1 Introduction

Photosynthesis is a fundamental biological process used by plants, algae and certain bacteria to convert sunlight into chemical energy. The initial step of photosynthesis involves absorption of light by the so-called light-harvesting antennae complexes, which transmit the excitation energy among molecules to a reaction center [1]. It has been found that the excitation energy absorbed by antenna toward the reaction center occurs with a high efficiency over 90% [2, 3]. Understanding the intrinsic mechanisms of such efficient and directional energy transport in light-harvesting complex has attracted great interest. These researches are not only of fundamental interest but also have a potential impact on new artificial light-harvesting technologies [4,5,6,7,8,9]. Recently, ultrafast optics and nonlinear spectroscopy experiments have started to reveal that the coherence between electronic excitonic states can contribute beneficially to the high efficiency in photosynthetic systems [3, 10, 11]. For example, Engel et al. [11] and Panitchayangkoon et al. [12] have shown that the electronic coherence persists a relatively long-time in Fenna–Matthews–Olson (FMO) complex of green sulfur bacteria at both cryogenic (77 K) and ambient (300 K) temperatures. For LH2 (light-harvesting complex 2) in purple bacteria, a long-time quantum coherence in the excited states of bacteriochlorophylls (BChl) has also been found experimentally [14, 14, 15]. These experimental studies have generated numerous theoretical works [16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35] in understanding the role of coherence in excitation energy transfer (EET) processes. Corresponding to different interactions of the pigments, different approaches have been proposed to study the single-exciton energy transfer. Typical quantum theories include the quantum network model [36,37,38,39,40,41], the small-polaron master equation approach [42,43,44], the generalized Bloch–Redfield equation [45,46,47], the non-Markovian quantum jump method [48,49,50], the renormalization group methods [51], and the hierarchical equation of motion [52, 53].

Despite the great experimental and theoretical advances, the fundamental physical mechanism of energy transfer in photosynthetic complexes is not yet fully understood. For example, the effect of noise on EET has been studied extensively. Contrary to conventional view that noise always destroys the efficiency of quantum processes, it was shown that noise (especially phase damping) has ability to enhance the rate of EET when compared to the closed system evolution [39, 40]. At the same time, Ishizaki et al. [53] have discussed the energy transport of the FMO complex with different initial excitations. Moreover, a number of studies have addressed the influence of excitation conditions. For example, Mancal et al. [54] have considered the effects of different light properties on the photo-induced exciton dynamics in a molecular system. Similarly, Chenu et al. [55] have shown that the influence of the excitation rate in the energy transfer efficiency of molecular aggregates, and Dodin et al. [56] have shown the dynamics of a system driven by weak incoherent excitation. Although it has been known since the 1990s that closely packed chromophores lead in enhanced transport [57], few studies have considered the influence of the initial entanglement of the donors on the excitation EET. Moreover, most studies have assumed that the donor molecules have completed the process of energy absorption and the initial exciton is supposed in only one of the donor molecules. This condition may not be satisfied precisely under experimental or natural conditions. From this viewpoint, it is still necessary to consider a more realistic absorption process in photosynthesis system. Motivated by the above open questions, in this paper we turn to model systems in order to systematically investigate the effect of the initial entanglement on EET under the external driving fields.

The structure of this paper is as follows. In Sect. 2, we give a brief introduction of the model we use to simulate the absorption and energy transfer in photosynthetic complexes. Here, we use a Markovian master equation with local dephasing and dissipation terms to explore the dynamics of the exciton transport process. In Sect. 3, unlike most of the previous schemes, we utilize the external driving fields to simulate the energy absorption process in photosynthetic complexes. Thus, the network model includes not only the energy transfer process but also the absorption process. Without loss of generality, we consider two different kinds of external driving fields for comparison: continuous driving field and Gaussian-type driving field. Because the total energy of the system will vary with time, the original energy transfer efficiency may not be suitable for our scheme. Here, we introduce a rescaled efficiency to measure the energy transferred to the sink with respect to the total energy of the system. In particular, we explore the dynamics of excitons in specific model systems with two-donor or three-donor molecules. It is shown that the initial entanglement between the donors plays an important role in the transfer efficiency and energy absorption process. Section 4 summarizes the main results of this paper and makes some conclusions.

