⇦Chinese Physics Letters, 2020, Vol. 37, No. 6, Article code 060501 Influence of Quantum Feedback Control on Excitation Energy Transfer * Xiao-Lan Zong (宗晓岚)1, Wei Song (宋伟)1**, Ming Yang (杨名)2, Zhuo-Liang Cao (曹卓良)1 Affiliations 1Institute for Quantum Control and Quantum Information, and School of Physics and Materials Engineering, Hefei Normal University, Hefei 230601, China 2School of Physics and Material Science, Anhui University, Hefei 230601, China Received 26 February 2020, online 26 May 2020 *Supported by the National Natural Science Foundation of China (Grant Nos. 11904071 and 11374085), the Key Program of the Education Department of Anhui Province (Grant Nos. KJ2017A922, KJ2019A0725, and KJ2018A0486), the Anhui Provincial Natural Science Foundation (Grant Nos. 1908085QA40, 1708085MA12, and 1708085MA10), and the Discipline Top-Notch Talents Foundation of Anhui Provincial Universities (Grant Nos. gxbjZD2017024 and gxbjZD2016078).
**Corresponding author. Email: wsong1@mail.ustc.edu.cn Citation Text: Zong X L, Song W, Yang M and Cao Z L 2020 Chin. Phys. Lett. 37 060501    Abstract Excitation energy transfer (EET) plays a vital role in many areas of physics and biology processes. Here we address the role of quantum-jump-based feedback control in the efficiency of EET through a chain model. Usually, the decoherence caused by dissipative noise is detrimental to the transfer efficiency. We demonstrate that feedback control can always enhance the efficiency of EET and the dependence of different feedback controls is also discussed in detail. In addition, we investigate the strategy to enhance the efficiency of EET in the Fenna–Matthews–Olson complex as a prototype for larger photosynthetic energy transfer systems. DOI:10.1088/0256-307X/37/6/060501 PACS:05.60.Gg, 03.65.Yz, 03.67.-a, 71.35.-y © 2020 Chinese Physics Society Article Text The phenomenon of excitation energy transfer (EET) has been one of the most important physical processes occurring in both natural and artificial systems during recent years. In particular, the EET in photosynthetic complexes has attracted much research interest, because of the recent observation of the long-lived quantum coherence in the Fenna–Matthews–Olson (FMO) complex of green sulfur bacteria[1,2] and marine cryptophyte algae[3,4] by two-dimensional electronic spectroscopy. The FMO complex[5] is one of the most well-characterized systems and it is a trimer in which each of the subunits has seven bacteriochlorophyll-a (BChl-a) molecules. This molecular system is a small water-soluble protein unique to green sulfur bacteria and is responsible for passing the excitation energy from the light harvesting complex to the photosynthetic reaction center. It functions as an intermediate candidate that transfers the exciton energy with almost 100$\%$ efficiency. The experimental evidence for coherent EET in the FMO complex has motivated numerous theoretical works to model the excited state dynamics of chromophores in a surrounding protein scaffold, such as the quantum network model,[6–12] the generalized Bloch–Redfield equation,[13–15] the non-Markovian quantum jump method,[16–18] the renormalization group methods,[19] and the hierarchical equation of motion.[20,21] Especially, it was suggested that dephasing does not always reduce the efficiency of EET and the interaction with the environment can assist EET. However, in the previous theoretical reports, the environmental noise is limited to local dephasing noise. In this Letter we mainly consider the dissipative decoherence caused by spontaneous emission. It is well known that quantum feedback control is a promising method to prolong coherence in quantum systems.[22–24] We intend to explore the effect of quantum feedback control on the efficiency of EET. To illustrate our idea we first investigate the single-excitation energy transfer in a chain model with three sites coupled to environments. Here we only consider the weak coupling case and use a Markovian master equation with dissipation terms to explore the dynamics of the exciton transport process. The dependence of different feedback controls on the efficiency of EET is discussed in detail. Then we focus on the influence of quantum feedback control on the efficiency of EET in the FMO complex. We compare the effects for different initial states and show that quantum-jump-based feedback control can always enhance the efficiency of EET. Finally, we summarize our results and draw some conclusions.
cpl-37-6-060501-fig1.png
Fig. 1. Schematic representation of a chain with three sites. Each site of the chain is coupled to its neighbor and the exciton is transferred from site 1 to 3 and finally trapped by the sink.
