Recently, several works have analysed the efficiency of photosynthetic complexes in a transient scenario and how that efficiency is affected by environmental noise. Here, following a quantum master equation approach, we study the energy and excitation transport in fully connected networks both in general and in the particular case of the Fenna–Matthew–Olson complex. The analysis is carried out for the steady state of the system where the excitation energy is constantly “flowing” through the system. Steady state transport scenarios are particularly relevant if the evolution of the quantum system is not conditioned on the arrival of individual excitations. By adding dephasing to the system, we analyse the possibility of noise-enhancement of the quantum transport.

Introduction

In the last years, quantum transport in photosynthetic complexes has become an interesting field of study and debate. An important part of this research focusses on the excitation transfer from the antennae that harvest the sunlight to the reaction centre (RC) where the photosynthetic process takes place. More concretely, for the Fenna–Matthew–Olson (FMO) complex of green sulfur bacteria, empirical evidence suggests that such transport is coherent even at room temperature [1]–[3]. These experiments show that the transient behaviour takes place on time scales much shorter than the decoherence time due to the environment. Thus, most of the recent analysis has focussed on the single-excitation scenario in the transient regime obtained after pulsed photoexcitation [4]–[12].

Actually, there is a vivid debate about the validity of the single-excitation picture for modelling the photosynthetic process in vivo. Photosynthesis in nature is a continuous process of absorption of energy from a radiation field. As there is no specific measurement mechanism that determines when the quanta of energy are effectively absorbed, some authors have argued that the photosynthetic complex and the radiation field should evolve to a steady state where the energy is constantly flowing through the system [13], [14]. Some of the conclusions of [14], regarding the importance of a steady state picture, are summarized in the following paragraph: `The classical picture of the photon as a particle incident on the molecule, repeatedly initiating dynamics, also assumes a known photon arrival time. This too is incorrect and inconsistent with the quantum analysis insofar as no specific arrival time can be presumed unless the experiment itself is designed to measure such times'. Also, it has been shown that some conclusions regarding the presence of entanglement in this kind of system rely on the assumption that the system is excited by a single excitation Fock state. This state cannot be obtained just by weak illumination, and changing this assumption for a more realistic one changes dramatically the conclusions [15]. These arguments makes it reasonable to analyse the natural photosynthetic processes also in other regimes, such as a steady state scenario.

Moreover, quantum transport in a non-equilibrium steady state is an active field in condensed matter physics. For ordered systems composed of qubits or harmonic oscillators, it has been shown that it is possible to violate Fourier's law and thus achieve an infinite thermal conductivity in the absence of noise [16], [17]. This ballistic transport turns into a diffusive one, with finite conductivity, if noise is added to the system as a dephasing channel, reducing therefore the energy transfer. That fact highlights the importance of the interaction with a dephasing environment for the energy transfer. The analysis of quantum transport can contribute to the design of artificial light-harvesting systems that are more efficient and robust [18].

Recently, quantum transport in photosynthetic complexes has been analysed through different models with different measures of the efficiency, principally in the single-excitation regime. In [10], the dynamics of the FMO complex was analysed by the use of a Markovian Redfield equation and by a generalized Bloch–Redfield equation [19]. The measure of efficiency that they use is the average time that a single excitation spends in the network before being absorbed by the sink. The results show that the Redfield approach correctly describes the dynamics of the system, but also that it fails to determine the optimal dephasing ratio that minimizes the trapping time. Moreover, this approach gives the unphysical results of a zero trapping time in the limit of strong dephasing . An analogous model was considered in [9], with the difference that the efficiency was quantified by the population of the sink in the long time limit. Finally, Scholak et al. [11], [12] have studied this problem in the absence of a sink, in such a way that the only incoherent dynamics was due to the presence of a dephasing environment. Here, the index for the quantification of the efficiency was the highest probability of finding the excitation in the outgoing qubit in a time interval , with being related to the estimate of the duration of the excitation transfer in real systems. The authors conclude that the addition of noise can increase the efficiency, but mainly in configurations that initially performed poorly. Despite their differences, all these papers coincide in analysing only the transient behaviour, and not the steady state, and they use very different indexes for quantifying the efficiency of the system.

In this paper, we analyse the energy transfer in quantum networks and, specifically, in the FMO complex in a steady state. We show that the excitations move coherently through the system also in this regime. The addition of a dephasing environment reduces, but does not destroy, the coherent transport. We also analyse the change in efficiency due to such an environment. The model we consider here is based on a quantum network connected to a thermal bath, to model the absorption of energy from the radiation field, and to a sink, that delivers the energy quanta to the reaction centre. As a particular case, we analyse the FMO complex and similar fully connected networks. In this scenario, the system evolves to a non-equilibrium steady state, where all the observables remain constant. A similar framework has already been used to analyse entanglement in light-harvesting complexes in the transient regime [20].

As has been discussed before, several indexes of the efficiency are usually applied in order to calculate the efficiency of these kinds of system. Also, for the complete photosynthetic procces itself, there is an important difference between analysing it by the use of quantum efficiency, that is, the average number of absorbed photons that finally give rise to photosynthetic products, and the energy efficiency. The second one is considered a more appropriate measure for comparing the efficiency of photosynthetic complexes with artificial light harvesting systems and for analysing the global procedure [18]. Because of that, we will use the energy transfer per unit of time, that is, the power, as our principal index of the efficiency of the systems. This measure will be compared with the excitation transfer, which corresponds to the quantum efficiency. We will show that, in general, they behave in a very different way, especially under the effect of noise.

The present paper is organized as follows. In the next section, we introduce the details of the model and of the master equation which describes its dynamics. In Section III, we introduce two indexes for evaluating the efficiency, and we perform an analytical comparison between the two of them. Uniform and general networks are analysed in Section IV, while in Section V we focus our attention on the FMO complex and related Hamiltonians. Finally, in Section VI some conclusions are drawn.