Non-Markovianity in photosynthetic reaction centers: a noise-induced quantum coherence perspective

Zibo Wang; Zibo Wang; Antonio V. Lim; Imran M. Mirza

doi:10.1364/OPTCON.459740

1. Introduction

The origins of the conceptual development of quantum biology can be traced back to the early days of quantum mechanics [1]. However, it is only in the recent decades, studies in chemistry, biology, and physics have theoretically proposed and experimentally indicated that specific types of biological processes may be taking advantage of quantum effects to achieve high levels of performance [2]. Some of the captivating examples in this regard include magnetoreception-based navigation in some species of birds (for example, homing pigeons) [3,4], long-range quantum tunneling of electrons in protein complexes/redox chains [5,6], Dyson and Wright quantum theory of olfaction based on molecular vibrations [7,8], and the presence of quantum coherence in photosynthesis enabling higher yield in light-harvesting complexes [9,10], etc.

In the present work, we focus on the intriguing problem of photosynthesis-based light-harvesting complexes (protein-based Fenna–Matthews–Olson protein as a preliminary example). There in the so-called photosystem Type-II reaction centers [11] two Chlorophyll molecules (Chl-a) absorb solar photons while acting as the electron donor. Followed by that, the electron donor absorbed the solar photons which then causes the electron transfer to the Pheophytin (Pheo) molecule which acts as the electron acceptor, as shown in Fig. 1. As a result of this photon exchange, charge separation is generated in this tri-pigment setup which leads to the conversion of sunlight energy into chemical energy. In the last decade or so, quantum optical models have been reported to mimic the process of sunlight-to-chemical energy conversion [12,13] in photosystems. In these models, the coupling between the photosynthetic reaction centers with their immediate surroundings (environment) is modeled as Markovian in nature essentially meaning that the environments fails to keep any record/memory of its coupling with the reaction centers.

Fig. 1. Schematic representation of a photosynthetic reaction center in type-II photosystems.

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As an example, Dorfman et al. studied this problem under the Markov and Born approximations (the Born approximation assumes a weak type of system-bath interaction) and discussed how a typical photosynthesis process can be modeled as a five-level quantum heat engine (QHE) dissipatively coupled to high and low temperate reservoirs [14,15]. They showed that the coupling of certain discrete energy transitions (in the five-level system) with a common Markovian environment generates noise-induced coherence in the system that can aid increasing photosynthetic yield by 27% compared with no coherence case. The basic idea of noise-induced coherence relies on the fact that if we treat the process of photosynthesis as a biological quantum heat engine and calculate the current-voltage characteristics of the engine, it turns out that these characteristics lead to defining a higher value of engine efficiency in the presence of coherence. Since this coherence is generated through the interaction of discrete atomic energy levels with a common reservoir and it doesn’t require any external coherent drive therefore it is called noise-induced coherence (for further details, we direct the interested reader to Ref. [14]). However, in reality (see for instance Ref. [16]), the interaction of the reaction centers with the surrounding protein environment can be weak but still can keep track of the memory effects which severely challenges the validity of the Markov approximation. Therefore, to perform a more realistic treatment of the problem, in the present work we go beyond the Markov approximation and study the dynamics of photosynthetic complexes under memory-full/non-Markovian conditions. Here we would like to mention some of the works relevant to non-Markovianity in photosynthetic reaction centers (see, for instance, [17–19]).

In particular, we derive the time convolutionless (TCL) type non-Markovian master equation [20] for five-level quantum systems and apply it to analyze the light absorption and exciton transfer in photosynthesis. As one of the key results, we find that non-Markovian environments with a Lorentzian spectral density can generate and (more importantly) control the noise-induced coherence in the system. This opens the possibility of increasing the efficiency of the photosynthesis process in a transient manner which is not possible to achieve with traditional Markovian environments.

