Methods Of Quantum Dissipative Dynamics For Low Temperature Environments

The thesis focuses on the development and applications of quantum dissipation theory for low temperature environments.This is achieved by proposing a new method to decompose the Bose and Fermi functions:the Fano spectrum decomposition(FSD)scheme.The dissipative interactions of system-environment often have profound in-fluence on the thermodynamics and on the system dynamics.The underlying physics becomes particularly intriguing when the system-environment coupling is strong and the environmental modulation has a long-time memory.As one of the most popular methods for invetigating the quantum dissipative dynamics,the hierarchical equations of motion(HEOM)method has a well-known limitation:its applicability to systems at low temperatures is largely restrained by the massive numerical computation cost,because too many exponential functions are required to accurately characterize the non-Markovian memory of the reservoir environment.The new FSD method decomposes the Fermi or Bose function into a high-temperature reference and a low-temperature correction.While the former can be efficiently decomposed via the standard Pade spec-trum decomposition(PSD),the latter can be accurately described by several modified Fano functions.The results of FSD method show that it converges overwhelmingly faster than the standard PSD method.Notably,the low-temperature correction supports further a recursive and scalable extension to access the near-zero temperature regime.Then the FSD-based HEOM formalism is established for both bosonic and fermionic environments.The accuracy and efficiency of the FSD-based HEOM are exemplified by calculating low-temperature dissipative dynamics of a spin-boson model and the dy-namic and static properties of a single-orbital Anderson impurity model in the Kondo regime.The thesis is organized as follows:Chapter 1 reviews the background of this thesis.Firstly,I will briefly introduce the concept of open systems.Then,I will introduce the history of the development of the various quantum dissipation theories for studying open quantum systems,especially the HEOM method and the stochastic equation of motion(SEOM)method,I will also dis-cuss some representative and simple model open quantum systems.At last,an important physical phenomenon in low temperature regime-the Kondo effect is discussed.In Chapter 2,I will introduce the HEOM formalism for the open quantum systems.Firstly,the mathematical construction of HEOM is presented.If the reservoir environ-ment satisfies Gaussian statistics,the HEOM formalism is in principle formally rigor-ous.Moreover,the derivation of the HEOM formalism and the various spectrum de-composition schemes are presented.This is followed by the discussion of the techiques to reduce computational cost and improve the numerical efficiency.In Chapter 3,I will present a new method to decompose the distribution functions at low-temperature regime-the FSD method.The FSD method yields an efficient and accurate sum-over-poles expansion for the Fermi and Bose functions,which over-comes the problem of slow convergence in conventional methods.The Bose and Fermi distribution functions are decomposed into two parts in the frequency domain:A high-temperature reference which is treated efficiently by the standard PSD scheme,and a low-temperature correction which is comprised of a number of modified Fano functions.Then I present several sets of parameters for FSD.One can directly use the parameters without actually redoing the numerical fitting.The low-temperature correction supports further a recursive and scalable extension to access the near-zero temperature regime.At low temperatures,the superiority of FSD over the PSD scheme is quite impressive,and is demonstrated by numerical calculations.Moreover,profound physical consid-erations and extensive practicality of FSD are discussed.At last,I present a modified model of spectral density for femionic reservoir to simulate the energy gap of supercon-ductors,which paves the way to broaden the applicability of FSD.In Chapter 4,I establish the FSD-based HEOM formalisms for both bosonic and fermionic environments.With the FSD scheme,the reservoir correlation functions are accurately expanded by a small number of polynomial-exponential functions.The size of the memory basis set is fully reduced,especially in the ultra-low-temperature regime.The FSD scheme indeed allows the HEOM method to access unprecedentedly low temperatures and yield highly accurate results.In particular,the FSD-based HEOM method promises a high accuracy for the long-time dissipative dynamics as well as the stationary-state properties.I have also encountered the asymptotic instability problem,in which case,the instability is supposed to be alleviated by employing a more accurate FSD scheme.Chapter 5 concludes the thesis,and provides prospects for future development and applications of quantum dissipation methods.