2 Quantum network model

The quantum network we consider includes N connected sites, and each site is modeled as a spin-1/2 particle. The particles are interacting with each other excitations of which can be exchanged between lattice sites by hopping. The quantum evolution of the N-site system can be described by the so-called Frenkel exciton Hamiltonian:

$$\begin{aligned} H_{S}=\sum ^{N}_{j=1}\varepsilon _{j}\sigma ^{+}_{j}\sigma ^{-}_{j}+\sum _{j\ne l}V_{jl}\left( \sigma ^{+}_{j}\sigma ^{-}_{l}+\sigma ^{+}_{l}\sigma ^{-}_{j}\right) , \end{aligned}$$
(1)

where \(\sigma ^{+}_{j}\) and \(\sigma ^{-}_{j}\) are the raising and lowering operators for site j, \(\sigma ^{+}_{j}=|1\rangle _{j}\langle 0|\) and \(\sigma ^{-}_{j}=|0\rangle _{j}\langle 1|\). \(\varepsilon _{j}\) is the on-site energy of site j, and \(V_{jl}\) represents the coupling strength between site j and site l.

From the point of actual environment, we assume that all sites are susceptible simultaneously to two distinct types of noise processes: a dissipative process that transfers the excitation energy in site j to the environment with rate \(\Gamma _j\) and a dephasing process that randomizes the phase of local excitations at rate \(\gamma _j\). We expect that the aforementioned two processes can be described by a Markovian master equation with local dephasing and dissipation terms. So the dynamics of the network is described by a master equation:

$$\begin{aligned} \frac{\mathrm{d}\rho }{\mathrm{d}t}=-\frac{i}{\hbar }[H,\rho ]+L_{\mathrm{diss}}(\rho )+L_{\mathrm{deph}}(\rho )+L_{s}(\rho ). \end{aligned}$$
(2)

The first item on the right-hand side in Eq. (2) denotes the free evolution of N sites. The dissipative process will lead to energy loss, and the corresponding Lindblad superoperators \(L_{\mathrm{diss}}\) [38, 58, 59] can be written as:

$$\begin{aligned} L_{\mathrm{diss}}(\rho )=\sum ^{N}_{j=1}\Gamma _{j}\left[ 2\sigma ^{-}_{j}\rho \sigma ^{+}_{j}-\left\{ \sigma ^{+}_{j}\sigma ^{-}_{j},\rho \right\} \right] , \end{aligned}$$
(3)

where \(\Gamma _{j}\) denotes the energy transfer rate from site j to the environment and \(\{A, B\}\) is an anticommutator. The pure dephasing process (with rate \(\gamma _{j}\)) can be described by:

$$\begin{aligned} L_{\mathrm{deph}}(\rho )=\sum ^{N}_{j=1}\gamma _{j}\left[ 2\sigma ^{+}_{j}\sigma ^{-}_{j}\rho \sigma ^{+}_{j}\sigma ^{-}_{j}-\left\{ \sigma ^{+}_{j}\sigma ^{-}_{j},\rho \right\} \right] , \end{aligned}$$
(4)

which destroys the phase coherence of superposition state in the system. In order to describe the total transfer of excitation, we designate an additional site named sink which is populated by an irreversible decay process from a chosen site k. The subindex ‘sink’ emphasizes that no population can escape from this site. The irreversible transfer of excitations from site N to the sink (numbered s) is modeled by a Lindblad operator [36]:

$$\begin{aligned} L_{s}(\rho (t))=\Gamma _{s}\left[ \sigma ^{+}_{s}\sigma ^{-}_{N}\rho (t)\sigma ^{+}_{N}\sigma ^{-}_{s}-\frac{1}{2}\left\{ \sigma ^{+}_{N}\sigma ^{-}_{s}\sigma ^{+}_{s}\sigma ^{-}_{N},\rho (t)\right\} \right] , \end{aligned}$$
(5)

where \(\Gamma _{s}\) is the trapping rate.

