量子耗散动力学的级联主方程理论
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摘要
做为量子统计力学的中心问题,量子耗散在现代科学的很多领域中起着至关重要的作用。在环境满足高斯统计的前提下,Feynman–Vernon影响泛函路径积分是严格的理论方方法法。但是因为计算量太大,至今仍只能应应用用用于于个别简单体系。采用环境统计谱密度函数的参数化模型,通过对影响泛函路径积分表达式求时间的的导导数,可以推导建立级联形式耦合的微分运动方程组。该方程组通过一系列辅助密度算符依环境量子统计分布函数的分解基底层层展开和耦合,由此综合考虑体系―环境相互作用强度、环境涨落的记忆时间、非谐性和多体作用等效应。这一方方法法较路径积分分方方方法法提高了计算效率,且更方便于于应应应用用到各种具体动力学的计算,但是计算量仍仍然然很大。本论文致力于发展数值高效的非微扰级联联量量子主方程理论,同时针对有限温度下的任意体系,提供预估理论模模拟拟准确性的判据。
第一章介绍量子耗散动力学的级联运动方程组理论(hierarchical equationsof motion, HEOM)。首先我们选取一系列辅助的影响生成泛函,通过对影响泛函路径积分求时间的导数构建级联微分运动方程,其中辅助影响生成泛函的选取与环境统计相关关函函数的具体形式紧密相关。接下来,我们介绍级联运动方程组的截断处理方案,以及剩余(residue)修正准则。最后我们对级联运动方程组的总体结构进行重新分析,提出新的标度处理方案,从而可以运用过滤传播子法,提高级联方程的计算效率。该级联运动方程组理论,是普适的量子耗散理论方方法法,它可以非微扰地处理任意温度下的非马尔可夫量子耗散过程,并且适用用于于于有有含时外场驱动动的的情况。
第二章,我们发展了一种近似的级联量子主方程方法(hierarchical quantummaster equation, HQME)。该方法是在对Drude环境统计模型的传统半经典处理方法加以改进的基础上得到,最终所得的HQME方程也可以看作是传统随机Liouville方程的修正。虽然形式上看,我们所做的只是很简单的一项修正,但是改进后的方程不仅动力学准确性得到明显提高,而且方程的适用性范围也被极大地扩宽;更加突出的是,该修正并不会引起计算量的增加。我们以二能级电荷转移模型为研究体系,在该体系,HQME方程还相当于修正的Zusman方程。在HQME方程的基础上,利用连分数格林函数方方法法,能够推导出电荷转移体系解析的速率和平衡态布居表达式,从而实现全参数空间内正定性的扫描。最后,我们通过与严格(HEOM)理论对比动力学计算结果,提出关于该近似HQME理论适用性范围的一个简单判据。
在第三章,仍旧针对Drude模型,我们发展了最优化的双指数级联运动方程理论,该理论相对于第二章可以看作是升级的HQME方法,适用参数范围更广,约化体系动力学演化更加准确。同时,理论依然具备一个方便的简单判据。我们分别以二能级电荷转移体系的动力学演化和二聚体激子模型的时―频分辨瞬态光谱为研究对象,做了大量的数值计算测试,并与严格的HEOM理论计算结果进行比较。数值结果显示,双指数HQME理论在其有效判据区域内,不仅准确地描述了约化密度矩阵的的动动力学学行行为,而且对于非线性响应也给出了准确的结果。
上述章节中所使用的严格HEOM方法是基于环境统计分布玻色―爱因斯坦函数的传统Matsubara谱分解(Matsubara spectral decomposition, MSD)方案构建,简称为MSD–HEOM。在第四章,我们应用Pade′谱分解(Pade′spectraldecomposition, PSD)方案来建立级联运动方程组,称之为PSD–HEOM。此理论方方法法与MSD–HEOM相比更加数值高效,而且在Drude模型仍旧提供了评估理论模模拟拟准确性的简单判据,使得精度事先可可控控,无须经过多次计算来检查结果收敛性,因而在实际应用中会极大地节省计算时间。并且,第二、三章分别介绍的两套HQME方程实际上就是该理论系统的最低阶和次低阶代表。为了考察PSD–HEOM的数值效率,我们选取自旋―玻色体系为例,计算该体系低温下的演化动力学,发现PSD–HEOM具有很高的计算效率,并且所提供的判据也非常有效。
第五章对本论文工作做出总结,并讨论未来理论的发展方向和具体应用前景。
As a central topic in quantum statistical mechanics, dissipation plays an importantrole in almost all fields of modern science. For Gaussian bath, exact quantum dissipa-tion theory (QDT) can be formulated with Feynman–Vernon in?uence functional pathintegral. However, it can only be carried out in a few simple systems due to the expen-sive numerical cost. Alternatively, an exact hierarchical equations of motion (HEOM)formalism can be constructed on the basis of a calculus-on-path-integral algorithm, viathe auxiliary in?uence generating functionals related to the interaction bath correla-tion functions in a parametrization expansion form. The HEOM couples the primaryreduced system density operator to a set of auxiliary density operators which accountfor systematically the system–bath coupling strength, memory time of bath ?uctuation,anharmonicity, and many–body interactions. The HEOM formalism has the advantagein both numerical efficiency and applications to various systems. However, its numer-ical cost is still expensive for large systems. This thesis aims at numerically efficienthierarchical quantum master equation (HQME) approach, which is also supported byversatile criterions to estimate in advance its accuracy for general systems.
In Chapter 1, we introduce the background of the exact HEOM formalism, whichis constructed on the basis of a calculus-on-path-integral algorithm, via the auxiliaryin?uence generating functionals related to the interaction bath correlation functions ina parametrization expansion form. Proposed further is the principle of residue correc-tion, not just for truncating the infinite hierarchy, but also for incorporating the smallresidue dissipation. Finally, we propose an efficient method to propagate the HEOMbased on a reformulation of the original HEOM formalism and the incorporation of afiltering algorithm that automatically truncates the hierarchy with a preselected toler-ance. HEOM constitutes a systematic, nonperturbative approach to quantum dissipativedynamics with non-Markovian dissipation at an arbitrary finite temperature in the pres-ence of time-dependent field driving.
In Chapter 2, we propose a HQME approach, which is rooted in an improvedsemiclassical treatment of Drude bath, beyond the conventional high temperature ap-proximations. It leads to the new theory a simple but important improvement overthe conventional stochastic Liouville equation theory, without extra numerical cost. Itsbroad range of validity and applicability is extensively demonstrated with two-level electron transfer model systems, where the new theory can be considered as the mod-ified Zusman equation. For this system, we can derive analytical rate and equilibriumdensity matrix expressions on the basis of the HQME–equivalent continued fractionLiouville-space Green’s function method. We can then explore the positivity propertyof HQME over the entire parameters space. Finally, we also propose a criterion to esti-mate the performance of HQME by comparing the dynamic results with exact HEOM.
In Chapter 3, we develop a biexponential theory of Drude dissipation via HQME.It is an advanced HQME, aiming at a numerically efficient non-Markovian quantumdissipation propagator, with the support of a convenient criterion to estimate in advanceits accuracy for general systems. Compared to its low level, single-exponential coun-terpart (chapter 2), the present theory remarkably improves the applicability range overall-parameter space, as tested critically with electron transfer and frequency-dispersedtransient absorption of exciton dimer model systems. The numerical demonstrationsshow that the advanced HQME approach can give accurate description for both thetime evolution of density matrix elements and nonlinear response functions.
The involved exact HEOM in the above chapters is constructed based on the Mat-subara spectral decomposition (MSD) of Bose–Einstein function. In Chapter 4, weimplement the Pade′spectrum decomposition scheme (PSD), to establish the corre-sponding PSD–HEOM, together with a convenient criterion of accuracy in advance,with the Drude model. The PSD is qualified to be the best sum-over-poles scheme forthe exponential series expansion of bath correlation functions. The above two HQMEtheories proposed in chapter 2 and 3 are just the special low–order cases of the presentPSD–HEOM. The performance and efficiency of PSD–HEOM is exemplified with achallenging benchmark spin-boson system.
In Chapter 5, we conclude the thesis, and discuss about the future work and appli-cations.
