  Conventionally the mobility edge (ME) separating extended states from
localized ones is a central concept in understanding Anderson localization
transition. The critical state, being delocalized and non-ergodic, is a third
type of fundamental state that is different from both the extended and
localized states. Here we study the localization phenomena in a one dimensional
periodically kicked quasiperiodic optical Raman lattice by using fractal
dimensions. We show a rich phase diagram including the pure extended, critical
and localized phases in the high frequency regime, the MEs separating the
critical regions from the extended (localized) regions, and the coexisting
phase of extended, critical and localized regions with increasing the kicked
period. We also find the fragility of phase boundaries, which are more
susceptible to the dynamical kick, and the phenomenon of the reentrant
localization transition. Finally, we demonstrate how the studied model can be
realized based on current cold atom experiments and how to detect the rich
physics by the expansion dynamics. Our results provide insight into studying
and detecting the novel critical phases, MEs, coexisting quantum phases, and
some other physics phenomena in the periodically kicked systems.
