Indefinite causal order in cavity quantum electrodynamics

<jats:p>Indefinite causal order (ICO) has the potential to be a new resource for quantum information processing. In most of its experiments, ICO has been investigated in a photonic platform. Here, we investigate ICO in a cavity quantum electrodynamics system composed of two cavities. Our results show that ICO can create highly entangled states of two distant cavity fields that never interact directly independent of the initial (excited or ground) state of the atom. These entangled states can have the form of one- or two-photon NOON states. We show that ICO can interchange one photon between both cavities without changing the state of the atom, something that is impossible to achieve for two cavities in well-defined order. Furthermore, the vacuum Rabi oscillations either disappear or lose their sinusoidal form with ICO. Our results show the potential that ICO can offer in the paradigm of light-matter interaction for coherently controlling atom-field observables.</jats:p>