  Superfluid stiffness $\rho_s$ is a defining characteristic of the
superconducting state, allowing phase coherence and supercurrent. It is
accessible experimentally through the penetration depth. Coexistence of
$d$-wave superconductivity with other phases in underdoped cuprates, such as
antiferromagnetism (AF) or charge-density waves (CDW), may drastically alter
$\rho_s$. To shed light on this physics, the zero-temperature value of
$\rho_s=\rho_{zz}$ along the $c$-axis was computed for different values of
Hubbard interaction $U$ and different sets of tight-binding parameters
describing the high-temperature superconductors YBCO and NCCO. We used Cellular
Dynamical Mean-Field Theory for the one-band Hubbard model with exact
diagonalization as impurity solver and state-of-the-art bath parametrization.
We conclude that Mott physics plays a dominant role in determining the
superfluid stiffness on the hole-doped side of the phase diagram. On the
electron-doped side, antiferromagnetism wins over superconductivity near
half-filling. But upon approaching optimal electron-doping, homogeneous
coexistence between superconductivity and antiferromagnetism causes the
superfluid stiffness to drop sharply. Hence, on the electron-doped side, it is
competition between antiferromagnetism and $d$-wave superconductivity that
plays a dominant role in determining the value of $\rho_{zz}$ near
half-filling. At large overdoping, $\rho_{zz}$ behaves in a more BCS-like
manner in both the electron- and hole-doped cases. We comment on some
qualitative implications of these results for the superconducting transition
temperature.
