量子耗散理论及在光谱中的应用
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摘要
凝聚相中的分子体系受到其所处周围环境(热库)的影响,体系要经历一个不可逆(退相干或弛豫)的过程达到一个平衡态,这一动力学过程就称为耗散。量子耗散不仅指能量的耗散,也包括量子力学相干性的耗散,即所说的退相干作用。耗散的研究与系统的约化描述(即体系与环境的区分)紧密关联。耗散理论的研究内容就是在环境作用下的约化体系的运动规律。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,例如核磁共振、量子光学、分子光谱、凝聚态物理和化学物理。本论文对量子耗散方程理论的构造有了一定的进展。另外光谱为探测和了解分子结构,分子之间的相互作用力及动力学提供了丰富的信息。本论文应用量子耗散理论对光谱动力学进行了研究。
本论文的理论方法是基于香港科技大学严以京教授组里近年来发展的量子耗散理论-级联耦合运动方程组。对该理论的构造需要对热库相关函数进行指数展开。热库相关函数满足涨落-耗散定理,包含热库谱密度函数,以及玻色-爱因斯坦分布函数,简称为玻色函数。相关函数的指数展开可等价于对谱密度函数和玻色函数的极点求和(SOP)。玻色函数传统的也是目前最常用的展开方案是松原谱分解(MSD)方案。但是它最大的问题是展开收敛非常慢,所构造的级联方程维度非常高。香港科技大学严以京教授组最近的工作中,应用Pade谱分解(PSD)方案来建立级联耦合运动方程组。至今为止,玻色函数最好的极点求和(SOP)展开方案是PSD。本文我们为PSD-HEOM的构造选择最佳的热库统计基组,并将其应用到了实际计算。
在第一章介绍量子耗散动力学的一些理论背景知识,包括体系的约化描述,线性响应理论及相关函数理论。重点讨论涉及到的关键概念与涨落-耗散定理。
在第二章,我们回顾了香港科技大学严以京教授组里对影响泛函的路径积分求导建立级联耦合运动方程,应用Pade谱分解方案来建立级联耦合运动方程组。我们为PSD-HEOM的热库谱密度函数采用MBO模型,对常用的Drude热库以及分子内的谐振子作了统一的描述。它们都满足高斯统计,所以可并入环境来处理。进一步推导了级联耦合运动方程组的Drude热库耗散和改进的半经典Drude耗散下的级联量子主方程。
在第三章,我们用第二章推导的HEOM和HQME这两种方法计算了Drude热库耗散二能级模型体系的线性稳态吸收和发射光谱：对于相同的热库截止频率γ,随着热库重整化能λ的变大,稳态的吸收和发射谱线变宽,强度减小,斯托克斯位移变大。说明热库对体系的作用强烈,所以电子能达到较高的振动能级,因此斯托克斯位移大；比较可看出用HQME和HEOM两种方法得到的谱线结果相差很小。对于相同的热库重整化能λ,随着热库截止频率γ的变大,稳态的吸收和发射谱线变窄,斯托克斯位移变小。γ越大说明热库越快达到平衡状态,即热库对体系作用减弱,所以电子不能达到较高的振动能级,因此斯托克斯位移变小并且标定了HQME的适用范围。计算光谱时注意要正确考虑体系-热库的初始耦合,可以从HEOM/HQME的稳态解得出。进一步讨论用完备二阶量子耗散理论方法来计算线性光谱。
在第四章我们用HEOM计算了Drude热库耗散二能级模型体系的泵浦-探测和荧光光谱。详细地给出了用到的非线性光谱理论的关键量-场缀饰响应函数的高效数值计算方法,此算法是基于场缀饰响应函数的Schrodinger/Heisenberg的混合表征,从而使得响应函数的计算中二维时间上的演化几乎可以简化为两个一维时间上的演化,这样大大减少了计算量。
1.泵浦-探测顺序情况下二能级模型体系的瞬态受激发射光谱：
(1)短泵浦场,频谱宽、波包在激发态上弛豫,所以受激辐射光谱会有红移现象出现λ→-λ。
(2)长泵浦场,泵浦光只能选择一个窄频激发,窄频波包在激发态上弛豫,因此有受激辐射光谱的展宽和红移现象出现0→-λ。
2.泵浦-探测场相干情况下二能级模型体系的瞬态受激发射光谱：
(1)短泵浦场,时间td=0,即泵浦场和探测场重叠最大时,从图中可以看出受激辐射谱线有负值存在。实际上吸收和发射的概念只对顺序部分有效,所以在相干情况下求出的吸收光谱称为激发Raman部分,而受激发射光谱称为受激发射Raman部分更好。受激辐射谱线红移现象依然存在λ→-λ。
(2)长泵浦场,受激辐射谱线的展宽和红移现象依然存在0→-λ。
3.在六个不同延迟时间比较了自发辐射SE(ω,td)和受激辐射α-(ω,td)。受激发射探测的是局域时间的动力学,而自发辐射包含记忆。由于耗散抑制了记忆效应,SE(ω,td)和α_(ω,td)只有在短时间内不同,时间比较长时差别不大。在时间很长时,SE(ω,td)和.α_(ω,td)与稳态发射谱相同。
在第五章我们用HEOM理论计算了二能级单体和二聚体的瞬态泵浦-探测吸收光谱。在计算单体时环境包含了溶剂化和分子内的光学声子振动的影响,所以环境采用MBO模型。实验上可以用泵浦-探测来测量光学声子,在这里我们把分子内的光学声子放在环境里来处理。并且说明了辅助密度算符没有改变约化密度算符的跃迁动力学性质,但是反应了HEOM中体系-环境耦合相干性的动力学行为。在计算二聚体时,仅考虑了Drude环境耗散。这里不仅展示了空穴的吸收和粒子的辐射,并且展示了激发态粒子的再吸收。
在第六章,我们总结了本论文,并对今后在量子耗散理论及其应用方面的工作进行了展望。
我们在这里重点强调HEOM是目前,在任意温度、高斯热库影响下处理严格的量子耗散动力学最有效公式。HEOM的构造依赖于热库记忆时间尺度按照彼此的权重以级联的方式解开,而热库的多时间性使得其分解方式的非唯一性。提出的PSD-MBO方案是目前最好的。我们也核实了分解所需的最小热库基组的判据。正如ADOs的总数所担心的,目前HEOM相当于体系-环境相干空间的完备组态相互作用。过滤算法对Drude耗散来说已得到成功的证明,需要进一步发展到包含一般的MBO情况。
