. Introduction This paper presents a unified formulation for treating the seemingly unrelated problems of control and inversion of molecular motion. In the case of molecular control,*the goal is to design a laser field capable of achieving a desired molecular objective, such as dissociation. The design process involves an inversion of the imposed physical objective, to find a portion of the Hamiltonian (e.g., the electric field e(t) appearing in the dipole interaction). In the second problem of inverting laboratory data,2 the goal is, once again, to extract a portion of the Hamiltonian (i.e., typically, a potential surface V(x) or dipole function ~ ( x ) ) Thus, . inverse dynamics is a conceptual point of commonality between the problems of control and inversion of molecular motion. To go further than the simple observation of commonality above, the problems of (1) optical field design for control and (2) the determination of potentials from laboratory data must be put on an equivalent mathematical footing. A simple principle enabling this connection is through the tracking of molecular m ~ t i o nwhich ,~ is schematically illustrated in Figure 1. Tracking most generally refers to following an observable trajectory O(t) = (0),which is the expectation value of a Hermitian operator 0. In the electric field e(t) design process for molecular control, the track O(t)would be specified a priori and inverted to yield the field. In a similar fashion, the potential surface V(x)could be obtained from an inversion of a laboratory observed track O(t). Perhaps the simplest conceptual case corresponds to 0 = x being the position operator in configuration space. Thus, for control design purposes, the goal is to find a control field c(t),such that the track follows an excursion over the potential surface, leading out the desired exit channel A or B in Figure 1. Using the same principle, the mean position might be available as laboratory data (e.g., through ultrafast X-ray or electron diffraction observations), and the track might be inverted to determine the underlying potential surface. In the latter case, it is physically attractive to think of the track as the excursion of a scout over the surface, and the interrogation of the scout, along its path, provides the sought-after information for inversion. Building on the concept of a physical track O(t) for control or inversion permits the construction of a common algorithm for these dual purposes. Subtle differences will certainly arise in this common algorithm for the two purposes, but the essential features are the same. The point of this paper is to explicitly @
Abstract published in Advance ACS Abstracts, August 15, 1995.
Figure 1. Schematic illustration of a wave packet track evolving on a potential surface. For control the goal is to find the field ~ ( tthat ) may steer the track out either product channel, A or B. For inversion the track is observed in the laboratory and the goal is to determine the potential in the regions scouted out by the track.
show that a common framework exists for the control and inversion of molecular motion. The full consequences of this algorithmic commonality remain to be seen, but it is suggested that the two areas of study could even be combined in a relayed sequence of experiments for potential inversion and control (i.e., control calls for knowledge of the potential surface, which in turn, could be obtained from the inversion of laboratory data). Some preliminary exploration of tracking for molecular control has already o ~ c u r r e d ,and ~ ? ~similarly, in the case of inversi~n.~?~ However, the precise parallel between these distinct areas was not evident in this earlier work. The goal is to present common algorithms between these two problems. Detailed illustrations for each application will be presented elsewhere. The organization of this paper is as follows. In section 2 tracking control and inversion are reviewed and formulated. Detailed comparisons and correspondences are given in section 3, while section 4 presents some concluding remarks.
