  We explore the defect formation and universally critical dynamics in
two-dimensional (2D) two-component Bose-Einstein condensates(BECs) subjected to
two types of potential traps: a homogeneous trap and a harmonic trap.We focus
on the non-equilibrium universal dynamics of the miscible-immiscible phase
transition with both linear and nonlinear quenching types.Although there exists
spatial independence of the critical point, we find that the inhomogeneity of
trap doesn't affect the phase transition of system and the critical exponents
can still be explained by the homogeneous Kibble-Zurek mechanism. By analyzing
the Bogoliubov excitations, we establish a power-law relationship between the
domain correlation length, the phase transition delay, and the quench
time.Furthermore, through real-time simulations of phase transition dynamics,
the formation of domain defects and the delay of phase transition in
non-equilibrium dynamics are demonstrated, along with the corresponding
universal scaling of correlation length and phase transition delay for various
quench time and quench coefficients, which align well with our analytical
predictions.Our study confirms that the universality class of two-component
BECs remains unaffected by dimensionality, while the larger nonlinear
coefficients effectively suppress non-adiabatic excitations, offering a novel
perspective for addressing adiabatic evolution.
