Quantum electrodynamics (QED) provides a highly accurate description of phenomena involving the interaction of atoms with light. We argue that the quantum theory describing the interaction of cold atoms with a vibrating membrane--quantum acoustodynamics (QAD)--shares many issues and features with QED. Specifically, the adsorption of an atom on a vibrating membrane can be viewed as the counterpart to QED radiative electron capture. A calculation of the adsorption rate to lowest-order in the atom-phonon coupling is finite; however, higher-order contributions suffer from an infrared problem mimicking the case of radiative capture in QED. Terms in the perturbation series for the adsorption rate diverge as a result of massless particles in the model (flexural phonons of the membrane in QAD and photons in QED). We treat this infrared problem in QAD explicitly to obtain finite results by regularizing with a low-frequency cutoff that corresponds to the inverse size of the membrane. Using a coherent state basis for the soft phonon final state, we then sum the dominant contributions to derive a new formula for the multiphonon adsorption rate of atoms on the membrane that gives results that are finite, nonperturbative in the atom-phonon coupling, and consistent with the KLN theorem. For micromembranes, we predict a reduction with increasing membrane size for the low-energy adsorption rate. We discuss the relevance of this to the adsorption of a cold gas of atomic hydrogen on suspended graphene.
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