2. Theoretical description

In Fig. 2(a) we represent the energy-level diagram for our five-level donor-acceptor complex. The thermodynamics cycle begins with the absorption of thermal photons from the high-temperature reservoir at temperature $T_h$ with the excitation of electrons in the donor levels. Upon de-excitation, the electrons transfer to the acceptor molecules with emission of photons into a heat sink with temperature $T_c$. Useful work is extracted out between the ground state $g_A$ and excited states $e_A$ of the acceptor molecules. In Fig. 2(b) we draw the energy-level diagram at the reaction center level. Therein, the state $|{g}\rangle$ specifies the ground state for the entire molecular complex. The state $|{e}\rangle$ (split into two levels $|{e_1}\rangle$ and $|{e_2}\rangle$) shows the single donor molecule excited state. $|{m_1}\rangle$ is the meta-stable state with the charge being separated and transferred to the acceptor molecul followed by the creation of a hole in the donor molecule. The other meta-stable energy state $|{m_2}\rangle$ describes the ionized state with the photon supplied to the sink and the useful work being performed. Finally, the system undergoes the transition $|{m_2}\rangle\rightarrow |{g}\rangle$ through the spontaneous emission.

Fig. 2. Energy-level configuration for (a) the acceptor-donors complex, and (b) the photosynthesis reaction center in its entirety. In Fig. 2(a) $e_{D_1}$ and $e_{D_2}$ ($g_{D_1}$, $g_{D_2}$) stand for the excited states (ground states) of two donor molecules. These molecules form a pair of chlorophylls that plays a key role in the charge separation mechanism in the reaction center.

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3. Non-Markovian master equation for damped few-level atoms

Next, to describe the non-Markovian dynamics of the reaction centers we apply the master equations technique. Typically, a projection operator framework is adopted which leads to two types of equations namely the Nakajima-Zwanzig master equation (which has an integrodifferential form) [21,22] and the time-convolutionless (TCL) master equation (which is a local in time differential equation) [23]. Keeping in view the need for a numerical solution for our five-level QHE problem, we focus on the TCL master equation which takes the following form for the case of a driven dissipative two-level system [24] (here $\hbar =1$):

(1)$$\begin{aligned} &\frac{d\hat{\rho}(t)}{dt}={-}i[\hat{H}_S+\hat{H}_L,\hat{\rho}]+\sum_{\lbrace s={+},-\rbrace}\Big[C^2_s\gamma(t)\Big(\hat{S}_{{-}s}\hat{\rho}\hat{S}_s-\frac{1}{2}\lbrace\hat{S}_s\hat{S}_{{-}s},\hat{\rho}\rbrace\Big)+sC_sC_{0}\gamma(t)\Big(\hat{S}_{{-}s}\hat{\rho}\hat{S}_z\\ &+\hat{S}_z\hat{\rho}\hat{S}_s\Big)\Big]+C^2_0\gamma(t)[\hat{S}_z\hat{\rho}\hat{S}_z-\frac{1}{2}\lbrace \hat{S}_z\hat{S}_z,\hat{\rho}\rbrace]+C^2_0\gamma(t)[\hat{S}_+\hat{\rho}\hat{S}_+{+}\hat{S}_+\hat{\rho}\hat{S}_-]+C_0[\frac{\gamma(t)}{2}\lbrace \hat{S}_x,\hat{\rho}\rbrace\\ &+i\lambda\lbrace \hat{S}_x,\hat{\rho} \rbrace],\end{aligned}$$

where $C_{\pm }=\frac {\omega _{eg}+\Delta }{2\omega _{eg}}$, and $C_0=\frac {\Omega }{2\omega _{eg}}$ with $\omega _{eg}$, $\Delta$, and $\Omega$ being the atomic transition frequency, laser detuning and the Rabi frequency, respectively. The operators $\hat {S}_{+}$ and $\hat {S}_{-}$ stand for the raising and lowering operator for the two-level atom, respectively. $\hat {H}_S$ is the system Hamiltonian while $\hat {H}_L$ represents the Lamb Hamiltonian given by

