The Geometric Phase Of The Open Quantum System And The Quantum Phase Transformation System

The basic spirit of the quantum mechanics is clear and concise, and has been tested frequently from Schrodinger times. On theoretical aspect, the fusion with other concepts leads to a huge knowledge system. Certainly, for most of the quantum device designs and of the quantum material researches, only nonrelativistic quantum mechanics should be considered. This paper was divided in two parts, the first part relating to open quantum systems included the first and the second chapters, while the second part relating to many-body quantum systems included the third and the forth chapters.The most valuable characteristics in the quantum world are coherence and entanglement, but which are very fragile as the decoherence caused by the coupling with the environment. In the quantum information or quantum chemistry domain, the master equations from a spin-boson model have been used to deal with these problems. In the spin-boson model, the spin-part is the system we concerned, the free-boson reservoir can be considered as the environment. The model can describe for example a simple two-well potential, a double quantum dots, superconducting qubit and different biomolecular systems. A formal solution of the model can be obtained by using Feynman-Vernon path integral method, however the realistic treatment should be based on a perturbative approach with the system-environment coupling, projection operator method. stochastic Schrodinger equation or quantum jump approach.In the first part, we generally describe the open quantum system and introduce some basic concepts. We give an example which can be exactly solved. A Markovian Lindblad master equation and a time-convolutionless non-Markovian master equation has been introduced. We also introduce the stochastic Schrodinger equation and the quantum-jump approach based on Monte Carlo Wave-Function. Our concrete work is in the chapter2. With the aid of a time-convolutionless non-Markovian master equation we analysis the decoherence of the Berry phase, and point out that the low frequency noise indeed has a bigger impact on the measure of the Berry phase than the high frequency noise, which is qualitatively consistent with the recent experiment. Then, we use a Bloch-Redfield master equation to investigate the decoherence of the AA phase gates, and compare them with equivalent dynamic phase gates. Quantum phase transition occurs at zero temperature, different phases can be distinguished by singularities of the energy or the wave-function of the ground state. Used order-parameters include Fidelity, Geometric phase, Entanglement, Riemann tensors, Topological order and Quantum order. Nowadays the cold atoms in optical lattices and the NMR quantum simulator can be used to model some common many-body systems, thus can verify some results in theoretical aspect. Preparation and manipulation for many-body system relates to quantum phase transition dynamics, two concert problems include:one is to calculate the excitation when the system crosses the phase transition point, another one is to analysis how a excitative state relaxes to a thermodynamic equilibrium state or a non-thermodynamic steady state.In second part of the paper, we firstly introduce the basic concepts of the quantum phase transition and the quantum phase transition dynamics in chapter3. The quantum phase transition properties of the XY spin-chain, the Kitaev model, the Hubbard model have also been introduced. In chapter4we discuss some questions about the Berry phase of and the suddenly quench of the XY spin chain. Our concert work is to calculate the survival probability of a XY spin chain after undergoing a rotation of ever spin. We found that there are three kinds of period times, and that when every spin undergoes a periodic rotation the survival probability actually reflect quantum phase transition. We also analysis the relation of the Loschmidt echo and the quantum phase transition when the Kitaev model has being suddenly quenched.