  Due to their proximity to an antiferromagnetic phase and to the mysterious
pseudogap, underdoped cuprates have attracted great interest in the high Tc
community for many years. A central issue concerns the role of quantum and
thermal fluctuations of the phase of the superconducting order parameter. The
evolution of superfluid density ns with temperature and doping is a powerful
probe of this physics. Here, we report superfluid density measurements on
underdoped Bi2Sr2CaCu2O8+x (Bi-2212) films at much lower dopings than have been
achieved previously, and with excellent control on doping level - Tc ranges
from Tc,min ~ 6K to Tc,max ~ 80K in steps of about 5K. Most famous studies on
Bi-2212 like angle-resolved photoemission and scanning probe microscopy are
surface-sensitive while superfluid density measurements are bulk-sensitive. We
find that strong two-dimensional quantum fluctuations are evident in the
observed linear scaling of Tc with ns(0) when Tc is below about 45 K, which
contrasts with three-dimensional quantum fluctuations evident in the square
root scaling, Tc $\propto \sqrt$ns(0), seen in the much less anisotropic
cuprate, YBa2Cu3O7 (YBCO). On the other hand, consistent with YBCO, ns(T) in
severely underdoped Bi-2212 loses its strong downward curvature near Tc,
becoming quasi-linear without any obvious critical behavior near Tc. We argue
that the quasi-linear T dependence arises from thermal phase fluctuations,
although the current theory needs modification in order to understand some
features.
