  There have been concerted efforts in recent years to realize the next
generation of clocks using alkaline earth atoms in an optical lattice. Assuming
that the atoms are independent, such a clock would benefit from a $\sqrt{N}$
enhancement in its stability, associated with the improved signal-to-noise
ratio of a large atom number $N$. An interesting question, however, is what
type of atomic interactions might affect the clock dynamics, and whether these
interactions are deleterious or could even be beneficial. In this work, we
investigate the effect of dipole-dipole interactions, in which atoms excited
during the clock protocol emit and re-absorb photons. Taking a simple system
consisting of a 1D atomic array, we find that dipole-dipole interactions in
fact result in an open quantum system exhibiting critical dynamics, as a set of
collective excitations acquires a decay rate approaching zero in the
thermodynamic limit due to subradiance. A first consequence is that the decay
of atomic excited population at long times exhibits a slow power-law behavior,
instead of the exponential expected for non-interacting atoms. We also find
that excitations among the atoms exhibit fermionic spatial correlations at long
times, due to the microscopic properties of the multi-excitation subradiant
states. Interestingly, these properties cannot be captured by mean-field
dynamics, suggesting the strongly interacting nature of this system. We finally
characterize the time-dependent frequency shift in the atomic frequency
measurement, and find that it is dominated by the interaction energy of
subradiant states at long times. Furthermore, we show that the decay of the
clock signal displays at long times a non-exponential behavior, which might be
useful to improve the uncertainty limit with which the atomic frequency can be
resolved.
