Effects of system-bath coupling on a photosynthetic heat engine: A polaron master-equation approach

Stimulated by suggestions of quantum effects in energy transport in photosynthesis, the fundamental principles responsible for the near-unit efficiency of the conversion of solar to chemical energy became active again in recent years. Under natural conditions, the formation of stable charge-separation states in bacteria and plant reaction centers is strongly affected by the coupling of electronic degrees of freedom to a wide range of vibrational motions. These inspire and motivate us to explore the effects of the environment on the operation of such complexes. In this paper, we apply the polaron master equation, which offers the possibilities to interpolate between weak and strong system-bath coupling, to study how system-bath couplings affect the exciton-transfer processes in the Photosystem II reaction center described by a quantum heat engine (QHE) model over a wide parameter range. The effects of bath correlation and temperature, together with the combined effects of these factors are also discussed in detail. We interpret these results in terms of noise-assisted transport effect and dynamical localization, which correspond to two mechanisms underpinning the transfer process in photosynthetic complexes: One is resonance energy transfer and the other is the dynamical localization effect captured by the polaron master equation. The effects of system-bath coupling and bath correlation are incorporated in the effective system-bath coupling strength determining whether noise-assisted transport effect or dynamical localization dominates the dynamics and temperature modulates the balance of the two mechanisms. Furthermore, these two mechanisms can be attributed to one physical origin: bath-induced fluctuations. The two mechanisms are manifestations of the dual role played by bath-induced fluctuations depending on the range of parameters. The origin and role of coherence are also discussed. It is the constructive interplay between noise and coherent dynamics, rather than the mere presence or absence of coherence or noise, that is responsible for the optimal heat engine performance. In addition, we find that the effective voltage of QHE exhibits superior robustness against the bath noise as long as the system-bath coupling is not very strong.

Figure 1

(a) Arrangement of six core-pigments in the PSII RC. It consists of a special pair ${\mathrm{P}}_{\mathrm{D}1},{\mathrm{P}}_{\mathrm{D}2}$, two accessory chlorophylls $\mathrm{Ch}{\mathrm{l}}_{\mathrm{D}1}, \mathrm{Ch}{\mathrm{l}}_{\mathrm{D}2}$, and two pheophytins $\mathrm{Ph}{\mathrm{e}}_{\mathrm{D}1}, \mathrm{Ph}{\mathrm{e}}_{\mathrm{D}2}$. (b) Exciton- and charge-transfer processes for ${\left(\mathrm{Ch}{\mathrm{l}}_{\mathrm{D}1}\mathrm{Ph}{\mathrm{e}}_{\mathrm{D}1}\right)}^{*}$ pathway (Path 1). (i) The lowest energy configuration with all six pigments in neutral ground state after an electron has been replenished at the special pair. (ii) Exciton state ${\left({\mathrm{P}}_{\mathrm{D}1}\right)}^{*}$ after absorption of a photon. (iii) Exciton state ${\left(\mathrm{Ch}{\mathrm{l}}_{\mathrm{D}1}\mathrm{Ph}{\mathrm{e}}_{\mathrm{D}1}\right)}^{*}$ promoted by exciton transfers from ${\left({\mathrm{P}}_{\mathrm{D}1}\right)}^{*}$. (iv) Primary charge-transfer state ${\mathrm{Chl}}_{\mathrm{D}1}^{+}{\mathrm{Phe}}_{\mathrm{D}1}^{-}$ after $\mathrm{Ch}{\mathrm{l}}_{\mathrm{D}1}$ as the electron donor rapidly loses an electron to the nearby electron acceptor molecule $\mathrm{Ph}{\mathrm{e}}_{\mathrm{D}1}$. (v) Secondary charge-transfer state ${\mathrm{P}}_{\mathrm{D}1}^{+}{\mathrm{Phe}}_{\mathrm{D}1}^{-}$ after the positive and negative charges are spatially separated. (vi) Positively charged state ${\mathrm{P}}_{\mathrm{D}1}^{+}{\mathrm{Phe}}_{\mathrm{D}1}^{}$ after an electron has been released from the system to perform work.

Figure 2

Schemes of the QHE model based on PSII RC. HTB denotes the high-temperature photon bath from sunlight, while LTB1, LTB2, and LTB3 stand for the low-temperature phonon baths attributed to molecular vibrational degrees of freedom. HTB induces the transition from the ground state $|0\rangle$ to the single-exciton states $|1\rangle$. The exciton transfer denoted by the transition between $|1\rangle$ and $|2\rangle$ with interpigment coupling $J$ is subjected to LTB3. LTB1 induces the transition from $|2\rangle$ to the charge-separated state $|\alpha \rangle$ with the excess energy radiated as a phonon, phenomenologically representing the primary and secondary charge-transfer process in which the positive and negative charges are spatially separated. Then the electron is released from state $|\alpha \rangle$ resulting in a current from $|\alpha \rangle$ to $|\beta \rangle$. LTB2 induces the transition from the positively charged state $|\beta \rangle$ to the ground state $|0\rangle$ which brings the electron back to the system with emission of a phonon with excess energy.

Figure 3

Franck–Condon factor $\kappa$ as a function of temperature $T$ and system-bath coupling $\gamma$ with cross-correlation coefficient $c=0$.

Figure 4

Coherence of exciton-dimer system as a function of temperature $T$ and system-bath coupling $\gamma$. (a) Real part of off-diagonal density-matrix element: $A_{12}=\mathrm{Re}\left[{\rho }_{12}\right]$. (b) Imaginary part of off-diagonal density-matrix element: $B_{12}=\mathrm{Im}\left[{\rho }_{12}\right]$. The cross-correlation coefficient $c=0$.

Figure 5

Delocalization length of the exciton-dimer system as a function of temperature $T$ and system-bath coupling $\gamma$. The cross-correlation coefficient $c=0$.

Figure 6

Steady-state (a) $j-V$ and (b) $P-V$ characteristics of QHE as a function of system-bath coupling strength $\gamma$ with cross-correlation coefficient $c=0$ and temperature $T=300$ K.

Figure 7

Steady-state (a) $j-V$ and (b) $P-V$ characteristics of QHE as a function of bath cross-correlation coefficient $c$ [defined by Eq. (7)] with system-bath coupling strength $\gamma =1$ and temperature $T=300$ K.

Figure 8

Steady-state (a) $j-V$ and (b) $P-V$ characteristics of QHE as a function of temperature $T$ with the system-bath coupling strength $\gamma =1$ and cross-correlation coefficient $c=0$.

Figure 9

The power generated by QHE as a function of cross-correlation coefficient $c$ and system-bath coupling strength $\gamma$.

Figure 10

The power generated by QHE as a function of temperature $T$ and system-bath coupling strength $\gamma$ for the uncorrelated bath ($c=0$).

Figure 11

The power generated by QHE as a function of temperature $T$ and cross-correlation coefficient $c$ with the system-bath coupling strength $\gamma =1$.

Figure 12

Illustrations of the effects of $\gamma , c$, and $T$ that we explored. $\gamma$ and $c$ are incorporated in the effective system-bath coupling $y$ determining which effect dominates the exciton-transfer process: noise-assisted transport or dynamical localization. The temperature $T$ modulates the balance of the two effects. The high efficiency of the heat engine requires the constructive interplay between noise and coherent dynamics.

Figure 13

Effective voltage of QHE as a function of temperature $T$ and system-bath coupling strength $\gamma$ for the uncorrelated bath ($c=0$) with different parameter ranges for panels (a) and (b).