第一章介绍量子耗散动力学的级联运动方程组理论(hierarchical equationsof motion, HEOM)。首先我们选取一系列辅助的影响生成泛函,通过对影响泛函路径积分求时间的导数构建级联微分运动方程,其中辅助影响生成泛函的选取与环境统计相关关函函数的具体形式紧密相关。接下来,我们介绍级联运动方程组的截断处理方案,以及剩余(residue)修正准则。最后我们对级联运动方程组的总体结构进行重新分析,提出新的标度处理方案,从而可以运用过滤传播子法,提高级联方程的计算效率。该级联运动方程组理论,是普适的量子耗散理论方方法法,它可以非微扰地处理任意温度下的非马尔可夫量子耗散过程,并且适用用于于于有有含时外场驱动动的的情况。
第二章,我们发展了一种近似的级联量子主方程方法(hierarchical quantummaster equation, HQME)。该方法是在对Drude环境统计模型的传统半经典处理方法加以改进的基础上得到,最终所得的HQME方程也可以看作是传统随机Liouville方程的修正。虽然形式上看,我们所做的只是很简单的一项修正,但是改进后的方程不仅动力学准确性得到明显提高,而且方程的适用性范围也被极大地扩宽;更加突出的是,该修正并不会引起计算量的增加。我们以二能级电荷转移模型为研究体系,在该体系,HQME方程还相当于修正的Zusman方程。在HQME方程的基础上,利用连分数格林函数方方法法,能够推导出电荷转移体系解析的速率和平衡态布居表达式,从而实现全参数空间内正定性的扫描。最后,我们通过与严格(HEOM)理论对比动力学计算结果,提出关于该近似HQME理论适用性范围的一个简单判据。
在第三章,仍旧针对Drude模型,我们发展了最优化的双指数级联运动方程理论,该理论相对于第二章可以看作是升级的HQME方法,适用参数范围更广,约化体系动力学演化更加准确。同时,理论依然具备一个方便的简单判据。我们分别以二能级电荷转移体系的动力学演化和二聚体激子模型的时―频分辨瞬态光谱为研究对象,做了大量的数值计算测试,并与严格的HEOM理论计算结果进行比较。数值结果显示,双指数HQME理论在其有效判据区域内,不仅准确地描述了约化密度矩阵的的动动力学学行行为,而且对于非线性响应也给出了准确的结果。
上述章节中所使用的严格HEOM方法是基于环境统计分布玻色―爱因斯坦函数的传统Matsubara谱分解(Matsubara spectral decomposition, MSD)方案构建,简称为MSD–HEOM。在第四章,我们应用Pade′谱分解(Pade′spectraldecomposition, PSD)方案来建立级联运动方程组,称之为PSD–HEOM。此理论方方法法与MSD–HEOM相比更加数值高效,而且在Drude模型仍旧提供了评估理论模模拟拟准确性的简单判据,使得精度事先可可控控,无须经过多次计算来检查结果收敛性,因而在实际应用中会极大地节省计算时间。并且,第二、三章分别介绍的两套HQME方程实际上就是该理论系统的最低阶和次低阶代表。为了考察PSD–HEOM的数值效率,我们选取自旋―玻色体系为例,计算该体系低温下的演化动力学,发现PSD–HEOM具有很高的计算效率,并且所提供的判据也非常有效。
第五章对本论文工作做出总结,并讨论未来理论的发展方向和具体应用前景。
As a central topic in quantum statistical mechanics, dissipation plays an importantrole in almost all fields of modern science. For Gaussian bath, exact quantum dissipa-tion theory (QDT) can be formulated with Feynman–Vernon in?uence functional pathintegral. However, it can only be carried out in a few simple systems due to the expen-sive numerical cost. Alternatively, an exact hierarchical equations of motion (HEOM)formalism can be constructed on the basis of a calculus-on-path-integral algorithm, viathe auxiliary in?uence generating functionals related to the interaction bath correla-tion functions in a parametrization expansion form. The HEOM couples the primaryreduced system density operator to a set of auxiliary density operators which accountfor systematically the system–bath coupling strength, memory time of bath ?uctuation,anharmonicity, and many–body interactions. The HEOM formalism has the advantagein both numerical efficiency and applications to various systems. However, its numer-ical cost is still expensive for large systems. This thesis aims at numerically efficienthierarchical quantum master equation (HQME) approach, which is also supported byversatile criterions to estimate in advance its accuracy for general systems.
In Chapter 1, we introduce the background of the exact HEOM formalism, whichis constructed on the basis of a calculus-on-path-integral algorithm, via the auxiliaryin?uence generating functionals related to the interaction bath correlation functions ina parametrization expansion form. Proposed further is the principle of residue correc-tion, not just for truncating the infinite hierarchy, but also for incorporating the smallresidue dissipation. Finally, we propose an efficient method to propagate the HEOMbased on a reformulation of the original HEOM formalism and the incorporation of afiltering algorithm that automatically truncates the hierarchy with a preselected toler-ance. HEOM constitutes a systematic, nonperturbative approach to quantum dissipativedynamics with non-Markovian dissipation at an arbitrary finite temperature in the pres-ence of time-dependent field driving.
In Chapter 2, we propose a HQME approach, which is rooted in an improvedsemiclassical treatment of Drude bath, beyond the conventional high temperature ap-proximations. It leads to the new theory a simple but important improvement overthe conventional stochastic Liouville equation theory, without extra numerical cost. Itsbroad range of validity and applicability is extensively demonstrated with two-level electron transfer model systems, where the new theory can be considered as the mod-ified Zusman equation. For this system, we can derive analytical rate and equilibriumdensity matrix expressions on the basis of the HQME–equivalent continued fractionLiouville-space Green’s function method. We can then explore the positivity propertyof HQME over the entire parameters space. Finally, we also propose a criterion to esti-mate the performance of HQME by comparing the dynamic results with exact HEOM.
In Chapter 3, we develop a biexponential theory of Drude dissipation via HQME.It is an advanced HQME, aiming at a numerically efficient non-Markovian quantumdissipation propagator, with the support of a convenient criterion to estimate in advanceits accuracy for general systems. Compared to its low level, single-exponential coun-terpart (chapter 2), the present theory remarkably improves the applicability range overall-parameter space, as tested critically with electron transfer and frequency-dispersedtransient absorption of exciton dimer model systems. The numerical demonstrationsshow that the advanced HQME approach can give accurate description for both thetime evolution of density matrix elements and nonlinear response functions.
The involved exact HEOM in the above chapters is constructed based on the Mat-subara spectral decomposition (MSD) of Bose–Einstein function. In Chapter 4, weimplement the Pade′spectrum decomposition scheme (PSD), to establish the corre-sponding PSD–HEOM, together with a convenient criterion of accuracy in advance,with the Drude model. The PSD is qualified to be the best sum-over-poles scheme forthe exponential series expansion of bath correlation functions. The above two HQMEtheories proposed in chapter 2 and 3 are just the special low–order cases of the presentPSD–HEOM. The performance and efficiency of PSD–HEOM is exemplified with achallenging benchmark spin-boson system.
In Chapter 5, we conclude the thesis, and discuss about the future work and appli-cations.
引文
[1] R. P. Feynman and F. L. Vernon, Jr. The theory of a general quantum systeminteracting with a linear dissipative system. Ann. Phys., 24:118–173, 1963.
[2] A. G. Redfield. The theory of relaxation processes. Adv. Magn. Reson., 1:1–32,1965.
[3] R. Kubo, M. Toda, and N. Hashitsume. Statistical Physics II: NonequilibriumStatistical Mechanics. Springer-Verlag, Berlin, 2nd Ed., 1985.
[4] U. Weiss. Quantum Dissipative Systems. World Scientific, Singapore, 1999. 2nded. Series in Modern Condensed Matter Physics, Vol. 10.
[5] H. Kleinert. Path Integrals in Quantum Mechanics, Statistics, Polymer Physics,and Financial Markets. World Scientific, Singapore, 2006. 4th ed.
[6] M. B. Plenio and P. L. Knight. The quantum-jump approach to dissipative dynam-ics in quantum optics. Rev. Mod. Phys., 70:101–144, 1998.
[7] T. Dittrich, P. Ha¨nggi, G. L. Ingold, B. Kramer, G. Scho¨n, and W. Zwerger. Quan-tum Transport and Dissipation. Wiley-VCH, Weinheim, 1998.
[8] M. Thorwart, M. Grifoni, and P. Ha¨nggi. Strong coupling theory for tunneling andvibrational relaxation in driven bistable systems. Ann. Phys., 293:15–66, 2001.
[9] H. P. Breuer and F. Petruccione. The Theory of Open Quantum Systems. OxfordUniversity Press, New York, 2002.
[10] H. Grabert, P. Schramm, and G. L. Ingold. Quantum brownian motion: The func-tional integral approach. Phys. Rep., 168:115–207, 1988.
[11] A. O. Caldeira and A. J. Leggett. Quantum tunnelling in a dissipative system.Ann. Phys., 149:374–456, 1983. 153, 445(Erratum) (1984).
[12] A. J. Leggett, S. Chakravarty, A. T. Dorsey, Matthew P. A. Fisher, Anupam Garg,and W. Zwerger. Dynamics of the dissipative two-state system. Rev. Mod. Phys.,59:1–85, 1987. 67, 725-726(Erratum) (1995).
[13] Y. J. Yan. Quantum fokker-planck theory in a non-gaussian-markovian medium.Phys. Rev. A, 58:2721–32, 1998.
[14] Y. J. Yan, F. Shuang, R. X. Xu, J. X. Cheng, X. Q. Li, C. Yang, and H. Y. Zhang.Unified approach to the bloch-redfield theory and quantum fokker-planck equa-tions. J. Chem. Phys., 113:2068–2078, 2000.
[15] R. X. Xu and Y. J. Yan. Theory of open quantum systems. J. Chem. Phys.,116:9196–9206, 2002.
[16] Y. J. Yan and R. X. Xu. Quantum mechanics of dissipative systems. Annu. Rev.Phys. Chem., 56:187–219, 2005.
[17] R. X. Xu, P. Cui, X. Q. Li, Y. Mo, and Y. J. Yan. Exact quantum master equationvia the calculus on path integrals. J. Chem. Phys., 122:041103, 2005.
[18] N. Makri. Numerical path integral techniques for long time dynamics of quantumdissipative systems. J. Math. Phys., 36:2430–2457, 1995.
[19] Y. Tanimura and R. Kubo. Time evolution of a quantum system in contact with anearly gaussian-markovian noise bath. J. Phys. Soc. Jpn., 58:101–114, 1989.
[20] Y. Tanimura and P. G. Wolynes. Quantum and classical fokker-planck equationsfor a guassian-markovian noise bath. Phys. Rev. A, 43:4131–4142, 1991.
[21] A. Ishizaki and Y. Tanimura. Quantum dynamics of system strongly coupled tolow temperature colored noise bath: Reduced hierarchy equations approach. J.Phys. Soc. Jpn., 74:3131, 2005.
[22] Y. Tanimura. Stochastic liouville, langevin, fokker-planck, and master equationapproaches to quantum dissipative systems. J. Phys. Soc. Jpn., 75:082001, 2006.
[23] J. S. Shao. Decoupling quantum dissipation interaction via stochastic fields. J.Chem. Phys., 120:5053–56, 2004.
[24] Y. A. Yan, F. Yang, Y. Liu, and J. S. Shao. Hierarchical approach based on stochas-tic decoupling to dissipative systems. Chem. Phys. Lett., 395:216–21, 2004.
