Theoretical Study On Quantum Response And Its Applications In Quantum Open System

Quantum mechanics was constructed at the beginning of 20th century to explore the law of motion for microparticles. It enables people to know and understand the behavior and law of microworld. Simultaneously it propels the development of quantum optics, condensed matter theory, quantum information, quantum chemistry and so on. Recently with the rapid development of quantum information technology, the study of the fundamental quantum the-ory and the realization in laboratory were highly valued. When one deals with real problems, the influence of the environment on the evolution of the system is inevitable, which leads to the decoherence of the quantum system. In this thesis, we mainly focus on the issue of open quantum systems. We concretely study quantum response theory and its application in open quantum systems. We give the general formula of susceptibility for nonlinear response. Then we extend it to non-Markovian open quantum systems. We reveal the crossover of suscep-tibility between non-Markovian and Markovian regime. This has great value of reference to quantum optics and condensed matter physics. There are seven chapters in this thesis, which follows:In Chapter 1, the basic concepts of quantum mechanics and research status related to this work are presented. Then we show the research methods and research progresses for exploring open quantum systems, and clarify the importance and research basis of response theory for open quantum systems.In Chapter 2, the projective operator methods and the second-order perturbative meth-ods to derive the master equation is presented. Quantum Langevin equation and non-Markovian input-output relation are derived in Heisenberg and Schrodinger pictures, re-spectively. Then we introduce the mean-field approximation and BBGKY hierarchy approx-imation which are frequently used in the research of open quantum systems. Finally, Kubo formula for linear response and the derivation of the Hall conductance are exhibited.In Chapter 3, we develop nonlinear response theory for open quantum system. The nonlinear susceptibility is defined, and any order of perturbation is reached. Then we apply it to the derivation of Hall conductance of open quantum system. The result reveals that the transition points of topological phase are robust against the environment.In Chapter 4, based on the perturbative methods in open quantum systems, we develop the linear response theory for open quantum systems. Then the Hall conductance for two-band system coupling to the environment is derived. With the decoherence induced by environment, Hall conductance for open quantum system can be written as two parts, one is contribution derived from closed system, the other is the correction induced by environment. It is pointed that with respect to the Hall conductance, there is no topological invariant in open quantum system.In Chapter 5, we study the response of two-band model to quantized field. The Hall conductance of two-band model subjected to quantized filed is defined. We present our theory by two-band model with the spin-orbit coupling. The result shows that quantum fluctuation suppresses Hall conductance driven by the classical electrical field. Quantum field (fluctuations of the classical field) has small effect on the Hall conductance on average.In Chapter 6, considering non-Markovian effect, we derive the response formula for open quantum system under the time-dependence external field. It is pointed that by manipulating the spectrum density of the environment, one can control crossover of the susceptibility of system from non-Markovian to Markovian regime. To apply such non-Markovian response theory, we derive Hall conductance of two-band model coupling to the environment.Finally, we present the conclusion and outlook based on this work.