  Overdoped high-temperature cuprate superconductors have been widely believed
to be described by the physics of d-wave BCS-like superconductivity. However,
recent measurements indicate that as the doping is increased, the superfluid
density decreases smoothly to zero rather than increasing as expected by BCS
theory in the absence of disorder. Here, we combine time-domain THz
spectroscopy with kHz range mutual inductance measurements on the same
overdoped La$_{2-x}$Sr$_{x}$CuO$_{4}$ films to determine both the superfluid
and the uncondensed carrier density as a function of doping. A significant
fraction of the carriers remains uncondensed in a wide Drude-like peak even as
$T\rightarrow0$, which, when taken with the linear-in-temperature superfluid
density, is inconsistent with existing theories for the role of disorder in
suppressing the superfluid density in a d-wave superconductor. Our almost eight
orders of magnitude in measurement frequency range gives us a unique look at
the low frequency spectral weight distribution, which may suggest the presence
of quantum phase fluctuations as the critical doping is approached.
