  We carry out a theoretical investigation to clarify the dynamic property of
photo-created nano-sized ferroelectric cluster observed in the paraelectric
BaTiO3 as a real time correlation of speckle pattern between two soft X-ray
laser pulses, at just above the paraelectric-ferroelectric phase transition
temperature. Based on a model with coupled soft X-ray photon and ferroelectric
phonon mode, we study the time dependence of scattering probability by using a
perturbative expansion approach. The cluster-associated phonon softening as
well as central peak effects are well reproduced in the phonon spectral
function via quantum Monte Carlo simulation. Besides, it is found that the time
dependence of speckle correlation is determined by the relaxation dynamics of
ferroelectric clusters. Near the transition point, cluster excitation is
stable, leading to a long relaxation time. While, at high temperature, cluster
structure is subject to the thermal fluctuation, ending up with a short
relaxation time.
