  We examine the third-order non-linear optical response of a one-dimensional
Mott insulator coupled with phonons. The Mott insulator is described by an
extended Hubbard-Holstein model. The third harmonic generation (THG) of the
model is calculated by dynamical density-matrix renormalization group. We find
that the electron-phonon interaction enhances the intensity of THG if the
on-site Coulomb interaction is comparable to the band width, but the
enhancement is small for a realistic parameter value of a typical Mott
insulator Sr$_2$CuO$_3$. The low-energy spin excitation does not contribute to
the optical response if the electron-phonon coupling is neglected. We find that
the THG spectrum can detect the spin excitations even for no electron-phonon
coupling. The introduction of the electron-phonon coupling leads to a slight
increase of low-energy weight in THG without peak structure through
phonon-assisted spin excitation. To fully understand the phonon-assisted spin
excitation, we additionally calculate the linear susceptibility of a doped
Hubbard-Holstein model, showing a spectral distribution different from that at
half filling.
