Quantum Dissipative Dynamics: Hierarchical Quantum Master Equation Approach

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.