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In a quantum measurement setting, it is known that environment-induceddecoherence theory describes the emergence of effectively classical features ofthe quantum system-measuring apparatus composite system when the apparatus isallowed to interact with the environment. In [E.A. Galapon {\it EPL} {\bf 113}60007 (2016)], a measurement model is found to have the feature of inducingexact decoherence at a finite time via one internal degree of freedom of theapparatus provided that the apparatus is decomposed into a pointer and aninaccessible probe, with the pointer and the probe being in momentum-limitedinitial states. However, an issue can be raised against the model: while thefactorization method of the time evolution operator used there is formallycorrect, it is not completely rigorous due to some unstated conditions on thevalidity of the factorization in the Hilbert space of the model. Furthermore,no examples were presented there in implementing the measurement scheme inspecific quantum systems. The goal of this paper is to re-examine the model andconfirm its features independently by solving the von Neumann equation for thejoint state of the composite system as a function of time. This approachreproduces the joint state obtained in the original work, leading to the sameconditions for exact decoherence and orthogonal pointer states when therequired initial conditions on the probe and pointer are imposed. We illustratethe exact decoherence process in the measurement of observables of a spin-1/2particle and a quantum harmonic oscillator by using the model.

Exact Decoherence and Orthogonal Pointer States Brought by One Degree of Freedom: von Neumann Equation Approach and Examples