  In this paper we demonstrate the necessity of including the generally omitted
collective mode contributions in calculations of the Meissner effect for
non-uniform superconductors. We consider superconducting pairing with non-zero
center of mass momentum, as is relevant to high transition temperature
cuprates, cold atoms, and quantum chromodynamic superconductors. For the
concrete example of the Fulde-Ferrell phase we present a quantitative
calculation of the superfluid density, showing the collective mode
contributions are not only appreciable but that they derive from the amplitude
mode of the order parameter. This latter mode (related to the Higgs mode in a
charged system) is generally viewed as being invisible in conventional
superconductors. However, our analysis shows that it is extremely important in
pair-density wave type superconductors, where it destroys superfluidity well
before the mean-field order parameter vanishes.
