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Time-dependent pointer states of the quantized atom-field model in a nonresonance regime and consequences regarding the decoherence of the central system

We consider the quantized atom-field model and for the regime that $\hat{H}_{\cal E}\ll\hat{H}_{\cal S}\ll\hat{H'}$ (but $\hat{H}_{\cal E}\neq0$ and $\hat{H}_{\cal S}\neq0$); where $\hat{H}_{\cal E}$, $\hat{H}_{\cal S}$ and $\hat{H'}$ respectively represent the self Hamiltonians of the environment and the system, and the interaction between the system and the environment. Considering a single-mode quantized field we obtain the time-evolution operator for the model. Using our time-evolution operator we calculate the time-dependent pointer states of the system and the environment (which are characterized by their ability not to entangle with states of another subsystem) by assuming an initial state of the environment in the form of a Gaussian package in position space. We obtain a closed form for the offdiagonal element of the reduced density matrix of the system and study the decoherence of the central system in our model. We will show that for the case that the system initially is not prepared in one of its pointer states, the offdiagonal element of the reduced density matrix of the system will decay with a decoherence time which is inversely proportional to the square root of the mass of the bosonic field particles.