Abstract
When dealing with systemreservoir interactions in an open quantum system, such as a photosynthetic lightharvesting complex, approximations are usually made to obtain the dynamics of the system. One question immediately arises: how good are these approximations and in what ways can we evaluate them? Here, we propose to use entanglement and a measure of nonMarkovianity as benchmarks for the deviation of approximate methods from exact results. We apply two frequentlyused perturbative but nonMarkovian approximations to a photosynthetic dimer model and compare their results with that of the numericallyexact hierarchy equation of motion (HEOM). This enables us to explore both entanglement and nonMarkovianity measures as means to reveal how the approximations either overestimate or underestimate memory effects and quantum coherence. In addition, we show that both the approximate and exact results suggest that nonMarkonivity can, counterintuitively, increase with temperature and with the coupling to the environment.
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Introduction
Modelling and understanding the nonequilibrium dynamics of open quantum systems is a ubiquitous problem in physics, chemistry and biology^{1,2,3,4,5,6,7,8,9}. In such systems, the environment is usually composed of a huge number of microscopic constituents, an exact description of which is challenging. One can invoke intensive computational techniques, such as pathintegral formalisms^{1,2,10,11}, Monte Carlo algorithms^{12}, the hierarchy equations of motion (HEOM)^{13,14,15,16}, the reactioncoordinate method^{17,18} and others, to explicitly and exactly propagate the quantum state of a complete systemenvironment model. However, a common drawback of these exact numerical solutions is their demanding computational resource requirements, which can scale badly depending on the spectral density of the environment being modelled, the number of independent baths the system is coupled to, or the complexity of the system itself.
To simplify the problem and gain useful physical insights, approximations are usually made to reduce the system dynamics to that of a relatively few degrees of freedom. In that regard, much effort has been devoted to develop quantum master equations (QMEs) which describe these reduced degrees of freedom in various limits. Redfield theory^{19} provides one with QME based on (together with a secular approximation) a secondorder perturbation approximation in the systemenvironment coupling. For the strongcoupling limit, Förster theory^{20,21,22} adopts a diffusionrate equation^{23} to describe the incoherent transport phenomenon. Nevertheless, these conventional Markovian QME treatments cannot capture the memory effects of the bath.
In order to take into account the memory effects, many attempts at improving these Markovian QMEs have been made. The secondorder timeconvolution (TC2)^{24} equation is known as a chronologicalordering prescription^{25} or timenonlocal equation^{26,27}. It is a direct generalization of Redfield QME without performing the Markov and secular approximations. The secondorder timelocal (TL2) equation is another frequently used QME, sometimes called a partialtimeordering prescription^{25} or timeconvolutionless equation. Some works suggest that TL2 shows better performance than TC2 at numerically approximating exact results^{28}. Nevertheless, their respective domains of applicability have not been thoroughly investigated yet.
In each QME model (TC2, TL2), certain approximations and simplifications are introduced to obtain solvable equations. To investigate the deviation of each approximate QME model from the exact results, we first compare the explicit dynamics of these two approximative QMEs with that of the HEOM. The HEOM approach is considered to be numerically exact for the models with the DrudeLorentz spectral density function investigated here. For more general bath models and dynamics at low temperatures, the need to truncate at a certain level of the hierarchy equations can lead to errors and thus the exactness of the HEOM approach requires further scrutiny in such cases^{29,30,31}. We focus on the intermediate systemenvironment coupling regime, which has proven to be the most challenging and relevant to the dynamics in realistic systems such as the photosynthetic FennaMatthewsOlson complex. Notably, the intermediate regime is also the one at which the region of validity of most approximations breaks down. Both approximate methods are perturbative in the systembath coupling, but can in principle harbor memory effects of the environment.
Recently, much effort has been devoted to the quantification of memory effects^{32,33,34,35} which has subsequently been studied in the context of various physical systems^{36,37,38}. To investigate how well the models we study here capture the memory effect, we utilize the concept of the ChoiJamiołkowski isomorphism^{39,40} to encode complete information on the dynamics of the system into the entanglement with an ancilla. By comparing the time evolution of the entanglement between system and ancilla and an associated measure of nonMarkovianity^{33}, one can find out to what extent the memory effects and coherence predicted by each approximate QME deviates from being numerically exact. Our results suggest that entanglement and nonMarkovianity provide a useful benchmark for the performance of such approximative treatments, providing a more finegrained insight into the deviation from exact results than quantities like the fidelity alone.
