  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.