We would like to start our investigation by considering a chain model with three two-level systems coupled to environments, as shown in Fig. 1. Each site of the chain only coupled to its neighbor and we use $\left| i \right\rangle$ to denote the excitation at site $i$, which can hop from one site to the other. Site 1 corresponds to the chlorosome antenna (donor), and the last site 3 is connected to the sink $|s\rangle$, which denotes that no population can escape from this site. This chain model can be regarded as a simplified form of the FMO complex, a network of seven coupled sites, each of which can be treated as a two-level system. In the weak-coupling limit, the dynamics of the total system density matrix $\rho$ is described by the Markovian master equation:[6,9,10] $$\begin{align} \frac{{d\rho }}{{dt}} = - \frac{i}{\hbar }\left[ {H,\rho } \right] + L_{\rm diss} \left(\rho \right) + L_{\rm sink} \left(\rho \right),~~ \tag {1} \end{align} $$ where $H = \sum\limits_{i = 1}^3 {\varepsilon _i \sigma _i^ + \sigma _i^ - } + V_{12} \left({\sigma _1^ + \sigma _2^ - + \sigma _2^ + \sigma _1^ - } \right) + V_{23} \left({\sigma _2^ + \sigma _3^ - + \sigma _3^ + \sigma _2^ - } \right)$ with $\sigma _i^ {+}=\left| i \right\rangle \left\langle 0 \right|$ and $\sigma_i^ {-}=\left| 0 \right\rangle \left\langle i \right|$ are the raising and lowering operators of the system, and $\left| 0 \right\rangle$ represents the zero exciton state of the system. The site energy and two-body coupling strength are represented by $\varepsilon _i$ and $V _{ij}$, respectively. Here we only consider the dissipative noise and the corresponding Lindblad super-operator $L_{\rm diss}$ can be written as $$\begin{alignat}{1} L_{\rm diss} \left(\rho\right) = \sum\limits_{j = 1}^3 {{\varGamma} _j } \left[ {2\sigma _j^ - \rho \sigma _j^ + - \left\{ {\sigma _j^ + \sigma _j^ - ,\rho } \right\}} \right].~~ \tag {2} \end{alignat} $$ The irreversible transfer of excitations from site 3 to the sink (numbered s) is modeled by a Lindblad operator: $$\begin{alignat}{1} \!\!\!\!\!\!\!\!L_{\rm sink} \left(\rho \right) = {\varGamma} _{\rm s} \left[ {2\sigma _{\rm s}^ + \sigma _3^ - \rho \sigma _3^ + \sigma _{\rm s}^ - - \left\{ {\sigma _3^ + \sigma _{\rm s}^ - \sigma _{\rm s}^ + \sigma _3^ - ,\rho } \right\}} \right],~~ \tag {3} \end{alignat} $$ where ${\varGamma} _{\rm s}$ denotes the trapping rate. The efficiency of EET is defined by the population transferred to the sink from the site 3: $$\begin{align} P_{\rm sink} = \rho _{\rm sink} \left(\infty \right) = 2{\varGamma} _{\rm s} \int_0^\infty {\rho _{33} \left(t \right)dt}.