[25] J. S. Shao. Stochastic description of quantum open systems: Formal solution andstrong dissipation limit. Chem. Phys., 322:187–192, 2006.
[26] J. T. Stockburger and C. H. Mak. Stochastic liouvillian algorithm to simulatedissipative quantum dynamics with arbitrary precision. J. Chem. Phys., 110:4983–5, 1999.
[27] J. T. Stockburger and H. Grabert. Exact c-number representation of non-markovian quantum dissipation. Phys. Rev. Lett., 88:170407, 2002.
[28] W. T. Strunz and T. Yu. Convolutionless non-markovian master equations andquantum trajectories: Brownian motion. Phys. Rev. A, 69:052115, 2004.
[29] T. Yu. Non-markovian quantum trajectories versus master equations: Finite-temperature heat bath. Phys. Rev. A, 69:062107, 2004.
[30] P. Ha¨nggi and G. L. Ingold. Fundamental aspects of quantum brownian motion.Chaos, 15:026105, 2005.
[31] Bloch F. Generalized theory of relaxation. Phys. Rev., 105:1206, 1957.
[32] Redfield AG. The theory of relaxation processes. Adv. Magn. Reson., 1:1, 1965.
[33] Pollard WT, Felts AK, and Friesner RA. The redfield equation in condensed-phasequantum dynamics. Adv. Chem. Phys., 93:77, 1996.
[34] Kohen D, Marson CC, and Tannor DJ. Phase space approach to theories of quan-tum dissipation. J. Chem. Phys., 107:5236, 1997.
[35] Caldeira AO and Leggett AJ. Path integral approach to quantum brownian motion.Physica., 121.
[36] Caldeira AO and Leggett AJ. Quantum tunneling in a dissipative system. Ann.Phys., 149:374, 1983.
[37] R. X. Xu, Y. J. Yan, Y. Ohtsuki, Y. Fujimura, and H. Rabitz. Optimal control ofquantum non-markovian dissipation: Reduced liouville-space theory. J. Chem.Phys., 120:6600–6608, 2004.
[38] Y. Mo, R. X. Xu, P. Cui, and Y. J. Yan. Correlation and response functions withnon-markovian dissipation: A reduced liouville-space theory. J. Chem. Phys.,122:084115, 2005.
[39] N. Makri. Quantum dissipative dynamics: a numerically exact methodology. J.Phys. Chem. A, 102:4414–27, 1998.
[40] H. Kleinert and S. V. Shabanov. Quantum langevin equation from forward-backward path integral. Phys. Lett. A, 200:224–32, 1995.
[41] L. Dio′si and W. T. Strunz. The non-markovian stochastic Schro¨dinger equationfor open systems. Phys. Lett. A, 235:569–73, 1997.
[42] W. T. Strunz, L. Dio′si, and N. Gisin. Open system dynamics with non-markovianquantum trajectories. Phys. Rev. Lett., 82:1801–5, 1999.
[43] H-P. Breuer, B. Kappler, and F. Petruccione. Stochastic wave-function method fornon-markovian quantum master equations. Phys. Rev. A, 59:1633–1643, 1999.
[44] H. P. Breuer. Exact quantum jump approach to open systems in bosonic and spinbaths. Phys. Rev. A, 69:022115, 2004.
[45] J. T. Stockburger and H. Grabert. Non-markovian quantum state diffusion. Chem.Phys., 268:249–56, 2001.
[46] B. L. Hu, J. P. Paz, and Y. Zhang. Quantum brownian motion in a general environ-ment: Exact master equation with nonlocal dissipation and colored noise. Phys.Rev. D, 45:2843–61, 1992.
[47] R. Karrlein and H. Grabert. Exact time evoluation and master equations for thedamped harmonic oscillator. Phys. Rev. E, 55:153–164, 1997.
[48] R. X. Xu and Y. J. Yan. Dynamics of quantum dissipation systems interactingwith bosonic canonical bath: Hierarchical equations of motion approach. Phys.Rev. E, 75:031107, 2007.
[49] J. S. Jin, S. Welack, J. Y. Luo, X. Q. Li, P. Cui, R. X. Xu, and Y. J. Yan. Dynamicsof quantum dissipation systems interacting with Fermion and Boson grand canon-ical bath ensembles: Hierarchical equations of motion approach. J. Chem. Phys.,126:134113, 2007.
[50] Q. Shi, L. P. Chen, G. J. Nan, R. X. Xu, and Y. J. Yan. Efficient hierarchicalliouville space propagator to quantum dissipative dynamics. J. Chem. Phys.,130(8):084105, 2009.
[51] Axel Thielmann, Matthias H. Hettler, Ju¨rgen Ko¨nig, and Gerd Scho¨n. Cotunnelingcurrent and shot noise in quantum dots. Phys. Rev. Lett., 95:146806, 2006.
[52] C. Meier and D. J. Tannor. Non-markovian evolution of the density operator inthe presence of strong laser fields. J. Chem. Phys., 111:3365–3376, 1999.
[53] P. Han, R. X. Xu, B. Q. Li, J. Xu, P. Cui, Y. Mo, and Y. J. Yan. Kinetics andthermodynamics of electron transfer in debye solvents: An analytical and nonper-turbative reduced density matrix theory. J. Phys. Chem. B, 110:11438–43, 2006.
[54] P. Han, R. X. Xu, P. Cui, Y. Mo, G. Z. He, and Y. J. Yan. Electron transfer theoryrevisit: Quantum solvation effect. J. Theore. & Comput. Chem., 5:685–92, 2006.
[1] R. P. Feynman and F. L. Vernon, Jr. The theory of a general quantum systeminteracting with a linear dissipative system. Ann. Phys., 24:118–173, 1963.
[2] U. Weiss. Quantum Dissipative Systems. World Scientific, Singapore, 2008. 3rded. Series in Modern Condensed Matter Physics, Vol. 13.
[3] H. Kleinert. Path Integrals in Quantum Mechanics, Statistics, Polymer Physics,and Financial Markets. World Scientific, Singapore, 2009. 5th ed.
[4] Y. Tanimura. Stochastic liouville, langevin, fokker-planck, and master equationapproaches to quantum dissipative systems. J. Phys. Soc. Jpn., 75:082001, 2006.
[5] Y. Tanimura and R. Kubo. Time evolution of a quantum system in contact with anearly gaussian-markovian noise bath. J. Phys. Soc. Jpn., 58:101–114, 1989.
[6] A. Ishizaki and Y. Tanimura. Quantum dynamics of system strongly coupled tolow temperature colored noise bath: Reduced hierarchy equations approach. J.Phys. Soc. Jpn., 74:3131, 2005.
[7] R. X. Xu, P. Cui, X. Q. Li, Y. Mo, and Y. J. Yan. Exact quantum master equationvia the calculus on path integrals. J. Chem. Phys., 122:041103, 2005.
[8] R. X. Xu and Y. J. Yan. Dynamics of quantum dissipation systems interactingwith bosonic canonical bath: Hierarchical equations of motion approach. Phys.Rev. E, 75:031107, 2007.
[9] J. S. Jin, X. Zheng, and Y. J. Yan. Exact dynamics of dissipative electronic systemsand quantum transport: Hierarchical equations of motion approach. J. Chem.Phys., 128:234703, 2008.
[10] R. Kubo. Stochastic liouville equations. J. Math. Phys., 4:174–183, 1963.
[11] R. Kubo. A stochastic theory of line shape. Adv. Chem. Phys., 15:101, 1969.
[12] D. Abramavicius, B. Palmieri, D. V. Voronine, F. Sˇanda, and S. Mukamel. Coher-ent multidimensional optical spectroscopy of excitons in molecular aggregates;quasiparticle versus supermolecule perspectives. Chem. Rev., 109:2350–2408,2009.
[13] W. Zhuang, T. Hayashi, and S. Mukamel. Coherent multidimensional vibrationalspectroscopy of biomolecules: Concepts, simulations, and challenges. Angew.Chem. Int. Ed., 48:3750–3781, 2009.
[14] L. D. Zusman. Outer-sphere electron transfer in polar solvents. Chem. Phys.,49:295–304, 1980.
[15] L. D. Zusman. The theory of transitions between electronic states. application toradiationless transitions in polar solvents. Chem. Phys., 80:29–43, 1983.
[16] A. Garg, J. N. Onuchic, and V. Ambegaokar. Effect of friction on electron transferin biomolecules. J. Chem. Phys., 83:4491–503, 1985.
[17] D. Y. Yang and R. I. Cukier. The transition from nonadiabatic to solvent controlledadiabatic electron transfer: Solvent dynamical effects in the inverted regime. J.Chem. Phys., 91:281–292, 1989.
[18] P. A. Frantsuzov. Chem. Phys. Lett., 267:427, 1997.
[19] P. A. Frantsuzov. J. Chem. Phys., 111:2075, 1999.
[20] M. Thoss, H. B. Wang, and W. H. Miller. Self-consistent hybrid approach for com-plex dystems: Application to the spin-boson model with debye spectral density.J. Chem. Phys., 115:2991–3005, 2001.
[21] J. S. Cao and Y. J. Jung. Spectral analysis of electron transfer kinetics. i. symmet-ric reactions. J. Chem. Phys., 112:4716–22, 2000.
[22] Y. J. Jung and J. S. Cao. Spectral analysis of electron transfer kinetics. ii. J. Chem.Phys., 117:3822–36, 2002.
[23] L. Mu¨hlbacher and R. Egger. Crossover from nonadiabatic to adiabatic electrontransfer reactions: Multilevel blocking monte carlo simulations. J. Chem. Phys.,118:179–191, 2003.
[24] J. Ankerhold and H. Lehle. Low temperature electron transfer in strongly con-densed phases. J. Chem. Phys., 120:1436, 2004.
