We report terahertz (THz) second harmonic generation (SHG) in superconductors (SC) with inversion symmetric equilibrium states that forbid even-order nonlinearities. Such SHG signal is observed in single-pulse emission by periodic driving with a multi-cycle THz electric field tuned below the SC energy gap and vanishes above the SC critical temperature. We explain the microscopic physics by a dynamical symmetry breaking principle at sub-THz-cycle by using quantum kinetic modeling of the interplay between strong THz-lightwave nonlinearity and pulse propagation. The resulting non-zero integrated pulse area inside the SC drives lightwave nonlinear supercurrents due to sub--cycle Cooper pair acceleration, in contrast to d.c.-biased superconductors, which can be controlled by the bandstructure and the THz pump field.
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