Quantum dissipation refers to the reduced dynamics of a system embedded in environment (bath). The latter consists of macroscopic degrees of freedom whose effects on the system should be treated in a statistical manner. Consequently, the system of primary interest undergoes energy relaxation and dephasing processes, evolving eventually to the thermal equilibrium state. The key quantity in quantum dissipation theory is the reduced density operator. The development of quantum dissipation theory has involved diversified fields of research, such as nuclear magnetic resonance, quantum optics, molecular spectroscopy, condensed phase physics, and chemical physics. Optical spectroscopy provides a powerful tool in the detection and understanding of molecular structures, interactions, and dynamics. We present the efficient quantum dissipation theory to study spectroscopies in the thesis.
In the thesis, We exploit the hierarchical equations of motion (HEOM) formalism of quantum dissipation theory by Prof. Yan's group in Hong Kong University of Science and Technology as a foundation. Nevertheless, an explicit HEOM construction relies on the choice of the statistical environment basis set that expands the environment correlation functions into its complex memory conponents. The environment correlation function is dictated by the fluctuation-dissipation theorem, involving the Fourier transform of the product of the spectral density and Bose Function. We request the best statistical environment basis set for an efficient HEOM construction and its practical applications. The conventional HEOM construction involves the Matsubara spectrum decomposition (MSD) for the Bose function. However, MSD is notorious for its slow convergence. The resulting HEOM is rather expensive and limited largely to simple systems with Drude dissipations. We have shown recently that the Pade spectral decomposition (PSD) is the best sum-over-poles (SOP) scheme for the Bose function.
In Chapter 1, we introduce the theoretical background of QDT, including the reduced system description, the correlation and response functions versus linear response theory, with emphasis on key concepts and fluctuation-dissipation theorem.
In chapter 2, we summarize the establishment of the exact and nonperturbative HEOM of QDT, via the calculus on the influence functional path integral. We implement the PSD scheme to establish the corresponding PSD-HEOM. For environment spectral density, we adopt the multiple Brownian oscillators (MBO) model. It can describe optically actively vibronic coupling via underdamped Brownian oscillator mode, and also energy fluctuation via strongly overdamped Drude dissipation. We also propose a hierarchical quantum master equation (HQME) approach, which adopts a modified semiclassical Drude model.
