## 3 Quantum Modelling of FMO Quantum Excitation
### Superoperator Version of Master Equations  
In order to take computational advantage, master equations as (18), (23) and still (28) can be written simpler by switching them into the so called superoperator approach. In fact, by applying the rule:  
\[(U\rho V)_{ij}=\sum_{k,l}U_{kl}\rho_{kl}V_{lj}=\sum_{k,l}(U\otimes V^{T})_{ij, kl}\rho_{kl} \tag{29}\]  
in each term of (18), (23) and (28), we can jump \(\rho\) on the right through other operators by switch it into a supervector of size \(N^{2}\) (if \(\rho\) is a \(N\times N\) matrix). The remaining operators in the expressions become comprised in a superoperator \(\kappa^{(M)}\) (\(M=L,R\) or \(H\) for Lindblad, Redfield and HEOM, respectively) of size \(N^{2}\times N^{2}\) obtained through the tensor product \(\otimes\) as:  
\[\dot{\vec{\rho}}_{S}=\kappa^{(M)}\vec{\rho}_{S} \tag{30}\]  
Note that this single transformation allows to transform any of the three quantum master equations into a linear single matrix differential equation, easy to implement computationally despite its size. Appendix D develops the operative process applied to the HEOM method, thus providing a tool to translate it into a matrix differential equation.  
Numerical Solving of HEOM Models and Analysis of Physical Measures to Feature Parametrically the Quantum Regime in the FMO Complex  
As it was previously mentioned in Section 2.2 and they are illustrated in Figure 4, there are eight BChl molecules embedded in the protein scaffold of each FMO monomer. Even though this has been known and confirmed since over a decade [1; 11; 12], most of the publications related to the theoretical calculations of energy dynamics within the FMO have only considered the seven originally identified BChl sites [14; 41; 43; 47]. This is due to the fact that some models were published before the eight BChl was located, but also more recent studies continue considering only seven BChls in their models under the premises that the eighth BChl is weakly bound to each monomer and is relatively far away from the other seven pigments [47]. We will come back to analyze briefly this fact at the end of the article. It is believed that this BChl increases the overall energy-transfer efficiency of the complex acting as a bridge between the baseplate pigments and the core BChls [12]. In this study, we deliberately decided to consider first only the core seven BChls as we aim to compare our analysis with those previously reported. Such tracks consider the BChls 1 or 6 as detonators of dynamics. As we will see, those initial conditions rise two different process, each one corresponding to each side (left-right) of interactions depicted in Figure 6. On other report, we have already considered the eighth BChl in the dynamics following the entire model [52]. Thus, after to present the different approaches and lines of research in the study of FMO complex dynamics in the quantum regime, we arrive at the main topic of our analysis, the parametric behavior of such dynamics. The extensive analysis has been developed departing from the initial condition \(\rho(0)=|1\rangle\langle 1|\) instead \(\rho(0)=|6\rangle\langle 6|\) also, because the first is considered as the main entry of excitation energy for many authors. Nevertheless, in the final section, we will come back briefly to this other pathway considering in addition the case of eight BChls.