Three-time correlation function, single

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. 

Sikorsky and Romiszowski [172,173] have recently presented a dynamic MC study of a three-arm star chain on a simple cubic lattice. The quadratic displacement of single beads was analyzed in this investigation. It essentially agrees with the predictions of the Rouse theory [21], with an initial t scale, followed by a broad crossover and a subsequent t dependence. The center of masses displacement yields the self-diffusion coefficient, compatible with the Rouse behavior, Eqs. (27) and (36). The time-correlation function of the end-to-end vector follows the expected dependence with chain length in the EV regime without HI consistent with the simulation model, i.e., the relaxation time is proportional to l i+2v The same scaling law is obtained for the correlation of the angle formed by two arms. Therefore, the model seems to reproduce adequately the main features for the dynamics of star chains, as expected from the Rouse theory. A sim-... 

We concentrate on the role of quantum interference in the correlation of photons emitted from a coherently driven V-type atom, recently analyzed by Swain et al. [58]. We calculate the normalized second-order two-time correlation function g (R, t R, t + x) for the fluorescent field emitted from a three-level V-type atom driven by a coherent laser field coupled to both atomic transitions. The fluorescence field is observed by a single detector located at a point R = RR, where R is the unit vector in the direction of the observation. 

Dynamic information such as reorientational correlation functions and diffusion constants for the ions can readily be obtained. Collective properties such as viscosity can also be calculated in principle, but it is difficult to obtain accurate results in reasonable simulation times. Single-particle properties such as diffusion constants can be determined more easily from simulations. Figure 4.3-4 shows the mean square displacements of cations and anions in dimethylimidazolium chloride at 400 K. The rapid rise at short times is due to rattling of the ions in the cages of neighbors. The amplitude of this motion is about 0.5 A. After a few picoseconds the mean square displacement in all three directions is a linear function of time and the slope of this portion of the curve gives the diffusion constant. These diffusion constants are about a factor of 10 lower than those in normal molecular liquids at room temperature. 

Fig. 5.23 Time evolution of the three functions investigated for PIB at 390 K and Q=0.3 A"h pair correlation function (empty circle) single chain dynamic structure factor (empty diamond) and self-motion of the protons (filled triangle). Solid lines show KWW fitting curves. (Reprinted with permission from [187]. Copyright 2003 Elsevier)...

Figure 8.9 Time-resolved fluorescent lifetime analysis of Cy3 attached to double-stranded DNA (Iqbal et al., 2008b). Fluorescent decay curve for Cy3 attached to a 16 bp DNA duplex, showing the experimental data and the instrument response function (IRF), and the fit to three exponential functions (line). The decay curve was generated using time-correlated single-photon counting, after excitation by 200 fs pulses from a titanium sapphire laser at 4.7 MHz.

The intensity correlation function C t) is well fitted over more than three decades, from a delay time of 1 /s up to more than 1 ms with values of the parameters similar to those given above (Fig. 4). In particular, the deviations from the single exponential decay are well accounted for, at least in the time domain of interest. It is clear that a purely stretched exponential decay can be observed only at extremely long time where the amplitude of the correlation function is much less than the noise. 

Using the experimental setup (Fig. 15) described in the last paragraph, the fluorescence intensity correlation function was measured for a single terrylene molecule in p-terphenyl over nine orders of magnitude in time by the authors group [27]. The experimental trace in Fig. 16 clearly displays the characteristic features of photon antibunching and photon bunching. The onset of Rabi oscillations is clearly visible, too. We now want to discuss separately these three effects for different systems and what can be learned from such measurements. 

A typical decay curve of the correlation function in the ps time regime for a single pentacene molecule in p-terphenyl (site 0 ) is shown in Fig. 19(a) on a linear time scale [36]. The decay rate can be well approximated by a single exponential (Eq. (15)). A plot of A versus I is displayed in Fig. 19(b) for three different pentacene molecules. The drawn lines are fits of Eq. (16) to the experimental data. For pentacene we have kii > ki and at high intensity A —> kij,/ . Thus, from Fig. 19(b) it can be seen that the population rate ki-i varies strongly from molecule to molecule. These... 

In the second type of experiment that measures single molecule spectral dynamics one performs repeated fluorescence excitation scans of the same molecule. In each scan the line shape is described as above, but now there is the possibility that the center frequency of the line will change from scan to scan because of slow fluctuations. Thus one can measure the center frequency as a function of time, producing what has been called a spectral diffusion trajectory. This trajectory can, in principle, be characterized completely by the spectral diffusion kernel of Eqs. (16) and (19), but of course it must be understood that only the slow Kj < 1 /t) TLSs contribute. In fact, the experimental trajectories are really too short to be analyzed with this spectral diffusion kernel. Instead, it is useful [11, 12] to consider three simpler characterizations of the spectral diffusion trajectories the frequency-frequency correlation function in Eq. (14), the distribution of frequencies from Eq. (15), and the distribution of spectral jumps from Eq. (21). For this application of the theoretical results, in all three of these formulas j should be replaced by s, the labels for the slow TLSs. 

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