Decoherence and localization in quantum two-level systems

We study and compare the decoherent histories approach, the environment-induced decoherence and the localization properties of the solutions to the stochastic Schrödinger equation in quantum-jump simulation and quantum-state diffusion approaches, for a quantum two-level system model. We show, in particular, that there is a close connection between the decoherent histories and the quantum-jump simulation, complementing a connection with the quantum-state diffusion approach noted earlier by Diósi, Gisin, Halliwell and Percival. In the case of the decoherent histories analysis, the degree of approximate decoherence is discussed in detail. In addition, the various time scales regarding the decoherence and localization are discussed. By using the von Neumann entropy, we also discuss the predictability and its relation to the upper bounds of degree of decoherence.