## 3 Quantum Modelling of FMO Quantum Excitation
### LHCs As Open Quantum Systems
#### Lindblad Approach to FMO Complexes Modelling  
Lindblad master equation in its diagonalized operators form:  
\[\dot{\rho}_{S}=-\frac{i}{\hbar}[H_{S},\rho_{S}]+\frac{1}{\hbar^{2}}\sum_{ \alpha}\left(L_{\alpha}\rho_{S}L_{\alpha}^{\dagger}-\frac{1}{2}\{\rho_{S},L_{ \alpha}^{\dagger}L_{\alpha}\}\right) \tag{18}\]  
represents the interaction of one quantum system with an external bath through the generic operators \(S_{j}\) or \(L_{\alpha}\) as algebraic basis of the systems (see Appendix B). For the FMO complexes, despite \(L_{\alpha}\) still comprising the physical information, it becomes limited due to the impossibility to reproduce faithfully more details of the system and bath interaction. A representative proposal performed by [47] in order to adjust this master equation to the FMO problem has been described by a reduced density matrix \(\rho\) for the system of BChls. By including terms in the Schrodinger-Liouville equation for the decoherence due to environmental interaction, recombination of excitation on a common ground state, and transfer of excitation to a sink (explicitly representing the reaction center):  
\[\dot{\rho}=-i\left[H,\rho\right]+\mathcal{L}_{deph}(\rho)+\mathcal{L}_{sink}( \rho)+\mathcal{L}_{recomb}(\rho) \tag{19}\]captures the basic environmental effects used in several previous studies [28; 30; 31]. By introducing directly some operators in the site basis, it has been shown that the description of the Lindblad equation accounts for the main features of the dynamics [30; 36]:  
\[{\cal L}_{deph}(\rho)=2\gamma\sum_{n=1}^{N}\left(|n\rangle\langle n|\rho|n \rangle\langle n|-\frac{1}{2}\{|n\rangle\langle n|,\rho\}\right) \tag{20}\]  
where a site-independent dephasing rate is given by \(\gamma\) and \(\{,\}\) is the anticommutator. Similarly, an irreversible transfer of excitation from the \(m-\)th chromophore to the sink \(|s\rangle\) (an additional subsystem in the modeling, while \(m\) commonly will come \(m=3,4\) BChls) is represented by:  
\[{\cal L}_{sink}(\rho)=2\kappa\sum_{m}\left(|s\rangle\langle m|\rho|m\rangle \langle s|-\frac{1}{2}\{|m\rangle\langle m|,\rho\}\right) \tag{21}\]  
where \(\kappa\) denotes the sink rate. The irreversible loss of excitation due to recombination is given by an analogous term:  
\[{\cal L}_{recomb}(\rho)=2\Gamma\sum_{n=1}^{N}\left(|0\rangle\langle n|\rho|n \rangle\langle 0|-\frac{1}{2}\{|n\rangle\langle n|,\rho\}\right) \tag{22}\]  
with a site-independent recombination rate \(\Gamma\) with the ground state (represented by \(|0\rangle\)). Note that the need for an additional sink \(|s\rangle\) and the ground state \(|0\rangle\) should include two additionally dimensions giving a total of \(N+2\) sites in that model [47]. Note that each additional term in the modeling fits to the form of the generic term \(\frac{1}{\hbar^{2}}\sum_{\alpha}\left(L_{\alpha}\rho_{S}L_{\alpha}^{\dagger} -\frac{1}{2}\{\rho_{S},L_{\alpha}^{\dagger}L_{\alpha}\}\right)\) in the Lindblad equation through the single Lindblad diagonal operators given in the basis \(\{|0\rangle,|s\rangle,|1\rangle,...,|N\rangle\}\). This theoretical model based on the Lindblad equation has been successful in capturing the main properties of the evolution of the system [47]. The relevant environmental parameters are almost universal in the related literature: the dephasing strength \(\gamma\), recombination rate \(\Gamma\) and the sink rate \(\kappa\). Standard values are inferred from experiments, the sink rate \(\kappa=1\,\mathrm{ps}^{-1}\), the recombination rate \(\Gamma=1\,\mathrm{ns}^{-1}\) and the dephasing rate at room temperature \(\gamma=300\,\mathrm{cm}^{-1}\) given in their traditional spectroscopy unities transforming between \(\mathrm{cm}^{-1}\) to \(\mathrm{s}^{-1}\) via the factor \(200\pi\epsilon\) or between \(\mathrm{cm}^{-1}\) to \(\mathrm{J}\) through the factor \(200\pi\hbar c\). The dephasing rate, being a product of the temperature and the derivative of the spectral density, can be estimated by using the experimentally determined parameters of the spectral density [13; 47].