J . Phys. Chem. 1993, 97, 2320-2333
2320
Optical Control of Molecular Dynamics: Liouville-Space Theory YiJing Yan, Richard E. GiUilan, Robert M. Whitnell, and Kent R. Wilson' Department of Chemistry, University of California, San Diego, La Jolla, California 92093-0339
Shad Mukamel Department of Chemistry, University of Rochester, Rochester, New York 14627 Received: September 18, 1992; In Final Form: December I , 1992
We lay some theoretical foundations to deal with the experimental realities faced in controlling molecular dynamics with tailored light fields: the nonideality of the light, the mixed rather than pure quantum-state nature of matter, and environmental and solvent effects. The optimal control of molecular dynamics using light fields is formulated in terms of the density matrix in Liouville space, generalizing existing wave-function-based formulations. This formulation allows the inclusion of mixed states, so that thermal and other nonpure quantum states of matter can be treated, as well as reduced descriptions useful in studying dense gas and condensed phases. In addition, it allows for general constraints and arbitrary coherent and partially coherent radiation fields and provides a unified picture for quantum, semiclassical, and classical molecular dynamics. For weak fields, the calculation simplifies and is given in terms of a molecular response function which itself does not depend upon the field. The solution of an eigenequation then directly gives the globally optimal field and the yield with respect to the target. As a demonstration, we explicitly consider a two electronic surface displaced harmonic oscillator molecular system in a Brownian oscillator solvent at finite temperatures, including nuclear and electronic solvation effects. Numerical illustrations are presented for the quantum control of thermal samples using phase-locked and random-phase light fields in the presence of solvent.
toreaction. This localization has been used successfullytocontrol the reaction yield in the Xe 1 2 system.lO (Similar behavior has Dudley Herschbach, to whom this issue of The Journal of been observed in the multiphoton ionization of Na2 to produce Physical Chemistry is dedicated, is justly celebrated for his N a z + . ~ ~ JIn2 )the frequency regime, recent experiments by Crim pioneering experimental and theoretical work, particularly in the and co-workersI3-'5 and by Zare and co-workerst6have shown area of gas-phase chemical reaction molecular dynamics as studied that photoexcitation of either the OH or the OD bond prior to by crossed molecular beams. By thus observing nature, a much the reaction of H HOD leads to selective enhancement in the deeper understanding has been achieved of the fundamental product H2 OD or HD + OH, respectively. However, attempts dynamical processes by which chemical reactions take place. to use a singlemonochromatic laser beam to achievebond-selective A new era may now be opening, as Graham Fleming has so photochemistry have generally been disappointing. This failure well expressed it,l when we may be able to *change from being is due to the fact that the state prepared by monochromatic a voyeur to being a participant in chemical reactions" by guiding excitation is usually a superposition of more than one degenerate them with specially tailored light fields. If such an era is to channel, which leads in an uncontrolled way to a variety of arrive, i.e., if such optical control of chemical reaction dynamics products. In order to overcome this fundamental difficulty, is to become a common experimental tool, several experimental various theoretical and experimental schemes have been proposed to use other properties of light, such as phase and quantum realities must be faced, understood, and successfully mastered. First, we must recognize that the universe of experimentally interference, amplitude and temporal shape.17-29 realizable tailored light fields is a restricted one, in which Tannor and Rice30 have proposed a pump-dump control perfection is a goal, but imperfection (variation among laser shots, scheme, based on the Franck-Condon transition between two phase randomness among subpulses, partial temporal and spatial electronicsurfaces. Chemical selectivity on theground electronic coherence,and variation of intensityacross the sample) is a reality. surface was proposed by controlling the dump delay time with Second, we must learn to deal with realistic material samples. respect to the pump pul~e.3~JlThe experiments by Zewail and Pure quantum states of matter are not easily prepared, and we co-workers mentioned above are similar but use a pumppump will need to learn to control mixed quantum-state systems, for control configuration.I0 The second pump field in this experiment example, thermal equilibrium systems. Third, many chemical controls the timing of the excitation of the molecular wave packet, reactions of interest do not occur in the splendid isolation of a which is evolving on the excited surface, onto the reactive region dilute gas, but rather in liquid solution and at interfaces and in the final excited surface. In the Tannor-Rice control scheme, surfaces. Thus, solvent and other environmental effects need to the molecular dynamics between pump and dump are uncontrolled be considered. This article lays some theoretical foundations for and the wave packet may spread extensively out of the target dealing with each of these experimental realities: imperfection Franck-Condon region. Paramonov and Manz and their