Continuum eigenstate

The essential principle of coherent control in the continuum is to create a linear superposition of degenerate continuum eigenstates out of which the desired process (e.g., dissociation) occurs. If one can alter the coefficients a of the superposition at will, then the probabilities of processes, which derive from squares of amplitudes, will display an interference term whose magnitude depends upon the a,. Thus, varying the coefficients a, allows control over the product properties via quantum interference. This strategy forms the basis for coherent control scenarios in which multiple optical excitation routes are used to dissociate a molecule. It is important to emphasize that interference effects relevant for control over product distributions arise only from energetically degenerate states [7], a feature that is central to the discussion below. 

Figure 1.8 The probability density of several continuum eigenstates of the Hamiltonian in Eq. (24) plotted on the baseline of their corresponding energy. The potential is also plotted for convenience. Note that most continuum states (dashed lines) are delocalized and have a very small amplitude inside the potential well between the two barriers, whereas there are continuum functions that are localized inside the well. The localized eigenstate (solid line) is the same as shown in Figure 1.1.

Looking now at the expansion in Eq. (4), the wave packet ir x,t) will contain contributions from these two "groups" of continuum eigenstates, which will depend on the expansion coefficients C(E) given in Eq. (5) and can now be separated to... 

The states excited-state potential energy surface (labeled 6 ),... 

Let us denote the excited state after initial excitation by x,) with quantum numbers denoted by the composite index i, which is assumed to be in a particular vibrational state v,. The continuum eigenstate of the full Hamiltonian H is denoted as whose quantum numbers are given by the composite index /. The probability per unit energy that the system will end up in the continuous state / is given by... 

A continuum eigenstate of the total Hamiltonian h with eigenvalue E and degeneracy index jS will then be written as a superposition of the unperturbed eigenstates... 

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