Absence of the superconducting collective excitations in the photoexcited nonequilibrium state of underdoped YBa$_2$Cu$_3$O$_y$

Recent observation of the light-induced superconducting (SC)-like transient response in the $c$-axis optical conductivity far above the SC transition temperature $T_{\mathrm{c}}$ in underdoped YBa$_2$Cu$_3$O$_y$ (YBCO) has attracted great attention in the field of high-$T_{\mathrm{c}}$ superconductors. Since then, various theoretical and experimental studies have been devoted to elucidating its microscopic origin. One prominent fingerprint of the light-induced superconductivity is the emergence of $1/\omega$-like spectral behavior in the imaginary part of the optical conductivity in the terahertz (THz) frequency range. However, the spectral profile can also be described by the Drude response of the quasiparticles (QPs) with a substantially low scattering rate. To circumvent this critical ambiguity, we investigated the photoinduced nonequilibrium state in underdoped YBCO samples using the nonlinear THz optical responses originating from the SC collective excitations: the THz Kerr effect that reflects the Higgs mode and SC QPs responses, and the ac-driven Josephson current. Upon the near-infrared (NIR) photoexcitation above $T_{\mathrm{c}}$, the $1/\omega$-like spectral behavior in the imaginary part of the optical conductivity emerges, whereas the THz nonlinear responses associated with those collective modes were not observed. These results indicate that the NIR-pump induced state exhibiting the $1/\omega$-like response above $T_{\mathrm{c}}$ is distinct from the long-range ordered SC state in equilibrium. According to these observations, the possible origin of the irregularly coherent charge carrier response along the $c$-axis is discussed.