2. Tracking of Molecular Motion Regardless of whether tracking is being utilized for molecular control or inversion, the input is a time series O(t),t > 0. In the case of control design, there is wide latitude in the choice of
Oo22-3654/95/2099- 13731 $09.00/0 0 1995 American Chemical Society
13732 J. Phys. Chem., Vol. 99, No. 37, 1995
Lu and Rabitz
the operator 0 and, hence, the time series. However, in the case of laboratory data for inversion, practical limitations can enter. Considering a data time series for inversion, it is natural to think in terms of ultrafast pump-probe experiments’ as a source of laboratory data. However, traditional and presently more plentiful high-quality spectroscopic data may also be manipulated into a synthetic track by performing a temporal Fourier transform, utilizing knowledge of transition intensities and spectral lines. In essence, the time variable acts as either (1) an organizer of the data for inversion or (2) an index of the sought-after track for control purposes. These notions will be expressed for the dual purposes below, in a fashion which draws as close a parallel between them as possible. Both algorithms in sections 2.1 and 2.2 have the same analogous three steps. 2.1. Tracking for Molecular Control. Molecular tracking to design control fields has many aspects, and a full elaboration of the subject will not be presented here. Rather, a particular perspective, distinct from the earlier work?.4 will be formulated, as it has the closest parallel to tracking for data inversion in section 2.2. Suppose that the control objectives are to achieve (Oj(r))at a target time T by following some prescribed tracking trajectories yj(t) to the latter end point. In the case of exact tracking we demand
(1)
(Oj(t))= (W(t)lOjlV(t))= Y j ( d
where Oj are operators associated with the physical observables, j = 1, 2, ...ND, and Ily(t)) is the molecular wave function satisfying the Schrodinger equation
(2)
+
Here HO = T V is the field-free Hamiltonian. According to the Heisenberg equation of motion, the observable (Oj(t)) satisfies d if#Oj(t))
= ([Oj,HoI)- d t ) ( [ O j , ~ l )
(3)
where [A, B] = AB - BA. Integrating this equation over time yields the following integral equation for the control field:
J‘K(t,t‘) ~ ( t ’ dr’ )
+
= g(t)
(8)
where
(9) g(t) =
C WJ)gj(0
(10)
j
Equation 8 is a Volterra equation of the second kind, which is the regularized version of the Volterra equation of the first kind in eq 4. If a is sufficiently large in magnitude, then the kernel in eq 8, K(t,t‘) ad(t - r‘), is diagonally dominant and thus well-conditioned. Only if a is close to 0 do we then go back to the ill-conditioned case. For this reason the kemel K(t,t’) ad(t - t‘)is called a stabilized kernel. In practice, an optimal a is chosen at each time step. We also notice that a superposition of all the dynamics up to the time t is used to determine the control field in eq 8. A key step in the minimization of eq 7 is the utilization of only the explicit appearance of the field. Although the kernel in eq 9 implicitly depends on the field, this aspect of the problem is treated as an issue of self-consistency in the next step below. Step A in the algorithm for control tracking consists of formally solving for the field from eq 8. That is, the stabilized kernel K ( t , l ) ad(t - t’) in eq 8 is inverted to yield the field in terms of the prescribed tracking information in g(t). This inversion step A is referred to as formal since K(t,t‘) and g(t) depend implicitly on the unknown field ~ ( t )through , the wave function. We know the wave function must satisfy Schrodinger’s equation:
+
+
+
where the formally inverted field c(t,W) from eq 8 is shown to have an implicit dependence on the wave function. Despite the nonlinearity implied through the field, eq 11 may be numerically integrated forward in time, to yield a quantum mechanical state Iq(t))consisted with the formally inverted field from eq 8. This solution process in eq 11 is step B in the algorithm. Once Ily(t)) is obtained, it may be substituted in eq 8 for one final inversion, as step C, to yield the control field
40. JrK(j,t’) Q’) dt‘ = gj(t)
(4)
where
Equation 4 is a Volterra equation of the first kind, encountered in many inverse problemse8 When the kemel in eq 4 is not well-conditioned, the solution is typically unstable. We can stabilize the solution by balancing the demands of tracking and the desire for a nonsingular control field. This can be done through minimizing the following cost function:
J = I IhrK(j,r’)E(t’) dt‘
- gj(r)l I:
+ a Jrc(r‘)*
dr‘
(7)
where the norm is with respect to the tracking sequence j = 1, 2, ...ND and a is a positive weight. Minimization of J with respect to c(t) leads to the following equation:
