## 3 Quantum Modelling of FMO Quantum Excitation
### Construction of the Entire Hamiltonian to Model FMO Complex  
Because of the experimental analysis, each BChl\({}_{i}\) becomes in two most common excitation states denoted by \(|0_{i}\rangle\) (the ground state) and \(|1_{i}\rangle\) (the first excited state). Such most common states for each single BChl set a two level restricted system for each BChl. Together, the tensor product of them could set a basis for the entire set of \(N\) BChls (\(N=7,8\)). This notable property has raised the attention of quantum information science on the complex. Nevertheless, as it has been proved [41], the most common global state corresponds to the superposition of single excited BChls, \(|k\rangle\equiv|0_{1}0_{2}...1_{k}...0_{N}\rangle\), called the occupation or site basis. Then, \(H_{S}\) is written as:  
\[H_{S} = \sum_{i=1}^{N}E_{i}|i\rangle\langle i|+\sum_{1\leq i,j\leq N}J_{ ij}(|i\rangle\langle j|+|j\rangle\langle i|) \tag{2}\] \[= \sum_{i=1}^{N}\epsilon_{i}|\epsilon_{i}\rangle\langle\epsilon_{i}| \tag{3}\]  
where \(|\epsilon_{i}\rangle\) are the eigenstates of energy in the BChls system, the excitonic basis. Those Hamiltonians have been reported in the literature for \(N=7\)[13] and \(N=8\)[1], both are included in Appendix A. While, the structure surrendering such BChls, the protein structure scaffolding such chemical structures (Figure 5B) has a lower level of interactions but still meaningful on the BChls. For instance, as we will discuss below, characteristic wavelengths exciting the set of BChls through dipole-dipole interactions represent energies of order of \(1.2\times 10^{4}\) cm\({}^{-1}\) which are compared with the reorganization corrections due to the interactions with the protein scaffolding in the order of just 35\(\sim\)65 cm\({}^{-1}\) (using the common spectroscopy units of energy in cm\({}^{-1}\) discussed in the following subsections). Thus, it is commonly approached as a phononic medium. It means quantum vibratory modes on the mechanical structure exchange energy with the set of BChls. Due to the complexity of such a scaffolding structure, it is normally modeled as a continuum medium of oscillators working additionally as a thermal bath, \(B\), which barely reproduces the whole phenomena present there. The Hamiltonian for the bath \(H_{B}\) is then considered as a set of oscillators of mass \(m_{\alpha}\) for each BChl. In terms of their positions \(q_{k,\alpha}\) and momenta \(p_{k,\alpha}\)[46], their global Hamiltonian is:  
\[H_{Bk}=\frac{1}{2m_{\alpha}}\sum_{\alpha}(p_{k,\alpha}^{2}+m_{\alpha}^{2} \omega_{\alpha}^{2}q_{k,\alpha}^{2}) \tag{4}\]which is written in terms of the modes \(\omega_{\alpha}\) of such oscillators. Each oscillator has the possibility to switch the BChl state \(|k\rangle\) in \(H_{S}\). Due to the interaction with each BChl, such oscillators become displaced into a new equilibrium position \(q^{\prime}_{k,\alpha}=q_{k,\alpha}-d_{k,\alpha}\). Thus, under such an interaction:  
\[H_{Bk} \rightarrow \frac{1}{2m_{\alpha}}\sum_{\alpha}(p_{k,\alpha}^{2}+m_{\alpha}^{2 }\omega_{\alpha}^{2}(q_{k,\alpha}-d_{k,\alpha})^{2}) \tag{5}\] \[=\frac{1}{2m_{\alpha}}\sum_{\alpha}(p_{k,\alpha}^{2}+m_{\alpha}^{ 2}\omega_{\alpha}^{2}q_{k,\alpha}^{2})+\frac{1}{2}\sum_{\alpha}\hbar\omega_{ \alpha}d^{\prime 2}_{k,\alpha}-\sum_{\alpha}\hbar\omega_{\alpha}d^{\prime}_{k, \alpha}\frac{q_{k,\alpha}}{\lambda_{\alpha}}\] \[\equiv H_{Bk}+H_{\text{reloc}k}+H_{S-Bk} \tag{6}\]  
where the following relations were considered: \(m_{\alpha}c^{2}=\hbar\omega_{\alpha},c=\omega_{\alpha}\lambda_{\alpha}\) (for the mass, the frequency and the wavelength of each \((k,\alpha)\) phononic mode). Together, \(d^{\prime}_{k,\alpha}=d_{k,\alpha}/\lambda_{\alpha}\) is the dimensionless displacement from each equilibrium configuration. Then, the first term represents the bath energy. The second term is a reorganization Hamiltonian modifying the diagonal elements of \(H_{S}\). Third term is the interaction between the system and the bath (with a linear response).