~~ \tag {4} \end{align} $$ In Ref. [6], the authors have demonstrated the phenomena of dephasing assisted transport of excitations for non-uniform chains. However, the dephasing noise cannot enhance the transport of excitations for the uniform chains as shown in Ref. [6]. In this Letter we only consider the dissipative noise caused by spontaneous emission. In this case the transfer efficiency decreases with the increase of dissipative rate. We first consider the case in which feedback operation is applied on the donor site 1, and the corresponding quantum-jump-based feedback master equation can be derived from the general measurement theory:[24,25] $$\begin{alignat}{1} \frac{{\partial \rho }}{{\partial t}} =\,& - \frac{i}{\hbar }\left[ {H,\rho } \right] + {\varGamma} _1 \left[ {2U_{\rm f1} \sigma _1^ - \rho \sigma _1^ + U_{\rm f1}^† - \left\{ {\sigma _1^ + \sigma _1^ - ,\rho } \right\}} \right] \\ &+ {\varGamma} _2 \left[ {2\sigma _2^ - \rho \sigma _2^ + - \left\{ {\sigma _2^ + \sigma _2^ - ,\rho } \right\}} \right] \\ &+ {\varGamma} _3 \left[ {2\sigma _3^ - \rho \sigma _3^ + - \left\{ {\sigma _3^ + \sigma _3^ - ,\rho } \right\}} \right] \\ &+ {\varGamma} _{\rm s} \left[ {2\sigma _{\rm s}^ + \sigma _3^ - \rho \sigma _3^ + \sigma _{\rm s}^ - - \left\{ {\sigma _3^ + \sigma _{\rm s}^ - \sigma _{\rm s}^ + \sigma _3^ - ,\rho } \right\}} \right],~~ \tag {5} \end{alignat} $$ where the operator $U_{\rm f1} = e^{iH_{\rm f1} } \otimes I_2 \otimes I_3$. In order for the feedback to be Markovian, the feedback operator $U_{\rm f}$ must act immediately after a detection and the control operator is applied only after a detection click, i.e., a quantum jump occurs. Here we assume $H_{\rm f}$ can be decomposed in terms of the three Pauli matrices $H_{\rm f}=A_{x}\sigma_{x}+A_{y}\sigma_{y}+A_{z}\sigma_{z}$ ($A_{x}$, $A_{y}$ and $A_{z}$ are real numbers). Thus the feedback control operator $U_{\rm f}$ can be represented by the following unitary transformation: $$ U_{\rm f}=e^{iH_{\rm f}} =\cos|{\boldsymbol A}|+i\frac{\sin|{\boldsymbol A}|}{|{\boldsymbol A}|}{\boldsymbol A}\cdot{\boldsymbol \sigma},~~ \tag {6} $$ where ${\boldsymbol \sigma}=(\sigma_{x},\sigma_{y},\sigma_{z})$, and ${\boldsymbol A}=(A_{x},A_{y},A_{z})$ represents the amplitude of $\sigma_{x}$, $\sigma_{y}$ and $\sigma_{z}$ control. The physical meaning of the feedback operator $U_{\rm f}$ corresponds to rotate the Bloch vector of the two-level system with a certain angle around the ${\boldsymbol n}$ axis in the Bloch sphere.
cpl-37-6-060501-fig2.png
Fig. 2. (a) The dynamics of population in the sink $P_{\rm sink}$ as a function of $t$ with different feedback amplitudes $A_{x1}$. (b) The population in the sink $P_{\rm sink}$ as a function of $t$ and $A_{x1}$, where amplitudes $A_{x1}$ range from $0$ to $\pi$. The other parameters are taken as $\varepsilon _1 = \varepsilon _2 = \varepsilon _3=20$, $V_{12}=V_{23}= 5$, ${\varGamma}_1={\varGamma}_2={\varGamma}_3=1.5$, and ${\varGamma}_{\rm s}=10$.
