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We consider a spin-boson Hamiltonian which is generalized such that theHamiltonians for the system ($\hat{H}_{\cal S}$) and the interaction with theenvironment ($\hat{H}_{\rm int}$) do not commute with each other. Considering asingle-mode quantized field in exact resonance with the tunneling matrixelement of the system, we obtain the time-evolution operator for our model.Using our time-evolution operator we calculate the time-dependent pointerstates of the system and the environment (which are characterized by theirability not to entangle with each other) for the case that the environmentinitially is prepared in the coherent state. We show that our solution for thepointer states of the system and the environment is valid over a length of timewhich is proportional to $\bar{n}$, the average number of bosons in theenvironment. We also obtain a closed form for the offdiagonal element of thereduced density matrix of the system and study the decoherence of the centralsystem in our model. We show that for the case that the system initially isprepared in one of its pointer states, the offdiagonal element of the reduceddensity matrix of the system will be a \emph{sinusoidal function} with a slowdecaying envelope which is characterized by a decay time proportional to$\bar{n}$; while it will experience a much faster decoherence, when the systeminitially is \emph{not} prepared in one of its initial pointer states.

Time-dependent pointer states of the generalized spin-boson model and consequences regarding the decoherence of the central system