  A rapid restoration of the bath state is usually required to induce Markovian
dynamics for an open quantum system, which typically can be realized only in
limits such as weak system-bath coupling and infinitely large bath. In this
work, we investigate the Markovianity of a qubit system coupled to a
single-qubit bath with the qubit bath being continuously refreshed by quantum
cooling. A surprising result is that there exists a finite threshold for the
cooling rate at which the system transitions from non-Markovian dynamics to
Markovian dynamics, which is in sharp contrast to the usual understanding that
Markovian dynamics is an asymptotic behavior under the Born-Markov
approximation. We also study the time correlation of the bath, and find that
the decay rate of bath time correlation is of the same order as the system
evolution speed. This suggests that quantum Markovian dynamics can exist beyond
the usual short bath correlation limit.
