  Time-dependent speckle pattern of paraelectric barium titanate observed in a
soft x-ray laser pump-probe measurement is theoretically investigated as a
correlated optical response to the pump and probe pulses. The scattering
probability is calculated based on a model with coupled soft x-ray photon and
ferroelectric phonon mode. It is found that the speckle variation is related
with the relaxation dynamics of ferroelectric clusters created by the pump
pulse. Additionally, critical slowing down of cluster relaxation arises on
decreasing temperature towards the paraelectric-ferroelectric transition
temperature. Relation between critical slowing down, local dipole fluctuation
and crystal structure are revealed by quantum Monte Carlo simulation.
