Geometric phase for a nonstatic coherent light wave: nonlinear evolution harmonized with the dynamical phase

Properties of the geometric phase for a nonstatic coherent light wave arisen in a static environment are analyzed from various angles. The geometric phase varies in a regular nonlinear way, where the center of its variation increases monotonically with time. This extraordinary outcome is due to the effects of the periodic wave collapse and expansion on the evolution of the geometric phase. Harmonization of such a geometric-phase evolution with the dynamical phase makes the total phase evolve in a unique pattern that depends on the degree of nonstaticity. The total phase exhibits a peculiar behavior for the case of extreme nonstaticity, which is that it precipitates periodically in its evolution. While this consequence is owing to a strong response of the geometric phase to the wave nonstaticity, its evolved scale in the coherent state is mostly larger compared to that in the Fock states. For a simple case where the wave nonstaticity disappears, our description of the geometric phase recovers to the well-known conventional one which no longer undergoes periodical change. Whereas the familiar dynamical phase is just related to the expectation value of the Hamiltonian, the geometric phase that we have managed reflects a delicate nonstaticity difference in the evolution of quantum states.