Four-time correlation function

This is an example of the way to proceed to evaluate the four-time correlation function. 

In fact, as earlier pointed out, the four-time correlation function is the sum of two probabilities, the first being that t and t4 are in the same theoretical laminar region, and the second being that one or more random events occur between t2 and f(, but no event occurs between t and t2 and between t4 and t4, either. The second term of Eq. (212) takes this condition into account. In fact, p(23) ensures that a random event takes place between t2 and tj, p( 12123) establishes that no event occurs between t and L. given the fact one or more events occur between 2 and f (, while, finally, p(34 12,23) establishes that no event occurs between t4 and f4 under the condition that no event occurs between t and t2 and one or more events occur between t2 and ff. It is evident that the product of these three probabilities is the probability that t and t2 belong to the same theoretical laminar region, so do f ( and f4, while one or more events occur between t2 and t4. In conclusion, Eq. (212) affords an exact expression for the four-time correlation function. 

To make our discussion more transparent, it is useful to express the four-time correlation function in terms of the two-time correlation function. The exact expression of Eq. (212) is not suitable for this important purpose. For this reason we make recourse to the approximation... 

In the previous sections, we derived general correlation function expressions for the nonlinear response function that allow us to calculate any 4WM process. The final results were recast as a product of Liouville space operators [Eqs. (49) and (53)], or in terms of the four-time correlation function of the dipole operator [Eq. (57)]. We then developed the factorization approximation [Eqs. (60) and (63)], which simplifies these expressions considerably. In this section, we shall consider the problem of spontaneous Raman and fluorescence spectroscopy. General formal expressions analogous to those obtained for 4WM will be derived. This will enable us to treat both experiments in a similar fashion and compare their information content. We shall start with the ordinary absorption lineshape. Consider our system interacting with a stationary monochromatic electromagnetic field with frequency w. The total initial density matrix is given by... 

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