(2)$$\hat{H}_L=\lambda(t)[C^2_+\hat{\sigma}_-\hat{\sigma}_+{+}C^2_+\hat{\sigma}_-\hat{\sigma}_-{+}C^2_0\hat{\sigma}^2_z].$$

We note that Eq. (2) represents the Lamb Hamiltonian which originates due to the coupling of our five-level atomic system with the vacuum fluctuations of the environment. The Lamb shift has been observed in a variety of systems in recent years, for instance, in ultracold atomic gas mixtures [25] and superconductor–insulator–normal-metal tunnel junctions [26] etc. In the last two equations the time-dependent parameters $\gamma (t)$ and $\lambda (t)$ which have the interpretation of atomic decay rate and Lamb shift are related to the transition rate $\Gamma (t)$ through $\Gamma (t)=\gamma (t)/2+i\lambda (t)$, while $\Gamma (t)$ itself is given by

(3)$$\Gamma(t)=\int_0^tdt'\int d\omega'J(\omega')\exp[i(\omega_L-\omega')t'].$$

Note the subtle but important point that in the Markovian Lindblad master equations for two-level quantum systems both atomic decay rate and Lamb shifts were time-independent [20]. For our photosynthesis model, coherence is not laser-assisted; rather the coupling of the excited donor states with a common ground state indirectly produces noise-induced coherence through a Fano-like process [14]. Therefore, we ignore the driving laser, and after some mathematically involved calculations (for details see our thesis [27]), we extend the aforementioned theory to a five-level situation in the interaction picture. Finally, we find the following TCL non-Markovian master equation

(4)$$\begin{aligned} \frac{d\tilde{\rho}(t)}{dt}&=\sum_{i,j=1}^5\gamma_{ij}(t)\bigg[(\bar{n}_{i}+1)\Big(\hat{S}_i^\dagger \hat{S}_j\tilde{\rho}(t)+\tilde{\rho}\hat{S}_i^\dagger \hat{S}_j-\hat{S}_j\tilde{\rho}(t)\hat{S}_j^\dagger{-}\hat{S}_j\tilde{\rho} \hat{S}_i^\dagger\Big)\\ &+\bar{n}_{i}\Big(\hat{S}_j\hat{S}_j^\dagger\tilde{\rho}+\tilde{\rho}(t) \hat{S}_j\hat{S}_j^\dagger{-}\hat{S}_i^\dagger\tilde{\rho}\hat{S}_j-\hat{S}_j^\dagger\tilde{\rho}\hat{S}_j\Big)\bigg],\end{aligned}$$

where $\hat {S}_j (\hat {S}^\dagger _j)$ represents the lowering (raising) operator for the $j$th atomic transition in our five-level system. Also, we have ignored the Lamb shift. Note that, unlike the two-level atom case, we have now included the non-zero temperature of the baths (represented through the Bosonic occupation number $\bar {n}_i$) to incorporate the thermal aspect of the photosynthesis process. The time-dependent transition rate for the present case takes the form

(5)$$\gamma_{ij}(t)=\int_0^t dt'\int d\omega'J(\omega')\mathrm{e}^{i(\omega_{ij}-\omega')(t-t')}.$$

As pointed out above, the transition rate in the non-Markovian case becomes time-dependent and varies as a function of the environmental spectral density function $J(\omega )$. This, in complete contrast to the Markovian case, leads to a more realistic situation where the environment now keeps a record of the memory effects. Proceeding further, we calculate the time-dependent transition rate $\gamma _{ij}(t)$ and, for that, we assume the following Lorentzian form of the density function

(6)$$J(\omega)=\mathcal{N}^{{-}1}\pi\gamma[1+\gamma^{{-}1}(\omega-\omega_0)^2],$$

where $\omega _0$ represents the frequency at which the density function’s peak resides, $\gamma$ shows the width and, $\mathcal {N}$ is the normalization constant. We remark that this form of the spectral density is experimentally feasible and can be mimicked by a single-mode optical cavity with imperfect mirrors [28]. To perform the time integral in Eq. (5) we consider the following complex function $f(z)$ which we assume to be holomorphic on a contour $\mathrm {C}$ (see Fig. 3)

