The Geometric Phase In A Quantum Optical System

In moden physics the geometric phase has been extensively studied, and generalized in various directions. It is not only of interest from a fundamental point of view, for example, it turns out to be deeply rooted with topologicalstates of matter, and it may turn out to have important applications in quantum computing. In a strict sense, however, the Berry phase has been studied only in a semiclassical context until now. This means that the geometric evolution of a quantum system is studied under the effect of a time varying classical field. However, this field itself has never been quantized. Thus, the effects of the vacuum field on the geometric evolution are unknown. Many effects in quantum optics such as quantum jumps, collapses, and revivals of the Rabi oscillations, can be explained only by considering a quantum field, showing the importance of field quantization in the complete description of physical systems. Moreover, in quantum mechanics several interesting effects are observed due to the interaction of quantum systems with the vacuum (spontaneous emission, lamb shift). This thesis is about the the gometric phase in a quantum optical system.Chapter one introduces the geometric phase in semiclassical context. In 1984, Berry showed that the eigenstate of a quantum system acquires a purely geometric feature (called the Berry phase) in addition to the usual dynamical phase when it is varied adiabatically and eventually brought back to its initial form. Then Aharonov and Anandan generalized the Berry phase to the AA phase which is acquired when an arbitrary state of a quantum system does a cyclic evolution. We calculate the geometric phase in a two-level system, and find that the gometric phase of this system is a half of the solid angle subtended by the cyclic loop in the Bloch or Poincare sphere traversed by the corresponding quantum state.Chapter two introduces the Berry phase in a quantum optical system and analyses its origin. In 2002,I. Fuentes-Guridi et al. modified the archetype of Berry phases, namely, a spin-1=2 particle in a classical magnetic field. The external classical field was replaced by a fully quantum one,then applied a unitary transformation U(φ)= exp(-iφaa), to the Hamiltonian of the system, when φ varies adiabatically from 0 to 2π the Berry phase is generated. In the beginning, they believe that this phase is determined by the vacuum photon fluctuation. However, this conclusion is due to the misunderstanding of the vacuum state and the Berry phase. In 2012, Jonas Larson questioned this conclusion. He argues that this Berry phase is just a result of the rotating wave approximation. However, his argument comes entirely from the improper semiclassical approximation. After studying their work, we find a proof of exixtence of the Berry phase in the Rabi model by contradiction and find that this phase is proportional to the average photon number. So it does not origin from the vacuum photon fluctuation or the rotating wave approximation. Then we introduce the variational method to calculate the Berry phase in the Rabi model, and generalize our result to a three-level atom in the quantized light field beyond the rotating wave approximation.Chapter three introduces the genuine vacuum-induced gometric phase. This phase is generated from a cyclic evolution whose initial and final state are both the vacuum state of the quantized light field. Specifically, in two-level system we find a general cyclic and adiabatic evolution. If this system is set initially in the vacuum state, the geometric phases acquired during the evolution can be called the vacuum-induced gometric phases. And we find these phases are related to the average photon number exited during the evolution. Photon exited from the vacuum field can only be explained with a quantized field. In this sense, the exixtence of the vacuum-induced gometric phases could be regard as a proof of the field quantization. Hopefully, the effects of the genuine vacuum-induced gometric phases proposed in my thesis can be tested experimentally in future.