In chapter 3, we use HEOM and HQME, which are introduced in chapter 2 to calculate the linear absorption and emission spectrum of a two-level system (TLS). For the same bath cut-off frequency y, along with the bigger solvent reorganizationλ, the linear absoption and emission spectrum get wider, strength get smaller, the spectroscopic Stokes shift get bigger. The reason is the effect of bath on the system is intense, the electron can transfer to the higher vibrational energy levels, so the Stokes shift get larger. Compare to the HQME and HEOM, the discrepancy seems not to matter. For the same solvent reorganizationλ, along with the bigger bath cut-off frequency y, the linear absoption and emission spectrum get narrower, the spectroscopic Stokes shift get smaller. The bigger bath cut-off frequency y is, the more quickly bath get equilibrium state. The reason is the effect of bath on the system is weak, the electron can only transfer to the lower vibrational energy levels, so the Stokes shift get smaller. We also propose a criterion to estimate the performance of HQME. For the TLS system, the initial steady state solution of HEOM can be obtained analytically. If without considering the initial system-bath coupling, the response function is zero. We demonstrate the linear spectroscopies by the complete second-order correlated driving-dissipation equations (CODDE).
In chapter 4, we use HEOM to demonstrate the transient stimulated and spontaneous emissions spectroscopies of a two-level system. We present in detail a highly efficient numerical method to evaluate the key quantity in our spectroscopic theory—field-dressed response function. Our newly developed numberical method is based on a mixed Schrodinger/Heisenberg picture of the field-dressed response function, which reduced the two-dimensional time-grid problem almost to two one-dimensional time-grid problems, so it costs little time to calculate.
1. Pump-probe field on the sequential instance:
(1) Shot time pump field, wave packet will have relaxation on excited state. So the stimulated emissionα_(ω),td) get red shitλ→-λ.
(2) Long time pump field, pump light will choose a narrow frequency to excitated. So the narrow wave packet will have relaxation on excited state. The stimulated emissionα_ (ω>td) get wide and red shit 0→-λ.
2. Pump-probe field on the coherent instance:
(1) shot time pump field, note that the stimulated emissionα_[ω,td) has some significant negative values in some frequency region. In fact, the concept of being emissive may only be physical correct for the sequentialα_S (ω,td) contribution. This implies thatα_C(ω,td) may be more properly called as the stimulated emission Raman contribution. The negative values inα_C(ω,td) thus indicate the underlying conherent Raman processes. But the stimulated emissionα_(ω,td) get red shitλ→-λ.
(2) long time pump field, have the same conclusion as shot time pump field. The stimulated emissionα_(ω,td) get wide and red shit 0→-λ.
3. At six representing delay times comparison among the ordinary fluorescence SE(ω,td) and the stimulated emissionα_(ω,td). SE andα_ are only different in the short time regime, but evolve to be identical as the results of dissipation that suppresses the memory effect. At the long time regime both SE andα_ coincide with the stationary emission spectrum.
In Chapter 5, we use HEOM to study the transient absorption spectroscopies in the pump-probe scenario of some models monomer and dimer exciton systems. For a monomer, the vibrational versus electronic coherence will be demonstrated, so we adopt the multiple Brownian oscillators model for environment spectral density. In the experiment we can measure optically active phonon by the pump-probe, we also demonstrate the HEOM approach to correlated system-environment coherence. Turn to a dimer, we only consider the Drude spectral density. While S_ (td) remains as the single-exciton state particle emission, S+(td) contains now not just the ground-state hole absorption, but also the single-exciton state particle reabsorption.
In Chapter 6, we summarize the thesis, and comment on the future work concerning both the theoretical and application aspects of quantum dissapation theory.
We like to emphasis here that HEOM is by far the most numerically tractable formalism for exact quantum dynamics under arbitrary Gaussian bath influence at finite temperature. The explicit HEOM construction depends on the way of decomposing bath influence into complex memory components. This is about the statistical bath basis set representation of HEOM quantum dissipation theory. The proposed PSD-MBO scheme is likely the best. We have also verified the established minimum bath basis set criterions. In most of the numerical demonstrations presented in this thesis, only one PSD pole is sufficient, while about five MSD poles are required in the same cases. As the total number of ADOs is concerned, the present HEOM resembles a full configuration interaction formalism in the system-bath coherence space. An efficient filtering algorithm which exists for Drude dissipations needs to be developed further to include the general MBO cases.
