The quantum dynamics of ultra-cold atoms inside dark optical traps (known as "atom-optics billiards") is studied using a new spectroscopic technique. In an experimental realization of a "gedanken experiment" proposed by Asher Peres in 1984 [Phys. Rev. A30, 1610] the decay of quantum correlations of perturbed high-lying states (with <n> up to 10^8) are measured when the underlying classical motion is regular, chaotic or mixed. For weak perturbations we observe partial quantum revivals, reflecting non-universal features of the perturbation and for strong perturbations we observe a regime where the correlation decay times are independent in the perturbation strength.