Terahertz nonlinear optics is a viable method to interrogate collective phenomena in quantum materials spanning ferroelectrics, charge-density waves, and superconductivity. In superconductors this includes the Higgs amplitude and Josephson phase modes. We have investigated the nonlinear $c$-axis response of optimally doped ${\mathrm{La}}_{1.85}{\mathrm{Sr}}_{0.15}\mathrm{Cu}{\mathrm{O}}_4$ using high-field THz time domain spectroscopy at field strengths up to $\sim 80$ kV/cm. With increasing field, we observe a distinct redshift of the Josephson plasma edge and enhanced reflectivity (above the plasma edge) arising from third harmonic generation. The nonmonotonic temperature-dependent response is consistent with nonlinear drive of the Josephson plasma mode (JPM) as verified with comparison to theoretical expectations. Our results add to the understanding that, using THz light, the JPM (in addition to the Higgs mode) provides a route to interrogate and control superconducting properties.

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• Superconductivity
• High-temperature superconductors
• Strongly correlated systems
• Superconductors
• High-harmonic generation
• Terahertz spectroscopy
• Terahertz techniques