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In this paper we demonstrate the necessity of including the generally omittedcollective mode contributions in calculations of the Meissner effect fornon-uniform superconductors. We consider superconducting pairing with non-zerocenter of mass momentum, as is relevant to high transition temperaturecuprates, cold atoms, and quantum chromodynamic superconductors. For theconcrete example of the Fulde-Ferrell phase we present a quantitativecalculation of the superfluid density, showing the collective modecontributions are not only appreciable but that they derive from the amplitudemode of the order parameter. This latter mode (related to the Higgs mode in acharged system) is generally viewed as being invisible in conventionalsuperconductors. However, our analysis shows that it is extremely important inpair-density wave type superconductors, where it destroys superfluidity wellbefore the mean-field order parameter vanishes.

Collective mode contributions to the Meissner effect: Fulde-Ferrell and pair-density wave superfluids