  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.
