We study the nonequilibrium dynamics of bosons in a two-dimensional optical lattice after a sudden quench from the superfluid phase to the free-boson regime. The initial superfluid state is described approximately using both the Bogoliubov theory and the Gaussian variational principle. The subsequent time evolution remains Gaussian, and we compare the results from each approximation of the initial state by examining different aspects of the dynamics. First, we analyze the entanglement entropy and observe that, in both cases, it increases linearly with time before reaching a saturation point. This behavior is attributed to the propagation of entangled pairs of quantum depletions in the superfluid state. Next, we explore the fate of particle-number symmetry, which is spontaneously broken in the superfluid phase. To do so, we use the entanglement asymmetry, a recently introduced observable that enables us to track symmetry breaking within a subsystem. We observe that its evolution varies qualitatively depending on the theory used to describe the initial state. However, in both cases, the symmetry remains broken and is never restored in the stationary state. Finally, we assess the time it takes to reach the stationary state by evaluating the quantum fidelity between the stationary reduced density matrix and the time-evolved one. Interestingly, within the Gaussian variational principle, we find that an initial state further from the stationary state can relax more quickly than one closer to it, indicating the presence of the recently discovered quantum Mpemba effect. We derive the microscopic conditions necessary for this effect to occur and demonstrate that these conditions are never met in the Bogoliubov theory.
