General Principles of Optimal Control Theory

Dynamics is introduced as an additional constraint. In our case this is the require-i ment that T be a solution of the time-dependent Schrodinger equation, 

The problem of maximizing the objective J subject to the above constraints can be transformed into an unconsbained problem by using Lagrange multipliers. According to this standard procedure, we multiply Eq. (4.11) by an unknown number X and Eq. (4.12) by an unknown two-component state vector  

The optimum field, defined via the requirement SJ — 0 is then related to the abovef solutions and to X as 

This procedure was applied to a number of cases by Kosloff et al. [104] who studied a model for tire theoretical 

There are now direct experimental confirmations [41-43], to be discussed in Section 13.3, of the effect of the pulse shaping. In addition, the role of the phase j-1 between two pulses, predicted in both OCT [104,119] and in CC studies [94,96,1R J. has been confirmed experimentally by Fleming et al. [137,138], Girard et al. [JO- j 142], Kinrot et al, [143], and Warmuth et al. [144] in the so-called wave paiket interferometry experiments [145], For example, Fleming et al, [137,138] [and Warmuth et al. [144] describe experiments where the fluorescence from I2 in the F t state is influenced by constructive or destructive interference between two wave pad.-, j ets induced by two-phase related excitation pulses. This study relates to a large voliuiiejy of work on wave packet interferometry in atoms [146-148], as well as to various. femtochemistry experiments, where similar effects were seen in absorption [106] . jj 

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