  Laser cooled and quantum degenerate atoms are widely being pursued as quantum
simulators that may explain the behavior of strongly correlated material
systems, and as the basis of today's most precise sensors. A key challenge
towards these goals is to understand and control coherent interactions between
the atoms. Here, we observe long-range exchange interactions mediated by an
optical cavity, which manifest as tunable spin-spin interactions on the pseudo
spin-1/2 system composed of the millihertz linewidth clock transition in
strontium. We observe the so-called one axis twisting dynamics, the emergence
of a many-body energy gap, and signatures of gap protection of the optical
coherence against certain sources of decoherence. These effects manifest in the
output of a pulsed, superradiant laser operating on the millihertz linewidth
transition. Our observations will aid in the future design of versatile quantum
simulators that take advantage of the unique control and probing capabilities
of cavity QED and the rich internal structure of long-lived Sr atoms. They also
open a route for the next generation of atomic clocks that utilize quantum
correlations for enhanced metrology.
