Geometric Phase And Related Problems In Quantum Physics

This dissertation studies on the basic problems of quantum theory, which consistsof geometric phase, quasi-exactly solvable problem and optic demonstration. First, s-ince the discovery of geometric phase, it has significant influence over both theoreticaland applied realm. On one hand, its extensive study of theoretical facets includes adia-batic and cyclic evolutions, nonadiabatic and noncyclic generalization, mixed states,off-diagonal ones, open systems and its connections with differential geometry andgauge theory. On the other hand, its applications have permeated through all branchesof physics, which range among molecular physics, condense matter as well as quan-tum information and computation. Second, exactly and quasi-exactly solvable modelsare very important in physics not only from a theoretical point of view but also fromthe experimentalist’ perspective, since calculational results and experimental outcomescan be compared unambiguously. Third, theoretical results of special relativity, such aslength contraction and time dilation, are far beyond our daily experience and hard to beobserved in experiments with low speed. So demonstration of its outcomes by virtue ofoptics experiments is very meaningful.This paper contains four parts. In the first part (chapter1), the necessary conceptsabout geometric phase and quasi-exactly solvable problem are retrospected. And somesimple and concrete models are demonstrated in order to get a further understanding ofthecorrespondingconcepts. ThesecondpartiscomposedofChapters24, whichcon-centrates on geometric phase for bipartite and tripartite system together with nonlinearsystem. The third part focuses on two-dimensional hydrogen with a linear potential in amagnetic field, which is a typical quasi-exactly solvable model. The fourth part dwellson demonstration of additional law of relativistic velocities by use of squeezed light.In chapter2, we research on geometric phases for the system of several interact-ing spins, which have been researched intensively. However, the studies are main-ly focused on the adiabatic case (Berry phase), so it is necessary for us to study thenon-adiabatic counterpart (Aharonov and Anandan phase). we analyze both the non- degenerateanddegenerateAharonovandAnandanphasesfortripartiteLipkin-Meskov-Glick type model, which has many application in Bose-Einstein condensates and entan-glement theory. Furthermore, in order to calculate degenerate geometric phases, theFloquet theorem and decomposition of operator are generalized.In chapter3, we focus on the geometric phase for nonlinear system. The geomet-ric phases for standard coherent states which are widely used in quantum optics haveattracted a large amount of attention. Nevertheless, few physicists consider about thecounterparts of nonlinear coherent states, which are useful in the description of the mo-tion of a trapped ion. In this paper, the nonunitary and noncyclic geometric phases fortwo nonlinear coherent and one squeezed states are formulated respectively. Moreover,some of their common properties are discussed respectively, such as gauge invariance,nonlocality and non-linear effects. The nonlinear functions have dramatic impacts onthe evolution of the corresponding geometric phases. They speed the evolution up ordown. So this property may have application in controlling or measuring geometricphase. For the squeezed case, when the squeezed parameter râ†'∞, the limiting valueof the geometric phase is also determined by nonlinear function at a given time and cir-cular frequency. In addition, the geometric phases for standard coherent and squeezedstates are obtained under a particular condition. When the time evolution undergoesa period, their corresponding cyclic geometric phases are achieved as well. And thedistinction between the geometric phases of the two coherent states maybe regarded asa geometric criterion.In chapter4, we explore geometric phases of entangled coherent states and someof their properties. In chapter1, a better and elegant expression of geometric phase forcoherent state is derived. It is used to obtain the explicit form of the geometric phase forentangled coherent states, several interesting results followed by considering differentcases for the parameters. The effects of entanglement and harmonic potential on thegeometric phase are discussed.The above chapters making up part2mainly concern on geometric phases. Part3concentrates on a quasi-exactly solvable problem, which is only illustrated in chapter5.In this chapter, the two-dimensional hydrogen with a linear potential in a magnetic fieldis solved by two different methods, namely series solutions and sl(2) algebra. Further- more the connection between the model and an anharmonic oscillator is investigated bymethods of KS transformation.Part4dwells on demonstration of additional law of relativistic velocities basedon squeezed light, of which main ideas are contained in chapter6. Special relativityis foundation of many branches of modern physics, of which theoretical results are farbeyondourdailyexperienceandhardtorealizedinkinematicexperiments. However,itsoutcomes could be demonstrated by making use of convenient substitute, i.e. squeezedlight in present paper. Squeezed light is very important in the field of quantum opticsand the corresponding transformation can be regarded as the coherent state of SU(1,1).In this paper, the connection between the squeezed operator and Lorentz boost is builtunder certain conditions. Furthermore, the additional law of relativistic velocities andthe angle of Wigner rotation are deduced as well.