  Phase transitions in nonequilibrium dynamics of many body quantum systems,the
so-called dynamical phases transition (DPTs), play an important role for
understanding various dynamical phenomena observed in different branches of
physics.In general, there have two types of DPTs, the first one refers to the
phase transition that is characterized by distinct evolution behaviors of a
physical observable, while the second one is marked by the nonanalyticities in
the rate function of the initial state survival probability. Here, we focus on
such DPTs from both quantum and semiclassical perspectives in a spinor
Bose-Einstein condensate (BEC), an ideal platform to investigate nonequilibrium
dynamics.By using the sudden quench process, we demonstrate that the system
exhibits both types of DPTs as the control parameter quenches through the
critical one, referring to as the critical quench. We show analytically how to
determine the critical quenches by means of the semiclassical approach and
carry out a detailed examination on both semiclassical and quantum signatures
of two types of DPTs. Moreover, we further reveal that the occurrence of DPTs
is closely connected to the separatrix in the underlying classical system. Our
findings provide more insights into the properties of DPTs and verify the
usefulness of semiclassical analysis for understanding DPTs in quantum systems
with well-defined semiclassical limit.