(7)$$f(z)=\frac{\mathrm{e}^{{-}2\pi i\gamma z}}{z^2+t^2}.$$

To integrate $f(z)$ we focus on three possible cases

• Case-(I) $\gamma =0$ (the trivial scenario): $\int_\mathrm{R}\frac{1}{x^2+t^2}=\left.\frac{1}{t}\arctan(x)\right|_{x={-}\infty}^{x=\infty}=\frac{\pi}{t}$
• Case-(II) $\gamma <0$: As shown in Fig. 3, we consider a semicircle containing the pole at $it$. As the radius of the semi-circle grows and approaches infinity, we can ignore the arc contribution that leaves us with the integration carried out on the straight line. Consequently, Cauchy’s residue theorem gives us $\int_\mathrm{R}f(z)dz = 2\pi i\left(\frac{1}{2it}\mathrm{e}^{2\pi\gamma t}\right)=\frac{\pi}{t}\mathrm{e}^{{-}2\pi|\gamma| t}.$
• Case-(III) $\gamma >0$: Similar to the above $\gamma <0$ case leading to the same result. The main difference will now be to draw the contour below the $\mathrm {R}$-axis such that the pole point $-it$ can be enclosed by the contour.

Putting everything together, we observe

(8)$$\frac{\mathcal{N}}{\pi\gamma}\int\frac{1}{1+\left(\frac{\omega'-\omega_0}{\gamma}\right)}\mathrm{e}^{i(\omega_0-\omega')(t-t')}d\omega' =\mathcal{N}\mathrm{e}^{-\gamma(t-t')},$$

which leads to the following temporal integral

(9)$$\gamma(t)=\mathcal{N}\int_0^tdt'\mathrm{e}^{-\gamma(t-t')} =\gamma(1-\mathrm{e}^{-\gamma t}).$$

This type of time-dependent decay rate is known as Ornstein-Uhlenbeck noise in the literature [29,30]. We note that in the long-time limit i.e. as $t\rightarrow \infty$, the non-Markovian decay rate $\gamma (t)$ reduces to its Markovian counterpart $\gamma$. Extending this result to the time-dependent transition rate $\gamma _{ij}(t)$ between energy states $|{i}\rangle$ and $|{j}\rangle$ we can write

(10)$$\gamma_{ij}(t)=\gamma_{ij}(1-\mathrm{e}^{-\gamma_{ij}t}),$$

where (as before) $\gamma _{ij}$ is the Markovian decay rate for the transition $|{i}\rangle\longleftrightarrow |{j}\rangle$. Next, in Table 1 we summarize the set of parameters used in three possible regimes of an open quantum system treatment of photosynthesis. We refer to systems as being in an overdamped (underdamped) regime if the energy separation between the states $|{e_1}\rangle$ and $|{e_2}\rangle$ is smaller (larger) than these states’ inverse lifetimes. In the case that the energy gap is comparable to the inverse lifetime, we call such a scenario intermediately damped. Furthermore, for simplicity, we assume the temperature of the cold reservoirs to be almost zero compared to the hot reservoir temperature; i.e., we set $\bar {n}_{1c}\approx 0$, $\bar {n}_{2c}\approx 0$, and $\bar {n}_{m_2c}\approx 0$. Note that the set of parameters described in Table 1 is experimentally feasible for type II photosystems as reported in Ref. [31].

Fig. 3. A semi-circle contour for the integration of $f(z)$ on the complex plane.

