Quantum Dissipative Dynamics: Theory And Application

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.