## 3 Quantum Modelling of FMO Quantum Excitation
### Parametric Dynamics of Populations during the BChls Excitation  
The interest on the parametric analysis of some remarkable properties of the quantum regime lays on the precise knowledge for those parameters, whose combination sets notable differences in such quantum properties. Moreover, a set of well differentiated values for those parameters are reported in the literature for several species and strains, some of them exhibiting tiny differences in their compared behavior. Thus, parametric differences are becoming important for issues as: bio-inspired resources in quantum information, genetic manipulation of bacteria modifying such behavior, development of control on the FMO complex as technological resource, etc. Thus, a more deep understanding of the parametric dependence on the FMO complex quantum behavior is beginning a new era to manipulate biological resources for quantum applications. To the best of our knowledge, no previous analysis has been performed with such extent on those parameters despite interesting analysis has been performed as in [55] with the purpose of getting control on this nano-structure for quantum processing purposes. The phononic interactions with the bath are crucial, they are resonance modes whose wavelengths are not easily observed directly from the experiment instead to its associated reorganization energy \(\lambda_{k}\) in (5). An interesting issue about this parameter is the fact that while the absorption wavelength peaks \(\lambda_{\mathrm{absorption}}\) for the BChls in the FMO complex of green sulfur bacteria ranges approximately from \(830\,\mathrm{nm}\) to \(840\,\mathrm{nm}\) as they are observed in the spectroscopy absorption spectrum [56,57], instead, the characteristic excitonic wavelength \(\lambda_{\alpha}\) for the largest phononic frequency (\(\alpha=0\)) can be estimated as \(\lambda_{0}\approx\frac{2\pi c}{\omega_{0}}=\frac{2\pi\hbar c}{2\lambda_{k}}= 142.9\,\mathrm{\SIUnitSymbolMicro m}\) (for \(\lambda_{k}=35\,\mathrm{cm}^{-1}\) changing the spectroscopy unities discussed previously), a very larger value compared with the characteristic absorption wavelengths but still meaningful to produce the decoherence, thus limiting the time of the process then re-initializing the process. Figure 7 shows some emblematic results obtained for the population evolution of each of the 7 BChls being present in the FMO monomer simply plotted against time, as it is customary in the most works [38; 43; 51]. With an excitation initially located only in the BCh1 \(\rho(0)=|1\rangle\langle 1|\), Figure 7 shows only the first 4 ps despite simulations considered more extended time periods until the equilibrium. It corresponds to the period immediately after excitation energy is captured by coupled BChls 1 and 2. It is important to note that there is another possible path through BChl 6 initial excitation (see Figure 6), but here, we are specifically modeling an excitation coming through BChl 1. One can appreciate the excitation declines in pigments 1 and 2 while it is transferred towards BChls 3 and 4, in accordance with the energy flows presented in Figure 6. Lower temperatures yield longer sustained coherent oscillation periods although these are not directly related with the population oscillations seen close to \(t=0\). All plots shown in Figure 7 belong to the fixed parameter \(\gamma=50\,\mathrm{cm}^{-1}\) contrasting the stabilization times between \(\lambda=35\,\mathrm{cm}^{-1}\) and \(\lambda=65\,\mathrm{cm}^{-1}\) at three distant temperatures (\(77\,\mathrm{\SIUnitSymbolCelsius},185\,\mathrm{\SIUnitSymbolCelsius}\) and \(293\,\mathrm{\SIUnitSymbolCelsius}\)) which presented general variations around of \(1/100\,\mathrm{ps}\) advisable in the length of the initial excitation time. In the following section we gather all simulations to get information about the parametric behaviour of those processes inside of each FMO monomer.