[25] D. V. Dodin. Problems of coherent description of electron transfer reactions.Chem. Phys., 325:257–264, 2006.
[26] M.-L. Zhang, S.-S. Zhang, and E. Pollak. Theory of electron transfer in the pres-ence of dissipation. J. Chem. Phys., 119:11864, 2003.
[27] M.-L. Zhang, S.-S. Zhang, and E. Pollak. Low temperature extension of thegeneralized zusman phase space equations for electron transfer. J. Chem. Phys.,120:9630, 2004.
[28] Q. Shi, L. P. Chen, G. J. Nan, R. X. Xu, and Y. J. Yan. Electron transfer dynamics:Zusman equation versus exact theory. J. Chem. Phys., 130:164518, 2009.
[29] P. Han, R. X. Xu, B. Q. Li, J. Xu, P. Cui, Y. Mo, and Y. J. Yan. Kinetics andthermodynamics of electron transfer in debye solvents: An analytical and nonper-turbative reduced density matrix theory. J. Phys. Chem. B, 110:11438–43, 2006.
[30] Q. Shi, L. P. Chen, G. J. Nan, R. X. Xu, and Y. J. Yan. Efficient hierarchicalliouville space propagator to quantum dissipative dynamics. J. Chem. Phys.,130(8):084105, 2009.
[31] X. Zheng, J. S. Jin, S. Welack, M. Luo, and Y. J. Yan. Numerical approach to time-dependent quantum transport and dynamical kondo transition. J. Chem. Phys.,130(16):164708, 2009.
[32] Y. Chen, R. X. Xu, H. W. Ke, and Y. J. Yan. Electron transfer theory revisit:Motional narrowing induced non-markovian rate processes. Chin. J. Chem. Phys.,20:438–44, 2007.
[33] R. X. Xu, Y. Chen, P. Cui, H. W. Ke, and Y. J. Yan. The quantum solvation,adiabatic versus nonadiabatic, and markovian versus non-markovian nature ofelectron-transfer rate processes. J. Phys. Chem. A, 111:9618–26, 2007.
[34] S. Jang and J. S. Cao. Nonadiabatic instanton calculation of multistate electrontransfer reaction rate: Interference effects in three and four states systems. J.Chem. Phys., 114:9959–68, 2001.
[35] Y. C. Cheng and G. R. Fleming. Dynamics of light harvesting in photosynthesis.Annu. Rev. Phys. Chem., 60:241–262, 2009.
[36] A. Ishizaki and G. R. Fleming. Theoretical examination of quantum coherence ina photosynthetic system at physiological temperature. Proc. Natl. Acad. Sci. USA,106:17255–60, 2009.
[37] Y. A. Yan, F. Yang, Y. Liu, and J. S. Shao. Hierarchical approach based on stochas-tic decoupling to dissipative systems. Chem. Phys. Lett., 395:216–21, 2004.
[38] Y. Zhou and J. S. Shao. Solving the spin-boson model of strong dissipation with?exible random-deterministic scheme. J. Chem. Phys., 128:034106, 2008.
[1] R. P. Feynman and F. L. Vernon, Jr. The theory of a general quantum systeminteracting with a linear dissipative system. Ann. Phys., 24:118–173, 1963.
[2] U. Weiss. Quantum Dissipative Systems. World Scientific, Singapore, 2008. 3rded. Series in Modern Condensed Matter Physics, Vol. 13.
[3] H. Kleinert. Path Integrals in Quantum Mechanics, Statistics, Polymer Physics,and Financial Markets. World Scientific, Singapore, 2009. 5th ed.
[4] Y. Tanimura and R. Kubo. Time evolution of a quantum system in contact with anearly gaussian-markovian noise bath. J. Phys. Soc. Jpn., 58:101–114, 1989.
[5] Y. Tanimura. Nonperturbative expansion method for a quantum system coupledto a harmonic-oscillator bath. Phys. Rev. A, 41:6676–87, 1990.
[6] A. Ishizaki and Y. Tanimura. Quantum dynamics of system strongly coupled tolow temperature colored noise bath: Reduced hierarchy equations approach. J.Phys. Soc. Jpn., 74:3131, 2005.
[7] Y. Tanimura. Stochastic liouville, langevin, fokker-planck, and master equationapproaches to quantum dissipative systems. J. Phys. Soc. Jpn., 75:082001, 2006.
[8] J. S. Shao. Decoupling quantum dissipation interaction via stochastic fields. J.Chem. Phys., 120:5053–56, 2004.
[9] Y. A. Yan, F. Yang, Y. Liu, and J. S. Shao. Hierarchical approach based on stochas-tic decoupling to dissipative systems. Chem. Phys. Lett., 395:216–21, 2004.
[10] R. X. Xu, P. Cui, X. Q. Li, Y. Mo, and Y. J. Yan. Exact quantum master equationvia the calculus on path integrals. J. Chem. Phys., 122:041103, 2005.
[11] R. X. Xu and Y. J. Yan. Dynamics of quantum dissipation systems interactingwith bosonic canonical bath: Hierarchical equations of motion approach. Phys.Rev. E, 75:031107, 2007.
[12] J. S. Jin, X. Zheng, and Y. J. Yan. Exact dynamics of dissipative electronic systemsand quantum transport: Hierarchical equations of motion approach. J. Chem.Phys., 128:234703, 2008.
[13] Xiao Zheng, Jin Shuang Jin, and Y. J. Yan. Dynamic electronic response of aquantum dot driven by time–dependent voltage. J. Chem. Phys., 129:184112,2008.
[14] Y. J. Yan and R. X. Xu. Quantum mechanics of dissipative systems. Annu. Rev.Phys. Chem., 56:187–219, 2005.
[15] A. Croy and U. Saalmann. Partial fraction decomposition of the fermi function.Phys. Rev. B, 80(7):073102, 2009.
[16] Alexander Croy and Ulf Saalmann. Propagation scheme for nonequilibrium dy-namics of electron transport in nanoscale devices. Phys. Rev. B, 80(24):245311,2009.
[17] J. Xu, R. X. Xu, M. Luo, and Y. J. Yan. Hierarchical theory of quantum dissipa-tion: Partial fraction decomposition scheme. Chem. Phys., 370:109–114, 2010.
[18] Jie Hu, Rui Xue Xu, and Yi Jing Yan. Pade′spectrum decomposition of fermifunction and bose function. J. Chem. Phys., 133(10):101106, 2010.
[19] Q. Shi, L. P. Chen, G. J. Nan, R. X. Xu, and Y. J. Yan. Efficient hierarchicalliouville space propagator to quantum dissipative dynamics. J. Chem. Phys.,130(8):084105, 2009.
[20] J. Xu, R. X. Xu, and Y. J. Yan. Exact quantum dissipative dynamics under externaltime-dependent driving fields. New J. Phys., 11:105037, 2009.
[21] X. Zheng, J. S. Jin, S. Welack, M. Luo, and Y. J. Yan. Numerical approach to time-dependent quantum transport and dynamical kondo transition. J. Chem. Phys.,130(16):164708, 2009.
[22] R. X. Xu, B. L. Tian, J. Xu, Q. Shi, and Y. J. Yan. Hierarchical quantum masterequation with semiclassical drude dissipation. J. Chem. Phys., 131(21):214111,2009.
[23] R. Kubo. Stochastic liouville equations. J. Math. Phys., 4:174–183, 1963.
[24] R. Kubo. A stochastic theory of line shape. Adv. Chem. Phys., 15:101, 1969.
[25] F. Shuang, C. Yang, and Y. J. Yan. Dynamical semigroup fokker-planck equationapproach to transient absorption and ?uorescence upconversion spectroscopies. J.Chem. Phys., 114:3868–79, 2001.
[26] Y. Mo, R. X. Xu, P. Cui, and Y. J. Yan. Correlation and response functions withnon-markovian dissipation: A reduced liouville-space theory. J. Chem. Phys.,122:084115, 2005.
[27] S. Mukamel. The Principles of Nonlinear Optical Spectroscopy. Oxford Univer-sity Press, New York, 1995.
[1] R. P. Feynman and F. L. Vernon, Jr. The theory of a general quantum systeminteracting with a linear dissipative system. Ann. Phys., 24:118–173, 1963.
[2] H. Kleinert. Path Integrals in Quantum Mechanics, Statistics, Polymer Physics,and Financial Markets. World Scientific, Singapore, 2009. 5th ed.
[3] U. Weiss. Quantum Dissipative Systems. World Scientific, Singapore, 2008. 3rded. Series in Modern Condensed Matter Physics, Vol. 13.
[4] Y. J. Yan and R. X. Xu. Quantum mechanics of dissipative systems. Annu. Rev.Phys. Chem., 56:187–219, 2005.
[5] M. A. Nielsen and I. L. Chuang. Quantum Computation and Quantum Informa-tion. Cambridge University Press, New York, 2000.
[6] S. Datta. Quantum Transport: Atom to Transistor. Cambridge University Press,New York, 2005.
[7] G. S. Engel, T. R. Calhoun, E. L. Read, T. K. Ahn, T. Mancˇal, Y. C. Cheng, R. E.Blankenship, and G. R. Fleming. Evidence for wavelike energy transfer throughquantum coherence in photosynthetic systems. Nature, 446:782–6, 2007.
[8] E. Collini, C. Y. Wong, K. E. Wilk, P. M. G. Curmi, P. Brumer, and G. D. Scholes.Coherently wired light-harvesting in photosynthetic marine algae at ambient tem-perature. Nature, 463:644–7, 2010.
[9] Y. Tanimura. Nonperturbative expansion method for a quantum system coupledto a harmonic-oscillator bath. Phys. Rev. A, 41:6676–87, 1990.
[10] Y. Tanimura. Stochastic liouville, langevin, fokker-planck, and master equationapproaches to quantum dissipative systems. J. Phys. Soc. Jpn., 75:082001, 2006.
