  A BEC interacting with an optical field via a feedback mirror can be a
realisation of the quantum Hamiltonian Mean Field (HMF) model, a paradigmatic
model of long-range interactions in quantum systems. We demonstrate that the
self-structuring instability displayed by an initially uniform BEC can evolve
as predicted by the quantum HMF model, displaying quasiperiodic "chevron"
dynamics for strong driving. For weakly driven self-structuring, the BEC and
optical field behave as a two-state quantum system, regularly oscillating
between a spatially uniform state and a spatially periodic state. It also
predicts the width of stable optomechanical droplets and the dependence of
droplet width on optical pump intensity. The results presented suggest that
optical diffraction-mediated interactions between atoms in a BEC may be a route
to experimental realisation of quantum HMF dynamics and a useful analogue for
studying quantum systems involving long-range interactions.
