  As a prototype of a disordered superconductor we consider the attractive
Hubbard model with on-site disorder. We solve the Bogoljubov-de-Gennes
equations on two-dimensional finite clusters at zero temperature and evaluate
the electromagnetic response to a vector potential. We find that the standard
decoupling between transverse and longitudinal response does not apply in the
presence of disorder. Moreover the superfluid density is strongly reduced by
the relaxation of the phase of the order parameter already at mean-field level
when disorder is large. We also find that the anharmonicity of the phase
fluctuations is strongly enhanced by disorder. Beyond mean-field, this provides
an enhancement of quantum fluctuations inducing a zero-temperature transition
to a non-superconducting phase of disordered preformed pairs. Finally, the
connection of our findings with the glassy physics for extreme dirty
superconductors is discussed.