Assume that site 1 is initially prepared in excitation state. In Fig. 2, the feedback control applied on site 1 only contains the $\sigma_{x}$ component $H_{\rm f1}=A_{x1}\sigma_{x1}$. The feedback amplitude $A_{x}$ influences the evolution of the system with a period of $\pi$, which is consistent with the results in Ref. [24]. It should be noted that $\sigma_{z}$ has no effect on the evolution of EET due to $e^{iA_{z}\sigma_{z}} \sigma_{-}\rho_{\rm s}\sigma_{+}e^{-iA_{z}\sigma_{z}}=\rho_{\rm s}$. Figure 2(a) shows the dependence of efficiency of EET for different feedback amplitudes $A_{x}$. We further plot the evolution of population in the sink $P_{\rm sink}$ as a function of $t$ and feedback control parameters $A_{x}$. The numerical simulation shows that the optimal value of transfer efficiency is obtained for $A_{x}=\pi/2$, which is plotted in Fig. 2(b) with the black line. Then we consider the case in which the feedback amplitudes $A_{x}$ and $A_{y}$ are both nonzero. Figure 3 shows that the evolution of population in the sink $P_{\rm sink}$ varies with different $A_{x}$ and $A_{y}$, and it is interesting to see that the value of population increases with the difference between the amplitudes $A_{x}$ and $A_{y}$. Numerical simulation shows that the optimal value of $P_{\rm sink}$ is obtained for $A_{x1}=\pi/2$ and $A_{y1}=0$. This result indicates that feedback control $\sigma_{x}$ has a better effect than the combination of $\sigma_{x}$ and $\sigma_{y}$.
cpl-37-6-060501-fig3.png
Fig. 3. The population in the sink $P_{\rm sink}$ as a function of $t$ with different $A_{x1}$ and $A_{y1}$. The feedback control is a combination of $\sigma_{x}$ and $\sigma_{y}$ applied on site 1.
cpl-37-6-060501-fig4.png
Fig. 4. The population in the sink $P_{\rm sink}$ as a function of $t$ with different feedback controls. The red curve represents the case with feedback control applied on site 1. The blue and green curves correspond to the cases with feedback control applied on sites 1 and 2.
cpl-37-6-060501-fig5.png
Fig. 5. The evolution of $P_{\rm sink}$ as a function of $t$ in the detuned case. The other parameters are $\varepsilon _1 = \varepsilon _3=20$, $\varepsilon _2 =2$, $V_{12}=V_{23}= 5$, ${\varGamma}_1={\varGamma}_2={\varGamma}_3=1.5$, and ${\varGamma}_{\rm s}=10$.
So far, we have only considered the case that the feedback control is applied on site 1. What would happen if the feedback control is applied on sites 1 and 2 simultaneously. As shown in Fig. 4, the transfer efficiency reaches a larger value if we apply feedback control on both sites. This result is not surprising since feedback control on more sites means stronger suppression of dissipative noise. Finally, we generalize our results to the non-uniform case with parameters are chosen as $\varepsilon _1 = \varepsilon _3 =20$, $\varepsilon _2 =2$, and the other parameters are the same as those in the uniform case. Here, site 2 is strongly detuned from its neighboring sites $1$ and $3$. Figure 5 shows the dependence of the population in the sink $P_{\rm sink}$ in the detuned case. In comparison with Fig. 2(a), we can see a significant decrease of the maximum population in the free evolution case while the maximum population almost have no change for the case with feedback control. This phenomenon shows that feedback control has better effect on enhancing efficiency of EET in the detuned case. Next we proceed to investigate if our feedback control strategy can be used to enhance the efficiency of EET in the FMO complex. The FMO is a trimer made of three identical subunits, each containing seven pigments. Due to the weak coupling between inter-subunit, we restrict our study to a single subunit. An additional eighth pigment was recently suggested by the recent crystallographic data.