量子耗散动力学的理论发展与应用
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摘要
本论文致力于量子耗散动力学两方面的研究。一方面是量子耗散理论的发展,我们系统介绍了通过影响泛函路径积分求导来建立严格的级联量子耗散方程的方法,给出了在不同方案下的：具体形式；另一方面是严格量子耗散理论的实际应用。这里我们涉及了几种常见的耗散模型体系,包括布居数转移体系、电荷转移体系和驱动布朗振子体系。另外,由于严格的量子耗散理论所需的计算量非常巨大,本论文也讨论了我们在数值算法方面所做的一些具体工作。本论文具体安排如下：
第一章介绍了量子耗散理论的一些背景知识,包括体系的约化描述,相关函数、响应函数及线性响应理论,重点讨论了其中涉及的一些关键概念以及涨落-耗散定理。
在第二章,我们首先回顾了影响泛函路径积分公式,然后通过求影响泛函路径积分时间导数的方法构建了级联耦合运动方程组(Hierarchical Equations of Motion, HEOM)。该方法非微扰地处理任意温度下的非马尔可夫量子耗散过程,并且适用于有含时外场驱动的情况。除此之外,我们还推导了与该级联方程组等价的格林函数的连分数方程。
在第三章,我们具体应用上述的HEOM方程研究了受激拉曼绝热转移过程中纯相位弛豫的影响。在具体应用上,HEOM方程包含一组已知定义的辅助密度算符(Auxiliary Density Operators, ADOs),这些辅助算符将体系-热库的耦合强度及记忆时间尺度以非微扰、级联的方式处理。为了实现HEOM理论的数值计算,我们提出了一种索引机制,该机制可以在复杂的级联结构中方便地实现下标序列和ADOs之间的对映关系,从而可以大大加快运算过程中的寻址操作。另一方面,我们对每一个ADO都重新做了标度,这样所有的ADOs都统一到和约化体系密度矩阵相同的误差范围内。在此基础上,我们可以采用一种有效的过滤方法,该方法大大减少参与运算的ADOs的数目。在做完这些准备工作后,我们利用该严格方法具体计算和分析了在一个简单的三能级受激拉曼绝热转移过程中,存在纯相位弛豫的耗散动力学问题,并将得到的严格结果同几种微扰理论的结果进行比较和标定。
上述HEOM是以玻色-爱因斯坦函数的Matsubara展开(Matsubara Spectral Decomposition, MSD)为基础的,简称为MSD-HEOMo在第四章,我们给出了玻色-爱因斯坦函数的部分分式分解方法(Partial Fraction Decomposition, PFD),并以此为基础构造了相应的HEOM方程,简称PFD-HEOM。PFD分解的一个特点就是它分解后得到的极点为复极点,这个特点使我们能够更有效、更准确地展开玻色-爱因斯坦分布函数。为了考察PFD-HEOM的数值效率,我们计算了自旋-玻色体系的耗散动力学演化,通过和MSD-HEOM比较,发现PFD-HEOM的计算效率明显优于MSD-HEOM,计算时间可以缩短一个数量级甚至更多。
在第五章,我们发展了一种近似的级联量子主方程方法(Hierarchical Quan-tum Master Equation, HQME)。该方法对Drude热库模型的传统半经典处理进行改进,所得到的HQME方程可以看作是对传统的随机Liouville方程理论的修正。虽然从形式上看只是很简单的一项修正,但是改进后的方程不仅提高了准确性而且也极大地扩宽了适用范围；更加难能可贵的是,该修正并不会导致计算量的增加。这在随后的对两能级电荷转移体系的耗散动力学研究中得到了验证。同时我们还推导了该电荷转移体系的严格、解析的速率表达式,其中用到了我们在前面提到的Liouville空间的连分数格林函数方法。最后,我们给出了该近似HQME理论的应用判则,它可以用来预估该理论在不同系统中的具体表现。
第六章中,我们通过Wigner相空间高斯波包演化方法构建了驱动布朗振子体系的严格量子主方程。该方程充分考虑了驱动和耗散之间的相关效应,并将这种相关效应归结为有效场修正。通过研究,我们发现在驱动场频率较低和热库记忆时间尺度为中等大小时,驱动和耗散之间的这种协同效应对体系的影响明显。
在第七章,我们总结了本论文,并着重讨论了作者接下来的工作计划。
The thesis comprises two major themes of quantum dissipative dynamics. One is the development of quantum dissipation theory (QDT). We summarize the estab-lishment of the exact and nonperturbative hierarchical equations of motion (HEOM) of QDT, via the calculus on the influence functional path integral. Different forms of HEOM on the basis of different decomposition/expansion schemes are presented. Another is the application of exact QDT in various dissipative systems, including pop-ulation/electron transfer systems and driven Brownian oscillators. Due to the expensive numerical cost of exact QDT, some special numerical implementation algorithms are also detailed. The thesis is organized as follows.
In Chapter 1, we introduce the theoretical background of QDT, including the re-duced system description, the correlation and response functions versus linear response theory, with emphasis on key concepts and fluctuation-dissipation theorem.
