Synchronization in two-level quantum systems

Recently, it was shown that dissipative quantum systems with three or more levels are able to synchronize to an external signal, but it was stated that it is not possible for two-level systems as they lack a stable limit cycle in the unperturbed dynamics. At the same time, several papers, demonstrate, under a different definition of what is synchronization, that the latter is possible in qubits, although in different models which also include other elements. We show how a quantum two-level system can be understood as containing a valid limit cycle as the starting point of synchronization, and that it can synchronize its dynamics to an external weak signal. This is demonstrated by analytically solving the Lindblad equation of a two-level system coupled to an environment, determining the steady state. This is a mixed state with contributions from many pure states, each of which provides a valid limit cycle. We show that this is sufficient to phase-lock the dynamics to a weak external signal, hence clarifying synchronization in two-level systems. We use the Husimi Q representation to analyze the synchronization region, defining a synchronization measure which characterizes the strength of the phase-locking. Also, we study the stability of the limit cycle and its deformation with the strength of the signal in terms of the components of the Bloch vector of the system. Finally, we generalize the model of the three-level system from in order to illustrate how the stationary fixed point of that model can be changed into a limit cycle similar to the one that we describe for the two-level system.