  Quantum noise correlations have been employed in several areas in physics
including condensed matter, quantum optics and ultracold atom to reveal
non-classical states of the systems. So far, such analysis mostly focused on
systems in equilibrium. In this paper, we show that quantum noise is also a
useful tool to characterize and study the non-equilibrium dynamics of one
dimensional system. We consider the Ramsey sequence of one dimensional,
two-component bosons, and obtain simple, analytical expressions of time
evolutions of the full distribution functions for this strongly-correlated,
many-body system. The analysis can also be directly applied to the evolution of
interference patterns between two one dimensional quasi-condensates created
from a single condensate through splitting. Using the tools developed in this
paper, we demonstrate that one dimensional dynamics in these systems exhibits
the phenomenon known as "prethermalization", where the observables of {\it
non-equilibrium}, long-time transient states become indistinguishable from
those of thermal {\it equilibrium} states.
