  Sum-frequency generation (SFG) spectroscopy provides a versatile method for
the investigation of non-centrosymmetric media and interfaces. Here, using
tunable picosecond infrared (IR) pulses from a free-electron laser, the
nonlinear optical response of 4H-SiC, a common polytype of silicon carbide, has
been probed in the frequency- and time-domain by infrared-visible vibrational
SFG spectroscopy. In the SFG spectra we observe a sharp resonance near the
longitudinal optical phonon frequency, arising from linear optical effects due
the epsilon-near-zero regime of the IR permittivity. In the time domain, the
build-up of the SFG intensity is linked to the free-induction decay of the
induced coherent IR polarization. When approaching the frequency of the phonon
resonance, a slower polarization dephasing is observed as compared to
off-resonant IR excitation. Thus, by introducing a temporal delay between the
IR and the visible up-conversion pulse we are able to demonstrate spectral
narrowing of the phonon SFG resonance, as corroborated by model calculations.
