  According to the Kibble-Zurek mechanism, there is a universal power-law
relationship between the defect density and the quench rate during a slow
linear quench through a critical point. It is generally accepted that a fast
quench results in a deviation from the Kibble-Zurek scaling law and leads to
the formation of a saturated plateau in the defect density. By adjusting the
quench rate from slow to very fast limits, we observe the varying quench
dynamics and identify a pre-saturated regime that lies between the saturated
and Kibble-Zurek regimes. This significant result is elucidated through the
adiabatic-impulse approximation first, then verified by a rigorous analysis on
the transverse Ising chain as well. As we approach the turning point from the
saturated to pre-saturated regimes, we notice a change in scaling laws and,
with an increase in the initial transverse field, a shrinking of the saturated
regime until it disappears. During another turning point from the Kibble-Zurek
to pre-saturated regimes, we observe an attenuation of the dephasing effect and
a change in the behavior of the kink-kink correlation function from a Gaussian
decay to an exponential decay. Finally, the coherent many-body oscillation
after quench exhibits different behaviors in the three regimes and shows a
significant change of scaling behavior between the S and PS regimes.
