  Recent experimental progress in coupling nanoscale conductors to
superconducting microwave cavities has opened up for transport investigations
of the deep quantum limit of light-matter interactions, with tunneling
electrons strongly coupled to individual cavity photons. We have investigated
theoretically the most basic cavity-conductor system with strong, single photon
induced non-local transport effects; two spatially separated double quantum
dots (DQD:s) resonantly coupled to the fundamental cavity mode. The system,
described by a generalized Tavis-Cummings model, is investigated within a
quantum master equation formalism, allowing us to account for both the
electronic transport properties through the DQD:s as well as the coherent,
non-equilibrium cavity photon state. We find sizeable non-locally induced
current and current cross-correlations mediated by individual photons. From a
full statistical description of the electron transport we further reveal a
dynamical channel blockade in one DQD lifted by photon emission due to
tunneling through the other DQD. Moreover, large entanglement between the
orbital states of electrons in the two DQD:s is found for small DQD-lead
temperatures.