Thus, the tracking leads to a simple algorithm for the control design, which can be formulated as an inverse problem. For this reason, this approach to control designs is sometimes called inverse contr01.~ In the earlier work on molecular t r a ~ k i n g , ~ , ~ the three-step A, B, C algorithm above was applied directly to the Heisenberg equation in eq 3. Application of the algorithm to the alternate eq 4-6 is slightly more complicated, but has the advantage that the field ~ ( tat) time t draws on full knowledge of dynamical evolution up to that time. The performance of steps A, B, and C does not call for iteration? as in traditional optimal c ~ n t r o l . ’ . ’ ~ -Although ’~ optimal control and tracking both rely on the minimization of a cost function, the central distinction resides in the subtle differences between the minimization processes. In optimal control, both the implicit and explicit functional dependence on the field are simultaneously taken into account upon minimization of the cost function. In tracking, only the explicit field dependence is included for minimization, and the implicit dependence is treated through the self-consistency process of assuring that the Schrodinger equation is satisfied. Applications of tracking for control have been made to diatomic and triatomic molecules, including bond selective dissociation. Details of such applications are available
J. Phys. Chem., Vol. 99, No. 37, 1995 13733
Control and Inversion and Molecular Dynamics e l ~ e w h e r e . ~The . ~ formulation of inverse tracking above is somewhat different from that presented earlier, as it provides the closest parallel to tracking for Hamiltonian identification in section 2.2 below. 2.2. Tracking for Inversion of Laboratory Data. The goal in inversion of laboratory data is to extract the potential V(x) and/or the dipole function p(x) from a series of observed laboratory tracking data. (oj(t))= (~j(t)IoIqj(t)) = yj(t)
where the norm is over the data (Le. the index j and possibly the time up to t) and a is a positive weight factor. Minimization of the functional with respect to f i x ) yields
(12)
Here, a single observable operator 0 is assumed (multiple operators also can be treated), along with a set of distinct dynamical evolutions corresponding to different initial conditions and/or applied fields q ( t ) ,j = 1, 2, ...ND. To invert the track and extract the molecular potential energy surface V(x)and/or dipole function p(x),the Heisenberg equation (3) is also employed. Equation 3 can be recast into the form of a Fredholm integral equation of the first kind for the potential or the dipole
LKO’J)~~~) = gj
The regularized solution of eq 13 is achieved by minimizing the following fun~tional:~
(13)
For the potential
where
As with eqs 7 and 8, the minimization of eq 16 is only with respect to the explicit dependence onfix). Equation 17 is the regularized version of eq 13 and is known as a Fredholm equation of the second kind. Step A in the data inverse tracking procedure consists of formally solving eq 17 forfix). In doing so, note that the data up to time t is involved, and the wave function up to that point will also be available. Step B in the inversion involves taking the formally extracted potential up to time t and substituting it into the Schrodinger equation 2 to obtain the wave function at time step t At. The sequence is then repeated with the formal inversion of step A carried out again using tracking data out to t At. This relay of A B steps is not an iteration but a sequential marching out in time over the data track to ever better refine the potential. Finally the potential and/or dipole is extracted to its fullest extent in step C from the last relay step. The potential and/or dipole is mapped out over all dynamically accessible regions sampled by the molecular evolution reflected in the laboratory data. An illustration of the relayed sequence of inversion steps is shown in Figure 2 with stimulated data (ref 15 discusses this case in full detail). As with inverse tracking for control, there are many variations on the data inversion algorithm stated a b ~ v e . ~Simulated .’~ data inversions already indicate that the algorithm is quite stable and capable of reliably extracting Hamiltonian information.
+
+
For the dipole function
Here T is the kinetic energy operator. Time dependence is implicitly understood to be present in the kernel and the inhomogeneity in eq 13. It is easy to generalize eq 13 to accommodate multidimensional potentials/dipoles.l 4 Again we have substituted (0j(t)) = yj(t) for exact tracking of the data. Unlike the inverse control problem where the tracking trajectories are given a priori for the design process, in the case of potentiaUdipole inversion5 the tracking trajectories are the time series of experimental data (0j(t)). Other than this distinction, eqs 4 and 13 have the same essential structure. Thus, we may once again present an analogous three-step algorithm A, B, C, for inversion purposes. The solution of the integral equation (13) has two characteristics. The first point is that the solution is generally not unique. This is due to the limited information available from the inevitably incomplete laboratory data. The second point is that the solution is likely not stable with respect to small changes of the data, which is known as ill-posedness.8 Both of these issues are common to virtually all inverse problems. However, we can use some fundamental physical requirements and boundary conditions to constrain the solution. For example, we may impose that the potential and dipole should be smooth functions and negligible asymptotically as x =. By incorporating this prior knowledge, the solution can be stabilized, which is referred to as a regularization process.
