  We theoretically investigate up-conversion process of entangled two photons
on a dimer molecule, which is coupled by a cavity or nanoscale metallic
structure. Within one-dimensional input-output theory, the propagators of the
photons are derived analytically and the up-conversion probability is
calculated numerically. It is shown that the coupling with the nanostructure
clearly enhances the process. We also find that the enhancement becomes further
pronounced for some balanced system parameters such as the quantum correlation
between photons, radiation decay rates and coupling between the nanostructure
and molecule. The non-monotonic dependencies are reasonably explained in view
of quantum interference between the coupled modes of the whole system. This
result can provide a guideline for nonlinear optical reactions by weak light of
a few photons.
