  Measurement science now connects strongly with engineering of quantum
coherence, many-body states, and entanglement. To scale up the performance of
an atomic clock using a degenerate Fermi gas loaded in a three-dimensional
optical lattice, we must understand complex many-body Hamiltonians to ensure
meaningful gains for metrological applications. In this work, we use a near
unity filled Sr 3D lattice to study the effect of a tunable Fermi-Hubbard
Hamiltonian. The clock laser introduces a spin-orbit coupling spiral phase and
breaks the isotropy of superexchange interactions, changing the Heisenberg spin
model into one exhibiting XXZ-type spin anisotropy. By tuning the lattice
confinement and applying imaging spectroscopy we map out favorable atomic
coherence regimes. With weak transverse confinement, both s- and p-wave
interactions contribute to decoherence and atom loss, and their contributions
can be balanced. At deep transverse confinement, we directly observe coherent
superexchange interactions, tunable via on-site interaction and site-to-site
energy shift, on the clock Ramsey fringe contrast over timescales of multiple
seconds. This study provides a groundwork for using a 3D optical lattice clock
to probe quantum magnetism and spin entanglement
