We give a fully analytical description of the dynamics of the quasi-probability distribution (Q-PD)
functions for any model describing a single two-level qubit interacting with a field. The Wigner function at time
(t= 0) and Q-function for a pure state as an initial field state are studied. But, in this paper, the temporal behavior
of the Q-PD functions for a coherent superposition state (SS) and a statistical mixture of coherent states (SM)
for the interaction between a single ion and laser field are investigated. It is found that, the temporal behavior
of the Q-PD functions is in very good agreement with its counterpart for the entanglement. If the curve of the
entanglement between the trapped ion (specially for the second red sideband) and laser field are known, we can
expect the shape of the Q-PD functions.
An analytical method to calculate the sub-entropies and entanglement for the mixed
state as an initial field is presented. Also, we investigate the effects of the atomic motion
and the field-mode structure on sub-entropies and phase properties of the coherent super
position state and a statistical mixture of coherent states as initial field states taking
into account different forms of the intensity-dependent coupling. The initial state, the
atomic motion and the field-mode structure play important roles in the time evolution
of the entropies, entanglement and phase properties.