Two-field Dynamics Generated By The Time-dependent Interaction With One Atom

One effective Hamiltonian is helpful for us to study the dynamical properties of one interaction system. The adiabatic elimination is very conventional to achieve the effective Hamiltonian. Comparatively, vivid form makes Frohlich transformation the priority to get the effective dynamics of one system and to be widely used in the study of quantum optics and atomic optics. The entanglement Entropy is one concept in quantum information to measure the correlation of two or more particles. It, of course, is the measurement of the ordering of one system. It is always to be used to analyze the maximal entangled states as one writes the explicit form of the entropy.The interaction between one atom with two-mode cavity field is discussed in this thesis. The coupling between the atom with the field is always dependent of time. That is because that the coupling is always connected with the coordinate, of course the coordinate connected with the time. As we are deriving the effective Hamiltonian the Frohlich transformation is used. It shows that the Hamiltonian can be divided into two parties according to the internal levels of the atom respectively. In another word, the Hamiltonian shows us the correlation between the two modes while the degree of freedom of atomic transition has been eliminated. As the specific parameters are appointed, the effectively dynamical evolution and the entanglement of two modes are given, as well as the entanglement entropy when the time is infinity. After that the exact evolution operator for the time-dependent effective system is given, therefore the coherent evolution of the two modes could be following.