Controlling Collective Phenomena via the Quantum State of Interaction Mediators: Changing the Criticality of Photon-Mediated Superconductivity via Fock States of Light

How are two-body scattering and the resulting collective phenomena affected by preparing the mediator of interactions in different quantum states? This question has recently become experimentally relevant in a specific nonrelativistic version of QED implemented within materials, where standard techniques of quantum optics are available for the preparation of desired quantum states of the photon mediating interactions between matter’s constituents. We develop the necessary nonequilibrium 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 mediator’s quantum state. This emergent structure becomes nontrivial for non-Gaussian quantum states of the mediator, and can dramatically affect scattering and collective phenomena. As a first application, we show that by preparing photons in pure Fock states one can enhance pair correlations, and even modify the criticality of the superconducting phase transition. Our results also reveal that the thermal mixture of Fock states regularizes the strong pair correlations present in each of its components, yielding the standard Bardeen-Cooper-Schrieffer criticality. Besides the above QED platform, ultracold atomic mixtures are among the most promising candidates for the experimental implementation of these ideas.