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Trimer superfluid induced by photoassocation on the state-dependent optical lattice

We use the mean-field method, the Quantum Monte-carlo method and the Density matrix renormalization group method to study the trimer superfluid phase and the quantum phase diagram of the Bose-Hubbard model in an optical lattice, with explicit trimer tunneling term. Theoretically, we derive the explicit trimer hopping terms, such as $a_i^{3\dagger}a_j^3$, by the Schrieffer-Wolf transformation. In practice, the trimer super\-fluid described by these terms is driven by photoassociation. The phase transition between the trimer super\-fluid phase and other phases are also studied. Without the on-site interaction, the phase transition between the trimer superfluid phase and the Mott Insulator phase is continuous. Turning on the on-site interaction, the phase transitions are first order with Mott insulators of atom filling $1$ and $2$. With nonzero atom tunneling, the phase transition is first order from the atom superfluid to the trimer superfluid. In the trimer superfluid phase, the win\-ding numbers can be divided by three without any remainders. In the atom superfluid and pair superfluid, the vorticities are $1$ and $1/2$, respectively. However, the vorticity is $1/3$ for the trimer superfluid. The power law decay exponents is $1/2$ for the non diagonal correlation $a_i^{\dagger 3} a_j^{3}$, i.e. the same as the exponent of the correlation $a_i^{\dagger}a_j$ in hardcore bosons. The density dependent atom-tunneling term $n_i^2a_i^{\dagger}a_j$ and pair tunneling term $n_ia_i^{\dagger2}a_j^2$ are also studied. With these terms, the phase transition from the empty phase to atom superfluid is first order and different from the cases without the density dependent terms. The ef\-fects of temperature are studied. Our results will be helpful in realizing the trimer superfluid by a cold atom experiment.