  Synchronizing a few-level quantum system is of fundamental importance to the
understanding of synchronization in the deep quantum regime. We investigate
quantum phase synchronization of a two-level system (qubit) driven by a
semiclassical laser field, in the presence of a general non-Markovian
dissipative environment. The phase preference of the qubit is demonstrated
through Husimi Q-function, and the existence of a limit cycle is also shown in
our system. Synchronization of the qubit is quantified using the shifted phase
distribution. The signature of quantum phase synchronization viz the Arnold
tongue is obtained from the maximal value of the shifted phase distribution.
Two distinct types of qubit dynamics is considered depending on the reservoir
correlation time being very short and a situation when bath correlation time is
finite. In the Markov regime of the environment, the phase preference of the
qubit goes away in the long time limit, whereas the long-time phase
localization persists in the non-Markovian regime. We also plot the maximum of
the shifted phase distribution in two ways: (a) by varying the detuning and
laser driving strength, and (b) by varying the system-bath coupling and laser
driving strength. Various system-environment parameters determine the
synchronization regions and the qubit phase synchronization is shown to be
enhanced in the non-Markov regime.
