Classical-driving-assisted quantum synchronization in non-Markovian environments

We study the quantum phase synchronization of a driven two-level system (TLS) coupled to a structured environment and demonstrate that quantum synchronization can be enhanced by the classical driving field. We use the Husimi $Q$-function to characterize the phase preference and find the in-phase and anti-phase locking phenomenon in the phase diagram. Remarkably, we show that the in-phase classical driving enables a TLS to reach stable anti-phase locking in the Markovian regime. However, we find that the synergistic action of classical driving and non-Markovian effects significantly enhances the initial in-phase locking. By introducing the $S$-function and its maximal value to quantify the strength of synchronization and sketch the synchronization regions, we observe the typical signatures of the hollowed Arnold tongue in the parameter regions of synchronization. In the hollowed Arnold tongue, the synchronization regions exist both inside and outside the tongue while unsynchronized regions only lie on the boundary line. We also provide an intuitive interpretation of the above results by using the quasimode theory.