本论文的理论方法是基于香港科技大学严以京教授组里近年来发展的量子耗散理论-级联耦合运动方程组。对该理论的构造需要对热库相关函数进行指数展开。热库相关函数满足涨落-耗散定理,包含热库谱密度函数,以及玻色-爱因斯坦分布函数,简称为玻色函数。相关函数的指数展开可等价于对谱密度函数和玻色函数的极点求和(SOP)。玻色函数传统的也是目前最常用的展开方案是松原谱分解(MSD)方案。但是它最大的问题是展开收敛非常慢,所构造的级联方程维度非常高。香港科技大学严以京教授组最近的工作中,应用Pade谱分解(PSD)方案来建立级联耦合运动方程组。至今为止,玻色函数最好的极点求和(SOP)展开方案是PSD。本文我们为PSD-HEOM的构造选择最佳的热库统计基组,并将其应用到了实际计算。
在第一章介绍量子耗散动力学的一些理论背景知识,包括体系的约化描述,线性响应理论及相关函数理论。重点讨论涉及到的关键概念与涨落-耗散定理。
在第二章,我们回顾了香港科技大学严以京教授组里对影响泛函的路径积分求导建立级联耦合运动方程,应用Pade谱分解方案来建立级联耦合运动方程组。我们为PSD-HEOM的热库谱密度函数采用MBO模型,对常用的Drude热库以及分子内的谐振子作了统一的描述。它们都满足高斯统计,所以可并入环境来处理。进一步推导了级联耦合运动方程组的Drude热库耗散和改进的半经典Drude耗散下的级联量子主方程。
在第三章,我们用第二章推导的HEOM和HQME这两种方法计算了Drude热库耗散二能级模型体系的线性稳态吸收和发射光谱：对于相同的热库截止频率γ,随着热库重整化能λ的变大,稳态的吸收和发射谱线变宽,强度减小,斯托克斯位移变大。说明热库对体系的作用强烈,所以电子能达到较高的振动能级,因此斯托克斯位移大；比较可看出用HQME和HEOM两种方法得到的谱线结果相差很小。对于相同的热库重整化能λ,随着热库截止频率γ的变大,稳态的吸收和发射谱线变窄,斯托克斯位移变小。γ越大说明热库越快达到平衡状态,即热库对体系作用减弱,所以电子不能达到较高的振动能级,因此斯托克斯位移变小并且标定了HQME的适用范围。计算光谱时注意要正确考虑体系-热库的初始耦合,可以从HEOM/HQME的稳态解得出。进一步讨论用完备二阶量子耗散理论方法来计算线性光谱。
在第四章我们用HEOM计算了Drude热库耗散二能级模型体系的泵浦-探测和荧光光谱。详细地给出了用到的非线性光谱理论的关键量-场缀饰响应函数的高效数值计算方法,此算法是基于场缀饰响应函数的Schrodinger/Heisenberg的混合表征,从而使得响应函数的计算中二维时间上的演化几乎可以简化为两个一维时间上的演化,这样大大减少了计算量。
1.泵浦-探测顺序情况下二能级模型体系的瞬态受激发射光谱：
(1)短泵浦场,频谱宽、波包在激发态上弛豫,所以受激辐射光谱会有红移现象出现λ→-λ。
(2)长泵浦场,泵浦光只能选择一个窄频激发,窄频波包在激发态上弛豫,因此有受激辐射光谱的展宽和红移现象出现0→-λ。
2.泵浦-探测场相干情况下二能级模型体系的瞬态受激发射光谱：
(1)短泵浦场,时间td=0,即泵浦场和探测场重叠最大时,从图中可以看出受激辐射谱线有负值存在。实际上吸收和发射的概念只对顺序部分有效,所以在相干情况下求出的吸收光谱称为激发Raman部分,而受激发射光谱称为受激发射Raman部分更好。受激辐射谱线红移现象依然存在λ→-λ。
(2)长泵浦场,受激辐射谱线的展宽和红移现象依然存在0→-λ。
3.在六个不同延迟时间比较了自发辐射SE(ω,td)和受激辐射α-(ω,td)。受激发射探测的是局域时间的动力学,而自发辐射包含记忆。由于耗散抑制了记忆效应,SE(ω,td)和α_(ω,td)只有在短时间内不同,时间比较长时差别不大。在时间很长时,SE(ω,td)和.α_(ω,td)与稳态发射谱相同。
在第五章我们用HEOM理论计算了二能级单体和二聚体的瞬态泵浦-探测吸收光谱。在计算单体时环境包含了溶剂化和分子内的光学声子振动的影响,所以环境采用MBO模型。实验上可以用泵浦-探测来测量光学声子,在这里我们把分子内的光学声子放在环境里来处理。并且说明了辅助密度算符没有改变约化密度算符的跃迁动力学性质,但是反应了HEOM中体系-环境耦合相干性的动力学行为。在计算二聚体时,仅考虑了Drude环境耗散。这里不仅展示了空穴的吸收和粒子的辐射,并且展示了激发态粒子的再吸收。
在第六章,我们总结了本论文,并对今后在量子耗散理论及其应用方面的工作进行了展望。
我们在这里重点强调HEOM是目前,在任意温度、高斯热库影响下处理严格的量子耗散动力学最有效公式。HEOM的构造依赖于热库记忆时间尺度按照彼此的权重以级联的方式解开,而热库的多时间性使得其分解方式的非唯一性。提出的PSD-MBO方案是目前最好的。我们也核实了分解所需的最小热库基组的判据。正如ADOs的总数所担心的,目前HEOM相当于体系-环境相干空间的完备组态相互作用。过滤算法对Drude耗散来说已得到成功的证明,需要进一步发展到包含一般的MBO情况。
