Research Of Berry Phases For The Strongly Coupled Systems In Semi-classical Approximation

Consider that in parameter space,a quantum system going through adiabatic evolution and in its eigenstate.When its wave function has gone through one period,a dynamic phase will be obtained,meanwhile,a geometric phase that reflects the topological properties of the states.The geometric phase is called the Berry phase.Berry phase has been experimentally proved,and is widely used in systems such as atomic physics,polarized light,etc.The Berry phase of the atom-cavity field interaction is discussed in details in this paper.In the range of strong coupling,we involve the full interaction between atoms and the cavity fields to describe the dynamics of the combined system.And the Rotating Wave Approximation has to be taken out.For the reason of the more difficulties in the analysis of the dynamics,we adopt the method of semi-classical description.That is,the cavity field is regarded to be a classical one.The Berry phase of these interacting systems are given as the adiabatic rotation of the field is considered.We analyze the energy surface of the single atom's interaction with a one-mode cavity in the two cases of the Rotating Wave Approximation and Non-Rotating Wave Approximation,as well as the properties of the Berry phases.In the condition of RWA,the orbit of the energy in the phase space is closed and there will be a conical intersection,which brings forward a non-zero Berry phase.On the contrary,the orbit of the energy in the case of Non-RWA can not be closed,which can not lead to a non-zero Berry phase.For the two atoms' interaction with a single mode field,it can be easily found that the maximal entangled states of the two-qubit system have zero Berry phase,and the product states have the non-zero Berry phase.Of course there are the zero Berry phases in the condition of the Non-RWA.We hope these researches can be expanded to the associated study of the quantum information,e.g.the construction of the phase gates.