  How are the scattering between the constituents of matter and the resulting
collective phenomena affected by preparing the force carriers in different
quantum states? This question has become experimentally relevant in a specific
non-relativistic version of QED implemented within materials, where standard
techniques of quantum optics are available for the preparation of desired
quantum states of the carrier photon. We develop the necessary non-equilibrium
approach for computing the vertex function and find that, in addition to the
energy and momentum structure of the scattering, a further structure emerges
which reflects the Hilbert-space distribution of the carrier quantum state.
This emergent structure becomes non-trivial for non-Gaussian quantum states of
the force carrier, and can dramatically affect interactions and collective
phenomena. As a first application, we show that by preparing photons in pure
Fock states one can enhance pair correlations, and even control the criticality
and universality class of the superconducting phase transition by the choice of
the number of photons. Our results also reveal that the thermal mixture of Fock
states regularises the strong pair correlations present in each of its
components, yielding the standard Bardeen-Cooper-Schrieffer criticality.

Quantum Physics;Mesoscale and Nanoscale Physics;Quantum Gases;Strongly Correlated Electrons;Superconductivity