[26] However, the eighth pigment is only loosely bound and it usually detaches from the others when the system is isolated from its environment to perform experiments. Thus we ignore this pigment and the dynamics of EET will be investigated by using the seven-site model. The subunit containing seven pigments can be modeled as a network of seven sites with site dependent coupling and site energies. We use the experimental Hamiltonian of FMO given in Ref. [27] with the matrix of the Hamiltonian in the following form: $$\begin{alignat}{1} H = \left(\!\!\!{\begin{array}{*{7}c} \! \!\!{215} &\!\!\!\!\!{ - 104.1}\!&\!\!\!\!\!{5.1} &\!\!\!\!{ - 4.3}\!&\!\!\!{4.7} \!&\!\!\!\!{ - 15.1}\!&\!\!\!{ - 7.8} \\ \!\!{ - 104.1} \!&\!\!\!\!\!{220.0}\!&\!\!\!\!\!{32.6}\!&\!\!\!\!\! {7.1}\!&\!\!\!{5.4}\!&\!\!\!\! {8.3}\!&\!\!\!{0.8} \\ \!\!{5.1} \!&\!\!\!\!\!{32.6}\!&\!\!\!\!\!{0.0}\!&\!\!\!\! { - 46.8}\!&\!\!\!{1.0} \!&\!\!\!\!{ - 8.1}\!&\!\!\!{5.1} \\ \!\!{ - 4.3}\!&\!\!\!\!\! {7.1}\!&\!\!\!\!\!\!{ - 46.8}\!&\!\!\! {125.0}\!&\!\!\!{ - 70.7}\!&\!\!\!\! { - 14.7}\!&\!\!\!{ - 61.5} \\ \!\!{4.7}\!&\!\!\!\!\!{5.4}\!&\!\!\!\!\!{1.0}\!&\!\!\!\! { - 70.7}\!&\!\!\!{450.0}\!&\!\!\!\!{89.7}\!&\!\!\!{ - 2.5} \\ \!\!{ - 15.1}\!&\!\!\!\!\! {8.3}\!&\!\!\!\!\!{-8.1} \!&\!\!\!\!{ - 14.7}\!&\!\!\!{89.7} \!&\!\!\!\!{330.0}\!&\!\!\!{32.7} \\ \!\!{ - 7.8}\!&\!\!\!\!\! {0.8}\!&\!\!\!\!\!{5.1} \!&\!\!\!\!{ - 61.5}\!&\!\!\!{ - 2.5} \!&\!\!\!\!{32.7}\!&\!\!\!{280.0} \\ \end{array}} \!\!\!\!\right) \\~~ \tag {7} \end{alignat} $$ in units of cm$^{-1}$ and we have shifted the zero energy by 12230  cm$^{-1}$ for convenience which does not affect the dynamics of EET in our system. In units of $\hbar = 1$, we have the rate 1 ps$^{-1} \equiv 5.3$ cm$^{-1}$. If we neglect the couplings weaker than 15 cm$^{-1}$ in the Hamiltonian (7), the transport in an individual monomer of FMO can be represented to the quantum network shown in Fig. 6. Similarly, we only consider the dissipative noise caused by spontaneous emission. Suppose that the dissipative rate for each site is the same as ${\varGamma}_j={0.3}/{1.88}$ cm$^{-1}$. The energy trapping rate from pigment 3 to the reaction center in the literature ranges from 0.25 ps$^{-1}$ to $4$ ps$^{-1}$.[7,27,28] We choose ${\varGamma}_{\rm s}=30/1.88$ cm$^{-1}$ corresponding to about $3$ ps$^{-1}$. Here the feedback controls are added on the pigments 1 and 6 simultaneously, and the initial state for our discussion is chosen as $\left| {\psi \left(0 \right)} \right\rangle = \left| 1 \right\rangle$. For simplicity, we suppose $A_{x1}=\pi/2,A_{y1}=0,A_{z1}=0$ and $A_{x6}=\pi/2,A_{y6}=0,A_{z6}=0$. As shown in Fig. 7(a), the feedback control is beneficial to enhance the efficiency of EET. As a comparison, we also plot the evolution of $P_{\rm sink}$ as a function of $t$ for the initial state $\left| {\psi \left(0 \right)} \right\rangle = {1 / {\sqrt 2 }}\left({\left| 1 \right\rangle + \left| 6 \right\rangle } \right)$, as shown in Fig. 7(b). The above results show that the effect of feedback control also depends on the initial state. In fact, the FMO complex can be regarded as a non-uniform network consisting of seven coupled sites, which is a generalization of the non-uniform chain model.
