## 3 Quantum Modelling of FMO Quantum Excitation
### Entanglement Generation Depending of \(T,\lambda_{k}\) and \(\gamma_{k}\)
This is due to the fast excitation is being transferred to these pairs coupled pigments (see Figure 6). Note how the increasing of \(\lambda\) slows and reduces the entanglement process particularly for larger temperatures as seen in Figure 11B. BChl 4 is an exception of such behavior particularly notable for the largest temperatures. Results shown in Figure 11 can be also traced back to Figure 6, where we can relate the entanglement effects with the energy transfer paths. Figure 11: Entanglement color chart for \(\gamma_{k}=50\,\mathrm{cm}^{-1}\) with (**A**) \(\lambda_{k}=35\,\mathrm{cm}^{-1}\), and (**B**) \(\lambda_{k}=65\,\mathrm{cm}^{-1}\). In the center, concurrence \(\mathcal{C}_{\{k\}}\) evolution for each BChl, and in the outer tracks \(\mathcal{C}_{\{kl\}}\) between different pairs in the crossing of each circular track and each angular sector. Time grows counterclockwise in \([0,2]\) ps on each angular sector. Temperature grows radially outwards on each track in \([77,347]\,^{\circ}\)K.  
Table 1 also reports the maximum concurrence that each BChl presents with the rest of the system \(\mathcal{C}_{\{k\}}\) at the lowest and highest temperature conditions during the evolution process (columns three to four for \(\gamma=50\,\mathrm{cm}^{-1},\lambda=35\,\mathrm{cm}^{-1}\) and six to seven for \(\gamma=50\,\mathrm{cm}^{-1},\lambda=65\,\mathrm{cm}^{-1}\)). Columns five and eight of the table enumerates for each BChl with which of other BChl pigments presents the highest value of concurrence \(\mathcal{C}_{\{k\}}\) (in parenthesis). Note the dominant entanglement with BChl 1 for \(\lambda=35\,\mathrm{cm}^{-1}\). Another important characterization in the parametric behavior of the FMO is given by the relation between the coherence \(C_{l_{1}}(\rho)\) and each concurrence \(\mathcal{C}_{\{k\}}\): \(C_{l_{1}}(\rho)=\sum_{k<l}\mathcal{C}_{\{k\}}\). In such terms, coherence could be seen partially as sustained by the entanglement between pairs. Thus, we perform an analysis to advise how the coherence is sustained by certain pair through the excitation time in the Figure 12. For several temperatures depicted in color in agreement with the color-bar besides, the evolution of \(C_{l_{1}}(\rho)\) (the first 2.5 ps) shows its decay transitioning from the entanglement between pairs \((1,2)\)--solid line--and \((3,4)\)--dashed line--as main providers of concurrence. Lower temperatures first slow such a transition except for higher ones, while a larger \(\lambda\) (see panels A and B) accelerates such a transition. As it was previously commented in the previous subsection, higher concurrences are present for lower temperatures. Thus, during the quantum regime, the evolution is dominated by two processes of entanglement in the system involving mainly to those pairs of BChls, despite clearly all of them share the entanglement process in a lower level in the multipartite interaction. Parameters as \(T\) and \(\lambda\) boost or slow each one of those main entanglement interactions, while the level of coherence becomes mainly dependent from \(T\) and any delay appears as function of those parameters as is observed in Figure 12, remarking the dominance for the process established on the left-side of scheme on Figure 6.