Quantum analog of classical Arnold tongues

I will review papers by Lee et. al related to synchronization of open quantum systems. Each system is nothing more than a simple harmonic oscillator, interacting with an environment. To start with, I present the essential facts about phase-locking and synchronization in simple dynamical system on torus. Next one will see how does the quatum limit cylce can appear, then we will see how does the classical synchronization domain (Arnold tongue) evolves into its quantum version. The talk is on physical level of rigour.