Quantum frequency locking and downconversion in a driven qubit-cavity system

We study a periodically driven qubit coupled to a quantized cavity mode. Despite its apparent simplicity, this system supports a rich variety of exotic phenomena, such as topological frequency conversion as recently discovered in Martin et al. [Phys. Rev. X 7, 041008 (2017)]. Here we report on a qualitatively different phenomenon that occurs in this platform, where the cavity mode's oscillations lock their frequency to a rational fraction r/q of the driving frequency Omega. This phenomenon, which we term quantum frequency locking, is characterized by the emergence of q-tuplets of stationary (Floquet) states whose quasienergies are separated by Omega/q, up to exponentially small corrections. The Wigner functions of these states are nearly identical, and exhibit highly regular and symmetric structure in phase space. Similarly to Floquet time crystals, these states underlie discrete time-translation symmetry breaking in the model. We develop a semiclassical approach for analyzing and predicting quantum frequency locking in the model, and use it to identify the conditions under which it occurs.