We review the latest advances in the generation of quantum light with nonlinear nanoresonators in metasurfaces, which act both as sources of quantum states and nanoantennas shaping the emitted photons. We outline a general quantum theory of spontaneous photon-pair generation in arbitrary nonlinear photonic structures, including nanoresonators and metasurfaces, which provides an explicit analytical solution for the photon state expressed through the classical Green function. We formulate the correspondence between the quantum photon-pair generation and classical sum-frequency process in nonlinear media, and discuss its application in various contexts, including waveguide circuits and nanostructures. We also discuss the first experimental results demonstrating photon-pair generation in a single nonlinear nanoantenna.

36 pages, 13 figures. arXiv admin note: text overlap with arXiv:1903.06956