-
-
3. Correspondence between Control and Inversion The analysis in the previous section demonstrates that there exists a complete three-step A, B, C correspondence between control and inversion, when considering both from a tracking perspective. The correspondence starts with the common statement of the problem summarized with the Hamiltonian given by H = T V - pc(t) and the track prescribed by y ( t ) = (v(t)lOl+(t)). The distinction between control and inversion resides in the assumptions and goals, as comparatively shown in Table 1. In carrying out operations A, B, and C, the common themes arise of (1) tracking, ( 2 ) regularization, and (3) wave packet propagation. Below is a discussion of some characteristics of these common themes. 3.1. Tracking. In both the control and data inversion problems a time series track is employed. In the case of control design, the tracks are given as prescribed trajectories of the observables. The choices of these trajectories are flexible, in the sense that they allow physical intuition to enter, through the form of the tracking trajectories that lead to the molecular objectives. Good input physical intuition on the track will result
+
Lu and Rabitz
13734 J. Phys. Chem., Vol. 99, No. 37, 1995
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in a well-behaved control field. Experience will likely play a role here, and the tracking design tools provide a systematic means to test and develop that intuition. In the case of potentiddipole inversion, the tracks are timedependent laboratory data. There is a special advantage to extracting potential and/or dipole functions from time series data. This advantage arises as the time variable acts to organize the inversion process. Conventional time-independent spectral inversion methods rely on the propeties of the molecular eigenstates that are determined globally by the underlying potential. In such an approach to inversion the necessary levelby-level quantum number assignment and/or level labeling of the eigenstates is difficult to obtain, especially for highly excited states in a polyatomic molecule. Errors in identifying the eigenstates or missing levels in the measurements will greatly affect the quality of the sought-after potential and/or dipole. In the tracking-based inversion algorithm we propose, the data are organized according to time, and no quantum assignment a n d or level labeling is needed. Furthermore, this algorithm permits the treatment of readily available high-quality spectroscopic data because time-dependent tracks can be synthesized from the timeindependent data. For example, the correlation function track y ( t ) can be synthesized from the Franck-Condon factors F, = l(v(0)lYl)12and energy spectra E,: y(t) =
C F, exp(-iE,t/h)
(20)
1
This synthesized time-dependent track y ( t ) can serve as the input for the inversion, following steps analogous to those in section 2.2. Note that the computation of the correlation function in eq 20 does not call for an explicit identification of the quantum indices but simply utilization of the observed correspondence between the Franck-Condon factors and the spectral levels.
TABLE 1: Parallel Structure of Molecular Tracking for Control and Inversion assumption ____
track
objective
~
control
V(x),p(x) known
y ( t ) prescribed a priori
E(?)
inversion
€ ( I ) observed
y ( t ) observed in the laboratory
V(x),p(x)
in the laboratory
3.2. Regularization. As demonstrated in section 2, both molecular control and Hamiltonian identification can be formulated as inverse problems. The striking similarity is revealed when we compare eqs 7-10 with eqs 16-19. The control field and the potentialldipole can be formally determined by solving linear integral equations. Both solutions must be regularized. In the case of control design the demand for minimal field intensity can serve as the source for regularization, while in Hamiltonian identification prior knowledge such as smoothness stabilizes the solution. In the process of regularization the weight a must be chosen. For the case of control the value of a is not crucial, and it may be chosen to balance the degree that the track is followed versus the field intensity. For Hamiltonian identificationan optimal a value can be determined by introducing an auxiliary minimization function involving a subset of the data. 3.3. Wave Packet Propagation. In both control design and data inversion we need to solve the Schrodinger equation. The control design and the potentialldipole inversion are achieved by a relay of tracking and wave packet propagation. Because the formally inverted control field or inverted potentialldipole from step A depends on the wave function, the resultant Schrodinger equation is highly nonlinear. The nonlinearity of the Schrodinger equation, in principle, allows for a variety of phenomena inherent in nonlinear systems. It would be of interest to explore the special characteristics of these equations.
