  Nonlinear responses of superconductors to intense terahertz radiation has
been an active research frontier. Using terahertz pump-terahertz probe
spectroscopy, we investigate the c-axis nonlinear optical response of a
high-temperature superconducting cuprate. After excitation by a single-cycle
terahertz pump pulse, the reflectivity of the probe pulse oscillates as the
pump-probe delay is varied. Interestingly, the oscillatory central frequency
scales linearly with the probe frequency, a fact widely overlooked in
pump-probe experiments. By theoretically solving the nonlinear optical
reflection problem on the interface, we show that our observation is well
explained by the Josephson-type third-order nonlinear electrodynamics, together
with the emission coefficient from inside the material into free space. The
latter results in a strong enhancement of the emitted signal whose physical
frequency is around the Josephson plasma edge. Our result offers a benchmark
for and new insights into strong-field terahertz spectroscopy of related
quantum materials.