cpl-37-6-060501-fig6.png
Fig. 6. The simplified network for the subunit of FMO complex. The sites 1 and 6 are closed to the chlorosome antenna corresponding to the donor and the exciton is transferred from sites 1 and 6 to site 3 through the network, and finally trapped by the reaction center.
cpl-37-6-060501-fig7.png
Fig. 7. The evolution of $P_{\rm sink}$ as a function of $t$ with different initial states: (a) $\left| {\psi \left(0 \right)} \right\rangle = \left| 1 \right\rangle$, (b) $\left| {\psi \left(0 \right)} \right\rangle = 1/{\sqrt 2 }\left({\left| 1 \right\rangle + \left| 6 \right\rangle } \right)$. The other parameters are ${\varGamma}_j={0.3}/{1.88}$ cm$^{-1}$ and ${\varGamma}_{\rm s}=30/1.88$ cm$^{-1}$.
In this work we use the chain model with three sites to explore the dynamics of the exciton transport process. Under the weak coupling limit, the system can be described by a Markovian master equation with dissipation terms. Our results demonstrate that, although dissipative noise is detrimental to the EET process, we can use the quantum-jump-based feedback control to enhance the transport of excitations. For comparison, the different feedback controls on the efficiency of EET are also discussed in detail. In particular, we further investigate the strategy to enhance the efficiency of EET in the FMO complex. We also compare the influence of different initial states on the dynamics of EET. Recently, the experimental quantum simulation of photosynthetic energy transfer using nuclear magnetic resonance (NMR) has been reported in Refs. [29,30], thus it is possible to simulate our results in the near future within NMR technology. Furthermore, our results may shed light on design of more efficient nanofabricated structures for quantum transport and optimized solar cells. References Evidence for wavelike energy transfer through quantum coherence in photosynthetic systemsLong-lived quantum coherence in photosynthetic complexes at physiological temperatureCoherently wired light-harvesting in photosynthetic marine algae at ambient temperatureElectronic coherence lineshapes reveal hidden excitonic correlations in photosynthetic light harvestingChlorophyll arrangement in a bacteriochlorophyll protein from Chlorobium limicolaDephasing-assisted transport: quantum networks and biomoleculesEnvironment-assisted quantum walks in photosynthetic energy transferLimits of quantum speedup in photosynthetic light harvestingHighly efficient energy excitation transfer in light-harvesting complexes: The fundamental role of noise-assisted transportNoise-assisted energy transfer in quantum networks and light-harvesting complexesCoherence in the B800 Ring of Purple Bacteria LH2Enhancing the absorption and energy transfer process via quantum entanglementA phase-space study of Bloch–Redfield theoryEfficient energy transfer in light-harvesting systems, I: optimal temperature, reorganization energy and spatial–temporal correlationsExcitonic energy transfer in light-harvesting complexes in purple bacteriaNon-Markovian Quantum JumpsOpen system dynamics with non-Markovian quantum jumpsNon-Markovian quantum jumps in excitonic energy transferEfficient Simulation of Strong System-Environment InteractionsStochastic Liouville, Langevin, Fokker–Planck, and Master Equation Approaches to Quantum Dissipative SystemsUnified treatment of quantum coherent and incoherent hopping dynamics in electronic energy transfer: Reduced hierarchy equation approachDynamical creation of entanglement by homodyne-mediated feedbackStabilizing entanglement by quantum-jump-based feedbackSuppressing decoherence and improving entanglement by quantum-jump-based feedback control in two-level systemsQuantum theory of continuous feedbackThe structural basis for the difference in absorbance spectra for the FMO antenna protein from various green sulfur bacteriaHow Proteins Trigger Excitation Energy Transfer in the FMO Complex of Green Sulfur BacteriaEfficiency of energy transfer in a light-harvesting system under quantum coherenceEfficient quantum simulation of photosynthetic light harvestingQuantum simulation of Fenna-Matthew-Olson(FMO) complex on a nuclear magnetic resonance(NMR) quantum computer
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