  Universal dynamics of spontaneous symmetry breaking is central to
understanding the universal behavior of spontaneous defect formation in various
system from the early universe, condensed-matter systems to ultracold atomic
systems. We explore the universal real-time dynamics in an array of coupled
binary atomic Bose-Einstein condensates in optical lattices, which undergo a
spontaneous symmetry breaking from the symmetric Rabi oscillation to the
broken-symmetry self-trapping. In addition to Goldstone modes, there exist
gapped Higgs mode whose excitation gap vanishes at the critical point. In the
slow passage through the critical point, we analytically find that the
symmetry-breaking dynamics obeys the Kibble-Zurek mechanism. From the scalings
of bifurcation delay and domain formation, we numerically extract two
Kibble-Zurek exponents $b_{1}=\nu/(1+\nu z)$ and $b_{2}=1/(1+\nu z)$, which
give the static correlation-length critical exponent $\nu$ and the dynamic
critical exponent $z$. Our approach provides an efficient way to simultaneous
determination of the critical exponents $\nu$ and $z$ for a continuous phase
transition.
