Dynamics and quantumness of excitation energy transfer through a complex quantum network

Understanding the mechanisms of efficient and robust energy transfer in organic systems provides us with insights for the optimal design of artificial systems. In this paper, we explore the dynamics of excitation energy transfer (EET) through a complex quantum network by a toy model consisting of three sites coupled to environments. We study how the coherent evolution and the noise-induced decoherence work together to reach efficient EET and illustrate the role of the phase factor attached to the coupling constant in the EET. By comparing the differences between the Markovian and non-Markovian dynamics, we discuss the effect of environment and the spatial structure of system on the dynamics and the efficiency of EET. A intuitive picture is given to show how the exciton is transferred through the system. Employing the simple model, we show the robustness of EET efficiency under the influence of the environment and elucidate the important role of quantum coherence in EET. We go further to study the quantum feature of the EET dynamics by quantumness and show the importance of quantum coherence from a different perspective. We calculate the energy current in the EET and its quantumness, and results for different system parameters are presented and discussed.

Figure 1

The schematic representation of the three-sites network. The exciton, initially in site $\left|1\right.〉$, is transferred to site $\left|2\right.〉$ and finally trapped by the sink $\left|3\right.〉$.

Figure 2

The transfer efficiency $P$: (a) $P$ vs $\gamma$ and $\Delta$, (b) $P$ vs $\Gamma$, (c) $P$ vs ${\Gamma }_{12}$ and ${\Gamma }_{21}$, and (d) $P$ vs ${\Gamma }_s$. Unless otherwise noted, the other parameters chosen are $\Gamma =1,\gamma =0.1,J=1,{\Gamma }_s=1,{\Gamma }_{12}={\Gamma }_{21}=1$, and $\Delta =0$. In (d), the parameters are as follows: $J=0,{\Gamma }_{12}={\Gamma }_{21}=1$ for thick orange-dashed, $J=0.5,{\Gamma }_{12}={\Gamma }_{21}=1$ for thin orange-dashed, $J=1.5,{\Gamma }_{12}={\Gamma }_{21}=1$ for thick green-dotted, $J=3,{\Gamma }_{12}={\Gamma }_{21}=1$ for thin green-dotted, $J=4,{\Gamma }_{12}={\Gamma }_{21}=1$ for black-solid, $J=4,{\Gamma }_{12}={\Gamma }_{21}=1.2$ for thick red-dot-dashed, ${\Gamma }_{12}={\Gamma }_{21}=1.4$ for thin red-dot-dashed.

Figure 3

The transfer efficiency $P$: (a) $P$ vs ${\delta }_1$ and ${\delta }_2$, (b) $P$ vs $p$, (c) $P$ vs $\alpha$ and $\beta$ $\left(\phi =\frac{\pi }{3}\right)$, (d) $P$ vs $\alpha$ and $\phi$ $\left(\beta =\frac{\pi }{3}\right)$, and (e) $P$ vs $\beta$ and $\phi$ $\left(\alpha =\frac{\pi }{3}\right)$. Unless otherwise addressed, the other parameters chosen are $\Gamma =1,\gamma =0.1,J=1,{\Gamma }_s=1,{\Gamma }_{12}={\Gamma }_{21}=1$, and $\Delta =0$.