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Table 1. Parameters used under different working regimes

View Table

4. Results in the over-, under-, and intermediate coupling regimes

In Fig. 4 we present the system dynamics in a panel format obtained from the numerical solution of our five-level TCL master equation (4). The relevant time scale of the problem is in femtoseconds (fs). The top (bottom) row of the panel represents the non-Markovian (Markovian) model, while three columns show three parameter regimes (overdamped, underdamped, and intermediate) summarized in Table 1. Out of various density matrix elements involved in the problem, we primarily focus on the element $\rho _{e_1e_2}$ which represents the quantum coherence generated between states $|{e_1}\rangle$ and $|{e_2}\rangle$ due to the coupling with a common bath via the ground state $|{g}\rangle$ channel. This so-called noise-induced coherence has been argued to be not a hinder but to aid the higher photosynthetic yield in the Markovian case [14]. Here we focus on the behavior of this coherence due to non-Markovian baths and ask the question which type of bath (Markovian or non-Markovian) results in a more efficient excitation energy transfer? As the initial conditions, we suppose

(11)$$\rho_{e_1e_1}=0.5,~\text{and}~\rho_{e_1e_2}=0.5,$$

with all other density matrix elements being zero in the beginning. From Fig. 4, we first notice that in both Markovian and non-Markovian cases the coherence starts from its initial value, and then gradually decreases as a function of time. Next, we observe two key results: (1) In all three working regimes, there is a considerable enhancement in the coherence magnitude $|\rho _{e_1e_2}|$ as we consider the non-Markovian case. (2) However, in the non-Markovian scenario we also observe the meta-stable state $|{m_1}\rangle$ population (i.e. $\rho _{m_1m_1}$) exhibit relatively slow build-up compared to its Markovian counterpart. This trend indicates that in the Markovian case even though a larger coherence can be generated, it takes a longer time for the charge separation to grow transiently as compared to the Markovian case. For example, if we select the transient time of $100fs$ then from Fig. 4 we notice that in the overdamped regime for non-Markovian case $\rho _{m_1m_1}\sim 0.05$ while for the corresponding Markovian case $\rho _{m_1m_1}\sim 0.2$. We notice that the same behavior extends down to undercoupled and critical coupling regimes as well.

Fig. 4. Top (bottom) row shows the population and coherence dynamics in non-Markovian (Markovian) case. Plots pair ((a), (b)), ((c), (d)), and ((e), (f)) represents the overdamped, underdamped, and intermediately damped regimes, respectively.

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In Fig. 5 we report the time evolution of real and imaginary parts of $\rho _{e_1e_2}$. The dotted magenta, dotted-dashed magenta, dashed blue, and solid blue curves represent the Markovian $\mathrm {Re}[\rho _{e_1e_2}]$, Markovian $\mathrm {Im}[\rho _{e_1e_2}]$, non-Markovian $\mathrm {Re}[\rho _{e_1e_2}]$, and non-Markovian $\mathrm {Im}[\rho _{e_1e_2}]$ cases, respectively. Inspired by the Fig. 4(c) and 4(d), in Fig. 5 we have selected the underdamped regime which allows the survival of higher coherence in the long-time limit of non-Markovian regime. Both $\mathrm {Re}[\rho _{e_1e_2}](t)$, $\mathrm {Im}[\rho _{e_1e_2}](t)$ curves confirm that the non-Markovian case not only maintains a higher coherence for all transient times, but also reaches a steady-state value (around $1500 fs$) that surpasses the corresponding Markovian case value. This behavior opens the possibility of further improvement in the photosynthetic yield due to non-Markovianity.

Fig. 5. Comparison of the temporal evolution of real and imaginary parts of coherence in the Markovian and non-Markovian cases in the underdamped regime.

