Excitation relaxation in a molecular chain and energy transfer at steady state

We consider the reduced dynamics of a molecular chain weakly coupled to a phonon bath. With a small and constant inhomogeneity in the coupling, the excitation relaxation rates are obtained in closed form. They are dominated by transitions between exciton modes lying next to each other in the energy spectrum. The rates are quadratic in the number of sites in a long chain. Consequently, the evolution of site occupation numbers exhibits longer coherence lifetime for short chains only. When external source and sink are added, the rate equations of exciton occupation numbers are similar to those obtained earlier by Fröhlich to explain energy storage and energy transfer in biological systems. There is a clear separation of timescale into a faster one pertaining to internal influence of the chain and phonon bath, and a slower one determined by external influence, such as the pumping rate of the source, the absorption rate of the sink, and the rate of radiation loss. The energy transfer efficiency at steady state depends strongly on these external parameters and is robust against a change in the internal parameters, such as temperature and inhomogeneity. Excitations are predicted to concentrate to the lowest energy mode when the source power is sufficiently high. In the site basis, this implies that when sustained by a high power source, a sink positioned at the center of the chain is more efficient in trapping energy than a sink placed at its end. Analytic expressions of energy transfer efficiency are obtained in the high power and low-power source limit. Parameters of a photosynthetic system are used as examples to illustrate the results.

Figure 1

$\overline{N}$ is the total number of excitations Eq. (43) at steady state. We use the set of reference parameters listed at the beginning of Sec. 7, where $J=100{\text{cm}}^{-1}$, ${\omega }_0=12,500{\text{cm}}^{-1}$, ${\gamma }_d=20{\text{ps}}^{-1}$, ${\gamma }_{\text{s}}=1{\text{ps}}^{-1}$, ${\gamma }_{\text{r}}=0.001{\text{ps}}^{-1}$, and $\eta =0.1$. The symbols denote $\circ =(77,0.001)$, $•=(77,10),$ and $\blacksquare =(300,10)$, where the numbers in brackets denote $(T\text{K},s{\text{ps}}^{-1})$. The sink is prepared at the end of the chain.

Figure 2

Similar symbols denote the same set of parameters. Filled (empty) symbols denote sink at the end (center) of the chain. In all the curves, $J=100{\text{cm}}^{-1}$, ${\omega }_0=12500{\text{cm}}^{-1}$, $T=77$ K, $\eta =0.1,$ and ${\gamma }_d=20{\text{ps}}^{-1}$. The symbols denote $\circ =(1,0.001,0.1)$, $\square =(1,0.01,0.1)$, $◊=(0.1,0.01,0.1)$, and $▵=(0.1,0.01,10)$, where the numbers in brackets refer to $({\gamma }_{\text{s}},{\gamma }_{\text{r}},s)$ in units of ${\text{ps}}^{-1}$.

Figure 3

In all the curves, $J=100{\text{cm}}^{-1}$, ${\omega }_0=12,500{\text{cm}}^{-1}$, $\eta =0.1,{\gamma }_d=20{\text{ps}}^{-1},{\gamma }_{\text{s}}=1{\text{ps}}^{-1}$, and ${\gamma }_{\text{r}}=0.001{\text{ps}}^{-1}$. The symbols denote $\circ =(5,1)$, $•=(5,10)$, $▵=(10,1)$, and $▴=(10,10)$, where the numbers in brackets denote $(\ell ,s{\text{ps}}^{-1})$.

Figure 4

In all the curves, ${\omega }_0=12500{\text{cm}}^{-1}$, $\eta =0.1,{\gamma }_d=20{\text{ps}}^{-1},{\gamma }_{\text{s}}=1{\text{ps}}^{-1}$, and ${\gamma }_{\text{r}}=0.001{\text{ps}}^{-1}$. Numbers in the following brackets refer to $(T\text{K},J{\text{cm}}^{-1},s{\text{ps}}^{-1})$. The symbols denote $•=(7.7,10,{10}^4)$, $\blacksquare =(77,100,{10}^4)$, $▴=(770,1000,{10}^4)$, $\circ =(0.77,10,{10}^3)$, $\square =(7.7,100,{10}^3)$, and $▵=(77,1000,{10}^3)$.

Figure 5

In all curves, $T=77$ K, $J=100{\text{cm}}^{-1}$, ${\omega }_0=12500{\text{cm}}^{-1}$, ${\gamma }_{\text{s}}=1{\text{ps}}^{-1}$, $\eta =0.1,{\gamma }_d=20{\text{ps}}^{-1}$, and $s=1{\text{ps}}^{-1}$. The four groups of curves arranged from the top to the bottom correspond to ${\gamma }_{\text{r}}=0.001,0.01,0.1,1{\text{ps}}^{-1}$, respectively. Within each group, there are five curves labeled by $(\circ ,\square ,◊,▵,▿)$ from the top to the bottom, corresponding to $\xi =0.001,0.01,0.1,1,10$, respectively.

Figure 6

In all curves, $T=77$ K, $J=100{\text{cm}}^{-1}$, ${\omega }_0=12500{\text{cm}}^{-1}$, $\eta =0.1,{\gamma }_d=20{\text{ps}}^{-1}$, ${\gamma }_{\text{s}}=1{\text{ps}}^{-1}$, and ${\gamma }_d=0.001{\text{ps}}^{-1}$. Source power $s$ is in units of ${\text{ps}}^{-1}$, plotted in natural logarithmic scale. Circles and squares denote chains with $\ell =7$ and 15, respectively. Filled and empty shapes label configurations with sinks prepared at the end of the chain and at its center, respectively.