Fast modulation regime

Q in the Slow Modulation Regime Q in the Fast Modulation Regime... 

In this section, we consider the fast modulation regime, v, E R. Usually, in this fast modulation regime the dynamics of the bath (here modeled with R) is so fast that only the long time limit of our solution should be considered [i.e., Eqs. (A.47) and (A.48)]. Hence, all our results below are derived in the limit of T 00, since the time dependence of Q is irrelevant. The fast modulation regime considered in this section includes case 3 (T V R) as the strong, fast modulation case and case 4 (v E R) as the weak, fast modulation case. 

It is well known that the line shape is Lorentzian [62] in the fast modulation regime (soon to be defined precisely). 

Now, we define the fast modulation regime considered in this work. When J 00 (and other parameters fixed) Q 0, so fluctuations become Poissonian, which is expected and in a sense trivial because the molecule cannot respond to the very fast bath, so the two limits R->- co and v = 0 (i.e., no interaction with the bath) are equivalent. It is physically more interesting to consider the case that R oo with Fy/F remaining finite, which is the standard definition of the fast modulation regime [60]. In this fast modulation regime, the well-known line shape is Lorentzian as given in Eq. (4.70) with a width F -I- F. ... 

Note that these two approximations, Eqs. (4.72) and (4.73), are not limited to the two-state jump model but are generally valid in the fast modulation regime. The frequency correlation function is given by... 

Figure 4.10. Case 3 (F v R) in the steady-state limit. Line shape and Q in the fast modulation regime are shown as functions of Parameters are chosen as v = 100 MHz, F = 1 MHz, n = r/10, R = 1-100 GHz, and T oo.

So far, we have considered four limiting cases (1) strong, slow, (2) weak, slow, (3) strong, fast, and (4) weak, fast case. Now, we consider cases 5 (v R E) and 6 (E R v). They are neither in the slow nor in the fast modulation regime according to our definition. 

Note that the same relation between Q and was also found to be vahd in one of the fast modulation regimes, Eq. (4.79) with T T. Figure 4.13 shows that in this case the limiting expressions approximate well the exact results. 

We investigate the overall effect of the bath fluctuation on the photon statistics for the steady-state case as the fluctuation rate R is varied from slow to fast modulation regime. To characterize the overall fluctuation behavior of the photon statistics, we define an order parameter q. 

Based on the approximation introduced in the text we can calculate Q in the fast modulation regime. Once the factorization of the three-time correlation functions is made in Eq. (4.72),, (s), the functions determining ... 

Since only the long time limit is relevant for the calculation of Q in the fast modulation regime we make expansions of (Wy and around... 

Note that Eq. (A.61) is valid once the factorization approximation is made irrespective of the second-order cumulant approximation. After performing a lengthy but straightforward algebra using Eqs. (A.59)-(A.61), we obtain the result of Q in the fast modulation regime given as Eq. (4.77). 

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