  The Markovianity of quantum dynamics is an important property of open quantum
systems determined by various ingredients of the system and bath. Apart from
the system-bath interaction, the initial state of the bath, etc., the dimension
of the bath plays a critical role in determining the Markovianity of quantum
dynamics, as a strict decay of the bath correlations requires an infinite
dimension for the bath. In this work, we investigate the role of finite bath
dimension in the Markovianity of quantum dynamics by considering a simple but
nontrivial model in which a central spin is isotropically coupled to a finite
number of bath spins, and show how the dynamics of the central spin transits
from non-Markovian to Markovian as the number of the bath spins increases. The
non-Markovianity is characterized by the information backflow from the bath to
the system in terms of the trace distance of the system states. We derive the
time evolution of the trace distance analytically, and find periodic
collapse-revival patterns in the information flow. The mechanism underlying
this phenomenon is investigated in detail, and it shows that the period of the
collapse-revival pattern is determined by the competition between the number of
the bath spins, the system-bath coupling strength, and the frequency detuning.
When the number of bath spins is sufficiently large, the period of the
collapse-revival structure as well as the respective collapse and revival times
increase in proportion to the number of the bath spins, which characterizes how
the information backflow decays with a large dimension of the bath. We also
analyze the effect of the system-bath interaction strength and frequency
detuning on the collapse-revival patterns of the information flow, and obtain
the condition for the existence of the collapse-revival structure. The results
are illustrated by numerical computation.
