  Optical nanoantennas have shown a great capacity for efficient extraction of
photons from the near to the far-field, enabling directional emission from
nanoscale single-photon sources. However, their potential for the generation
and extraction of multi-photon quantum states remains unexplored. Here we
demonstrate experimentally the nanoscale generation of two-photon quantum
states at telecommunication wavelengths based on spontaneous parametric
down-conversion in an optical nanoantenna. The antenna is a crystalline AlGaAs
nanocylinder, possessing Mie-type resonances at both the pump and the bi-photon
wavelengths and when excited by a pump beam generates photonpairs with a rate
of 35 Hz. Normalized to the pump energy stored by the nanoantenna, this rate
corresponds to 1.4 GHz/Wm, being one order of magnitude higher than
conventional on-chip or bulk photon-pair sources. Our experiments open the way
for multiplexing several antennas for coherent generation of multi-photon
quantum states with complex spatial-mode entanglement and applications in
free-space quantum communications and sensing.
