  The study of interaction between different degrees of freedom in solids is of
fundamental importance to understand the functionalities of materials. One
striking example of such interaction is the intertwined coupling or competition
between superconductivity (SC), charge density wave (CDW), pseudogap state
(PG), and other exotic phases in cuprate superconductors. Recent emergence of
nonlinear Terahertz (THz) third harmonic generation (THG) spectroscopy provides
a powerful tool for exploring the collective (Higgs) modes of superconductivity
order parameters, and its interaction with intertwined/competing phases. In
this study, we report on nonlinear THz THG spectroscopy of the
YBa$_2$Cu$_3$O$_{6+x}$ (YBCO) thin films with different doping. We identify a
characteristic temperature $T_{THG}$, below which third order suscepetility
$\chi^{(3)}$ emerges. Notably, the $T_{THG}$ is coincident with the crossover
temperature $T^*$ of pseudogap in a wide range doping of phase diagram. Upon
entering the superconducting state, THG increases sharply but exhibits an
abnormal dip feature near $T_c$ which is more clearly seen in optimally doped
sample. Strikingly, we observe a beating structure directly in the measured
real time waveform of THG signal. Fourier transformation of the time domain
waveform gives two separate modes below and above original THG frequency. The
observation strongly indicates that an additional mode, presumably Higgs mode,
appears at $T_c$ and couples to the mode already developed below $T^*$. The
strong coupling effect offers new insight into the interplay between
superconductivity and pseudogap. The result unambiguously suggests that the
pseudogap phase is not a precursor of superconductivity but represents a
distinct order.
