^{1,2}, Ye Xiong

^{3}and Yang Zhao

^{1}

### Abstract

We investigate the influence of static disorder and thermal excitations on excitonic energy transport in the light-harvesting apparatus of photosynthetic systems by solving the Schrödinger equation and taking into account the coherent hoppings of excitons, the rates of exciton creation and annihilation in antennas and reaction centers, and the coupling to thermally excited phonons. The antennas and reaction centers are modeled, respectively, as the sources and drains which provide the channels for creation and annihilation of excitons. Phonon modes below a maximum frequency are coupled to the excitons that are continuously created in the antennas and depleted in the reaction centers, and the phonon population in these modes obeys the Bose-Einstein distribution at a given temperature. It is found that the energy transport is not only robust against the static disorder and the thermal noise, but it can also be enhanced by increasing the randomness and temperature in most parameter regimes. Relevance of our work to the highly efficient energy transport in photosynthetic systems is discussed.

Support from the Singapore National Research Foundation through the Competitive Research Programme (CRP) under Project No. NRF-CRP5-2009-04 is gratefully acknowledged. This work was also supported in part by the State Key Programs for Basic Research of China (Grant No. 2011CB922102), and by National Foundation of Natural Science in China (Grant No. 61076094).

I. INTRODUCTION

II. MODEL AND FORMALISM

III. EFFECT OF STATIC DISORDER IN THE ABSENCE OF INTERACTION WITH PHONONS

IV. EFFECT OF THERMALLY EXCITED PHONONS

V. CONCLUSIONS

## Figures

Schematic for a network of exciton transport. The antenna (reaction center) sites are represented with green (magenta) circles which are interconnected forming a network, and creation (annihilation) of excitons is represented with dashed incoming (dotted outgoing) arrows attached to corresponding sites. The couplings between sites, represented with brown links in the network, correspond to hopping integrals J i, j in Hamiltonian H c , while the incoming and outgoing channels expressed by the arrows depict virtual semi-infinite chains in Hamiltonians H s and H d .

Schematic for a network of exciton transport. The antenna (reaction center) sites are represented with green (magenta) circles which are interconnected forming a network, and creation (annihilation) of excitons is represented with dashed incoming (dotted outgoing) arrows attached to corresponding sites. The couplings between sites, represented with brown links in the network, correspond to hopping integrals J i, j in Hamiltonian H c , while the incoming and outgoing channels expressed by the arrows depict virtual semi-infinite chains in Hamiltonians H s and H d .

Total transmission coefficient as a function of the photon energy. (a) w d = 0.002 eV, (b) w d = 0.006 eV. Other parameters are: ε d = 1.5 eV, w s = 0.001 eV, g d = 0.003 eV, g s = 0.001 eV, J 0 = 0.01 eV, W = 0.005 eV, and p = 0.3. For a given set of parameters, there is only a single peak corresponding to the global resonance of the entire source-network-drain system.

Total transmission coefficient as a function of the photon energy. (a) w d = 0.002 eV, (b) w d = 0.006 eV. Other parameters are: ε d = 1.5 eV, w s = 0.001 eV, g d = 0.003 eV, g s = 0.001 eV, J 0 = 0.01 eV, W = 0.005 eV, and p = 0.3. For a given set of parameters, there is only a single peak corresponding to the global resonance of the entire source-network-drain system.

Total exciton current as a function of drain density p for different values of g d . ε s = ε d , w d = 0.002 eV, w s = 0.001 eV, g s = 0.001 eV, J 0 = 0.01 eV, and W = 0.005 eV. For all combinations of g s and g d the curve of exciton current exhibits a single maximum at p c , pointing to an optimal p for energy transfer efficiency.

Total exciton current as a function of drain density p for different values of g d . ε s = ε d , w d = 0.002 eV, w s = 0.001 eV, g s = 0.001 eV, J 0 = 0.01 eV, and W = 0.005 eV. For all combinations of g s and g d the curve of exciton current exhibits a single maximum at p c , pointing to an optimal p for energy transfer efficiency.

Dependence of the total exciton current as a function of the degrees of disorder w d (=w s ) and W. For I(w s = w d ), W = 5 meV. For I(W), w d = 2 meV, w s = 1 meV, and ε s = ε d . Other parameters are: p = 0.3, ε s = 1.5 eV, g d = 0.003 eV, g s = 0.001 eV, and J 0 = 0.01 eV. The exciton current is enhanced by energetic fluctuations in the antennas and the reaction centers, but it is almost unaffected by disorder in the NN hopping integral of the pigment network.

Dependence of the total exciton current as a function of the degrees of disorder w d (=w s ) and W. For I(w s = w d ), W = 5 meV. For I(W), w d = 2 meV, w s = 1 meV, and ε s = ε d . Other parameters are: p = 0.3, ε s = 1.5 eV, g d = 0.003 eV, g s = 0.001 eV, and J 0 = 0.01 eV. The exciton current is enhanced by energetic fluctuations in the antennas and the reaction centers, but it is almost unaffected by disorder in the NN hopping integral of the pigment network.

Temperature dependence of the total exciton current for various amplitudes of energetic disorder in the sources and the drains. Other parameters are: p = 0.3, ε s = 1.5 eV, ε d − ε s = 2 meV, g d = 1 meV, g s = 0.5 meV, J 0 = 0.01 eV, W = 5 meV, ω c = 60 meV, and λ0 = 1 meV. For a given value of exciton-phonon coupling strength, e.g., λ0 = 1 meV, the exciton current increases with the increasing temperature for all amplitudes of energetic disorder.

Temperature dependence of the total exciton current for various amplitudes of energetic disorder in the sources and the drains. Other parameters are: p = 0.3, ε s = 1.5 eV, ε d − ε s = 2 meV, g d = 1 meV, g s = 0.5 meV, J 0 = 0.01 eV, W = 5 meV, ω c = 60 meV, and λ0 = 1 meV. For a given value of exciton-phonon coupling strength, e.g., λ0 = 1 meV, the exciton current increases with the increasing temperature for all amplitudes of energetic disorder.

Temperature dependence of the total exciton current for various strengths of exciton-phonon coupling. Other parameters are: p = 0.3, ε s = 1.5 eV, ε d − ε s = 2 meV, w d = w s = 10 meV, g d = 1 meV, g s = 0.5 meV, J 0 = 0.01 eV, W = 5 meV, and ω c = 60 meV. The exciton current is found to increase with the temperature for all exciton-phonon coupling strength except the narrow bracket from 6 meV to 7 meV.

Temperature dependence of the total exciton current for various strengths of exciton-phonon coupling. Other parameters are: p = 0.3, ε s = 1.5 eV, ε d − ε s = 2 meV, w d = w s = 10 meV, g d = 1 meV, g s = 0.5 meV, J 0 = 0.01 eV, W = 5 meV, and ω c = 60 meV. The exciton current is found to increase with the temperature for all exciton-phonon coupling strength except the narrow bracket from 6 meV to 7 meV.

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