of light fields, mixed quantum states of matter, and environmental co-workers32-3*have described a control scheme using a sequence and solvent effects. of pulses with optimized field strength, frequency, time delay, and duration for each subpulse. Using laser fields to control and manipulate the outcome of Brumer and Shapiro39-41 have proposed a coherent control chemical reactions is a long-standing goal of laser c h e m i ~ t r y . ~ - ~ theory, in which the yield of a molecular reaction is controlled, The first successful experiments have utilized control of the as the direct consequence of quantum interference effects, by temporal or frequency behavior of the molecules involved in a using weakcontinuous-wave (CW) light ofvariable relative phase, reaction. In the temporal case, it is possible with femtosecond amplitude, and polarization. This coherent control scheme has laser pulses to localize nuclear configurationsin a region favorable
I. Introduction
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0022-3654f 93/2097-232OSQ4.O0f 0
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0 1993 American Chemical Society
Optical Control of Molecular Dynamics
The Journal of Physical Chemistry, Vol. 97, No. 10, 1993 2321
been experimentally demonstrated by Elliott and c o - w o r k e r ~ ~ ~ , ~Illustrative ~ weak field optimal control numerical results for a two and by Gordon and c o - w o r k e r ~in ~ ~the . ~ resonance-enhanced ~ electronic surface, displaced harmonic oscillator are shown in multiphoton ionizationof Hg, HCl, and CO. In these experiments, section V, using thermal samples as well as a Brownian model two exciting light beams derived from the same laser are employed; to incorporate nuclear and electronic solvent effects. Finally, one providing a single-photon excitation route and the other a our results are summarized in section VI. three-photon route. The ionization signals vary as a function of the relative phase between these two excitation fields, as predicted 11. Green Function Formulation of Optimal Control in by theory.394' LiouviUe Space Rabitz and co-worker~~f-~~ have employed the general theory of optimal control to illustrate how the yield of the selected target Consider a material system characterized by the Hamiltonian state can be maximized by using optimally shaped fields. This H M which is coupled to an external radiation field c(t) via the is a strong field theory with unrestricted coherent control fields. system dipole operator D. The total Hamiltonian of the coupled Microscopic control can in principle be achieved, and at least in material-field system, in the electric dipole approximation, is a bound system, the molecular system can be directed from a given by given (wave function) pure state to a final pure target state at a specifiedtime.29.6 Application of the optimal control formalism to multisurface molecular systems has been made by Kosloff et The density matrix of the coupled system satisfies the Liouville a1.23 equation: Many of the optimal control results require light fields with complex shapes for optimal control. Several experimental techniques may prove useful in constructing such fields, including laser pulse shaping technology, which has been demonstrated in In eq 2 and hereafter, we set h = 1. f is the Liouville operator, frequency-phase variables by Weiner, Heritage, Nelson, and defined by the commutator in the above equation. The formal c o - w o r k e r ~and ~ ~ in ~ ~the time variable by Warren and cosolution of eq 2 is w o r k e r ~ . ~Scherer, ~ , ~ ' Fleming, and c o - w o r k e r ~ and ~ ~ -Fayer ~~ and co-workersss have recently developed phase-locked laser A t ) = Q ( v o ) P(to) (3) techniques in which the relative phase of two pulses can be controlled. In two-pulse phase-locked interference measurements The Liouville-space propagator, Q(t,to), i.e., the time-domain of the spontaneous light emission of 12, Scherer et al.s6 have Green function for the Liouville equation, is given by75.76 demonstrated the possibility of using the second pulse to interfere constructively or destructively with the excitation effects of the o(t,to)= exp+[-ii:d+ L(7)]I first. Recent nonlinear spectroscopic work with chirped laser pulses may provide another approach to tailor the laser field for E(-i)'JSfd7,,...Jr2di, 10 L ( T , , ) f ( 7 , ) (4) the optimal control of chemical event^.^^-^^ Warren and con=O " workers have demonstrated highly selective population inversion for a variety of cases, using amplitude- or frequency-swept pulses In eq4, exp,, defined in the second identity, denotes the positively in both NMR and optical spectroscopy.6244 time-ordered exponential ~perator;'~ that is that if 9 acts on an arbitrary operator from the left, the Liouville operators act at the The quantum molecular control theories developed so far have positively chronic or sequential order (in eq 4, 1 .,.1 71). The been based on the Schrddinger wave function formalism which requires the coupled quantum dynamics of all degrees of freedom expectation value of an arbitrary dynamical variable, or a and usually assumes that the system is initially in a pure state. Hermitian operator 2,for the system at time tf is given by However, experimentallypreparing a pure state is itself a difficult task. In this paper, we develop a molecular control theory based on the Liouville-space density matrix formalism. This allows the Equations 1-5 summarize the Liouville-space f ~ r m u l a t i o nof ~~~~* inclusion of mixed states (thermal ensembles, for example) and quantum dynamics and allow the calculation of any physical provides a unified theory for quantum, classical, or semiclassical observable for a given radiation field t ( t ) . In this article, we shall implementations of molecular dynamics, useful for complex use this formulation to look for a radiation field that will drive systems. the system to produce the best match for a given objective. The paper is organized as follows. In section 11, we present We shall hereafter denote A(tr) of cq 5 as the control target the general unrestricted field formulation of optimal control in and 2 as the target operator, which can be a dynamical variable, Liouvillespace. A closed self-consistentequation for the optimal a projection operator to a pure state, or a general wave packet field is obtained. The key quantity is a control kernel which itself in phase space. Our objective is to find a form of the external depends upon the control field. The calculation of the kernel can field, t ( ~ ) ,which optimizes our target, A(tr). We may wish to either be performed via two coupled forward propagations, or a impose certain penalties on the field c(7), for example, if it exceeds forward and a backward propagation in Liouvillespace. In section a certain energy or if it deviates from a particular ~hape.26.2~ To 111, we consider the Hilbert-space implementation of the control that end, let us introduce J(rf),which is a functional depending kernel and show that the present result recovers the previous on the external field c(7), wave function formalism in the pure quantum-state case. In section IV.A, we developa weak field control theory. The control kernel in this case is related to a molecular response function, which does not depend on the external field. This greatly simplifies The second term in eq 6 is the cost penalty function for the the computational procedure. The unrestricted globally optimal influence of the field,26-27characterized by a weight function weak field can be obtained by solving an eigenequation of a X ( T ) . In the case where the penalty is simply the constraint that Hermitian matrix, constructed from the molecular response the total incident energy has an upper bound in the optimization function, in which the eigenfunctionis the field and the eigenvalue process, the weight function X(7) reduces to a Lagrange multiplier is the yield with respect to the target. Application of the weak which then depends only on the total incident energy constraint, field results in a two electronic surface molecular system with ground- or excited-state control is presented in section 1V.B. but not on time 7.23*26+27 A properly constructed time-dependent
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The Journal of Physical Chemistry, Vol. 97, No. 10, 1993
weight function A( T ) can also allow discrimination against experimentally unrealisitic wave forms in the resulting control field.26v27For simplicity, we do not include other penalty terms in eq 6, such as the minimization of the expectation values of undesired observables; however, including such terms is a straightforward generalization.26 In previous presentations of molecular control theories, the dynamical equation, e.g., the Schrtkiinger equation, has usually been included explicitly in the control functional J(tr), as a constraint via a time-dependent Lagrange multiplier.23~26.~~ The final control equation is then derived via the variation of the control functional, bJ(tr),together with the dynamical equation constraint, with respect to the variation of the control field, & ( T ) . In the following, we present an alternative approach, in which the dynamical equation, or the Liouville equation (eq 2), is explicitly included via its Green function, $(?,to) (cf. eq 4). That is done by substituting eqs 3 and 5 into the equation for the control objective functional, eq 6. No Lagrange multiplier for the dynamical equation is introduced. The final equation can then be obtained via the perturbative expansion of the Green function around the control field c ( T ) , as follows. Assume that we know the optimal field t ( ~ which ) gives the optimal target A(tr). By varying the field by &(t) froman optimal formc(7) tot(7) + bt(~),ourtargetdeviat~fromalocalmdmum A(tf) to A(tr) bA(tr). The variation of eq 6 is then
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bJ(tf) = bA(tf)- 1:dr A(T)
(7")
Q(Tn,Tfi,)
saa6.l
( 7 1)
9 (71,to)
P(t0)1
(8)
where Sf (t) is the variation of the Liouville operator, induced , is defined by its action on by the deviation of the field ~ c ( T )and an arbitrary operator B 7 6
bL(r)B Gc(t)[D,B]I be(t)fDB
f(7)
(9) In eq 9,B is the Liouville dipole operator, defined by the identity on the right-hand side of eq 9 as the ordinary dipole operator, D. Equation 8 constitutes the perturbativeexpansion of thedeviation of an observable target about an arbitrary field ~ ( 7 ) .The nth termineq8dependsonbeinthenthorder. Sinceweareinterested in a material systemwhich is initiallyin a stationary or equilibrium state before the interaction with external fields, the initial time to in the above equations and also in the remaining parts of this paper will eventuallybe set to-=. Therefore, wedo not explicitly state the dependence on to of A(tr), SA(tf), etc. The necessary condition for a local optimum/extremum is that the first-order variation in the functional (eq 7) vanishes. To that end, let us consider bA(l)(tr),i.e., the n = 1 term in eq 8:
= K(T;tf)/A(T);
fo