Quantum dissipation refers to the reduced dynamics of a system embedded in environment (bath). The latter consists of macroscopic degrees of freedom whose effects on the system should be treated in a statistical manner. Consequently, the system of primary interest undergoes energy relaxation and dephasing processes, evolving eventually to the thermal equilibrium state. The key quantity in quantum dissipation theory is the reduced density operator. The development of quantum dissipation theory has involved diversified fields of research, such as nuclear magnetic resonance, quantum optics, molecular spectroscopy, condensed phase physics, and chemical physics. Optical spectroscopy provides a powerful tool in the detection and understanding of molecular structures, interactions, and dynamics. We present the efficient quantum dissipation theory to study spectroscopies in the thesis.
In the thesis, We exploit the hierarchical equations of motion (HEOM) formalism of quantum dissipation theory by Prof. Yan's group in Hong Kong University of Science and Technology as a foundation. Nevertheless, an explicit HEOM construction relies on the choice of the statistical environment basis set that expands the environment correlation functions into its complex memory conponents. The environment correlation function is dictated by the fluctuation-dissipation theorem, involving the Fourier transform of the product of the spectral density and Bose Function. We request the best statistical environment basis set for an efficient HEOM construction and its practical applications. The conventional HEOM construction involves the Matsubara spectrum decomposition (MSD) for the Bose function. However, MSD is notorious for its slow convergence. The resulting HEOM is rather expensive and limited largely to simple systems with Drude dissipations. We have shown recently that the Pade spectral decomposition (PSD) is the best sum-over-poles (SOP) scheme for the Bose function.
In Chapter 1, we introduce the theoretical background of QDT, including the reduced system description, the correlation and response functions versus linear response theory, with emphasis on key concepts and fluctuation-dissipation theorem.
In chapter 2, we summarize the establishment of the exact and nonperturbative HEOM of QDT, via the calculus on the influence functional path integral. We implement the PSD scheme to establish the corresponding PSD-HEOM. For environment spectral density, we adopt the multiple Brownian oscillators (MBO) model. It can describe optically actively vibronic coupling via underdamped Brownian oscillator mode, and also energy fluctuation via strongly overdamped Drude dissipation. We also propose a hierarchical quantum master equation (HQME) approach, which adopts a modified semiclassical Drude model.