Figure 4

Time-dependent decoherence rates from Eqs. (20), (21), (22), and (23), and time evolution of population on each site from Eq. (17) for Markovian and non-Markovian dynamics. The left panels [Figs. 4–4] show the time evolution of relaxation (upper panel) and dephasing (lower panel) rates. The parameters in the spectral density are ${\omega }_c$ = 50 ${\mathrm{cm}}^{-1}$ and $\lambda$ = 50 ${\mathrm{cm}}^{-1}$, as used in Ref. [35]. The relaxation rates (blue-dotted line for ${\omega }_{12}$ and green-dot-dashed for ${\omega }_{21})$ oscillate; they may take a negative value and converge to values in the Markovian limit (red-solid line for ${\omega }_{12}$, the purple-dashed line is for ${\omega }_{21})$. Similarly, the dephasing rates in the non-Markovian case (blue solid) start with zero and converge to these in the Markovian limit (red dashed). Figures 4–4 show the population on each site as a function of time: ${\rho }_{00}$ in purple-dashed, ${\rho }_{11}$ in blue-dotted, ${\rho }_{22}$ in green-dot-dashed, and ${\rho }_{33}$ in red-solid line. (e)–(h) are plotted for the Markovian case, and (i)–(l) for the non-Markovian case. The temperatures $T$ and phase $\phi$ change from figure to figure. The population on the reaction center ${\rho }_{33}$ with the same temperatures $T$ is also illustrated (red-solid line for $\phi =0$, orange-dashed for $\phi =\pi /2$, black-dot-dashed for $\phi =\pi )$. The Hamiltonian is taken to be the same as that in Ref. [13]. The dissipation and trapping rates we chose are ${\Gamma }_1$ = 0.1 ${\mathrm{ps}}^{-1},{\Gamma }_2$ = 0.1 ${\mathrm{ps}}^{-1}$, and ${\Gamma }_s$ = 10 ${\mathrm{ps}}^{-1}$.

Figure 5

Time evolution of the coherence elements (Re ${\rho }_{12}$ red-dashed line, Im ${\rho }_{12}$ blue-solid line) from Eq. (17) for Markovian and non-Markovian dynamics. The excitonic Hamiltonian is taken to be the form equal to that in Ref. [13]. The parameters are $\lambda$ = 50 ${\mathrm{cm}}^{-1},{\omega }_c$ = 50 ${\mathrm{cm}}^{-1}$, phase $\phi =0$, and room temperature $T$ = 300 K. The dissipation and trapping rates we chose are ${\Gamma }_1$ = 0.1 ${\mathrm{ps}}^{-1},{\Gamma }_2$ = 0.1 ${\mathrm{ps}}^{-1}$, and ${\Gamma }_s$ = 10 ${\mathrm{ps}}^{-1}$.

Figure 6

The transfer efficiency $P$ as a function of different temperatures, the phases $\phi$, and the main decoherence parameters (reorganization energy $\lambda$, cutoff ${\omega }_c)$ for Markovian (red-dashed) and non-Markovian (blue-solid) cases. The parameters are ${\omega }_c$ = 30 ${\mathrm{cm}}^{-1},\lambda$ = 30 ${\mathrm{cm}}^{-1}$, phase $\phi =0$, which is typical for some natural energy transfer systems such as chromophores in photosynthetic systems, and the temperature is $T$ = 300 K. The dissipation rate ${\Gamma }_{1,2}$, trapping rate ${\Gamma }_s$, and the excitonic Hamiltonian take the same values as in Fig. 4. In (b), the thick red-solid line indicates Markovian and the thin blue-solid line the non-Markovian with the initial state ${\rho }_{11}\left(0\right)=1$, and the red-dashed line indicates Markovian and the blue-dot-dashed non-Markovian dynamics with the initial state a superposition state $\left|\psi (t=0)\right.〉=\frac{1}{2}\left|1\right.〉+\frac{\sqrt{3}}{2}\left|2\right.〉$.

Figure 7

Energy current $〈j_{12}\left(t\right)〉$ (top) and corresponding quantumness $Q_{12}\left(\rho \right)$ (bottom) for the case of Markovian (red-solid line for $〈j_{12}\left(t\right)〉$ and blue-dashed for $Q_{12}\left(\rho \right))$ and non-Markovian (black-dashed line for $〈j_{12}\left(t\right)〉$ and green-solid for $Q_{12}\left(\rho \right))$ dynamics with different values of temperature $T$. Dissipation rate ${\Gamma }_{1,2}$, trapping rate ${\Gamma }_s$, and spectral density parameters $\lambda ,{\omega }_c$ in the excitonic Hamiltonian take the same values as in Fig. 4.