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5. Summary and conclusions

In summary, by questioning the validity of Markov approximation in the open quantum system treatment of photosynthetic reaction centers in this work, we studied the impact of memory-full environments (non-Markovianity) on the time evolution of noise-induced coherence and population dynamics in type-II photosystems. Therein, we model the photosystem-II as a five-level QHE and studied the dynamics of the photosynthetic process through a local-in-time non-Markovian master equation of the TCL type. As an example, we concentrated on the experimentally feasible Lorentzian spectral-shaped non-Markovian environments that lead to time-dependent transition rates. As the main finding of our calculations, we concluded that the Lorentzian environments can enhance the noise-induced coherence (in all working regimes under transient conditions and the underdamped regime under long-time situations) as compared to the corresponding Markovian case. Since it is already known in the literature that the noise-induced coherence can increase the power of quantum heat engines [32], this result can improve our understanding of the nearly perfect efficiency attained by nature in the photosynthesis processes. Additionally, this work may find applications in engineering next-generation efficient quantum devices that are bio-inspired.

In the end, we mention that the TCL master equations are typically applied in a short time interval (shorter than the time set by the inverse lifetime of atomic transition). However, one can understand our results plotted in Fig. 4 and Fig. 5 by dividing the entire time range into many consecutive short time intervals such that each internal defines the initial condition for the next interval. Combining all internal behavior one can obtain the complete plots reported in Fig. 4 and Fig. 5. Alternately, one can perform the numerical simulation of the total Hamiltonian and perform a comparison with the results obtained from the TCL master equation. We leave this as a future direction of this work.

Note Added. Note that the Refs. [17–19] were brought to our attention after the submission of our work and therefore, in our work, the model, the mathematical treatment adopted to solve the problem, and the discussion of the results remains original.

Appendix

A. Non-Markovian master equation derivation for two-level quantum systems

In this appendix, we outline the derivation of a certain type of non-Markovian master equation for two-level quantum systems known in the literature as the time-convolutionless (TCL) master equations [20]. Here we closely follow the treatment of the two-level problem reported in Ref. [24]. Later, as the novel part of this derivation, we extend this calculation to the five-level quantum systems which we then use to model photosynthetic reaction centers. We begin by considering a two-level atom with atomic transition frequency $\omega _{eg}$ which is driven by a laser with frequency $\omega _L$ and also coupled with a heat reservoir. In the open quantum system considerations, the total Hamiltonian can be decomposed into three parts, namely, the system Hamiltonian or $\hat {H}_S$, reservoir/bath Hamiltonian or $\hat {H}_B$ and system-bath interaction Hamiltonian $\hat {H}_{SB}$. The mathematical forms of these Hamiltonians are given by

(12)$$\hat{H}_S=\frac{\hbar}{2}(\Delta\hat{S}_z+\Omega\hat{S}_x),$$

(13)$$\hat{H}_B=\hbar\sum_k\omega_k \hat{a}_k \hat{a}^\dagger_k,$$

(14)$$\hat{H}_{SB}=\hbar\sum_kg_k[e^{i\omega_Lt}\hat{S}^\dagger\hat{a}_k+e^{{-}i\omega_Lt}\hat{S}\hat{a}^\dagger_k].$$

Here $\Delta =\omega _{eg}-\omega _L$ is the laser-atom detuning frequency, $\Omega$ represents the Rabi frequency, $\omega _k$ is the $k$th mode frequency of the bath, and $g_k$ shows the interaction between the atom and the $k$th bath mode. $\hat {S}_x$ and $\hat {S}_z$ are the $x$th and $z$th spin operators while $\hat {a}_k$ is the annihilation operator for the $k$th bath mode. $\hat {S}$ and $\hat {S}^\dagger$ are the atomic lowering and raising operators, respectively. Note that in defining the interaction Hamiltonian we have applied the rotating wave approximation. The system Hamiltonian can be expressed in matrix form as follows

(15)$$\hat{H}_S=\frac{\hbar}{2}\begin{pmatrix}\Delta & \Omega\\ \Omega & -\Delta \end{pmatrix}.$$

The diagonalization of the above Hamiltonian produces the following eigensystem

(16)$$\text{Eigenvalues:}~~\lambda_\pm{=}\pm\frac{\hbar}{2}\sqrt{\Delta^2+\Omega^2},$$

