  Strongly interacting quantum many-body systems are fundamentally compelling
and ubiquitous in science. However, their complexity generally prevents exact
solutions of their dynamics. Precisely engineered ultracold atomic gases are
emerging as a powerful tool to unravel these challenging physical problems.
Here we present a new laboratory for the study of many-body effects: strongly
interacting two-level systems formed by the clock states in ${}^{87}$Sr, which
are used to realize a neutral atom optical clock that performs at the highest
level of optical-atomic coherence and with precision near the limit set by
quantum fluctuations. Our measurements of the collective spin evolution reveal
signatures of many-body dynamics, including beyond-mean-field effects. We
derive a many-body Hamiltonian that describes the experimental observation of
severely distorted lineshapes, atomic spin coherence decay, density-dependent
frequency shifts, and correlated quantum spin noise. These investigations open
the door to exploring quantum many-body effects and entanglement in quantum
systems with optical energy splittings, using highly coherent and precisely
controlled optical lattice clocks.
