## 3 Quantum Modelling of FMO Quantum Excitation
### Efficiency Depending of \(T,\lambda_{k}\) and \(\gamma_{k}\)  
The so called efficiency \(\eta\) of quantum energy transport inside of each monomer in the FMO is defined as the asymptotic sum of populations in BChls 3 and 4 due they are the responsible to transfer the excitation to the RC:  
\[\eta=\lim_{t\to\infty}\rho_{33}(t)+\rho_{44}(t) \tag{37}\]  
Figure 12: Evolution of coherence for the first 2.5 ps for \(\gamma=50\,\mathrm{cm}^{-1}\) as function of \(T\) in color, for (**A**) \(\lambda=35\,\mathrm{cm}^{-1}\), and (**B**) \(\lambda=65\,\mathrm{cm}^{-1}\) showing the transition from concurrence between pair \((1,2)\)—solid line—and \((3,4)\)—dashed line—as the main provider of coherence. Together, the time of transference is the time \(T_{t_{95\%}}\) that system uses to funnel the excitation to the exiting system at the 95% of its final equilibrium value (as instance on the BChl 3 or 4 connected to the RC). In the current work, we use the joint population of BChl 4 and BChl 3 to fix this time because they are the main populations remaining among the whole BChls while they are the responsible to transfer the excitation to the RC. Following the previous parametric analysis performed through HEOM calculations, we analyze \(\eta\) in Figure 13A as function of parameters \(T,\lambda\) and \(\gamma\) (\(\Delta_{\eta}^{2}=2.1\times 10^{-4}\)). While for the original data obtained from the simulations for \(\eta\) in the analysis region, its general range became \([0.51,0.99]\) ps, the contour surfaces exhibit approximated constant \(\eta\) values by fitting in agreement with the color scale below. As before, on them, green, yellow and red contour lines correspond to constant values of \(T\), \(\gamma\) and \(\lambda\), respectively. The black screens representing once more characteristic constant parameter values showing the contour lines of the corresponding efficiency contours. In the case of the bottom black screen for \(\lambda=65\) cm\({}^{-1}\), we appreciate an inflection point in the middle of the graph where efficiency appears to increase and decrease in crossing directions in the \(\gamma-T\) plane for \(\lambda=65\) cm\({}^{-1}\). Efficiency values increases if \(T\) decreases in the \(\gamma=50\) cm\({}^{-1}\) plane, but it is not the same behavior for lower values of \(\gamma\) where higher efficiencies are sustained at larger temperatures. In a related analysis, for the source data obtained from the simulations, the range of \(T_{t_{95\%}}\) in the region was \([3.11,20.82]\) ps, denoting a wide range for the parametric behavior. For the approximation by fitting the data (\(\Delta_{T_{t_{95\%}}}^{2}=3.0\times 10^{-1}\)), Figure 13B provides complementary information about the process comparing the transference times \(T_{t_{95\%}}\) through the parametric values. Once more, high values for \(\lambda\) appear to have little effect in transfer time, comparable with the behaviour observed for this same parameter in half-life coherence (Figure 9). Instead, \(T_{t_{95\%}}\) values increases with \(T\) and \(\gamma\). On the other hand, transfer times show a better performance (lower \(T_{t_{95\%}}\)) at lower values of \(T\) while are combined with small values for \(\gamma\). Parametric characterization of FMO complexes in the quantum regime becomes important to understand the multivariable functioning of those structures no depends monotonically on \(T\). In fact, FMO complexes are in the chemical mesoscale, thus exhibiting lots of variants in their  
Figure 13: (**A**) Efficiency \(\eta\) in the entire dynamic, and (**B**) transference time \(T_{t_{95\%}}\) as function of \(T,\lambda_{k}\) and \(\gamma_{k}\). structure.