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.
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