Time crystals in a shaken atom-cavity system

We demonstrate the emergence of a time crystal of atoms in a high-finesse optical cavity driven by a phase-modulated transverse pump field, resulting in a shaken lattice. This shaken system exhibits macroscopic oscillations in the number of cavity photons and order parameters at noninteger multiples of the driving period, which signals the appearance of an incommensurate time crystal. The subharmonic oscillatory motion corresponds to dynamical switching between symmetry-broken states, which are nonequilibrium bond ordered density wave states. Employing a semiclassical phase-space representation for the driven-dissipative quantum dynamics, we confirm the rigidity and persistence of the time crystalline phase. We identify experimentally relevant parameter regimes for which the time crystal phase is long lived, and map out the dynamical phase diagram. We compare and contrast the incommensurate time crystal with the commensurate Dicke time crystal in the amplitude-modulated case.