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As a prototype of a disordered superconductor we consider the attractiveHubbard model with on-site disorder. We solve the Bogoljubov-de-Gennesequations on two-dimensional finite clusters at zero temperature and evaluatethe electromagnetic response to a vector potential. We find that the standarddecoupling between transverse and longitudinal response does not apply in thepresence of disorder. Moreover the superfluid density is strongly reduced bythe relaxation of the phase of the order parameter already at mean-field levelwhen disorder is large. We also find that the anharmonicity of the phasefluctuations is strongly enhanced by disorder. Beyond mean-field, this providesan enhancement of quantum fluctuations inducing a zero-temperature transitionto a non-superconducting phase of disordered preformed pairs. Finally, theconnection of our findings with the glassy physics for extreme dirtysuperconductors is discussed.

Superfluid density and phase relaxation in superconductors with strong disorder