Abstract
We present a scheme to realize the (extended) Bose-Hubbard model in an $N$-coupled optomechanical system. By treating the cavities as intermediary and eliminating them adiabatically with the condition of large detuning or fast decay, we can obtain the effective Hamiltonian for the $N$ oscillators, with the regular terms in the Bose-Hubbard model, i.e., the pair tunnelings and the density-density interactions. Then we verify and provide the condition for our approximation with numerical results. Due to the existence of the pair tunnelings and the density-density interactions, we can investigate the density wave and supersolid phases in our model. Moreover, we also discuss the competition between the regular tunneling and the pair tunneling.
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