In performing this analysis we also discuss several interesting physical trends, including a counterintuitive increase of nonMarkovianity with both temperature and with the coupling strength to the environment. We attribute this increase to an enhancement of systemenvironment correlations when both the coupling and temperature are increased. Additionally, evidence from other studies^{41,42,43} suggests that nonMarkovian environments are capable of sustaining quantum coherence. The interplay of these factors finally results in the increase of nonMarkovianity with both temperature and coupling strength that we see in our results.
Results
The spinboson model
The spinboson model^{1} is one of the most extensively studied models of open quantum systems and is the one we employ here. It describes a spinorlike twostate system interacting with a bosonic environment. First, let us consider this standard model, which can be divided into three components
The system Hamiltonian, , is written as
where is the coherentcoupling term, which enables the tunneling between the two system quantum states, labeled as 1〉 and −1〉, with the energy level spacing ħω_{0}. Usually, one adopts the delocalized basis χ_{+}〉 and χ_{−}〉 (exciton), which is defined by the following eigenvalue problem
with .
The environment, , is usually modelled as a large collection of harmonic oscillators
where () is the creation (annihilation) operator of the environment mode k with angular frequency ω_{k}. For simplicity, a linear systemenvironment coupling, , is adopted throughout this work:
where g_{k} is the coupling constant between the environment mode k and the system. In most physical problems, the details of the microscopic description of g_{k} are not clear and one usually employs a spectral density function, J(ω) = ∑_{k}g_{k}^{2}δ(ω − ω_{k}), to characterize the coupling strength via the reorganization energy . The physical meaning of the spectral density function can be understood as the density of states of the environment, weighted by the coupling strengths. Moreover the way in which the environment modulates the dynamics of the system is described by the correlation function
The real part is related to the dissipation process, while the imaginary part corresponds to the response function.
The statistical properties of the entire system can be described by the total density matrix ρ_{tot}, which contains all the degrees of freedom of the system and environment. If the correlation between the system and environment is negligible, the Born approximation can be used and the total density matrix can be factorized into
where ρ_{sys}(t) describes the dynamics of the system and is the environment density matrix in thermal equilibrium at temperature T. Here, k_{B} is the Boltzmann constant and is the partition function.
One notes that when ω_{0}, J and λ are comparable, this makes the conventional perturbative treatment unreliable. In the following, we will adopt the two frequentlyused perturbative but nonMarkovian QME formalisms discussed in the introduction and compare their results with the exact one in the intermediatecoupling regime, as they both begin to break down and investigate ways in which to evaluate their accuracy.
Secondorder timeconvolution equation (TC2)
For the Hamiltonian defined above, the time evolution of the system density matrix ρ_{tot}(t) under the TC2 approximation is expressed as
The tilde symbol above an operator denotes the interaction picture with respect to . The interaction Hamiltonian in terms of the delocalized basis can be expressed as
where and μ, ν = χ_{+}, χ_{−}. Substituting Eq. (9) into (8) with the explicit expansion leads to a set of simultaneous integrodifferential equations of the density matrix elements ρ_{μ,ν}(t)
One notes that the memory effects are taken into account in terms of the convolution of the memory kernel f_{μ,ν}(t − τ). A detailed expression for this kernel is given in the Appendix.
To solve the simultaneous integrodifferential components of Eq. (10), we invoke the Laplace transformation and transform them into a set of algebraic equations. After carefully analyzing the properties of the poles, the conventional residual theorem enables one to accomplish the inverse Laplace transformation and move back from Laplace space into the timedomain.
Secondorder timelocal equation (TL2)
In the TL2 formalism, the system is considered to be sluggish, hence the bath feedback on the system dynamics can be neglected by approximating . This assumption is reasonable because it is impossible for a system to change its configuration instantaneously. Consequently the system density matrix should be pulled out from the integral to obtain the following QME
Similarly, substituting Eq. (9) into (11) with the explicit expansion leads to a set of simultaneous differential equations of the density matrix elements ρ_{μ,ν}(t)
The detailed expression of the memory kernel h_{μ,ν}(t − τ) is given in the Appendix. It should be emphasized that although ρ_{μ,ν}(t) is pulled out from the integral, Eq. (12) is capable of predicting a nonMarkovian dynamics because the time integral of h_{μ,ν}(t) results in timevarying coefficients in front of ρ_{μ,ν}(t). Whether or not such differential equations behave nonMarkovianly crucially depends on these timevarying coefficients.
