Classical master equation for excitonic transport under the influence of an environment

In a previous paper [Phys. Rev. E 83, 051911 (2011)] we have shown that the results of a quantum-mechanical calculation of electronic energy transfer (EET) over aggregates of coupled monomers can be described also by a model of interacting classical electric dipoles in a weak-coupling approximation, which we referred to as the realistic coupling approximation (RCA). The method was illustrated by EET on a simple linear chain of molecules and also by energy transfer on an arrangement of monomers corresponding to that of the Fenna-Matthews-Olson (FMO) complex relevant for photosynthesis. The study was limited to electronic degrees of freedom, since this is the origin of coherent EET in the quantum case. Nevertheless, more realistic models of EET require the inclusion of the decohering effects of coupling to an environment, when the molecular aggregate becomes an open quantum system. Here we consider the quantum description of EET on a linear chain and on the FMO complex, incorporating environment coupling and constructing the classical version of the same systems in the density matrix formalism. The close agreement of the exact quantum and exact classical results in the RCA is demonstrated and justified analytically in the RCA. This lends further support to the conclusion that the coherence properties of EET in the FMO complex are evident at the classical level and should not be ascribed as solely due to quantum effects.

Figure 1

Time-dependent populations of a linear chain when the excitation is localized initially on one monomer “0.” The transport is symmetric with respect to this monomer. Left: $\gamma /V=1$. [For comparison, also the analytic $\gamma =0$ result is shown as a gray curve (dash-dot-dot)]. Right: $\gamma =20V$. Solid black curve: exact quantum calculation. The colored curves are results from the classical calculations for different $\epsilon$. Orange, dash-dot: $\epsilon =40V$. Blue, intermediate dashes: $\epsilon =10V$. Red, dotted: $\epsilon =6V$. Green, long dashes: $\epsilon =1V$. Time is in units of $V/\hslash$.

Figure 2

Left column: Quantum coherences $|{\rho }_{0,1}|$, $|{\rho }_{0,2}|$, and $|{\rho }_{0,3}|$ of a linear chain when initially the excitation is localized on one monomer “0.” The dephasing rate is $\gamma =V$. Middle column: Differences between classical and quantum results for the case $\epsilon /\gamma =40$. Right column: Differences between classical and quantum results for the case $\epsilon /\gamma =6$. Note the different scalings in columns (b) and (c), indicated by the factors in the plots. Time is in units of $V/\hslash$.

Figure 3

(a) Populations of the BChls as function of time when excitation is initially localized on BChl 1 obtained from the full quantum mechanical calculations. (b) Differences between the quantum result and the classical calculation for the actual transition energies. (c) The transition energies are reduced by 12$$000 cm${}^{-1}$. Note the different scales of the $y$ axis and the scaling factors which are indicated in the figures.

Figure 4

As Fig. 3 but now for the coherences of BChl 3 with all the other BChls. Note that in the third row also the diagonal term, i.e., the population of BChl 3, is shown. Note the scaling factors in columns (b) and (c).