3 The effect of the initial entanglement on EET

Up to now, the processes of light-harvesting and energy transfer in photosynthesis have attracted much attention. However, in most of the currently existing network schemes, the authors only pay attention to the transfer efficiency by assuming the donors have completed the energy absorption process. Moreover, the system is supposed to be initialized with a single-exciton in the donor molecule. In this section, we assume that multiple donor molecules can absorb energy and utilize the external driving on initial sites to simulate the light-harvesting process. To study the role of initial state in the light-harvesting and energy transfer process, we consider a simple quantum network model with N donor molecules. As illustrated in Fig. 1, we consider only the nearest neighbor interactions and apply the continuous external driving fields on all donor sites to simulate the light-harvesting process. To be concrete, we suppose the state of the donors are initially two-qubit and three-qubit entangled states, respectively.

Fig. 1
figure 1

(Color online) The schematic representation of our quantum network model. The red arrows denote the continuous external driving fields applied on the input sites. The blue arrow between site N and sink S denotes an irreversible transfer of excitations from site N to the sink S

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3.1 The two-qubit initial entangled state case

Firstly, in the case of two-donor model, we consider three different kinds of initial states for comparison: product state \(|01\rangle _{12}\), non-maximally entangled state \(\sqrt{1/6}|01\rangle _{12}+\sqrt{5/6}|10\rangle _{12}\) and Bell state \(1/\sqrt{2}(|01\rangle _{12}+|10\rangle _{12})\). According to the following definition of total energy of the system, these three initial states correspond to the same initial energy. In order to simplify the calculations, we assume all sites suffer from the same dissipative and pure dephasing processes (\(\Gamma _{j}=\Gamma \), \(\gamma _{j}=\gamma \), \(\varepsilon _{j}=\varepsilon \), \(V_{ij}=V\)). The continuous driving Hamiltonian applied on donor site j can be expressed as:

$$\begin{aligned} H_{d}^{j}=\hbar \Omega (\sigma ^{+}_{j}e^{-i\omega _{L}t}+\sigma ^{-}_{j}e^{i\omega _{L}t}), \end{aligned}$$
(6)

where \(\Omega \) is the coupling constant between the donor and the driving field, \(\omega _{L}\) is the frequency of the driving field. We assume the detuning \(\delta \) to be the difference between the driving field frequency \(\omega _{L}\) and site transition frequency \(\omega _{s}\)(\(\omega _{s}=\varepsilon /\hbar \), in order to simplify the calculations, we can set \(\hbar =1\)). In the following, we investigate the near-resonant and large-detuning cases, respectively.

For the near-resonant case, we choose \(\delta =20~\mathrm{cm}^{-1}\) and the frequency of the driving field is \(100~\mathrm{cm}^{-1}\). Then, we have \(1 ~\mathrm{ps}^{-1}=5.3 ~\mathrm{cm}^{-1}\) (as in [33]). Based on Eqs.(1–5), when a train of external driving fields are applied on sites 1 and 2, the corresponding master equation for the system is given by:

$$\begin{aligned} \frac{\mathrm{d}\rho }{\mathrm{d}t}=-\frac{i}{\hbar }[H+H_{d},\rho ]+L_{\mathrm{diss}}(\rho )+L_{\mathrm{deph}}(\rho )+L_{s}(\rho ), \end{aligned}$$
(7)

with \(H=\varepsilon (\sigma _{1}^{+}\sigma _{1}^{-}+\sigma _{2}^{+}\sigma _{2}^{-}+\sigma _{3}^{+}\sigma _{3}^{-})+V(\sigma _{1}^{+}\sigma _{3}^{-}+\sigma _{3}^{+}\sigma _{1}^{-}+\sigma _{2}^{+}\sigma _{3}^{-}+\sigma _{3}^{+}\sigma _{2}^{-}+ \sigma _{1}^{+}\sigma _{2}^{-}+\sigma _{2}^{+}\sigma _{1}^{-})\) and \(H_{d}=H_{d}^{1}+H_{d}^{2}\). Without applying continuous external driving fields on initial sites, the population of sink could quantify the transport efficiency in the single-exciton subspace. By definition, the population at time T in the sink can be expressed as:

$$\begin{aligned} P_{\mathrm{sink}}=\Gamma _{s}\int _0^T \rho _{\mathrm{ee}}^{(N)}(t)\mathrm{d}t, \end{aligned}$$
(8)

where \(\rho _{ee}^{(N)}(t)\) denotes the probability that the site N is excited. As shown in Fig. 2a, the population of the sink will increase with time and the maximum value will approach 1.0 under the continuous external driving fields.

