Single-time correlation function, nonlinear

In this chapter, we developed a stochastic theory of single molecule fluorescence spectroscopy. Fluctuations described by Q are evaluated in terms of a three-time correlation function C iXi, X2, T3) related to the response function in nonlinear spectroscopy. This function depends on the characteristics of the spectral diffusion process. Important time-ordering properties of the three-time correlation function were investigated here in detail. Since the fluctuations (i.e., Q) depend on the three-time correlation function, necessarily they contain more information than the line shape that depends on the one-time correlation function Ci(ti) via the Wiener-Khintchine theorem. 

In the 2D-IR experiments there exists a useful simplification of the description of the spectra when the vibrational dynamics is in the separation of time-scales limit of the so-called Bloch dynamics. Then the correlations of the fluctuations of the various quasi-degenerate amide modes dominate the signals and the interpretations are quite straightforward and analytic. On the basis of experimental determinations of the correlation functions, we explore some of the sensitivities of the 2D-1R signals to the dynamic approximations. In addition, we discuss some of the important possible manipulations of the 2D-IR spectra that permit the display of essentially all possible third-order nonlinear responses from a single data set. 

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