  We study the non-equilibrium quench dynamics crossing a continuous phase
transition between the charge density wave (CDW) and supersolid (SS) phases of
a bosonic lattice gas with cavity-mediated interactions. When changing the
hopping amplitude in the Hamiltonian as a function of time, we investigate the
scaling behavior of the correlation length and vortex density with respect to
the quench time and find that there is a threshold of the quench rate
separating two distinct scaling regimes. When slowly varying the system below
that threshold, we find a power-law scaling as predicted by the Kibble-Zurek
mechanism (KZM). While considering fast quench above that threshold, a
deviation from the KZM prediction occurs, manifested by a saturation of the
defect density. We further show that such distinct scaling behaviors during
different dynamic procedures can be understood through comparing the relaxation
time and the quench rate.