Fig. 2
figure 2

(Color online) The near-resonant continuous external driving fields are applied on two-donor sites. a the population in the sink \(P_{sink}\) as a function of t; b the total energy of network \(E_{total}\) as a function of t; c the energy transfer efficiency \(\eta _{sink}\) as a function of t. We set \(\varepsilon =120~\mathrm{cm}^{-1}\), \(V=40~\mathrm{cm}^{-1}\), \(\Gamma =0.2~\mathrm{ps}^{-1}\), \(\Gamma _{S}=10~\mathrm{ps}^{-1}\), \(\gamma =0.2~\mathrm{ps}^{-1}\), \(\Omega =20~\mathrm{cm}^{-1}\), \(\omega _{L}=100~\mathrm{cm}^{-1}\)

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However, different from the fixed exciton case, the total energy of the system will increase with time due to the absorption of energy from the continuous external fields. In this sense, the total energy of the system at time T includes three parts: the energy transferred to the sink \(E_{sink}\), the energy dissipated into environment \(E_{dis}\) and the energy existing in sites \(E_{site}\). They can be calculated as follows:

$$\begin{aligned} E_{\mathrm{total}}= & {} E_{\mathrm{site}}+E_{\mathrm{dis}}+E_{\mathrm{sink}}, \end{aligned}$$
(9)
$$\begin{aligned} E_{\mathrm{site}}= & {} \varepsilon _{1}\int ^{T}_{0}\rho _{\mathrm{ee}}^{(1)}(t) \mathrm{d}t +\varepsilon _{2}\int ^{T}_{0}\rho _{ee}^{(2)}(t)\mathrm{d}t \nonumber \\&{ +\,\varepsilon _{3}\int ^{T}_{0}\rho _{\mathrm{ee}}^{(3)}(t) \mathrm{d}t,} \end{aligned}$$
(10)
$$\begin{aligned} E_{\mathrm{dis}}= & {} \varepsilon _{1}\int ^{T}_{0}\Gamma _{1} \rho _{\mathrm{ee}}^{(1)}(t)\mathrm{d}t+\varepsilon _{2}\int ^{T}_{0} \Gamma _{2}\rho _{\mathrm{ee}}^{(2)}(t)\mathrm{d}t \nonumber \\&+\,\varepsilon _{3}\int ^{T}_{0}\Gamma _{3}\rho _{\mathrm{ee}}^{(3)}(t)\mathrm{d}t, \end{aligned}$$
(11)
$$\begin{aligned} E_{\mathrm{sink}}= & {} \varepsilon _{3}\int ^{T}_{0}\Gamma _{s}\rho _{\mathrm{ee}}^{(3)}(t)\mathrm{d}t, \end{aligned}$$
(12)

where \(\varepsilon _{1}=\varepsilon _{2}=\varepsilon _{3}=\varepsilon \), \(\Gamma _{1}=\Gamma _{2}=\Gamma _{3}=\Gamma \), and \(\rho _{\mathrm{ee}}^{(j)}(t)\) denotes the probability that the site j is excited. As depicted in Fig. 2b, we can see that each site will continuously absorb energy from the external fields, leading to that the total energy of the system defined in Eq. (9) grows linearly with time. Moreover, the total energy will reach a larger value more quickly for the maximally entangled state case. It means that the initial entanglement of the donors has beneficial effect on the energy absorption process.

Furthermore, based on the definition of total energy, we define the following measure to quantify the transfer efficiency of the sink under continuous driving fields:

$$\begin{aligned} {\eta }_{\mathrm{sink}}=E_{\mathrm{sink}}/E_{\mathrm{total}}, \end{aligned}$$
(13)

which manifests the energy transfer efficiency from external driving to the sink. According to the numerical simulation, we can see clearly from Fig. 2c that the transfer efficiency will reach a higher value faster for the entangled states than the separable states, which indicates that the initial entanglement between the donors is helpful for improving the absorption efficiency of the sink.