In Chapter 2, we revisit the influence functional path integral formulation and construct the HEOM. It constitutes a systematic, nonperturbative approach to quantum dissipative dynamics with non-Markovian dissipation at an arbitrary finite temperature in the presence of time-dependent field driving. The well-known continued fraction Green's function formalism is also proposed for time-independent reduced Hamiltonian systems.
In Chapter 3, we apply the HEOM to study the dephasing effect on the stimulated Raman adiabatic passage (STIRAP). The HEOM couples the primary reduce density operator with a set of well-defined auxiliary density operators (ADOs), which resolve not just system-bath coupling strength but also memory. For the numerical implemen-tation of HEOM, we propose a convenient index scheme that allows an easy tracking of the coupled ADOs in the hierarchical equations. On the other hand, we scale ADOs individually to achieve a uniform error tolerance, as set by the reduced density oper-ator. An efficient filtering algorithm is then adopted, by which the effective number of ADOs is greatly reduced. Using HEOM, numerically exact studies are carried out on the dephasing effect on STIRAP. We also make assessments on several perturbative theories for their applicabilities in the present system of study.
The above HEOM is constructed on the basis of the Matsubara spectral decompo-sition (MSD) of Bose-Einstein function. In Chapter 4, we implement the partial frac- tion decomposition (PFD) scheme, and derive the corresponding HEOM. One feature of PFD scheme is the complex poles in the decomposition of Bose-Einstein function, which lead to not just the Bose function expansion more efficient and accurate, but also the HEOM construction more compact. The performance of the resulting PFD-HEOM is exemplified with spin-boson systems. We find it performs much better, about an order of magnitude faster, than the best available HEOM based on the MSD scheme.
In Chapter 5, we propose a hierarchical quantum master equation (HQME) ap-proach. The theoretical development is rooted in an improved semiclassical treatment of Drude bath, beyond the conventional high temperature or classical approximations. It leads to the new theory a simple but important improvement over the conventional stochastic Liouville equation theory, without extra numerical cost. Its broad range of validity and applicability is extensively demonstrated with two-level electron trans-fer model systems, where the new theory can be considered as the modified Zusman equation. For this system, we can derive an analytical rate expression on the basis of the aforementioned continued fraction Liouville-space Green's function formalism, together with the Dyson equation technique. Finally, we also propose a criterion to estimate the performance of HQME.
In Chapter 6, we construct an exact quantum master equation for a driven Brow-nian oscillator system via a Wigner phase-space Gaussian wave packet approach. It shows explicitly that the driving-dissipation correlation results in an effective field cor-rection that enhances the polarization. As the linear response and nonlinear dynamics are concerned, we demonstrate this cooperative effect is important in the low-frequency driving and intermediate bath memory region.
In Chapter 7, we conclude the thesis, and discuss some future work.
第一章介绍了量子耗散理论的一些背景知识,包括体系的约化描述,相关函数、响应函数及线性响应理论,重点讨论了其中涉及的一些关键概念以及涨落-耗散定理。
在第二章,我们首先回顾了影响泛函路径积分公式,然后通过求影响泛函路径积分时间导数的方法构建了级联耦合运动方程组(Hierarchical Equations of Motion, HEOM)。该方法非微扰地处理任意温度下的非马尔可夫量子耗散过程,并且适用于有含时外场驱动的情况。除此之外,我们还推导了与该级联方程组等价的格林函数的连分数方程。
在第三章,我们具体应用上述的HEOM方程研究了受激拉曼绝热转移过程中纯相位弛豫的影响。在具体应用上,HEOM方程包含一组已知定义的辅助密度算符(Auxiliary Density Operators, ADOs),这些辅助算符将体系-热库的耦合强度及记忆时间尺度以非微扰、级联的方式处理。为了实现HEOM理论的数值计算,我们提出了一种索引机制,该机制可以在复杂的级联结构中方便地实现下标序列和ADOs之间的对映关系,从而可以大大加快运算过程中的寻址操作。另一方面,我们对每一个ADO都重新做了标度,这样所有的ADOs都统一到和约化体系密度矩阵相同的误差范围内。在此基础上,我们可以采用一种有效的过滤方法,该方法大大减少参与运算的ADOs的数目。在做完这些准备工作后,我们利用该严格方法具体计算和分析了在一个简单的三能级受激拉曼绝热转移过程中,存在纯相位弛豫的耗散动力学问题,并将得到的严格结果同几种微扰理论的结果进行比较和标定。
上述HEOM是以玻色-爱因斯坦函数的Matsubara展开(Matsubara Spectral Decomposition, MSD)为基础的,简称为MSD-HEOMo在第四章,我们给出了玻色-爱因斯坦函数的部分分式分解方法(Partial Fraction Decomposition, PFD),并以此为基础构造了相应的HEOM方程,简称PFD-HEOM。PFD分解的一个特点就是它分解后得到的极点为复极点,这个特点使我们能够更有效、更准确地展开玻色-爱因斯坦分布函数。为了考察PFD-HEOM的数值效率,我们计算了自旋-玻色体系的耗散动力学演化,通过和MSD-HEOM比较,发现PFD-HEOM的计算效率明显优于MSD-HEOM,计算时间可以缩短一个数量级甚至更多。
在第五章,我们发展了一种近似的级联量子主方程方法(Hierarchical Quan-tum Master Equation, HQME)。该方法对Drude热库模型的传统半经典处理进行改进,所得到的HQME方程可以看作是对传统的随机Liouville方程理论的修正。虽然从形式上看只是很简单的一项修正,但是改进后的方程不仅提高了准确性而且也极大地扩宽了适用范围；更加难能可贵的是,该修正并不会导致计算量的增加。这在随后的对两能级电荷转移体系的耗散动力学研究中得到了验证。同时我们还推导了该电荷转移体系的严格、解析的速率表达式,其中用到了我们在前面提到的Liouville空间的连分数格林函数方法。最后,我们给出了该近似HQME理论的应用判则,它可以用来预估该理论在不同系统中的具体表现。
