  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, PRX 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.