[11] R. X. Xu, P. Cui, X. Q. Li, Y. Mo, and Y. J. Yan. Exact quantum master equationvia the calculus on path integrals. J. Chem. Phys., 122:041103, 2005.
[12] R. X. Xu and Y. J. Yan. Dynamics of quantum dissipation systems interactingwith bosonic canonical bath: Hierarchical equations of motion approach. Phys.Rev. E, 75:031107, 2007.
[13] J. S. Jin, X. Zheng, and Y. J. Yan. Exact dynamics of dissipative electronic systemsand quantum transport: Hierarchical equations of motion approach. J. Chem.Phys., 128:234703, 2008.
[14] Y. Tanimura and R. Kubo. Time evolution of a quantum system in contact with anearly gaussian-markovian noise bath. J. Phys. Soc. Jpn., 58:101–114, 1989.
[15] Q. Shi, L. P. Chen, G. J. Nan, R. X. Xu, and Y. J. Yan. Electron transfer dynamics:Zusman equation versus exact theory. J. Chem. Phys., 130:164518, 2009.
[16] R. X. Xu, B. L. Tian, J. Xu, Q. Shi, and Y. J. Yan. Hierarchical quantum masterequation with semiclassical drude dissipation. J. Chem. Phys., 131(21):214111,2009.
[17] A. Ishizaki and Y. Tanimura. Modeling vibrational dephasing and energy relax-ation of intramolecular anharmonic modes for multidimensional infrared spectro-scopies. J. Chem. Phys., 125:084501, 2006.
[18] A. Ishizaki and Y. Tanimura. Dynamics of a multimode system coupled to multi-ple heat baths probed by two-dimensional infrared spectroscopy. J. Phys. Chem.A, 111:9269–9276, 2007.
[19] L. P. Chen, R. H. Zheng, Q. Shi, and Y. J. Yan. Two-dimensional electronic spectrafrom the hierarchical equations of motion method: Application to model dimers.J. Chem. Phys., 132(2):024505, 2010.
[20] B. L. Tian, J. J. Ding, R. X. Xu, and Y. J. Yan. Bi-exponential theory of drude dissi-pation via hierarchical quantum master equation. J. Chem. Phys., 133(11):114112,2010.
[21] Xiao Zheng, Jin Shuang Jin, and Y. J. Yan. Dynamic electronic response of aquantum dot driven by time–dependent voltage. J. Chem. Phys., 129:184112,2008.
[22] X. Zheng, J. S. Jin, and Y. J. Yan. Dynamic coulomb blockade in single–leadquantum dots. New J. Phys., 10:093016, 2008.
[23] X. Zheng, J. S. Jin, S. Welack, M. Luo, and Y. J. Yan. Numerical approach to time-dependent quantum transport and dynamical kondo transition. J. Chem. Phys.,130(16):164708, 2009.
[24] Jie Hu, Rui Xue Xu, and Yi Jing Yan. Pade′spectrum decomposition of fermifunction and bose function. J. Chem. Phys., 133(10):101106, 2010.
[25] M. Thoss, H. B. Wang, and W. H. Miller. Self-consistent hybrid approach for com-plex dystems: Application to the spin-boson model with debye spectral density.J. Chem. Phys., 115:2991–3005, 2001.
[26] G. A. Baker Jr. and P. Graves-Morris. Pade′Approximants. Cambridge UniversityPress, New York, 1996. 2nd ed.
[27] Jin-Jin Ding, Jie Hu, Bao-Ling Tian, Rui-Xue Xu, and YiJing Yan. Optimal hi-erarchical theory for nonperturbative quantum dissipation dynamics. J. Chem.Phys., submitted, 2010.
[28] Q. Shi, L. P. Chen, G. J. Nan, R. X. Xu, and Y. J. Yan. Efficient hierarchicalliouville space propagator to quantum dissipative dynamics. J. Chem. Phys.,130(8):084105, 2009.
[29] A. Ishizaki and Y. Tanimura. Quantum dynamics of system strongly coupled tolow temperature colored noise bath: Reduced hierarchy equations approach. J.Phys. Soc. Jpn., 74:3131, 2005.
[30] J. Xu, R. X. Xu, and Y. J. Yan. Exact quantum dissipative dynamics under externaltime-dependent driving fields. New J. Phys., 11:105037, 2009.
[1] Green B. R. and Parson W. W., editors. Light-Harvesting Antennas in Photosyn-thesis. Kluwer, Dordrecht, 2003.
[2] G. D. Scholes. Long-range resonance energy transfer in molecular systems. Annu.Rev. Phys. Chem., 54:57–87, 2003.
[3] S. Jang, M. D. Newton, and R. J. Silbey. Multichromophoric fo˙ J. Phys. Chem.B, 111:6807–6814, 1961.
[4] Y. C. Cheng and G. R. Fleming. Dynamics of light harvesting in photosynthesis.Annu. Rev. Phys. Chem., 60:241–262, 2009.
[5] R. van Grondelle and V. I. Novoderezhkin. Energy transfer in photosynthesis:Experimental insights and qunatitive models. Phys. Chem. Chem. Phys., 8:793–807, 2006.
[6] G. S. Engel, T. R. Calhoun, E. L. Read, T. K. Ahn, T. Mancˇal, Y. C. Cheng, R. E.Blankenship, and G. R. Fleming. Evidence for wavelike energy transfer throughquantum coherence in photosynthetic systems. Nature, 446:782–6, 2007.
[7] H. Lee, Y. C. Cheng, and G. R. Fleming. Coherence dynamics in photosynthesis:protein protection of excitonic coherence. Science, 316:1462–1465, 2007.
[8] Ian P. Mercer et al. Instantaneous mapping of coherently coupled electronic tran-sitions and energy transfers in a photosynthetic complex using angleresolved co-herent optical wave-mixing. Phys. Rev. Lett., 102:057402, 2009.
[9] D. M. Jonas. Two-dimensional femtosecond spectroscopy. Annu. Rev. Phys.Chem., 54:425–463, 2003.
[10] T. Brixner, T. Mancal, I. V. Stiopkin, and G. R. Fleming. Phase-stabilized twodi-mensional electronic spectroscopy. J. Chem. Phys., 121:4221–4236, 2004.
[11] M. H. Cho. Coherent two-dimensional optical spectroscopy. Chem. Rev.,108:1331–1418, 2008.
[12] D. Abramavicius, B. Palmieri, D. V. Voronine, F. Sˇanda, and S. Mukamel. Coher-ent multidimensional optical spectroscopy of excitons in molecular aggregates;quasiparticle versus supermolecule perspectives. Chem. Rev., 109:2350–2408,2009.
[13] E. Collini, C. Y. Wong, K. E. Wilk, P. M. G. Curmi, P. Brumer, and G. D. Scholes.Coherently wired light-harvesting in photosynthetic marine algae at ambient tem-perature. Nature, 463:644–7, 2010.
[14] R. van Grondelle, J. P. Dekker, T. Gillbro, and V. Sundstro¨m. Biochem. Biophys.Acta, 1187:1, 1994.
[15] A. Ishizaki and G. R. Fleming. On the adequacy of the redfield equation andrelated approaches to the study of quantum dynamics in electronic energy transfer.J. Chem. Phys., 130:234110, 2009.
[16] A. Ishizaki and G. R. Fleming. Unified treatment of quantum coherent and inco-herent hopping dynamics in electronic energy transfer: Reduced hierarchy equa-tion approach. J. Chem. Phys., 130:234111, 2009.
[17] A. Ishizaki and G. R. Fleming. Theoretical examination of quantum coherence ina photosynthetic system at physiological temperature. Proc. Natl. Acad. Sci. USA,106:17255–60, 2009.
[2] A. G. Redfield. The theory of relaxation processes. Adv. Magn. Reson., 1:1–32,1965.
[3] R. Kubo, M. Toda, and N. Hashitsume. Statistical Physics II: NonequilibriumStatistical Mechanics. Springer-Verlag, Berlin, 2nd Ed., 1985.
[4] U. Weiss. Quantum Dissipative Systems. World Scientific, Singapore, 1999. 2nded. Series in Modern Condensed Matter Physics, Vol. 10.
[5] H. Kleinert. Path Integrals in Quantum Mechanics, Statistics, Polymer Physics,and Financial Markets. World Scientific, Singapore, 2006. 4th ed.
[6] M. B. Plenio and P. L. Knight. The quantum-jump approach to dissipative dynam-ics in quantum optics. Rev. Mod. Phys., 70:101–144, 1998.
[7] T. Dittrich, P. Ha¨nggi, G. L. Ingold, B. Kramer, G. Scho¨n, and W. Zwerger. Quan-tum Transport and Dissipation. Wiley-VCH, Weinheim, 1998.
[8] M. Thorwart, M. Grifoni, and P. Ha¨nggi. Strong coupling theory for tunneling andvibrational relaxation in driven bistable systems. Ann. Phys., 293:15–66, 2001.
[9] H. P. Breuer and F. Petruccione. The Theory of Open Quantum Systems. OxfordUniversity Press, New York, 2002.
[10] H. Grabert, P. Schramm, and G. L. Ingold. Quantum brownian motion: The func-tional integral approach. Phys. Rep., 168:115–207, 1988.
[11] A. O. Caldeira and A. J. Leggett. Quantum tunnelling in a dissipative system.Ann. Phys., 149:374–456, 1983. 153, 445(Erratum) (1984).
[12] A. J. Leggett, S. Chakravarty, A. T. Dorsey, Matthew P. A. Fisher, Anupam Garg,and W. Zwerger. Dynamics of the dissipative two-state system. Rev. Mod. Phys.,59:1–85, 1987. 67, 725-726(Erratum) (1995).