第六章中,我们通过Wigner相空间高斯波包演化方法构建了驱动布朗振子体系的严格量子主方程。该方程充分考虑了驱动和耗散之间的相关效应,并将这种相关效应归结为有效场修正。通过研究,我们发现在驱动场频率较低和热库记忆时间尺度为中等大小时,驱动和耗散之间的这种协同效应对体系的影响明显。
在第七章,我们总结了本论文,并着重讨论了作者接下来的工作计划。
The thesis comprises two major themes of quantum dissipative dynamics. One is the development of quantum dissipation theory (QDT). We summarize the estab-lishment of the exact and nonperturbative hierarchical equations of motion (HEOM) of QDT, via the calculus on the influence functional path integral. Different forms of HEOM on the basis of different decomposition/expansion schemes are presented. Another is the application of exact QDT in various dissipative systems, including pop-ulation/electron transfer systems and driven Brownian oscillators. Due to the expensive numerical cost of exact QDT, some special numerical implementation algorithms are also detailed. The thesis is organized as follows.
In Chapter 1, we introduce the theoretical background of QDT, including the re-duced system description, the correlation and response functions versus linear response theory, with emphasis on key concepts and fluctuation-dissipation theorem.
In Chapter 2, we revisit the influence functional path integral formulation and construct the HEOM. It constitutes a systematic, nonperturbative approach to quantum dissipative dynamics with non-Markovian dissipation at an arbitrary finite temperature in the presence of time-dependent field driving. The well-known continued fraction Green's function formalism is also proposed for time-independent reduced Hamiltonian systems.
In Chapter 3, we apply the HEOM to study the dephasing effect on the stimulated Raman adiabatic passage (STIRAP). The HEOM couples the primary reduce density operator with a set of well-defined auxiliary density operators (ADOs), which resolve not just system-bath coupling strength but also memory. For the numerical implemen-tation of HEOM, we propose a convenient index scheme that allows an easy tracking of the coupled ADOs in the hierarchical equations. On the other hand, we scale ADOs individually to achieve a uniform error tolerance, as set by the reduced density oper-ator. An efficient filtering algorithm is then adopted, by which the effective number of ADOs is greatly reduced. Using HEOM, numerically exact studies are carried out on the dephasing effect on STIRAP. We also make assessments on several perturbative theories for their applicabilities in the present system of study.
The above HEOM is constructed on the basis of the Matsubara spectral decompo-sition (MSD) of Bose-Einstein function. In Chapter 4, we implement the partial frac- tion decomposition (PFD) scheme, and derive the corresponding HEOM. One feature of PFD scheme is the complex poles in the decomposition of Bose-Einstein function, which lead to not just the Bose function expansion more efficient and accurate, but also the HEOM construction more compact. The performance of the resulting PFD-HEOM is exemplified with spin-boson systems. We find it performs much better, about an order of magnitude faster, than the best available HEOM based on the MSD scheme.
In Chapter 5, we propose a hierarchical quantum master equation (HQME) ap-proach. The theoretical development is rooted in an improved semiclassical treatment of Drude bath, beyond the conventional high temperature or classical approximations. It leads to the new theory a simple but important improvement over the conventional stochastic Liouville equation theory, without extra numerical cost. Its broad range of validity and applicability is extensively demonstrated with two-level electron trans-fer model systems, where the new theory can be considered as the modified Zusman equation. For this system, we can derive an analytical rate expression on the basis of the aforementioned continued fraction Liouville-space Green's function formalism, together with the Dyson equation technique. Finally, we also propose a criterion to estimate the performance of HQME.
In Chapter 6, we construct an exact quantum master equation for a driven Brow-nian oscillator system via a Wigner phase-space Gaussian wave packet approach. It shows explicitly that the driving-dissipation correlation results in an effective field cor-rection that enhances the polarization. As the linear response and nonlinear dynamics are concerned, we demonstrate this cooperative effect is important in the low-frequency driving and intermediate bath memory region.
In Chapter 7, we conclude the thesis, and discuss some future work.
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