Criterion for bosonic superfluidity in an optical lattice

We show that the current method of determining superfluidity in optical lattices based on a visibly ``sharp" bosonic momentum distribution $n({\bf k})$ can be misleading, for even a {\em normal} Bose gas can have a similarly ``sharp'' $n({\bf k})$. We show that superfluidity can be detected from the so-called ``visibility'' $(v)$ of $n({\bf k})$ $-$ that $v$ must be 1 within $O(N^{-2/3})$, where $N$ is the number of bosons. Many current experiments, however, have interpreted states with $v<1$ as superfluid. Such states are in fact normal, reflecting strong temperature effects in the system. These normal states, however, allow one to explore the physics in the quantum critical regime.