Ž . Chemical Physics 233 1998 191–205 Optimal pump-dump control: phase-locked versus phase-unlocked schemes YiJing Yan ) , Zhen Wen Shen, Yi Zhao Department of Chemistry, Hong Kong UniÕersity of Science and Technology, Kowloon, Hong Kong, China Received 6 November 1997 Abstract We report a unified theoretical framework for the study of the pump-dump control with either a single coherent field or a pair of phase-unlocked coherent fields in both the strong and weak response regimes, and in terms of both the Liouville-space density matrix dynamics and the Hilbert-space wave function evolution. Shown are also the close relations Ž . between the pump-dump control kernels in the phase-locked i.e. the single coherent field and the phase-unlocked control schemes in the strong response regime. These strong field control kernels can further be linearized in the case of pure state control in the weak pump-dump response regime. In this case, the optimal control theory reduces to the eigen problem of a certain specially constructed Hermitian matrix, from which even the globally optimal pump-dump control fields in either the phase-locked or the phase-unlocked control scheme can be identified. The common key quantity in both of the control Ž X . schemes is a Hilbert-space pump-dump control response function, B t ,t , which shares a great amount of information mutually with the optical resonant Raman spectroscopies. Numerical examples of pump-dump controlling I vibration onto 2 an eigenstate and onto a minimum uncertainty wave packet in the ground electronic X state are presented to further elucidate the control mechanisms in the phase-locked and phase-unlocked schemes in the weak response regime. q 1998 Elsevier Science B.V. All rights reserved. 1. Introduction Using light to actively control the outcomes of molecular events has been one of the central themes in laser chemistry for decades. The common feature in a variety of quantum active control methods is to exploit the temporal-spectral coherent nature of light fields to interfere either constructively or destruc- tively with the matter waves. From a theoretical point of view, a quantum dynamic observable associ- ates with an operator, and its experimental outcome relates to the operator’s expectation value. A control ) Corresponding author. E-mail: yan@chsg4.ust.hk. theory can therefore generally start by defining an ˆ appropriate target operator A and consider the de- ˆ ² Ž . < < Ž .: sired outcome as c t A c t , the expectation f f value at the target time t in the presence of control f Ž. field Et . However,various theoretical control methods are all constructed under different con- Ž. straints to find the field Et that optimizes the Ž . outcome of At . f Quantum control methods can be largely classi- fied into two categories. One is called the coherent control based on the direct interference among two or more independent photo-excitation paths. Brumer w x and Shapiro 1–3 have shown that, in a molecular system having a degenerate photo-excitation door- 0301-0104r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. Ž . PII: S0301-0104 97 00362-5