Spin squeezing dynamics and large-spin analogues in an optical lattice clock

Spin squeezing has been studied for decades as a means to overcome the so-called standard quantum limit for measurement precision. Despite numerous poof-of-principle experiments, however, spin squeezing has yet to push the state-of-the-art in any practical sensing application. We summarize a recent proposal to generate spin squeezing dynamics of two-level Sr-87 atoms in a 3D optical lattice clock, a world-class measurement system. Our proposal combines interactions and spin-orbit coupling to generate spin-squeezed states that are robust to typical sources of experimental noise. We then discuss generalizations of our protocol to the case of multilevel fermions with SU(n)-symmetric interactions, which have been experimentally realized with nuclear spin degrees of freedom. We show how the multilevel generalization of interactions, spin-orbit coupling, and external driving fields can be treated in a simple, unified form. Finally, we discuss prospects to explore the rich dynamics of interacting multilevel systems, such as multilevel spin squeezing and SYK-like models that may feature fast scrambling behavior.