Fig. 3
figure 3

(Color online) The large-detuning continuous external driving fields are applied on two-donor sites. a The population in the sink \(P_{sink}\) as a function of t; b the total energy of network \(E_{total}\) as a function of t; c the energy transfer efficiency \(\eta _{sink}\) as a function of t with \(\omega _{L}=12~\mathrm{cm}^{-1}\), and the other parameters we choose are the same as Fig. 2

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Until now, the continuous external driving field is assumed to be near-resonance, but in the real situation, there may exist a large detuning \(\delta \) between the driving field frequency \(\omega _{L}\) and site transition frequency \(\omega _{s}\). Therefore, it is of great importance to investigate the energy absorption efficiency of the sink in the far-off-resonant field case. Here, we choose the detuning \(\delta =108~\mathrm{cm}^{-1}\) with \(\omega _{L}=12~\mathrm{cm}^{-1}\). In Fig. 3a, we plot the population in the sink for the large-detuning field case. Similarly, Fig. 3a shows that the initial entanglement contributes a beneficial effect to the transfer efficiency. A comparison between Figs. 2b and 3b shows that the absorbed energy will be much smaller in the large-detuning case. Similarly, the energy transfer efficiency is smaller than that of the near-resonant case, as depicted in Fig. 3c.

Besides the continuous external driving field, the influence of a broadband field on the transfer efficiency and energy absorbtion process should be discussed. Here, we take the Gaussian-type driving field as an example. The Gaussian-type driving applied on donor site j can be described by the following Hamiltonian [31]:

$$\begin{aligned} H_{G}^{j}(t)=\frac{\hbar \Omega }{\sqrt{2\pi }\tau _{p}}e^{-t^{2}/2\tau _{p}^{2}}(\sigma ^{+}_{j}e^{-i\omega _{L}t}+\sigma ^{-}_{j}e^{i\omega _{L}t}), \end{aligned}$$
(14)

where \(\tau _{p}\) is the full width at half maximum of the pulse. In the following, we will show some numerical results for the near-resonance case. We suppose each donor is excited by a single Gaussian-type pulse with \(\tau _{p}=0.1/v\). From Fig. 4, we find that the initial entanglement still benefits the EET process with Gaussian-type external field. Compared with the results of continuous driving fields, we find that the total energy will increase within a finite time and reach a stable value finally. This is because a single Gaussian-type external field cannot supply energy all the time. Moreover, the effect of Gaussian-type driving is closely related to the width of the pulse. In order to clarify this point, in Fig. 5 we consider the pulse with \(\tau _{p}=2/v\). A comparison between Figs. 4 and 5 shows that the population of sink and the total system energy will reach a larger value with the increase of \(\tau _{p}\). This is consistent with the fact that a wide pulse driving field contains more energy.

Fig. 4
figure 4

(Color online) The near-resonant Gaussian-type external driving fields are applied on two-donor sites. a The population in the sink \(P_{sink}\) as a function of t; b the total energy of network \(E_{total}\) as a function of t; c the energy transfer efficiency \(\eta _{sink}\) as a function of t. We set \(\varepsilon =120~\mathrm{cm}^{-1}\), \(V=40~\mathrm{cm}^{-1}\), \(\Gamma =0.2~\mathrm{ps}^{-1}\), \(\Gamma _{S}=10~\mathrm{ps}^{-1}\), \(\gamma =0.2~\mathrm{ps}^{-1}\), \(\Omega =20~\mathrm{cm}^{-1}\), \(\omega _{L}=100~\mathrm{cm}^{-1}\), \(\tau _{p}=0.1/v\)

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Fig. 5
figure 5

(Color online) The near-resonant Gaussian-type external driving fields are applied on two-donor sites. a The population in the sink \(P_{sink}\) as a function of t; b the total energy of network \(E_{total}\) as a function of t; c the energy transfer efficiency \(\eta _{sink}\) as a function of t. We set \(\varepsilon =120~\mathrm{cm}^{-1}\), \(V=40~\mathrm{cm}^{-1}\), \(\Gamma =0.2~\mathrm{ps}^{-1}\), \(\Gamma _{S}=10~\mathrm{ps}^{-1}\), \(\gamma =0.2~\mathrm{ps}^{-1}\), \(\Omega =20~\mathrm{cm}^{-1}\), \(\omega _{L}=100~\mathrm{cm}^{-1}\), \(\tau _{p}=2/v\)

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3.2 The three-qubit initial entangled state case

