  Alkaline-earth (AE) atoms have metastable clock states with minute-long
optical lifetimes, high-spin nuclei, and SU($N$)-symmetric interactions that
uniquely position them for advancing atomic clocks, quantum information
processing, and quantum simulation. The interplay of precision measurement and
quantum many-body physics is beginning to foster an exciting scientific
frontier with many opportunities. Few particle systems provide a window to view
the emergence of complex many-body phenomena arising from pairwise
interactions. Here, we create arrays of isolated few-body systems in a
fermionic ${}^{87}$Sr three-dimensional (3D) optical lattice clock and use high
resolution clock spectroscopy to directly observe the onset of both elastic and
inelastic multi-body interactions. These interactions cannot be broken down
into sums over the underlying pairwise interactions. We measure
particle-number-dependent frequency shifts of the clock transition for atom
numbers $n$ ranging from 1 to 5, and observe nonlinear interaction shifts,
which are characteristic of SU($N$)-symmetric elastic multi-body effects. To
study inelastic multi-body effects, we use these frequency shifts to isolate
$n$-occupied sites and measure the corresponding lifetimes. This allows us to
access the short-range few-body physics free from systematic effects
encountered in a bulk gas. These measurements, combined with theory, elucidate
an emergence of multi-body effects in few-body systems of sites populated with
ground-state atoms and those with single electronic excitations. By connecting
these few-body systems through tunneling, the favorable energy and timescales
of the interactions will allow our system to be utilized for studies of
high-spin quantum magnetism and the Kondo effect.
