  Quantised sound waves -- phonons -- govern the elastic response of
crystalline materials, and also play an integral part in determining their
thermodynamic properties and electrical response (e.g., by binding electrons
into superconducting Cooper pairs). The physics of phonons and elasticity is
absent in simulators of quantum materials constructed of neutral atoms and
light: unlike real solids, traditional optical lattices are silent because they
are infinitely stiff. Optical-lattice realisations of crystals therefore lack
some of the central dynamical degrees of freedom that determine the
low-temperature properties of real materials. Here, we create an optical
lattice with phonon modes using a Bose-Einstein condensate (BEC) coupled to a
confocal optical resonator. Playing the role of an active quantum gas
microscope, the multimode cavity QED system both images the phonons and induces
the crystallisation that supports phonons via short-range, photon-mediated
atom-atom interactions. Dynamical susceptibility measurements reveal the phonon
dispersion relation, showing that these collective excitations exhibit a sound
speed dependent on the BEC-photon coupling strength. Our results pave the way
for exploring the rich physics of elasticity in quantum solids, ranging from
quantum melting transitions to exotic "fractonic" topological defects in the
quantum regime.
