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4.One clarification regarding this computation is necessary. In accord with the idea of phase control, the authors' intention is to vary without changing . They proposed to do so by varying the chirp parameter in the laser field [their Eq. (8)]. However, computing from their shows that it depends upon . Hence, dynamical effects in their Fig. 4 that result from changing the sign of are indeed evidence for phase control. However, dynamical effects resulting from varying are a result of changes to both the relative phases and to , and are hence not evidence of phase control. Alternate computations showing phase control have been obtained as well (R. Kosloff, private communication).
5.Additional terms with a similar relationship between time dependence and laser phase appear in the case where excitation is not between different electronic surfaces.
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8.Consider the case of . Then (in the basis of ) and , where operates on the space of the environment. It follows that . Barring unusual cases involving degeneracies of exactly canceling the off-diagonal components of (such as cases where the system dependence of is solely a function of ), this shows that in general .
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13.Interestingly, in model systems studied with a Redfield approach we found that we could generally construct different wavepackets “by hand” on an excited potential energy surface that displayed the desired phase dependent control behavior. However, these wavepackets could not be obtained via one-photon excitation from the ground state.

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Coherent control of observables using the phase properties of weak light that induces one-photon transitions is considered. Measurable properties are shown to be categorizable as either class A, where control is not possible, or class B, where control is possible. Using formal arguments, we show that phase control in open systems can be environmentally assisted.


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