In chapter 3, we use HEOM and HQME, which are introduced in chapter 2 to calculate the linear absorption and emission spectrum of a two-level system (TLS). For the same bath cut-off frequency y, along with the bigger solvent reorganizationλ, the linear absoption and emission spectrum get wider, strength get smaller, the spectroscopic Stokes shift get bigger. The reason is the effect of bath on the system is intense, the electron can transfer to the higher vibrational energy levels, so the Stokes shift get larger. Compare to the HQME and HEOM, the discrepancy seems not to matter. For the same solvent reorganizationλ, along with the bigger bath cut-off frequency y, the linear absoption and emission spectrum get narrower, the spectroscopic Stokes shift get smaller. The bigger bath cut-off frequency y is, the more quickly bath get equilibrium state. The reason is the effect of bath on the system is weak, the electron can only transfer to the lower vibrational energy levels, so the Stokes shift get smaller. We also propose a criterion to estimate the performance of HQME. For the TLS system, the initial steady state solution of HEOM can be obtained analytically. If without considering the initial system-bath coupling, the response function is zero. We demonstrate the linear spectroscopies by the complete second-order correlated driving-dissipation equations (CODDE).
In chapter 4, we use HEOM to demonstrate the transient stimulated and spontaneous emissions spectroscopies of a two-level system. We present in detail a highly efficient numerical method to evaluate the key quantity in our spectroscopic theory—field-dressed response function. Our newly developed numberical method is based on a mixed Schrodinger/Heisenberg picture of the field-dressed response function, which reduced the two-dimensional time-grid problem almost to two one-dimensional time-grid problems, so it costs little time to calculate.
1. Pump-probe field on the sequential instance:
(1) Shot time pump field, wave packet will have relaxation on excited state. So the stimulated emissionα_(ω),td) get red shitλ→-λ.
(2) Long time pump field, pump light will choose a narrow frequency to excitated. So the narrow wave packet will have relaxation on excited state. The stimulated emissionα_ (ω>td) get wide and red shit 0→-λ.
2. Pump-probe field on the coherent instance:
(1) shot time pump field, note that the stimulated emissionα_[ω,td) has some significant negative values in some frequency region. In fact, the concept of being emissive may only be physical correct for the sequentialα_S (ω,td) contribution. This implies thatα_C(ω,td) may be more properly called as the stimulated emission Raman contribution. The negative values inα_C(ω,td) thus indicate the underlying conherent Raman processes. But the stimulated emissionα_(ω,td) get red shitλ→-λ.
(2) long time pump field, have the same conclusion as shot time pump field. The stimulated emissionα_(ω,td) get wide and red shit 0→-λ.
3. At six representing delay times comparison among the ordinary fluorescence SE(ω,td) and the stimulated emissionα_(ω,td). SE andα_ are only different in the short time regime, but evolve to be identical as the results of dissipation that suppresses the memory effect. At the long time regime both SE andα_ coincide with the stationary emission spectrum.
In Chapter 5, we use HEOM to study the transient absorption spectroscopies in the pump-probe scenario of some models monomer and dimer exciton systems. For a monomer, the vibrational versus electronic coherence will be demonstrated, so we adopt the multiple Brownian oscillators model for environment spectral density. In the experiment we can measure optically active phonon by the pump-probe, we also demonstrate the HEOM approach to correlated system-environment coherence. Turn to a dimer, we only consider the Drude spectral density. While S_ (td) remains as the single-exciton state particle emission, S+(td) contains now not just the ground-state hole absorption, but also the single-exciton state particle reabsorption.
In Chapter 6, we summarize the thesis, and comment on the future work concerning both the theoretical and application aspects of quantum dissapation theory.
We like to emphasis here that HEOM is by far the most numerically tractable formalism for exact quantum dynamics under arbitrary Gaussian bath influence at finite temperature. The explicit HEOM construction depends on the way of decomposing bath influence into complex memory components. This is about the statistical bath basis set representation of HEOM quantum dissipation theory. The proposed PSD-MBO scheme is likely the best. We have also verified the established minimum bath basis set criterions. In most of the numerical demonstrations presented in this thesis, only one PSD pole is sufficient, while about five MSD poles are required in the same cases. As the total number of ADOs is concerned, the present HEOM resembles a full configuration interaction formalism in the system-bath coherence space. An efficient filtering algorithm which exists for Drude dissipations needs to be developed further to include the general MBO cases.
引文
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[27]Mukamel S. Superoperator representation of nonlinear response:Unifying quantum field and mode coupling theories[J]. Phys. Rev. E,2003,68:021111.
[28]Breuer H P, Petruccione F. The Theory of Open Quantum Systems[M]. New York:Oxford University Press,2002.
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