  Achieving efficient nonlinear optical frequency conversion in small volumes
is key for future on-chip photonic devices that would provide a higher-speed
alternative to modern electronics. However, the already intrinsically low
conversion efficiency severely limits miniaturization to nanoscale dimensions.
Here we demonstrate that gradient-field effects can provide for an efficient,
conventionally dipole-forbidden nonlinear response, offering a new approach for
enhanced nonlinear optics in nanostructures. We show that a {\em longitudinal}
nonlinear source current can dominate the third-order optical nonlinearity of
the free electron response in gold in the technologically important near-IR
frequency range where the nonlinearities due to other mechanisms are
particularly small. Using adiabatic nanofocusing to spatially confine the
excitation fields, from measurements of the $2\omega_1 - \omega_2$ four-wave
mixing response as a function of detuning $\omega_1 - \omega_2$, we find up to
$10^{-5}$ conversion efficiency with a gradient field contribution to
$\chi^{(3)}_{\mathrm{Au}}$ of up to $10^{-19}~\mathrm{m}^2 / \mathrm{V}^2$. The
results are in good agreement with theory based on plasma hydrodynamics. Our
results demonstrate an increase in nonlinear conversion efficiency with
decreasing sample size that can offset and even overcompensate the volume
decrease of conventional dipolar pathways. This will enable more efficient
nonlinear optical devices and frequency converters and facilitate the extension
of coherent multidimensional spectroscopies to the nanoscale.
