The Study Of A-A Phase In Photon And Spinnings Coupled System

The concept of geometric phase was first introduceded by Pancharatnam in his study of the interference light in distinct states of polarization in 1956. In 1984 its concept was extended to its quantal couterpart by Berry, who showed that the state of a quantum system acquires a purely geometric feature in addition to the usual dynamical phase when it is varied slowly and eventually brought back to its initial form.Berry's phase has been extensively studied and generalized in various directions. In 1987, Aharonov and Anandan have removen to"adiabatic process"exterior parameter, the quantal system acquires the phase of nonadiabatic cyclic evolution namely A-A phase. Later, J. Samuel and R. Bhandari base on the anterior work, extended the geometric phase to noncyclic and nanunitary evolutions. Geometric phase is an important conception in quantum mechanics. The thorough research is beneficial to the people to the quantum mechanics deeper level understanding.In this paper, based on the study of Berry's phase of the composite system with intersubsystem coupling, where one of the subsystems is driven by a quantized single mode of field by X.X.Yi[Phys.Rev.A 70,052106(2004)].We mainly study the nonadiabatic cyclic phase(A-A phase) of the mode.The Hamilton of the system is: H₀^2q=(ω/2)(σ₁^z+σ₂^z)+va⁺a+λ(σ₁⁺a+σ₁^-a⁺)+Jσ₁^zσ₂^zin the full quantized context, we introduce the phase shift operator U (φ): U(φ) = exp(-iφa⁺a),the Hamilton is given that after the phase shift operator is applied: H(φ)=U(φ)HU⁺(φ)=(ω/2)(σ₁^z+σ₂^z)+va⁺a+λ(e^iφσ₁⁺a+e^-iφσ₁^-a⁺)+Jσ₁^zσ₂^zWe calculate the total phases and dynamical phases of above Hamilton, acquires theirs A-A phases. When the system is in some specific eigen condition, nonadiabatic cyclic phases return to Berry phase.The results will be helpful to actual controls.