We study non-equilibrium quench dynamics across continuous phase transitions and cavity-mediated interactions between the charge density wave (CDW) and supersolid (SS) phases of boson lattice gases. When changing the hopping amplitude of the Hamiltonian as a function of time, we investigated the scaling behavior of correlation length and eddy density versus quench time and found that there is a quench rate threshold that separates the two distinct scaling regions. When we slowly vary the system below that threshold, we see power law scaling as predicted by the Kibble-Zurek mechanism (KZM). Considering fast quenching above that threshold, deviations from the KZM prediction occur and are manifested by defect density saturation. Furthermore, we show that such different scaling behaviors during different dynamic procedures can be understood by comparing relaxation times and quench rates.

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