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We find an exact analytical solution of the reduced density matrix from theFeynman-Vernon influence functional theory for a wave packet influenced by anenvironment containing a few discrete modes. We obtain two intrinsic energyscales relating to the time scales of the system and the environment. Differentrelationship between these two scales alters the overall form of the solutionof the system. We also introduce a decoherence measure for a single wave packetwhich is defined as the ratio of Schr\"odinger uncertainty over thedelocalization extension of the wave packet and characterizes thetime-evolution behavior of the off-diagonal reduced density matrix element. Weutilize the exact solution and the docherence measure to study the wave packetdecoherence dynamics. We further demonstrate how the dynamical diffusion of thewave packet leads to non-Markovian decoherence in such a microscopicenvironment.

Exact wave-packet decoherence dynamics in a discrete spectrum environment