[13] Y. J. Yan. Quantum fokker-planck theory in a non-gaussian-markovian medium.Phys. Rev. A, 58:2721–32, 1998.
[14] Y. J. Yan, F. Shuang, R. X. Xu, J. X. Cheng, X. Q. Li, C. Yang, and H. Y. Zhang.Unified approach to the bloch-redfield theory and quantum fokker-planck equa-tions. J. Chem. Phys., 113:2068–2078, 2000.
[15] R. X. Xu and Y. J. Yan. Theory of open quantum systems. J. Chem. Phys.,116:9196–9206, 2002.
[16] Y. J. Yan and R. X. Xu. Quantum mechanics of dissipative systems. Annu. Rev.Phys. Chem., 56:187–219, 2005.
[17] R. X. Xu, P. Cui, X. Q. Li, Y. Mo, and Y. J. Yan. Exact quantum master equationvia the calculus on path integrals. J. Chem. Phys., 122:041103, 2005.
[18] N. Makri. Numerical path integral techniques for long time dynamics of quantumdissipative systems. J. Math. Phys., 36:2430–2457, 1995.
[19] Y. Tanimura and R. Kubo. Time evolution of a quantum system in contact with anearly gaussian-markovian noise bath. J. Phys. Soc. Jpn., 58:101–114, 1989.
[20] Y. Tanimura and P. G. Wolynes. Quantum and classical fokker-planck equationsfor a guassian-markovian noise bath. Phys. Rev. A, 43:4131–4142, 1991.
[21] A. Ishizaki and Y. Tanimura. Quantum dynamics of system strongly coupled tolow temperature colored noise bath: Reduced hierarchy equations approach. J.Phys. Soc. Jpn., 74:3131, 2005.
[22] Y. Tanimura. Stochastic liouville, langevin, fokker-planck, and master equationapproaches to quantum dissipative systems. J. Phys. Soc. Jpn., 75:082001, 2006.
[23] J. S. Shao. Decoupling quantum dissipation interaction via stochastic fields. J.Chem. Phys., 120:5053–56, 2004.
[24] Y. A. Yan, F. Yang, Y. Liu, and J. S. Shao. Hierarchical approach based on stochas-tic decoupling to dissipative systems. Chem. Phys. Lett., 395:216–21, 2004.
[25] J. S. Shao. Stochastic description of quantum open systems: Formal solution andstrong dissipation limit. Chem. Phys., 322:187–192, 2006.
[26] J. T. Stockburger and C. H. Mak. Stochastic liouvillian algorithm to simulatedissipative quantum dynamics with arbitrary precision. J. Chem. Phys., 110:4983–5, 1999.
[27] J. T. Stockburger and H. Grabert. Exact c-number representation of non-markovian quantum dissipation. Phys. Rev. Lett., 88:170407, 2002.
[28] W. T. Strunz and T. Yu. Convolutionless non-markovian master equations andquantum trajectories: Brownian motion. Phys. Rev. A, 69:052115, 2004.
[29] T. Yu. Non-markovian quantum trajectories versus master equations: Finite-temperature heat bath. Phys. Rev. A, 69:062107, 2004.
[30] P. Ha¨nggi and G. L. Ingold. Fundamental aspects of quantum brownian motion.Chaos, 15:026105, 2005.
[31] Bloch F. Generalized theory of relaxation. Phys. Rev., 105:1206, 1957.
[32] Redfield AG. The theory of relaxation processes. Adv. Magn. Reson., 1:1, 1965.
[33] Pollard WT, Felts AK, and Friesner RA. The redfield equation in condensed-phasequantum dynamics. Adv. Chem. Phys., 93:77, 1996.
[34] Kohen D, Marson CC, and Tannor DJ. Phase space approach to theories of quan-tum dissipation. J. Chem. Phys., 107:5236, 1997.
[35] Caldeira AO and Leggett AJ. Path integral approach to quantum brownian motion.Physica., 121.
[36] Caldeira AO and Leggett AJ. Quantum tunneling in a dissipative system. Ann.Phys., 149:374, 1983.
[37] R. X. Xu, Y. J. Yan, Y. Ohtsuki, Y. Fujimura, and H. Rabitz. Optimal control ofquantum non-markovian dissipation: Reduced liouville-space theory. J. Chem.Phys., 120:6600–6608, 2004.
[38] Y. Mo, R. X. Xu, P. Cui, and Y. J. Yan. Correlation and response functions withnon-markovian dissipation: A reduced liouville-space theory. J. Chem. Phys.,122:084115, 2005.
[39] N. Makri. Quantum dissipative dynamics: a numerically exact methodology. J.Phys. Chem. A, 102:4414–27, 1998.
[40] H. Kleinert and S. V. Shabanov. Quantum langevin equation from forward-backward path integral. Phys. Lett. A, 200:224–32, 1995.
[41] L. Dio′si and W. T. Strunz. The non-markovian stochastic Schro¨dinger equationfor open systems. Phys. Lett. A, 235:569–73, 1997.
[42] W. T. Strunz, L. Dio′si, and N. Gisin. Open system dynamics with non-markovianquantum trajectories. Phys. Rev. Lett., 82:1801–5, 1999.
[43] H-P. Breuer, B. Kappler, and F. Petruccione. Stochastic wave-function method fornon-markovian quantum master equations. Phys. Rev. A, 59:1633–1643, 1999.
[44] H. P. Breuer. Exact quantum jump approach to open systems in bosonic and spinbaths. Phys. Rev. A, 69:022115, 2004.
[45] J. T. Stockburger and H. Grabert. Non-markovian quantum state diffusion. Chem.Phys., 268:249–56, 2001.
[46] B. L. Hu, J. P. Paz, and Y. Zhang. Quantum brownian motion in a general environ-ment: Exact master equation with nonlocal dissipation and colored noise. Phys.Rev. D, 45:2843–61, 1992.
[47] R. Karrlein and H. Grabert. Exact time evoluation and master equations for thedamped harmonic oscillator. Phys. Rev. E, 55:153–164, 1997.
[48] R. X. Xu and Y. J. Yan. Dynamics of quantum dissipation systems interactingwith bosonic canonical bath: Hierarchical equations of motion approach. Phys.Rev. E, 75:031107, 2007.
[49] J. S. Jin, S. Welack, J. Y. Luo, X. Q. Li, P. Cui, R. X. Xu, and Y. J. Yan. Dynamicsof quantum dissipation systems interacting with Fermion and Boson grand canon-ical bath ensembles: Hierarchical equations of motion approach. J. Chem. Phys.,126:134113, 2007.
[50] Q. Shi, L. P. Chen, G. J. Nan, R. X. Xu, and Y. J. Yan. Efficient hierarchicalliouville space propagator to quantum dissipative dynamics. J. Chem. Phys.,130(8):084105, 2009.
[51] Axel Thielmann, Matthias H. Hettler, Ju¨rgen Ko¨nig, and Gerd Scho¨n. Cotunnelingcurrent and shot noise in quantum dots. Phys. Rev. Lett., 95:146806, 2006.
[52] C. Meier and D. J. Tannor. Non-markovian evolution of the density operator inthe presence of strong laser fields. J. Chem. Phys., 111:3365–3376, 1999.
[53] P. Han, R. X. Xu, B. Q. Li, J. Xu, P. Cui, Y. Mo, and Y. J. Yan. Kinetics andthermodynamics of electron transfer in debye solvents: An analytical and nonper-turbative reduced density matrix theory. J. Phys. Chem. B, 110:11438–43, 2006.
[54] P. Han, R. X. Xu, P. Cui, Y. Mo, G. Z. He, and Y. J. Yan. Electron transfer theoryrevisit: Quantum solvation effect. J. Theore. & Comput. Chem., 5:685–92, 2006.
[1] R. P. Feynman and F. L. Vernon, Jr. The theory of a general quantum systeminteracting with a linear dissipative system. Ann. Phys., 24:118–173, 1963.
[2] U. Weiss. Quantum Dissipative Systems. World Scientific, Singapore, 2008. 3rded. Series in Modern Condensed Matter Physics, Vol. 13.
[3] H. Kleinert. Path Integrals in Quantum Mechanics, Statistics, Polymer Physics,and Financial Markets. World Scientific, Singapore, 2009. 5th ed.
[4] Y. Tanimura. Stochastic liouville, langevin, fokker-planck, and master equationapproaches to quantum dissipative systems. J. Phys. Soc. Jpn., 75:082001, 2006.
[5] Y. Tanimura and R. Kubo. Time evolution of a quantum system in contact with anearly gaussian-markovian noise bath. J. Phys. Soc. Jpn., 58:101–114, 1989.
[6] A. Ishizaki and Y. Tanimura. Quantum dynamics of system strongly coupled tolow temperature colored noise bath: Reduced hierarchy equations approach. J.Phys. Soc. Jpn., 74:3131, 2005.
[7] R. X. Xu, P. Cui, X. Q. Li, Y. Mo, and Y. J. Yan. Exact quantum master equationvia the calculus on path integrals. J. Chem. Phys., 122:041103, 2005.
[8] R. X. Xu and Y. J. Yan. Dynamics of quantum dissipation systems interactingwith bosonic canonical bath: Hierarchical equations of motion approach. Phys.Rev. E, 75:031107, 2007.
[9] J. S. Jin, X. Zheng, and Y. J. Yan. Exact dynamics of dissipative electronic systemsand quantum transport: Hierarchical equations of motion approach. J. Chem.Phys., 128:234703, 2008.
[10] R. Kubo. Stochastic liouville equations. J. Math. Phys., 4:174–183, 1963.
[11] R. Kubo. A stochastic theory of line shape. Adv. Chem. Phys., 15:101, 1969.
[12] D. Abramavicius, B. Palmieri, D. V. Voronine, F. Sˇanda, and S. Mukamel. Coher-ent multidimensional optical spectroscopy of excitons in molecular aggregates;quasiparticle versus supermolecule perspectives. Chem. Rev., 109:2350–2408,2009.
