J. Phys. Chem. 1993,97, 8874-8880
8874
Control of Coherent Wave Functions: A Linearized Molecular Dynamics View Liyang Shen,+Shenghua SM,*and Herschel Rabitz' Department of Chemistry, Princeton University, Princeton, New Jersey 08544 Received: March 19, 1993; In Final Form: June 2, I993
In this paper, the Schrijdinger equation is linearized with regard to a low-intensity controlling electric field. For such a linearized quantum dynamical system, the present work answers the issue of controllability and explicitly provides the control field. Starting in a particular eigenstate, the resultant necessary and sufficient conditions for controllability require that the system satisfy the following two criteria: (1) the N eigenstates of the field-free Hamiltonian superimposed to form the coherent final state must be nondegenerate and (2) the electric dipole transition moments from the initial state to each of the above eigenstates must be nonzero. The control field is obtained analytically in terms of N monochromatic electric fields, each of which has a frequency corresponding to the transitions of the field-free Hamiltonian. We show that the physical properties of the control field are not affected by the overall phase of the coherent wave function. Using Liz as an example, we investigate the control properties of creating specified coherent wave functions on the excited potential energy surface A'Z: by excitation from an initial state on the X'Z; surface. The numerical results suggest that the required control field is reasonable for laboratory realization.
I. Introduction Control of molecular dynamical processes has become an increasingly active subject of research. Stimulated by recent advances in both theory and experiment, a number of theoretical studies have demonstrated that many molecular dynamical processes can be effectively controlled or enhanced by utilizing optimally designed electromagnetic The rapid progress of laser pulse shaping techniques provides a potential capability for generating the required electric fields in the laboratory for control of a molecular system.9-12 With readily available laser intensities ( 1.0MW/cmZ), the interaction with a molecule is usually very weak. The molecular state generated by the laser field is a coherent superposition of the eigenstates of the field-freeHamiltonian. Such coherent states are of particular importance for the control of moleculardynamics, the study of intramolecular vibrational energy redistribution (IVR), femtosecond transition-state spectroscopy (FTS),and state-to-state rates.13.14 A laser-driven molecular dynamical process can be described by the time-dependent Schrddinger equation which is bilinear with respect to the field-molecule interaction and the molecular coherent state. Under a weak field (taken as electric here), the dynamical equation can be linearized by utilizing first-order perturbation theory. The purpose of this work is to investigate the control properties of a molecular dynamical system in the linear field regime. In general, there are two fundamental issues involved in the study of molecular control: controllability and obtaining the control field. The controllability is concerned with the necessary and sufficient conditions that a system must satisfy to be fully controllable. It is essential to address this issue first because any attempt to design a laser field to reach a target state would be meaningless if the state is not controllable. Next, if a target state is indeed achievable,the remaining issue is todetermine the control field that drives the system from its initial state to the desired final state. Many efforts have been made to address the controllability of various quantum dynamical systems. While the controllability N
* Present address: Biosym Technologies, Inc., San Diego, CA 92121.
Prcscnt address: Department of MacromolecularModeling, Bristol-Mym Squibb Pharmaceutical Research Institute, Princeton, NJ 08543. f
of a general quantum system is not fully understood, some special cases of molecular system controllability have been explored.'5-M For example, Huang, Tarn, and Clark studied the controllability of Schradinger's equation by considering it as an infinitedimensional bilinear system. However, their study does not, in general, lead to an explicit formula for the external control field which drives the system from the initial state to target state. On the other hand, the work presented here does, and this is a consequence of the linearity of the control system. To our knowledge, the finite dimensional coupled harmonic oscillator system is the only case for which both the necessary and sufficient conditions for controlling the bond stretch and momentum are completely understood.1g.M In this paper, we will address the controllabilityof coherentwave functionsof an arbitrary linearized quantum system, which is confined to the Hilbert space spanned by N eigenstates of the field-free Hamiltonian, and provide the complete necessary and sufficient conditions. The results are general and can be applied to any linearized quantum dynamical system. Formally, for an N-dimensional linear system, if the states and a control field are defined in the same space, either real or complex, the conditions for controllability and the explicit formulation of the control field are known.z1 However, in the present case, the wave functions are defined in a complex space, while the electric field is confined to being real. By making an appropriate transformation, we obtain the necessary and sufficient conditions for coherent wave function control and extract the control field solution for a general linearized quantum system. These conditions are explicitlyexpressed in terms of the energy structure of the field-free Hamiltonian and the electric dipole transition moments. We show that the control field can be decomposed into a series of monochromatic electric fields. This article is organized as follows. In section 11, we outline the procedure of linearizing the time-dependent SchrMinger equation under a weak electric field. Controllability of the linearized system is explored in section 111. The analytic expression for the control field is derived in section IV. As an illustration, in section V we study the control properties of generating specified coherent wave functions on an excited potential energy surface of Liz. Finally, section VI summarizes the conclusions that can be drawn from this work.
0 1993 American Chemical Society 0022-3654/93/209l-SS14~04.~0/0
The Journal of Physical Chemistry, Vol. 97, No. 35, 1993 8875
Control of Coherent Wave Functions
II. Linearization of the SchrcTdinger Equation
in terms of N eigenstates of
&, one writes N
The dynamics of a molecule under the influence of an external electric field, t(r), t E [0, T'J,is described by the time-dependent SchrGdinger equation
(11.9) where & represents the kth eigenstate of & and c k ( t ) is the corresponding coefficient. Substituting eqs 11.8 and 11.9 into eq 11.7, one readily obtains
where $ ( t ) is the molecular wave function at time t , while $(O) is the initial state. Here ko representsthe field-free Hamiltonian, and kl(t) is the interaction Hamiltonian which usually can be written as
A,(t)= -i&)
(11.1Oa) where c(t) and d are N-dimensional vectors and H is a N matrix, which are defined by
(11.2)
[C(t)]k Ck(t), 1 5 k 5 N
X
N
(1I.lOb)
where 5, is the molecular electric dipole operator along the direction of the applied external electric field. It is assumed that the molecule is initially in an eigenstate of );ro whose energy is (YO. Under the definition
dk = dk = (6kljiel$(o)), 1 5 k
N
(1I.lOd)
Here, E&is the eigenvalue of &corresponding to the kth eigenstate
& and d are the electric dipole transition moments from the
eq 11.1 is transformed into
initial state to the individual eigenstates of ko. The eigenvalue may be among the set Ek, k = 1, ..., N, or it may be a value outside this set (e.g., corresponding to a state on a different BornOppenheimer surface). (YO
where i i s a unit operator. Since there is only the absolute phase difference (rot between the states X ( t ) and $(t), eqs 11.1 and 11.4 describe the same dynamic system. Equation 11.4 is more convenient for the analysis of controllability. Equation 11.4 is bilinear with respect to X(t) and c ( t ) . With normally available laboratory laser field intensities,the magnitude of fi,(t) (i.e,, the interaction energy) is usually much smaller than that of ko. Furthermore, if the interaction duration T and the field intensity satisfy l$ZllT
