Parametrically amplified phase-incoherent superconductivity in YBa$_2$Cu$_3$O$_{6+x}$

The possibility of enhancing desirable functional properties of complex materials by optical driving is motivating a series of studies of their nonlinear terahertz response. In high-Tc cuprates, large amplitude excitation of certain infrared-active lattice vibrations has been shown to induce transient features in the reflectivity suggestive of non-equilibrium superconductivity. Yet, a microscopic mechanism for these observations is still lacking. Here, we report measurements of time- and scattering-angle-dependent second-harmonic generation in YBa$_2$Cu$_3$O$_{6+x}$, taken under the same excitation conditions that result in superconductor-like terahertz reflectivity. We discover a three-order-of-magnitude amplification of a 2.5-terahertz electronic mode, which is unique because of its symmetry, momentum, and temperature dependence. A theory for parametric three-wave amplification of Josephson plasmons, which are assumed to be well-formed below T$_c$ but overdamped throughout the pseudogap phase, explains all these observations and provides a mechanism for non-equilibrium superconductivity. More broadly, our work underscores the role of parametric mode mixing to stabilize fluctuating orders in quantum materials.