(17)$$\text{Eigenvectors:}~~|{\lambda_\pm}\rangle=\left(-\frac{\Omega}{\mp\sqrt{\Delta^2+\Omega^2}+\Delta}\right)|{e}\rangle+|{g}\rangle,$$

At this stage, for the sake of simplicity, we perform the following change of basis for the system Hamiltonian

\begin{array}{c} \left\{|{e}\rangle=\begin{pmatrix}1\\0\end{pmatrix},|{g}\rangle=\begin{pmatrix}0\\1\end{pmatrix}\right\},\text{or}\left\{|{\psi_\pm}\rangle={\pm}\frac{1}{\sqrt{2}}\begin{pmatrix}\sqrt{1\pm\sin\theta}\\\sqrt{1\mp\sin\theta}\end{pmatrix}\right\}\\ \downarrow\\ \left\{|{\psi_+}\rangle=\begin{pmatrix}1\\0\end{pmatrix},|{\psi_-}\rangle=\begin{pmatrix}0\\1\end{pmatrix}\right\}. \end{array}

The corresponding transformation matrix $\hat {U}$ takes the form

(18)$$U=\begin{pmatrix}\frac{1}{2}\sqrt{1+\sin\theta} -\frac{1}{2}\sqrt{1-\sin\theta}\\ \frac{1}{2}\sqrt{1-\sin\theta}-\frac{1}{2}\sqrt{1+\sin\theta}\end{pmatrix},$$

where the angle $\theta$ is defined below. In the transformed basis, the atomic raising operator $\hat {S}^\dagger$ turns into $U^\dagger \hat {S}^\dagger U$, which in the matrix form becomes

(19)$$\hat{S}^\dagger\rightarrow\frac{1}{2}\begin{pmatrix}-\sqrt{1-\sin^2\theta} & -1-\sin\theta\\-1+\sin\theta & \sqrt{1-\sin^2\theta}\end{pmatrix}.$$

We notice that after the transformation the condition $(\hat {S})^\dagger =\hat {S}^\dagger$ remains intact. Using the definition $\theta =\arctan (\Delta /\Omega )$, we express $\hat {S}^\dagger$ in the matrix form as

(20)$$\hat{S}^\dagger{=}\frac{\hbar}{2}\begin{pmatrix}-\frac{\Omega}{\omega} & \frac{-\omega-\Delta}{\omega}\\ \frac{-\omega+\Delta}{\omega} & \frac{\Omega}{\omega}\end{pmatrix}.$$

Next, we transform the interaction Hamiltonian into our new basis. Therein, with the application of the Baker-Hausdorff lemma, we find

(21)$$\begin{aligned}\widetilde{H}_{SB}(t)&=e^{it(\hat{H}_S+\hat{H}_B)/\hbar}\hat{H}_{SB}e^{{-}it(\hat{H}_S+\hat{H}_B)\hbar}\\ & = \hbar\left(\frac{e^{i\omega_Lt}}{2\omega}[e^{{-}i\omega t}(\Delta-\omega){\hat{S}}+e^{i\omega t}(\Delta+\omega){\hat{S}}^\dagger{+}\Omega{\hat{S}}_z]+h.c.\right)\left(\sum_ke^{{-}i\omega_kt}g_k\hat{a}_k+h.c.\right). \end{aligned}$$

Furthermore, we take the continuum limit on the environmental modes and define the coupling constant $g_k$ in terms of the environmental spectral density $J(\omega )$ as

(22)$$\sum_kg_k^2\rightarrow\int d\omega J(\omega).$$

This spectral density is then utilized in defining the transition rate $\Gamma (t)$ through the equation

(23)$$\Gamma(t)=\int_0^tdt'\int\omega'J(\omega')e^{i(\omega_L-\omega')t'}.$$

The transition rate $\Gamma (t)$ is physically significant due to its connection with the atomic decay rate $\gamma (t)$ and frequency shift $\lambda (t)$ i.e. $\Gamma (t)=\gamma (t)/2+i\lambda (t)$. Inserting the aforementioned Hamiltonian into the integro-differential form of the von-Neumann equation leads us to the following equation obeyed by the system density matrix $\hat {\rho }(t)$

