Theoretical Study On The Geometric Phase Of Quantum And Classical Hybrid System

With the birth and development of quantum mechanics and quantum information theory, quantum theory has played an important role in the progress of modern society. The cor-respondence between quantum mechanics and classical mechanics and the quantum-classical hybrid system have been developed in depth, which has revealed the difference and connection between quantum mechanics and classical mechanics; the concepts of cavity-QED, geomet-ric phase and quantum entanglement have been widely studied and successfully realized in experiment. In this thesis, the geometric phases of quantum and quantum-classical hybrid system have been studied theoretically, which deepens the understanding of the structure and geometric properties of quantum and classical mechanics. By using the mean field theory, the properties and applications of the nonlinear cavity QED system have been studied. This thesis consists of six chapters and the main contents are given in Chapter 3-6.In Chapters 1 and 2, a briefly introduction of the background and importance of this work is made and the research history of quantum mechanics, quantum information, geomet-ric phase, cavity-QED is reviewed. The concepts and principles, such as quantum geometric phase, classical geometric angle, quantum-classical correspondence and entanglement are p-resented in detail.In Chapter 3, the Berry phase and Hannay's angle in coupled quantum-classical hybrid systems are studied. To calculate uniformly the Berry phase and Hannay's angle, a one-form connection is introduced, through which the Berry phase and Hannay's angle of the hybrid system can be derived synchronously. The results show that the Berry phase has been changed sharply by the couplings between the quantum and classical subsystems, whereas the couplings have a small effect on the Hannay's angle. Furthermore, the Berry phase for a system of two coupled oscillators is calculated and the result is compared with that of the quantum-classical hybrid system. This work provides us an effective method for quantum-classical hybrid problem.In Chapter 4, the geometric phase in a scattering process is studied. The geometric phase in such a process is defined and calculated and dependences of the geometric phase on the spin-spin coupling constant and the barrier strengths are given. Possible observation of the geometric phase is suggested and discussed. This research provide a procedure to calculating the geometric phase in trace non-preserving evolutions.In Chapter 5, the nonlinear effect and the entanglement between two atoms in two coupled cavities are studied theoretically. The dynamics of the two coupled cavity fields with high-intensity fields inside is also studied numerically, the effects of atom-field coupling on the self-trapping as well as on the entanglement are also analyzed and discussed. In vacuum and high-intensity fields we calculate the concurrence of the two atoms in both theoretical and realistic situation and discuss the nonlinear effect on the atomic entanglement. The result shows that the nonlinear interaction can play a dominant role in entangling two atoms. It provides a new way for quantum information processing.In Chapter 6, with nonlinear effects taken into account, the transmission properties of photons in a one-dimensional coupled cavities are studied theoretically, with the cavity at the center of the cavity array being coupled to a two-level system. By using the traditional scattering theory and the mean-field approximation, we calculate the transmission rate of photons along the cavities and discuss the effect of nonlinearity and the cavity-atom coupling on the photon transport. The results show that the cavity-atom couplings affect the coherent transport of photons. The dynamics of such a system is also studied by numerical simulations and the effect of the atom-field detuning and nonlinearity on the dynamics is shown and discussed.Finally, the conclusions and discussion are given.