Quantum Fluctuations of Superconductivity in Critically Underdoped La$_{2-x}$Sr$_x$CuO$_4$

In the underdoped pseudogap regime of the high-temperature superconductors, one expects that due to low superfluid densities and short correlation lengths, superconducting fluctuations will be very significant for transport and thermodynamic properties. We have used THz time-domain spectroscopy (TTDS) to probe the fluctuations of superconductivity in extremely underdoped La$_{2-x}$Sr$_x$CuO$_4$ films close to and beyond the superconducting quantum critical point. On the approach to the transition from above, we find a significant range of quantum superconducting fluctuations that we quantify through a comparison of the low and high frequency phase stiffnesses. An explicit measure of these fluctuations through the quantity we propose as a ``quantum Debye-Waller factor'' shows that these fluctuations diverge and drive the transition.