  We report the realisation of a Hanbury-Brown and Twiss (HBT)-like experiment
with a gas of strongly interacting bosons at low temperatures. The regime of
large interactions and low temperatures is reached in a three-dimensional
optical lattice and atom-atom correlations are extracted from the detection of
individual metastable Helium atoms after a long free-fall. We observe a HBT
bunching in the non-condensed fraction of the gas whose properties strongly
deviate from the HBT signals expected for non-interacting bosons. In addition,
we show that the measured correlations reflect the peculiar quantum statistics
of atoms belonging to the quantum depletion and of the Bogoliubov phonons,
i.e., of collective excitations of the many-body quantum state. Our results
demonstrate that atom-atom correlations provide information about the quantum
state of strongly-interacting particles, extending the interest of HBT-like
experiments beyond the case of non-interacting particles.
