| In 1984, Berry found the existence of the Berry$^{,}$s quantum phase ( i.e., the adiabatic geometric phase) of wave function in the adiabatic quantum process. Berry$^{,}$s discovery opens up new opportunities for investigating the global properties of quantum evolution. It is now well known that the geometric phase appears in systems with the time-dependent Hamiltonian, or in systems whose Hamiltonian possesses the evolution parameters. Differing from the dynamical phase, the geometrc phase is immediately independent of dynamical quantities of systems such as the energy, frequency, velocity as well as coupling coefficiencies, and is only related to the geometric nature of the pathway along which the quantum systems evolves. This, therefore, implies that the geometric phase presents the topological or global properties of quantum systems in time-evolution process, and that it possesses the physical significance and can thus be applied to various fields of physics. Geometric phase attracts attentions of many physicists in many fields such as gravity theory, differential geometry, atomic and molecular physics, nuclear physics, quantum optics, condensed matter physics and molecular-reaction chemistry as well. |