Control of coherent wave functions: A linearized molecular dynamics view

In this paper, the Schrödinger 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 Li₂ as an example, we investigate the control properties of creating specified coherent wave functions on the excited potential energy surface A¹Σ_u ⁺ by excitation from an initial state on the X¹Σ_g ⁺ surface. The numerical results suggest that the required control field is reasonable for laboratory realization.