## 3 Quantum Modelling of FMO Quantum Excitation
### Parametric Dynamics of Populations during the BChls Excitation
Some comparative outcomes should be commented at this point in spite the most approaches go through the dynamics simulation presenting similar plots as those of Figure 7, with variants in the modeling, initial conditions, complementary quantum systems (as a generic ground state or a sink representing the RC as in [47]) or still the master equation being used. Thus, outcomes included in the Figure 7 are consistent with those presented in [51; 52] using HEOM method but exploring different initial conditions or considering insight parametric dependence with \(N=8\) BChls. Initial oscillations in the main interactive BChls 1 and 2 are notably present in those models. Equilibrium populations are comparable due they depend mainly from \(T\). Outcomes generated using Redfield master equation could undergo to the failure of \(\mathrm{Tr}(\rho)=1\) as it has been remarked in [36], while the HEOM method, under the correct convergence, exhibits a correct behavior inherited from is formulation [49]. As it was previously remarked, the use of Lindblad master equations comprising the main observable features in the FMO dynamics, successfully reproduces it in [47], not showing the extended initial oscillations as product the markovianity and closer to the Secular approximation compared in [36; 58] with the Lindblad formulation combined with Redfield in the analysis of a single dimer interaction inside of the FMO complex. Figure 7: Population evolution in \([0,4]\) ps for the 7 BChls in the FMO monomer with \(\gamma_{k}=50\,\mathrm{cm}^{-1}\). \(\lambda_{k}=35,65\,\mathrm{cm}^{-1}\) reported in columns, and \(T=77,185,293\,^{\circ}\mathrm{K}\) in rows. Note the differences in the coherence times, transference times and oscillation amplitudes. **(A)**\(\lambda_{k}=35\,\mathrm{cm}^{-1},T=77\,^{\circ}\mathrm{K}\); **(B)**\(\lambda_{k}=65\,\mathrm{cm}^{-1},T=77\,^{\circ}\mathrm{K}\); **(C)**\(\lambda_{k}=35\,\mathrm{cm}^{-1},T=185\,^{\circ}\mathrm{K}\); **(D)**\(\lambda_{k}=65\,\mathrm{cm}^{-1},T=185\,^{\circ}\mathrm{K}\); **(E)**\(\lambda_{k}=35\,\mathrm{cm}^{-1},T=293\,^{\circ}\mathrm{K}\); **(F)**\(\lambda_{k}=65\,\mathrm{cm}^{-1},T=293\,^{\circ}\mathrm{K}\). Quantum Populations Reached in the Equilibrium Depending of \(T\), \(\lambda\) and \(\gamma\)  
The population evolution data obtained from the parametric analysis were simulated until the stabilization time was reached (the time when system populations reach asymptotic values transferring all energy to the final BChls). Then, asymptotic populations \(\rho_{ii_{\infty}}\equiv\lim_{t\to\infty}\rho_{ii}(t)\) were fitted through the method of maximum likelihood to be represented by the surface contours shown in Figure 8, with complementary information reported in the Table 1. The average square distance between the simulation data and the outcoming approximation function falls in the range from the extreme values \(\Delta^{2}_{\rho_{22\infty}}=2.2\times 10^{-5}\) (best) to \(\Delta^{2}_{\rho_{33\infty}}=1.6\times 10^{-3}\) (worst). Unlike Figure 7, Figure 8 represents the final state of the system when this stabilization time was reached. Panels A, B and C, respectively correspond to populations \(\rho_{11}\), \(\rho_{33}\) and \(\rho_{44}\), showing the populations with the greatest range of variability (other populations with less variability are included together in the fourth panel, D-G). Their corresponding insets show alternatively the continuous parametric dependence on the parameters for each final state in color in the common scale 0 (red) to 1 (blue). BChl 3 is clearly the pigment more populated by the time of system stabilization in all cases. Each plot in Figure 8 exhibits a series of surface contours for each approximated function with the \(\rho_{ii_{\infty}\equiv\lim_{t\to\infty}\rho_{ii}(t)}\) values given in color in the side scale on the left. The green, yellow and red mesh lines drawn on each contour surface in the Figure 8 corresponds to cuts of constant values of \(T\) (green), \(\gamma\) (yellow) and \(\lambda\) (red), respectively.