## 3 Quantum Modelling of FMO Quantum Excitation
### LHCs As Open Quantum Systems
#### 3.3.3 HEOM Method Approach in FMO Complexes Models  
The Hierarchical Equations of Motion (HEOM) method [49] is a non-markovian approach including the features of previous models, thus considered better than the previous ones which are markovian. It is considered and constructed as \(D\) recursive equations thus considering \(D\) previous temporal stages for the memory of the bath. Then, such stages are labeled each one with a vector \(\mathbf{n}\) of dimension \(N\) considering each chromophore and its interaction with the bath. Thus, only \(\rho_{\mathbf{n}}\) with \(\mathbf{n}=(0,...,0)\) is the real density matrix of the system \(\rho_{S}\) while other \(\rho_{\mathbf{n}}\) are just auxiliary matrices regarding the memory. This set of matrices have an associated degree \(s=0,1,...,D\) corresponding to all vectors \(\mathbf{n}=(\mathbf{n}_{1},\mathbf{n}_{2},...,\mathbf{n}_{N})\) with \(0\leq\mathbf{n}_{k}\in Z^{+}\cup\{0\}\) such as \(\sum_{k=1}^{N}\mathbf{n}_{k}=s\). Thus, the following method gathers the main features than its markovian relatives, Redfield or Lindblad quantum master equations. In fact, it includes the main Hamiltonian \(H_{S}\), the relocalization correction, the trapping from BChls 3 and 4, together with other recursive non-markovian terms obtained from its derivation [49] which includes a treatment for the equilibrium:  
\[\dot{\rho_{\mathbf{n}}} = -\frac{i}{\hbar}[H_{S}+\sum_{k=1}^{N}\lambda_{k}|k\rangle\langle k |,\rho_{\mathbf{n}}]-\sum_{k=1}^{N}\mathbf{n}_{k}\gamma_{k}\rho_{\mathbf{n}}- r_{\text{trap}}(\{|3\rangle\langle 3|,\rho_{\mathbf{n}}\}+\{|4\rangle\langle 4|, \rho_{\mathbf{n}}\}) \tag{28}\] \[+i\sum_{k=1}^{N}\Delta_{k}([V_{k},\rho_{\mathbf{n}_{k+}}]+\mathbf{ n}_{k}([V_{k},\rho_{\mathbf{n}_{k-}}]-i\frac{\beta\hbar\gamma_{k}}{2}\{V_{k}, \rho_{\mathbf{n}_{k-}}\}))\]  
where \(k\) is the Boltzman constant and \(\beta=kT\). Together, \(V_{k}=|k\rangle\langle k|\) and \(\Delta_{k}=\frac{\lambda_{k}}{\beta\hbar^{2}\gamma_{k}}\) (note particularly the reorganization term). In addition, if \(\mathbf{n}\) is a vector of order \(s\), then \(\mathbf{n}_{k\pm}\) is the vector of order \(s\pm 1\) similar to \(\mathbf{n}\) with only their component \(\mathbf{n}_{k}\) increased (decreased) by one. \(\gamma_{k}\) is the bath cutoff frequency reflecting the non-markovian nature of the bath, the rate that information flows from the system to the bath [50]. Thus, \(\gamma_{k}\) comprises the interaction strengths between the bath and each BChl. The initial condition for such a system is the corresponding to \(\rho_{S}\) at \(t=0\) being unentangled with the bath (typically \(\rho_{S}(0)=|k\rangle\langle k|\) with \(k=1,6\) if \(N=7\) or \(k=8\) if \(N=8\) in agreement with the behavior of the set of BChls in each monomer). Last model has been considered by several authors [38; 51] for \(N=7\). Because it has been observed that BChls 1, 6 and 8, works as FMO antennas while BChls 3 and 4 drive the energy oscillations to the RC at the trapping rate, \(r_{\text{trap}}\). Thus, the first ones are considered as initial values in the equations and the second ones within a term in (28). While, other \(\rho_{\mathbf{n}}(0)\) for \(\mathbf{n}\
eq\mathbf{0}\) are typically settled as the zero matrix. In the same way, if \(\mathbf{n}=\mathbf{0}\) is considered in (28), note that \(\mathbf{n}_{k-}\) does not appear in the equation because the coefficient \(\mathfrak{n}_{k}=0\). Otherwise, if \(\mathbf{n}\) is of order \(D\) then we will take \(\mathbf{n}_{k+}=\mathbf{0}\) as cut-off. At this point, the main models to afford the computer simulations for the FMO have been considered to arrive in the central master equation used in our further development: HEOM method (still valid in the non-markovian regime). Those methods are used to supply complementary aspects not yet affordable with experimental techniques in order to extend their operative behavior under natural conditions not directly observed in the experimental realm.