Coupling Effect On The Berry Phase

Since its discovery the Berry phase has permeated through all branches of physics and found universal applications in various fields. By relaxing superfluous conditions such as periodicity and adiabatic evolution, the Berry phase was generalized to a much wider setting. For example, Aharonov and Anandan proved that if there is a periodic but nonadiabatic evolution of the time-dependent system, we can observe the geomet-ric phase, which is called the Aharonov-Anandan phase. Recently, physicist found that the Berry phase is closely related to topological insulator. And one of the promising applications of the Berry phase is applied to implement the logic gates in quantum computation, which has been proved to be robust against faults originating from the experimental parameter fluctuations.In the first chapter of the thesis, we introduce the discovery and development of the Berry phase. In the second chapter of the thesis, the adiabatic theory about the Berry phase is retrospected. By taking two-level system as an example, we calculate the Berry phase and realize the significance of the adiabatic theory. In the third chapter of the thesis, we choose a two-level system to investigate the Aharonov-Anandan phase and the geometric phase in open system.In the fourth chapter of my thesis, the centre contribution in this paper, we re-search the Berry phase in a composite quantum system, which consists of a two-level system(TLS) driven by a slowly varying classical field and interacting with a single quantized mode. In contrast to previous investigations, in this work, we focus on ex-ploring the coupling effect on the Berry phase. Based on the calculation and theoretical analysis, we find the coupling effece is closely akin to two relevant energy scales:one is the coupling constant, and the other is the level spacing between neighbouring instanta-neous eigenstates. As long as the coupling constant is small compared to the neighbour-ing levels spacing, it is found that the Berry phase change is proportional to the square of the coupling constant, but it is quite small. Interestingly, a qualitatively different be-haviour is observed if the level spacing is comparable with the coupling constant. In this situation, the coupling effect produces a significant change of the Berry phase which no longer meet the relationship to the square of the coupling constant, even with a tiny modulation of the parameters, such as the coupling constant, the quantized mode fre-quency, and the transition frequency. To understand the physical picture behind these simulated results, the Berry phase change is explored by presenting analytical expres-sions. These perturbation theory calculations recover the numerical simulations quite well. Here we provide an alternative approach to control the Berry phase change, which is contrasty to the reported method, i.e. by adjusting a classical field strength in exper-iments. As a result, we provide the theory foreshadowing for the practical application of the Berry phase.Overall, we introduce the geometric phase and explore the coupling effect on the Berry phase of a two-level system adiabatically driven by a rotating classical field and interacting with a single quantized mode. The interesting finding provides an alternative approach to tune the Berry phase around the resonant frequency, and holds the promising applications (i.e.quantum logic gate) in quantum computation.