  We explore the dynamics of spontaneous symmetry breaking in a homogeneous
system by thermally quenching an atomic gas with short-range interactions
through the Bose-Einstein phase transition. Using homodyne matter-wave
interferometry to measure first-order correlation functions, we verify the
central quantitative prediction of the Kibble-Zurek theory, namely the
homogeneous-system power-law scaling of the coherence length with the quench
rate. Moreover, we directly confirm its underlying hypothesis, the freezing of
the correlation length near the transition due to critical slowing down. Our
measurements agree with beyond mean-field theory, and support the previously
unverified expectation that the dynamical critical exponent for this
universality class, which includes the $\lambda$-transition of liquid $^4$He,
is $z=3/2$.
