Fate of the spatial-temporal order under quantum fluctuation

In a previous theoretical work [arXiv:2205.01461], T. Esslinger group proposed a scheme to realize a spatial-temporal lattice, which possesses dual periodicity on space and time, in a cavity-boson system pumped by a travelling wave laser. However, the prediction was made under the mean-field approximation. In this work, we investigate the dynamics beyond mean-field approximation. By including the fluctuation of the cavity field, we obtain a larger set of equations of motion. Numerical results show that the spatial-temporal lattice is melted in the mean-field level but survives in the quantum fluctuation.