[13] W. Zhuang, T. Hayashi, and S. Mukamel. Coherent multidimensional vibrationalspectroscopy of biomolecules: Concepts, simulations, and challenges. Angew.Chem. Int. Ed., 48:3750–3781, 2009.
[14] L. D. Zusman. Outer-sphere electron transfer in polar solvents. Chem. Phys.,49:295–304, 1980.
[15] L. D. Zusman. The theory of transitions between electronic states. application toradiationless transitions in polar solvents. Chem. Phys., 80:29–43, 1983.
[16] A. Garg, J. N. Onuchic, and V. Ambegaokar. Effect of friction on electron transferin biomolecules. J. Chem. Phys., 83:4491–503, 1985.
[17] D. Y. Yang and R. I. Cukier. The transition from nonadiabatic to solvent controlledadiabatic electron transfer: Solvent dynamical effects in the inverted regime. J.Chem. Phys., 91:281–292, 1989.
[18] P. A. Frantsuzov. Chem. Phys. Lett., 267:427, 1997.
[19] P. A. Frantsuzov. J. Chem. Phys., 111:2075, 1999.
[20] M. Thoss, H. B. Wang, and W. H. Miller. Self-consistent hybrid approach for com-plex dystems: Application to the spin-boson model with debye spectral density.J. Chem. Phys., 115:2991–3005, 2001.
[21] J. S. Cao and Y. J. Jung. Spectral analysis of electron transfer kinetics. i. symmet-ric reactions. J. Chem. Phys., 112:4716–22, 2000.
[22] Y. J. Jung and J. S. Cao. Spectral analysis of electron transfer kinetics. ii. J. Chem.Phys., 117:3822–36, 2002.
[23] L. Mu¨hlbacher and R. Egger. Crossover from nonadiabatic to adiabatic electrontransfer reactions: Multilevel blocking monte carlo simulations. J. Chem. Phys.,118:179–191, 2003.
[24] J. Ankerhold and H. Lehle. Low temperature electron transfer in strongly con-densed phases. J. Chem. Phys., 120:1436, 2004.
[25] D. V. Dodin. Problems of coherent description of electron transfer reactions.Chem. Phys., 325:257–264, 2006.
[26] M.-L. Zhang, S.-S. Zhang, and E. Pollak. Theory of electron transfer in the pres-ence of dissipation. J. Chem. Phys., 119:11864, 2003.
[27] M.-L. Zhang, S.-S. Zhang, and E. Pollak. Low temperature extension of thegeneralized zusman phase space equations for electron transfer. J. Chem. Phys.,120:9630, 2004.
[28] Q. Shi, L. P. Chen, G. J. Nan, R. X. Xu, and Y. J. Yan. Electron transfer dynamics:Zusman equation versus exact theory. J. Chem. Phys., 130:164518, 2009.
[29] P. Han, R. X. Xu, B. Q. Li, J. Xu, P. Cui, Y. Mo, and Y. J. Yan. Kinetics andthermodynamics of electron transfer in debye solvents: An analytical and nonper-turbative reduced density matrix theory. J. Phys. Chem. B, 110:11438–43, 2006.
[30] Q. Shi, L. P. Chen, G. J. Nan, R. X. Xu, and Y. J. Yan. Efficient hierarchicalliouville space propagator to quantum dissipative dynamics. J. Chem. Phys.,130(8):084105, 2009.
[31] X. Zheng, J. S. Jin, S. Welack, M. Luo, and Y. J. Yan. Numerical approach to time-dependent quantum transport and dynamical kondo transition. J. Chem. Phys.,130(16):164708, 2009.
[32] Y. Chen, R. X. Xu, H. W. Ke, and Y. J. Yan. Electron transfer theory revisit:Motional narrowing induced non-markovian rate processes. Chin. J. Chem. Phys.,20:438–44, 2007.
[33] R. X. Xu, Y. Chen, P. Cui, H. W. Ke, and Y. J. Yan. The quantum solvation,adiabatic versus nonadiabatic, and markovian versus non-markovian nature ofelectron-transfer rate processes. J. Phys. Chem. A, 111:9618–26, 2007.
[34] S. Jang and J. S. Cao. Nonadiabatic instanton calculation of multistate electrontransfer reaction rate: Interference effects in three and four states systems. J.Chem. Phys., 114:9959–68, 2001.
[35] Y. C. Cheng and G. R. Fleming. Dynamics of light harvesting in photosynthesis.Annu. Rev. Phys. Chem., 60:241–262, 2009.
[36] A. Ishizaki and G. R. Fleming. Theoretical examination of quantum coherence ina photosynthetic system at physiological temperature. Proc. Natl. Acad. Sci. USA,106:17255–60, 2009.
[37] Y. A. Yan, F. Yang, Y. Liu, and J. S. Shao. Hierarchical approach based on stochas-tic decoupling to dissipative systems. Chem. Phys. Lett., 395:216–21, 2004.
[38] Y. Zhou and J. S. Shao. Solving the spin-boson model of strong dissipation with?exible random-deterministic scheme. J. Chem. Phys., 128:034106, 2008.
[1] R. P. Feynman and F. L. Vernon, Jr. The theory of a general quantum systeminteracting with a linear dissipative system. Ann. Phys., 24:118–173, 1963.
[2] U. Weiss. Quantum Dissipative Systems. World Scientific, Singapore, 2008. 3rded. Series in Modern Condensed Matter Physics, Vol. 13.
[3] H. Kleinert. Path Integrals in Quantum Mechanics, Statistics, Polymer Physics,and Financial Markets. World Scientific, Singapore, 2009. 5th ed.
[4] Y. Tanimura and R. Kubo. Time evolution of a quantum system in contact with anearly gaussian-markovian noise bath. J. Phys. Soc. Jpn., 58:101–114, 1989.
[5] Y. Tanimura. Nonperturbative expansion method for a quantum system coupledto a harmonic-oscillator bath. Phys. Rev. A, 41:6676–87, 1990.
[6] A. Ishizaki and Y. Tanimura. Quantum dynamics of system strongly coupled tolow temperature colored noise bath: Reduced hierarchy equations approach. J.Phys. Soc. Jpn., 74:3131, 2005.
[7] Y. Tanimura. Stochastic liouville, langevin, fokker-planck, and master equationapproaches to quantum dissipative systems. J. Phys. Soc. Jpn., 75:082001, 2006.
[8] J. S. Shao. Decoupling quantum dissipation interaction via stochastic fields. J.Chem. Phys., 120:5053–56, 2004.
[9] Y. A. Yan, F. Yang, Y. Liu, and J. S. Shao. Hierarchical approach based on stochas-tic decoupling to dissipative systems. Chem. Phys. Lett., 395:216–21, 2004.
[10] R. X. Xu, P. Cui, X. Q. Li, Y. Mo, and Y. J. Yan. Exact quantum master equationvia the calculus on path integrals. J. Chem. Phys., 122:041103, 2005.
[11] R. X. Xu and Y. J. Yan. Dynamics of quantum dissipation systems interactingwith bosonic canonical bath: Hierarchical equations of motion approach. Phys.Rev. E, 75:031107, 2007.
[12] J. S. Jin, X. Zheng, and Y. J. Yan. Exact dynamics of dissipative electronic systemsand quantum transport: Hierarchical equations of motion approach. J. Chem.Phys., 128:234703, 2008.
[13] Xiao Zheng, Jin Shuang Jin, and Y. J. Yan. Dynamic electronic response of aquantum dot driven by time–dependent voltage. J. Chem. Phys., 129:184112,2008.
[14] Y. J. Yan and R. X. Xu. Quantum mechanics of dissipative systems. Annu. Rev.Phys. Chem., 56:187–219, 2005.
[15] A. Croy and U. Saalmann. Partial fraction decomposition of the fermi function.Phys. Rev. B, 80(7):073102, 2009.
[16] Alexander Croy and Ulf Saalmann. Propagation scheme for nonequilibrium dy-namics of electron transport in nanoscale devices. Phys. Rev. B, 80(24):245311,2009.
[17] J. Xu, R. X. Xu, M. Luo, and Y. J. Yan. Hierarchical theory of quantum dissipa-tion: Partial fraction decomposition scheme. Chem. Phys., 370:109–114, 2010.
[18] Jie Hu, Rui Xue Xu, and Yi Jing Yan. Pade′spectrum decomposition of fermifunction and bose function. J. Chem. Phys., 133(10):101106, 2010.
[19] Q. Shi, L. P. Chen, G. J. Nan, R. X. Xu, and Y. J. Yan. Efficient hierarchicalliouville space propagator to quantum dissipative dynamics. J. Chem. Phys.,130(8):084105, 2009.
[20] J. Xu, R. X. Xu, and Y. J. Yan. Exact quantum dissipative dynamics under externaltime-dependent driving fields. New J. Phys., 11:105037, 2009.
[21] X. Zheng, J. S. Jin, S. Welack, M. Luo, and Y. J. Yan. Numerical approach to time-dependent quantum transport and dynamical kondo transition. J. Chem. Phys.,130(16):164708, 2009.
[22] R. X. Xu, B. L. Tian, J. Xu, Q. Shi, and Y. J. Yan. Hierarchical quantum masterequation with semiclassical drude dissipation. J. Chem. Phys., 131(21):214111,2009.
[23] R. Kubo. Stochastic liouville equations. J. Math. Phys., 4:174–183, 1963.
[24] R. Kubo. A stochastic theory of line shape. Adv. Chem. Phys., 15:101, 1969.
[25] F. Shuang, C. Yang, and Y. J. Yan. Dynamical semigroup fokker-planck equationapproach to transient absorption and ?uorescence upconversion spectroscopies. J.Chem. Phys., 114:3868–79, 2001.
