Pertuebation Theory For A Nonlinear Two-level System

For a real quantum system, due to the complexity of the Hamilton, it is usually difficult to find exact solutions to the Schrodinger equation and master equation, even not solvable. Hence, using approximate method is necessary and the perturbation theory is an important tool for describing real quantum systems. However, the perturbation theory is proved to be available for linear quantum systems, to what extent or in which circumstance the perturbation theory can be applied for nonlinear systems is an open question to date.In this thesis, firstly, we introduce a nonlinear two-level system and a simple open two-level nonlinear quantum system and two kinds of the most commonly used perturbation theory-perturbation theory for a nondegenerate stationary states and perturbative expansion for the master equation, secondly, we use the two methods to calculate the dynamics of the two modes separately. The results show that,(1) A nonlinear two-level system, for R>>C and V>>C, a perturbative treatment of the nonlinearity C/2<ψ|σz|ψ>σz is adequate, and the smaller C is, the better. ForC>>R and V >> R, the ratio (r) of the nonlinear rate C to the tunneling coefficient V determines the validity of the perturbation theory. For small ratio r, the perturbation theory is available, otherwise it yields wrong results. Because of Eq. (3.13) has two real roots when C< V, while there are four real roots when C> V. For C >> V and R >> V, perturbation theory is not available due to the exist of the nonlinearity.(2) A nonlinear open two-level system, forC=0, perturbation theory is adequate. For small decoherenceγthe nonlinear constant C determines the validity of the perturbation theory. For small C, the perturbation theory is available, otherwise it yields wrong results. For R