Compared with bipartite entangled states, multipartite entangled states possess more advantages and the entanglement property of multipartite entangled state is much more complicated than bipartite entanglement. In this section, we consider the influence of three-qubit initial state on the energy transfer efficiency with continuous external driving fields. Similarly, we consider three initial states: product state \(|001\rangle _{123}\), bipartite entangled state \(1/\sqrt{2}|0\rangle _{1}(|01\rangle _{23}+|10\rangle _{23})\) and tripartite W state: \(1/\sqrt{3}(|001\rangle _{123}+|001\rangle _{123}+|001\rangle _{123})\). The master equation for three-qubit initial state system is similar to Eq. (7), where the ingredients are replaced by \(H=\varepsilon (\sigma _{1}^{+}\sigma _{1}^{-}+\sigma _{2}^{+}\sigma _{2}^{-}+\sigma _{3}^{+}\sigma _{3}^{-}+\sigma _{4}^{+}\sigma _{4}^{-})\), \(V=(\sigma _{1}^{+}\sigma _{4}^{-}+\sigma _{4}^{+}\sigma _{1}^{-}+\sigma _{2}^{+}\sigma _{4}^{-}+\sigma _{4}^{+}\sigma _{2}^{-}+ \sigma _{3}^{+}\sigma _{4}^{-}+\sigma _{4}^{+}\sigma _{3}^{-}+\sigma _{1}^{+}\sigma _{2}^{-}+\sigma _{2}^{+}\sigma _{1}^{-}+\sigma _{2}^{+}\sigma _{3}^{-} +\sigma _{3}^{+}\sigma _{2}^{-})\) and \(H_{d}=H_{d}^{1}+H_{d}^{2}+H_{d}^{3}\). For the case of the near-resonant driving, Fig.6a shows the population of sink \(P_{sink}\) for different initial states. Due to the continuous external energy input, the value of \(P_{sink}\) will increase with time and the maximum value will approach 1.0. In Fig. 6b, c, we also plot the total system energy \(E_{total}\) and the energy absorbtion efficiency \({\eta }_{sink}\), respectively. We can see that the population in the sink and the total absorbtion energy will reach a larger value more quickly in the tripartite entanglement case compared to the bipartite entanglement case. This mechanism shows that the multipartite entanglement between the donors has better effect on improving the excitation energy transfer than the bipartite entanglement. In addition, we investigate the large-detuning case and the similar conclusions can be summarized from Fig. 7.

Fig. 6
figure 6

(Color online) The near-resonant continuous external driving fields are applied on three-donor sites. a The population in the sink \(P_{sink}\) as a function of t; b the total energy of network \(E_{total}\) as a function of t; c the energy transfer efficiency \(\eta _{sink}\) as a function of t. We set \(\varepsilon =120~\mathrm{cm}^{-1}\), \(V=40~\mathrm{cm}^{-1}\), \(\Gamma =0.2~\mathrm{ps}^{-1}\), \(\Gamma _{S}=10~\mathrm{ps}^{-1}\), \(\gamma =0.2~\mathrm{ps}^{-1}\), \(\Omega =20~\mathrm{cm}^{-1}\), \(\omega _{L}=100~\mathrm{cm}^{-1}\)

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Fig. 7
figure 7

(Color online) The large-detuning continuous external driving fields are applied on three-donor sites. a The population in the sink \(P_{sink}\) as a function of t; b the total energy of network \(E_{total}\) as a function of t; c the energy transfer efficiency \(\eta _{sink}\) as a function of t with \(\omega _{L}=12~\mathrm{cm}^{-1}\), and the other parameters we choose are the same as in Fig. 6

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4 Conclusion

In conclusion, we investigate the influences of the initial state entanglement on the absorption and energy transfer processes. Unlike most of the previous schemes by assuming the donors have completed the energy absorption process, we utilized the external driving fields applied on the donors to simulate the light-harvesting process. Because the total energy of the whole system varies with time, we define a new measure to quantify the transfer efficiency of the sink under the external driving fields. In the case of two-donor model, the near-resonant and large-detuning external driving fields have been discussed, respectively. According to the numerical simulations, we demonstrate that the initial entanglement between the donors plays an important role in the energy absorbtion process and improves the transfer efficiency of the sink. For the three-donor case, the results show that the tripartite entanglement between the donors has better effect on improving the absorption efficiency than the bipartite entanglement. In one word, the initial entanglement plays an important role in EET in our quantum network model.