Comparisons with exact results
To illustrate the differences of the approximations explicitly, we apply these two QMEs to a photosynthetic dimer model, which has attracted considerable interest recently^{8,9,44,45,46,47,48,49,50}. We employ the DrudeLorentz spectral density function (the overdamped Brownian oscillator model)^{15,51}, J(ω) = (2λγ/π)[ω/(ω^{2} + γ^{2})], which has been widely used for a range of theoretical studies of this type of system^{46,47,48,49,50}. We use it here because it is convenient for the comparison with the HEOM. However, in reality, the spectral densities found in real photosynthetic systems tend to be much more complex^{46} and while the HEOM can be extended to model such environments it typically involves a substantial additional numerical overhead^{52}. As mentioned in the previous section, within the DrudeLorentz spectral density the reorganization energy, λ, characterizes the coupling strength to the environment, while the quantity γ determines the width of the spectral density. These two parameters have considerable influence on the dynamics of the system.
In Fig. 1, we show the system dynamics given by (a) TC2, (b) HEOM and (c) TL2 with varying λ and temperature T. The other parameters are fixed at ω_{0} = 70 cm^{−1}, J = 100 cm^{−1} and γ = 50 cm^{−1} (γ^{−1} = 106 fs). These parameters are typical in photosynthetic systems. The solid curves in each panel denote the populations of the χ_{+}〉 state with temperatures T = 300 K (black), 250 K (red) and 200 K (blue), respectively. It can be seen that, at higher temperatures, the population of the χ_{+}〉 state transfers to the χ_{−}〉 state faster than at lower temperatures, but there is always a crossing so that the thermal equilibrium population of the χ_{+}〉 state is larger at higher temperatures.
For small values of λ, the results of the two QME models show excellent agreement with that of the HEOM, indicating that both TC2 and TL2 perform well in the weak systemenvironment coupling regime and that the bath memory effect is insignificant at small λ. Moreover, the result of TC2 completely coincides with that of the HEOM for very small couplings. We show the comparison between TC2 (solid curve) and HEOM (dotdashed curve) methods in the inset of Fig. 1(a) for λ = 5 cm^{−1} and T = 250 K. This is in line with a recent comparative work in Ref. 53. When λ is increased, the TC2 population results exhibit vigorous beating and produce oscillatory curves up to 800 fs, which is absent in the HEOM result. We attribute these oscillations to the overestimation of the coherence by TC2. Apart from these beatings, the overall magnitude of the population of HEOM is quantitatively better approximated by TC2 than TL2. The TL2 model yields monotonicallydecaying population dynamics that tends to reach thermal equilibrium too rapidly. This leads to a significant overestimation of the population relaxation rate by TL2, especially at large λ. This overestimation of the population relaxation rate in Redfield theory has been reported previously^{54} and here we gain further insight into its origin by comparing to the TC2 results.
The dashed curves in Fig. 1 denote the absolute value of the offdiagonal elements of the system density matrix, i.e., the coherence between the χ_{+}〉 and χ_{−}〉 states. The results from the TC2 method manifestly show the overestimation of the coherence even if λ is small. When λ is increased, the overestimation of the coherence becomes quite pronounced. On the other hand, the coherence in the TL2 model decays more rapidly, leading to the sluggish dynamics discussed above. In summary, the coherence dynamics is better approximated by TL2 and the TC2 model may fail in approximating the true coherence for large λ. However, the overall population decay rate predicted by the TC2 is generally more correct than that of TL2. It is interesting to note that the TL2 model yields an exact QME for a pure dephasing spinboson model (i.e. J = 0)^{28} while the TC2 model underestimates the pure dephasing rate, which is in line with our findings here.
Benchmark of approximative QMEs
In the previous section, we analyzed how the coherence terms of the two approximations are qualitatively different from the HEOM exact results. However, those comparisons fail in providing an overall intuitive picture about which model performs better as they are basisdependent. In other words, it is possible that one model may perform better or worse than another depending on the bases used. In this section, we apply a measure of the nonMarkovianity to develop a basesfree benchmark which can quantitatively describe the performance of the approximate methods.
Entanglement and nonMarkovianity
Let us consider an isolated ancilla possessing the same degrees of freedom of the system and with which the system forms a maximally entangled initial state (see Fig. 2). If the system evolves according to a process , then the ChoiJamiołkowski isomorphism^{39,40} guarantees that the extended density matrix
contains all the necessary information on the dynamics of the system, where is the identity process acting on the ancilla. The entanglement, E(ρ_{sys,anc}), between the system and the ancilla is a physical quantity which is typically very sensitive to environmental effects.
