Optimal control theory for a target state distributed in time: optimizing the probe-pulse signal of a pump-probe-scheme

Optimal control theory (OCT) is formulated for the case of a two-color pump-probe experiment. The approach allows to calculate the pump-pulse shape in such a way that the probe-pulse absorption signal is maximized. Since the latter quantity is given by the time-averaged expectation value of a time dependent operator (the probe-pulse field-strength times the dipole operator) a version of OCT has to be used where the target state is distributed in time. The method is applied to a molecular three-level system with the pump-pulse driving the transition from the electronic ground state into the first-excited electronic state and the probe-pulse connecting the first-excited state with a higher lying electronic state. Depending on the probe-pulse duration, the vibrational wave packet becomes localized or at least highly concentrated in the Franck-Condon window for the transition into the higher-excited state. The dependence on the probe-pulse duration and on the delay time between the optimized pump-pulse and the probe-pulse is discussed in detail. The whole study demonstrates the feasibility of laser pulse induced temporal wave packet localization and the use of spectroscopic quantities as target states in experiments on femtosecond laser pulse control.