Control and Inversion and Molecular Dynamics
4. Conclusions We have demonstrated that under appropriate assumptions a parallel structure exists for the control and inversion of molecular dynamics. This connection provides a one-to-one correspondence between these two important areas of study. Physically, the connection arises since both problems aim to elucidate aspects of the molecular Hamiltonian underlying the dynamics. The introduction of molecular t r a ~ k i n g ~ - of ~-'~ objectives leads to their formulation in similar mathematical terms. As pointed out in Table 1, for molecular control design we assume that the potential and dipole are known, while for inversion we assume the driving field for probing the molecular potentialldipole is known.16 In a sense the two problems are artificially separated for the purpose of analysis. The two problems are actually embedded in the larger objective of designing a control field in order to gain the maximum amount of information about a molecular Hamiltonian. One might envision an ultimate joining together of the two problems in the laboratory based on their unified formulation shown in this paper.
Acknowledgment. The authors acknowledge support from the Army Research Office and the Office of Naval Research. References and Notes (1) Warren, W. S.; Rabitz, H.; Dahleh, M. Science 1993, 259, 1581. (2) Ho, T.-S.; Rabitz, H. J . Phys. Chem. 1993, 97, 13447. (3) Gross, P.; Singh, H.; Rabitz, H.; Mease, K.; Huang, G. M. Phys. Rev. A 1993, 47, 4593.
J. Phys. Chem., Vol. 99, No. 37, 1995 13735 (4) Nguyen-Dang, T. T.; Chatelas, C.; Tanguay, D. J . Chem. Phys. 1995, 102, 1528. (5) Lu, Z.-M.; Rabitz, H. Phys. Rev. A, in press. (6) Caudill, L. F.; Askar, A. Inverse Probl. 1994, 10, 1099. (7) Zewail, A. H. J . Phys. Chem. 1993, 97, 12427. Gruebele, M.; Zewail, A. H. Phys. Today 1990, May, 24. (8) Craig, I. J. D.; Brown, J. C. Inverse Problems in Astronomy; Adam Hilger: Bristol, 1986. (9) Unlike the conventional optimal control theory where one specifies only a final outcome, the inverse control theory requires a final outcome and the whole time history of the system. Thus, there is more information content built into the molecular track that allows for noniterative solutions of the control equation. The extra information is provided by physical insight in designing the molecular track. The traditional iterative optimal control formulation might be advantageous in cases where it is difficult to determine the molecular track which steers the system to the desired objective. (10) Shi, S.; Rabitz, H. J . Chem. Phys. 1990, 92, 364. (11) Yan, Y. J.; Gilligan, R. E.; Whitnell, R. M.; Wilson, K. R.; Mukamel, S. J. Phys. Chem. 1993, 97, 2320. (12) Tannor, D. J.; Rice, S . A. J . Chem. Phys. 1985,83,5013. Rice, S . A.; Tannor, D. J.; Kosloff, R. J. J . Chem. SOC., Faraday Trans. 1986, 2, 2423. (13) Amstrup, B.; Doll, J.; Sauerbrey, R.; Szabo, G.; Lorincz, A. Phys. Rev. A 1993, 48, 3830. (14) For multidimension potentials/dipoles, we can still achieve an equation in a form like eq 13. For instance ifAr,B,Q) is three-dimensional, then we can expand f as f = XI,,,, Jm(r) Y/,,,(O,Q),where Yc, are spherical harmonic functions. We get X),,, S K,,,,(j.r)J,,,(r)$ dr = g,, where K/,,,(j,r) = J sin B d0 d@K(j,r,O,Q)YI,(O,Q). (15) Lu, Z.-M.; Rabitz, H. To be published. (16) Kane, D.; Trebino, R. O p f . Left. 1993, 18, 823. JF'950563A
