Theoretical Study On The Dynamics And The Geometric Phases In Open Quantum Systems

With the development of quantum information (QI) theory, physical resources such as quantum entanglement and geometric phase have been studied extensively due to their wide applications in QI. The inevitable interaction between the systems in which these resources reside and the environment leads the systems to be regarded as open quantum systems. The environment may affect the quality and efficiency of the QI processing. This dissertation applies the master equation method to different systems, extends and develops the effective Hamiltonian approach and the Monte Carlo wave function method, which are usually used to solve the master equation. Moreover, the geometric phase and quantum entanglement in some special quantum systems are studied.This thesis consists of three parts. The first part is composed of Chaps.2 and 3. The main aim of these two chapters is to introduce the fundamental conceptions and meth-ods which are relative to our work. In Chap.2, fundamental conceptions such as quan-tum entanglement, quantum adiabatic approximation and geometric phase are introduced. Chap.3 is focused on the master equation (ME) method, including the Markovian ME in the Lindblad form, which is derived under the Born-Markovian approximation, the generalized Lindblad ME, which is based on the correlation projection superoperator techniques, the time-dependent Markovian ME and the post-Markovian ME.Part two is composed of Chaps.4 and 5, in which the effective Hamiltonian approach (EHA) and Monte Carlo wave function method (MCWFM) are developed and applied to solve different problems. Chap.4 begins with a detailed introduction about the EHA. This method can map the ME to a Schrodinger-like equation by introducing an ancilla, so the ME can be solved by virtue of the methods which are usually used to solve the Schrodinger equation. Based on EHA, two approximations methods for closed systems, the adiabatic approximation and the Born-Oppenheimer approximation, are firstly extended to the open quantum systems. Then the validity and invalidity conditions for different types of dissipations are established. Two examples are studied. The results show that different types of dissipations affect the validity condition differently, and our method can describe the dynamics in the regimes where the validity condition is satisfied. Next the adiabatical geometric phase and decoherence-free subspace in the form of EHA are redefined. In the aspect of solving the steady state of the ME, the nonequilibrium thermal entanglement in a three-qubit XX model is investigated. Both symmetric and nonsymmetric qubit-qubit coupling are considered. The results show that for nonsymmetric qubit-qubit couplings, the environment at different temperatures would benefit the entanglement. Finally, the EHA is extended to the non-Markovian case. A non-Markovian ME of a two-level system interacted with a two-band environment is considered as an example. The MCWFM or the quantum trajectory approach is a powerful tool to study dissipative dynamics governed by the Markovian ME, in particular for high-dimensional system. In Chap.5, the basic idea of this method is introduced, and then this method is extended to the non-Markovian case with the generalized Lindblad ME. Two examples for the illustration the method are presented and discussed. The results show that the method can correctly reproduce the non-Markovian dissipative dynamics for the system. In the end, the computational efficiency of this method is discussed.Chap.6 by itself consists of the third part of this thesis, in which the geometric phase of different systems is studied. In the first section, a two-level system interacted with non-Markovian environment is considered, and the non-Markovian effects on the geometric phase are studied. Three kinds of methods that describe the non-Markovian process, projection superoperator technique, memory kernel ME and post-Markovian ME, are used in the dis-cussions. The results show that, when the dissipation rate is large, the non-Markovian effects change the geometric phase strikingly. In the second section, the Loschmidt echo (LE) and Berry phase in a nonlinear system transported around a double degeneracy is studied. A witness of nonlinearity for the nonlinear system is proposed, and then a connection between the LE and the witness of nonlinearity is established. Its dependence on the parameters of the system is studied in the standard Landau-Zener model.Finally, the conclusions and discussion are given.