[26] Y. Mo, R. X. Xu, P. Cui, and Y. J. Yan. Correlation and response functions withnon-markovian dissipation: A reduced liouville-space theory. J. Chem. Phys.,122:084115, 2005.
[27] S. Mukamel. The Principles of Nonlinear Optical Spectroscopy. Oxford Univer-sity Press, New York, 1995.
[1] R. P. Feynman and F. L. Vernon, Jr. The theory of a general quantum systeminteracting with a linear dissipative system. Ann. Phys., 24:118–173, 1963.
[2] H. Kleinert. Path Integrals in Quantum Mechanics, Statistics, Polymer Physics,and Financial Markets. World Scientific, Singapore, 2009. 5th ed.
[3] U. Weiss. Quantum Dissipative Systems. World Scientific, Singapore, 2008. 3rded. Series in Modern Condensed Matter Physics, Vol. 13.
[4] Y. J. Yan and R. X. Xu. Quantum mechanics of dissipative systems. Annu. Rev.Phys. Chem., 56:187–219, 2005.
[5] M. A. Nielsen and I. L. Chuang. Quantum Computation and Quantum Informa-tion. Cambridge University Press, New York, 2000.
[6] S. Datta. Quantum Transport: Atom to Transistor. Cambridge University Press,New York, 2005.
[7] G. S. Engel, T. R. Calhoun, E. L. Read, T. K. Ahn, T. Mancˇal, Y. C. Cheng, R. E.Blankenship, and G. R. Fleming. Evidence for wavelike energy transfer throughquantum coherence in photosynthetic systems. Nature, 446:782–6, 2007.
[8] E. Collini, C. Y. Wong, K. E. Wilk, P. M. G. Curmi, P. Brumer, and G. D. Scholes.Coherently wired light-harvesting in photosynthetic marine algae at ambient tem-perature. Nature, 463:644–7, 2010.
[9] Y. Tanimura. Nonperturbative expansion method for a quantum system coupledto a harmonic-oscillator bath. Phys. Rev. A, 41:6676–87, 1990.
[10] Y. Tanimura. Stochastic liouville, langevin, fokker-planck, and master equationapproaches to quantum dissipative systems. J. Phys. Soc. Jpn., 75:082001, 2006.
[11] R. X. Xu, P. Cui, X. Q. Li, Y. Mo, and Y. J. Yan. Exact quantum master equationvia the calculus on path integrals. J. Chem. Phys., 122:041103, 2005.
[12] R. X. Xu and Y. J. Yan. Dynamics of quantum dissipation systems interactingwith bosonic canonical bath: Hierarchical equations of motion approach. Phys.Rev. E, 75:031107, 2007.
[13] J. S. Jin, X. Zheng, and Y. J. Yan. Exact dynamics of dissipative electronic systemsand quantum transport: Hierarchical equations of motion approach. J. Chem.Phys., 128:234703, 2008.
[14] Y. Tanimura and R. Kubo. Time evolution of a quantum system in contact with anearly gaussian-markovian noise bath. J. Phys. Soc. Jpn., 58:101–114, 1989.
[15] Q. Shi, L. P. Chen, G. J. Nan, R. X. Xu, and Y. J. Yan. Electron transfer dynamics:Zusman equation versus exact theory. J. Chem. Phys., 130:164518, 2009.
[16] R. X. Xu, B. L. Tian, J. Xu, Q. Shi, and Y. J. Yan. Hierarchical quantum masterequation with semiclassical drude dissipation. J. Chem. Phys., 131(21):214111,2009.
[17] A. Ishizaki and Y. Tanimura. Modeling vibrational dephasing and energy relax-ation of intramolecular anharmonic modes for multidimensional infrared spectro-scopies. J. Chem. Phys., 125:084501, 2006.
[18] A. Ishizaki and Y. Tanimura. Dynamics of a multimode system coupled to multi-ple heat baths probed by two-dimensional infrared spectroscopy. J. Phys. Chem.A, 111:9269–9276, 2007.
[19] L. P. Chen, R. H. Zheng, Q. Shi, and Y. J. Yan. Two-dimensional electronic spectrafrom the hierarchical equations of motion method: Application to model dimers.J. Chem. Phys., 132(2):024505, 2010.
[20] B. L. Tian, J. J. Ding, R. X. Xu, and Y. J. Yan. Bi-exponential theory of drude dissi-pation via hierarchical quantum master equation. J. Chem. Phys., 133(11):114112,2010.
[21] Xiao Zheng, Jin Shuang Jin, and Y. J. Yan. Dynamic electronic response of aquantum dot driven by time–dependent voltage. J. Chem. Phys., 129:184112,2008.
[22] X. Zheng, J. S. Jin, and Y. J. Yan. Dynamic coulomb blockade in single–leadquantum dots. New J. Phys., 10:093016, 2008.
[23] X. Zheng, J. S. Jin, S. Welack, M. Luo, and Y. J. Yan. Numerical approach to time-dependent quantum transport and dynamical kondo transition. J. Chem. Phys.,130(16):164708, 2009.
[24] Jie Hu, Rui Xue Xu, and Yi Jing Yan. Pade′spectrum decomposition of fermifunction and bose function. J. Chem. Phys., 133(10):101106, 2010.
[25] M. Thoss, H. B. Wang, and W. H. Miller. Self-consistent hybrid approach for com-plex dystems: Application to the spin-boson model with debye spectral density.J. Chem. Phys., 115:2991–3005, 2001.
[26] G. A. Baker Jr. and P. Graves-Morris. Pade′Approximants. Cambridge UniversityPress, New York, 1996. 2nd ed.
[27] Jin-Jin Ding, Jie Hu, Bao-Ling Tian, Rui-Xue Xu, and YiJing Yan. Optimal hi-erarchical theory for nonperturbative quantum dissipation dynamics. J. Chem.Phys., submitted, 2010.
[28] Q. Shi, L. P. Chen, G. J. Nan, R. X. Xu, and Y. J. Yan. Efficient hierarchicalliouville space propagator to quantum dissipative dynamics. J. Chem. Phys.,130(8):084105, 2009.
[29] A. Ishizaki and Y. Tanimura. Quantum dynamics of system strongly coupled tolow temperature colored noise bath: Reduced hierarchy equations approach. J.Phys. Soc. Jpn., 74:3131, 2005.
[30] J. Xu, R. X. Xu, and Y. J. Yan. Exact quantum dissipative dynamics under externaltime-dependent driving fields. New J. Phys., 11:105037, 2009.
[1] Green B. R. and Parson W. W., editors. Light-Harvesting Antennas in Photosyn-thesis. Kluwer, Dordrecht, 2003.
[2] G. D. Scholes. Long-range resonance energy transfer in molecular systems. Annu.Rev. Phys. Chem., 54:57–87, 2003.
[3] S. Jang, M. D. Newton, and R. J. Silbey. Multichromophoric fo˙ J. Phys. Chem.B, 111:6807–6814, 1961.
[4] Y. C. Cheng and G. R. Fleming. Dynamics of light harvesting in photosynthesis.Annu. Rev. Phys. Chem., 60:241–262, 2009.
[5] R. van Grondelle and V. I. Novoderezhkin. Energy transfer in photosynthesis:Experimental insights and qunatitive models. Phys. Chem. Chem. Phys., 8:793–807, 2006.
[6] G. S. Engel, T. R. Calhoun, E. L. Read, T. K. Ahn, T. Mancˇal, Y. C. Cheng, R. E.Blankenship, and G. R. Fleming. Evidence for wavelike energy transfer throughquantum coherence in photosynthetic systems. Nature, 446:782–6, 2007.
[7] H. Lee, Y. C. Cheng, and G. R. Fleming. Coherence dynamics in photosynthesis:protein protection of excitonic coherence. Science, 316:1462–1465, 2007.
[8] Ian P. Mercer et al. Instantaneous mapping of coherently coupled electronic tran-sitions and energy transfers in a photosynthetic complex using angleresolved co-herent optical wave-mixing. Phys. Rev. Lett., 102:057402, 2009.
[9] D. M. Jonas. Two-dimensional femtosecond spectroscopy. Annu. Rev. Phys.Chem., 54:425–463, 2003.
[10] T. Brixner, T. Mancal, I. V. Stiopkin, and G. R. Fleming. Phase-stabilized twodi-mensional electronic spectroscopy. J. Chem. Phys., 121:4221–4236, 2004.
[11] M. H. Cho. Coherent two-dimensional optical spectroscopy. Chem. Rev.,108:1331–1418, 2008.
[12] D. Abramavicius, B. Palmieri, D. V. Voronine, F. Sˇanda, and S. Mukamel. Coher-ent multidimensional optical spectroscopy of excitons in molecular aggregates;quasiparticle versus supermolecule perspectives. Chem. Rev., 109:2350–2408,2009.
[13] E. Collini, C. Y. Wong, K. E. Wilk, P. M. G. Curmi, P. Brumer, and G. D. Scholes.Coherently wired light-harvesting in photosynthetic marine algae at ambient tem-perature. Nature, 463:644–7, 2010.
[14] R. van Grondelle, J. P. Dekker, T. Gillbro, and V. Sundstro¨m. Biochem. Biophys.Acta, 1187:1, 1994.
[15] A. Ishizaki and G. R. Fleming. On the adequacy of the redfield equation andrelated approaches to the study of quantum dynamics in electronic energy transfer.J. Chem. Phys., 130:234110, 2009.
[16] A. Ishizaki and G. R. Fleming. Unified treatment of quantum coherent and inco-herent hopping dynamics in electronic energy transfer: Reduced hierarchy equa-tion approach. J. Chem. Phys., 130:234111, 2009.
[17] A. Ishizaki and G. R. Fleming. Theoretical examination of quantum coherence ina photosynthetic system at physiological temperature. Proc. Natl. Acad. Sci. USA,106:17255–60, 2009.
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