Two-time correlation functions of a two-level system influenced by a composite environment

To study the quantum regression theorem and the Leggett–Garg inequality, two-time correlation functions are calculated for a two-level system which is placed under the influence of a composite environment consisting of two subsystems. Two different configurations, I and II, are considered. In the configuration I, a two-level system of interest interacts with a thermal reservoir via another two-level system. In the configuration II, a relevant two-level system is influenced independently by another two-level system and a thermal reservoir. In both configurations, the thermal reservoir is assumed to have a sufficiently short correlation time. When an interacting nuclear-spin and electron-spin system is considered, the relevant system is a nuclear-spin (electron-spin) in the configuration I (II). It is shown that the quantum regression theorem for the relevant two-level system is always valid in the configuration II while it is not in the configuration I, regardless of whether the reduced time evolution is Markovian or not. Furthermore, it is found that the Leggett–Garg inequality can be violated in both configurations. The dependence of the violation on the parameters characterizing the open two-level system is investigated.