  Metasurfaces consisting of nano-scale structures are underpinning new
physical principles for the creation and shaping of quantum states of light.
Multi-photon states that are entangled in spatial or angular domains are an
essential resource for quantum imaging and sensing applications, however their
production traditionally relies on bulky nonlinear crystals. We predict and
demonstrate experimentally the generation of spatially entangled photon pairs
through spontaneous parametric down-conversion from a metasurface incorporating
a nonlinear thin film of lithium niobate. This is achieved through nonlocal
resonances with tailored angular dispersion mediated by an integrated silica
meta-grating, enabling control of the emission pattern and associated quantum
states of photon pairs by designing the grating profile and tuning the pump
frequency. We measure the correlations of photon positions and identify their
spatial anti-bunching through violation of the classical Cauchy-Schwartz
inequality, witnessing the presence of multi-mode entanglement. Simultaneously,
the photon-pair rate is strongly enhanced by 450 times as compared to
unpatterned films due to high-quality-factor metasurface resonances, and the
coincidence to accidental ratio reaches 5000. These results pave the way to
miniaturization of various quantum devices by incorporating ultra-thin
metasurfaces functioning as room-temperature sources of quantum-entangled
photons.
