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In this paper, we study the dynamics of the Bose-Hubbard model with thenearest-neighbor repulsion by using time-dependent Gutzwiller methods. Near theunit filling, the phase diagram of the model contains density wave (DW),supersolid (SS) and superfluid (SF). The three phases are separated by twosecond-order phase transitions. We study "slow-quench" dynamics by varying thehopping parameter in the Hamiltonian as a function of time. In the phasetransitions from the DW to SS and from the DW to SF, we focus on how the SForder forms and study scaling laws of the SF correlation length, vortexdensity, etc. The results are compared with the Kibble-Zurek scaling. On theother hand from the SF to DW, we study how the DW order evolves with generationof the domain walls and vortices. Measurement of first-order SF coherencereveals interesting behavior in the DW regime.

physical review. a/physical review, a

Out-of-equilibrium dynamics of multiple second-order quantum phase  transitions in extended Bose-Hubbard model: Superfluid, supersolid and  density wave