Decoherence and recoherence of beam in phase space

Using the particle distribution in phase space, we introduce a new approach to study the decoherence and recoherence of a kicked beam when betatron tunes are far from any resonance. With this method, the decoherence and the recoherence of a beam can be analyzed easily in both 2-dimensional and 4-dimensional cases with any form of the tune spread. We have also studied the decoherence of a beam initially trapped in a resonance island. Due to the tune modulation, the separatrix of island develops a chaotic layer and particles within the island can drift out of the island along the layer. Consequently, the beam decoheres gradually. The rate of decoherence is shown to be proportional to the modulation amplitude. As a function of the modulation frequency, it reaches a peak near the tune of trapped particles before decreasing to zero