Another related quantity is the degree of nonMarkovianity, . Recently, many efforts have been devoted to construct a proper measure of the nonMarkovianity^{32,33}. Rivas et al.^{33} combine the concept of the divisibility of a quantum process^{55,56} and the fact that no local completely positive (CP) operation^{40} can increase the entanglement E between a system and its corresponding ancilla
Consequently, Rivas et al.^{33} proposed that the degree of nonMarkovianity within a given time interval [0, t] can be estimated by
where
The nonMarkovianity of opensystem quantum dynamics can be evaluated at many different theoretical levels^{32,33,34,35,36} and the quantity is an extremely strict indicator of nonMarkovianity that measures the information exchange in time between the system and its environment. For to have a nonzero value, explicit environmental memory effects must be present.
Here we compare the time evolution of the entanglement, E_{t} and the corresponding degree of nonMarkovianity, , for the two approximate systembath models and show how they can provide an integrated picture as to what extent their dynamics deviate from the exact results.
Evaluating nonMarkovianity
To analyse the behavior of the nonMarkovianity in each method, in this section we will show how the concurrence, a wellknown measure for bipartite entanglement^{57}, between system and ancilla evolves in time and how the corresponding nonMarkovianity [Eq. (15)] depends on the physical parameters of the original spinboson model.
As an explicit visualization of the integrand in Eq. (15), in Fig. 3, we apply the measure to (a) TC2, (b) HEOM and (c) TL2 and show the time evolution of the concurrence for different values of λ at temperatures T = 300 K (black), 250 K (red) and 200 K (blue), respectively. The other parameters are ω_{0} = 70 cm^{−1}, J = 100 cm^{−1} and γ = 50 cm^{−1} (γ^{−1} = 106 fs). It can be seen that, when increasing the temperature and λ, the decoherence becomes more pronounced. Hence, the concurrence will die out earlier for larger λ and higher temperature. As shown in Fig. 3(a), except for λ = 5 cm^{−1}, which produces monotonicallydecreasing concurrence, the TC2 model produces oscillatory curves, in which a concurrence revival is exhibited around 100 fs and results in a finite degree of nonMarkovianity (shown later). A similar entanglement revival can also be seen in biomolecular systems^{58}. While in Fig. 3(b,c), HEOM and TL2 produce monotonicallydecreasing concurrence and generate no visible nonMarkovianity with this measure.
In Fig. 4, we show the corresponding measure of the nonMarkovianity, , calculated using the time evolution of the concurrence shown in Fig. 3(a). Only TC2, for larger λ values, leads to nonzero nonMarkovianity, while TC2 at λ = 5 cm^{−1}, HEOM and TL2 generate null results due to the monotonicallydecreasing concurrence. This comparison not only shows that the TL2 yields a better approximation to the HEOM dynamics, but also explicitly demonstrates the degree to which TC2 deviates from HEOM. We again attribute this deviation to the overestimation of coherence shown in Fig. 1. In addition, it can be seen in Fig. 4 that tends to increase with increasing λ and temperature. We will investigate this below in a regime where the HEOM results exhibit similar behavior.
Increase of nonMarkovianity with λ and temperature
The other two important parameters in our spinboson model are the level spacing ω_{0} and the bath relaxation time γ. The former affects to what extent the state χ_{+}〉 is delocalized, while the latter is related to the correlation time of the environment and is directly connected to the nonMarkovianity of the system.
In Fig. 5(a), we reduce ω_{0} to 40 cm^{−1} and fix the other parameters at λ = 5 cm^{−1}, γ = 50 cm^{−1} and T = 200 K. The reduction of ω_{0} leads to a manifest concurrence revival around 100 fs in the TC2 concurrence dynamics, a result of stronger delocalization and significant enhancement of the coherence effect. An analogous result can be seen in Ref. 37. In the mean time, the concurrence of the HEOM result is still monotonically decreasing. The TC2 model further overestimates this enhancement and ends up with finite nonMarkovianity within all range of temperatures shown in Fig. 5(b). The TL2 model predicts almostMarkovian results, besides the very small nonMarkovianity at low temperatures, again showing a better agreement with the HEOM exact results.
