  We theoretically study the non-linear response of interacting neutral bosonic
gas in a synthetically driven one-dimensional optical lattice. In particular,
we examine the bosonic analogue of electronic higher harmonic generation in a
strong time-dependent synthetic vector potential manifesting itself as the
synthetic electric field. We show that the vector potential can generate
reasonably high harmonics in the insulating regime, while the superfluid regime
exhibits only a few harmonics. In the insulating regime, the number of
harmonics increases with the increase in the strength of the vector potential.
This originates primarily due to the field-driven resonant and non-resonant
excitations in the neutral Mott state and their recombination with the ground
state. If the repulsive interaction between two atoms ($U$) is close to the
strength of the gauge potential ($A_0$), the resonant quasiparticle-quasihole
pairs on nearest-neighbor sites, namely dipole states are found to a play a
dominant role in the generating higher harmonics. However, in the strong-field
limit $A_0\gg U$, the nonresonant states where quasiparticle-quasihole pairs
are not on nearest-neighbor sites give rise to higher harmonics.