(24)$$\begin{aligned} \frac{d\hat{\rho}(t)}{dt}=&-i[\hat{H}_S+\hat{H}_L,\hat{\rho}(t)]+C^2_\pm\gamma(t)\left[\hat{S}_\mp\hat{\rho}(t)\hat{S}_\pm{-}\frac{1}{2}\{\hat{S}_\pm\hat{S}_\mp,\hat{\rho}(t)\}\right]\\ &+C^2_0\gamma(t)\left[\hat{S}_z\hat{\rho}(t)\hat{S}_z-\frac{1}{2}\{\hat{S}_z\hat{S}_z,\hat{\rho}(t)\}\right]\pm C_\pm C_0\gamma(t)\left[\hat{S}_\mp\hat{\rho}(t)\hat{S}_z+\hat{S}_z\hat{\rho}(t)\hat{S}_\pm\right]\\ &+C^2_0\gamma(t)\left[\hat{S}_+\hat{\rho}(t)\hat{S}_+{+}\hat{S}_-\hat{\rho}(t)\hat{S}_-\right]+C_0\left(\frac{\gamma(t)}{2}\{\hat{S}_x,\hat{\rho}(t)\}+i\lambda(t)[\hat{S}_x,\hat{\rho}(t)]\right),\end{aligned}$$

which, when written in a compact notation, is the time-convolutionless (TCL) non-Markovian master equation we have written as Eq. (1). Before closing this appendix section we make a few observations. First, note that in Eq. (24), for the sake of simplicity we have assumed zero occupation number (i.e. reservoir/bath’s absolute temperature has been set to zero or $T=0$). Secondly (as discussed in Sec. 3 as well), after a lengthy but straightforward calculation the above derivation can be extended to five-level atomic systems. Finally, the connection with the corresponding Markovian case can be made under the limit that the bath correlation time $\tau _c$ is much less than the system relaxation time $\tau _R$ i.e. $\tau _c\ll \tau _R.$ Under this case the upper integration limit in the temporal integrals can be extended to infinity which leads to defining constant or time-independent decay rates i.e. $\gamma =\lim _{t\rightarrow \infty }\gamma (t)$ as encountered in the Markovian case. Additionally, the $\tau _c\ll \tau _R$ condition allows us to drop the last three terms in Eq. (24) which produces the typical Lindblad form of the Markovian master equations.

Funding

Miami University College of Arts and Science and Physics Department.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Parameter/Regime	Over	Under	Intermediate
$Δ [c m^{- 1}]$	120	600	720
$γ_{e_{1} g} [c m^{- 1}]$	0.005	0.005	0.005
$γ_{e_{2} g} [c m^{- 1}]$	0.0016	0.005	0.005
$γ_{e_{1} m} [c m^{- 1}]$	140	35	280
$γ_{e_{2} m} [c m^{- 1}]$	18	35	280
$γ_{m_{1} m_{2}} [c m^{- 1}]$	$10^{- 10}$	$10^{- 10}$	$10^{- 10}$
$γ_{m_{2} g} [c m^{- 1}]$	200	50	300
$τ_{2}^{- 1} [c m^{- 1}]$	41	41	41
${\bar{n}}_{1 h}$	60,000	10,000	90,000
${\bar{n}}_{2 h}$	10,000	20,000	10,000

Non-Markovianity in photosynthetic reaction centers: a noise-induced quantum coherence perspective

Abstract

1. Introduction

2. Theoretical description

3. Non-Markovian master equation for damped few-level atoms

4. Results in the over-, under-, and intermediate coupling regimes

5. Summary and conclusions

Appendix

A. Non-Markovian master equation derivation for two-level quantum systems

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (5)

Tables (1)

Equations (27)

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