  Quantum systems exhibit particle-like or wave-like behaviour depending on the
experimental apparatus they are confronted by. This wave-particle duality is at
the heart of quantum mechanics, and is fully captured in Wheeler's famous
delayed choice gedanken experiment. In this variant of the double slit
experiment, the observer chooses to test either the particle or wave nature of
a photon after it has passed through the slits. Here we report on a quantum
delayed choice experiment, based on a quantum controlled beam-splitter, in
which both particle and wave behaviours can be investigated simultaneously. The
genuinely quantum nature of the photon's behaviour is tested via a Bell
inequality, which here replaces the delayed choice of the observer. We observe
strong Bell inequality violations, thus showing that no model in which the
photon knows in advance what type of experiment it will be confronted by, hence
behaving either as a particle or as wave, can account for the experimental
data.