In Fig. 6(a), γ is further reduced to 20 cm^{−1} (γ^{−1} = 265 fs) to investigate the effect of slow environments. As the spectral density function is narrower, the correlation time of the environment becomes long compared with the characteristic time of the system dynamics. Hence the information on the system dynamics is more likely to be retained in the environment and flow back into the system. This backflow of information in turn affects the behavior of the system and results in beating in the concurrence curves for all methods. As shown in Fig. 6(b), the TC2 model predicts a nonMarkovianity much larger than the exact results. On the other hand, the TL2 model predicts a nonMarkovianity in excellent agreement with the HEOM results, with only a small underestimation of the nonMarkovianity in this set of parameters.
The above comparisons exhibit an interesting tendency for to increase with λ and temperature. Several relevant theoretical and experimental works have reported^{41,42,43} that strong systemenvironment correlations are helpful for maintaining quantum coherence even at high temperatures. As a result, higher temperature may in turn activate more phonon modes in the environment without destroying the quantum coherence significantly. This provides more channels via which the system can interact with the environment. In the language of quantum information science, smaller γ and strong systemenvironment correlation may help to preserve the dynamical information; while larger λ and higher temperature may increase the possibility that this information can flow back into the system from the environment. Consequently, this increase of with larger temperature and λ is a result of the competition between the backflow of information and thermal fluctuations. Meanwhile, the magnitude of the concurrence is reduced by the stronger random fluctuations in the environment.
Discussion
In summary, we first investigate the dynamics of two perturbative secondorder QME methods, TC2 and TL2 and compare their results with the numericallyexact results calculated by HEOM. We find that TC2 can approximate the HEOM population better than TL2. However, a drawback of the TC2 model is its overestimation of the coherence. This drawback results in the TC2 model predicting too much beating behavior in the population dynamics and limits the accuracy of TC2. In constrast, the TL2 model predicts sluggish dynamics and loss of coherence faster than that of the exact HEOM. As a result, the population tends to reach thermal equilibrium too rapidly.
To further investigate the dynamics and establish a benchmark for the performance of perturbative QMEs, we combine the concept of ChoiJamiokowski isomorphism^{39,40}, entanglement with an ancilla^{57} and a measure of nonMarkovianity^{33} to provide a quantitative way to determine how much the coherence dynamics and memory effects are deviating from the exact result. This provides a deep physical insight on the effects of each parameter and a single quantity to determine how much the QME dynamics deviates from the exact results. Here we find that the nonMarkovian measure indicates that the TL2 approximates HEOM better than TC2 in terms of the coherence dynamics and memory effects for the dimer system studied here. In addition, while it is well understood that the reorganization energy λ and temperature enhance the effect of thermal fluctuations in the environment on the system, increasing these parameters can have surprising results. In particular, our results show that higher temperature increases information backflow from the environment, thus increasing the nonMarkovianity of the system dynamics, even though the concurrence itself undergoes faster decay. Note that photosynthetic systems and other molecular lightharvesting networks are in general far more complex than the models studied here^{4,5} and more general models should be considered for realistic systems^{59,60,61,62}. Nevertheless, the focus of this work is the physics revealed in the comparison of the theoretical methods and the application of the nonMarkovianity measure for revealing new physical insights. The theoretical methods examined here have often been applied to model real photosynthetic systems and the quantitative measures we employ are themselves model independent. The nonMarkovianity analysis proposed here could be easily used to investigate coherence dynamics in more complex systems and more general models. Therefore, these results could have important implications in the theoretical modeling of electronic coherence in photosynthetic systems^{8,9,47}.
Additional Information
How to cite this article: Chen, H.B. et al. Using nonMarkovian measures to evaluate quantum master equations for photosynthesis. Sci. Rep. 5, 12753; doi: 10.1038/srep12753 (2015).
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Acknowledgements
This work is supported partially by the National Center for Theoretical Sciences and Minister of Science and Technology, Taiwan, grant numbers NSC 1012628M006003MY3 and MOST 1032112M006 017 MY4. FN is partially supported by the RIKEN iTHES Project, MURI Center for Dynamic MagnetoOptics and a GrantinAid for Scientific Research (S). YCC thanks the Ministry of Science and Technology, Taiwan (Grant No. NSC 1002113M002008MY3).
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Y.C.C., Y.N.C. and F.N. designed this work. H.B.C. and N.L. carried out the calculations. All authors contributed to the discussions and the writing of the manuscript.
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Chen, HB., Lambert, N., Cheng, YC. et al. Using nonMarkovian measures to evaluate quantum master equations for photosynthesis. Sci Rep 5, 12753 (2015). https://doi.org/10.1038/srep12753
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DOI: https://doi.org/10.1038/srep12753
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