  We realize a spin-orbit interaction between the collective spin precession
and center-of-mass motion of a trapped ultracold atomic gas, mediated by spin-
and position-dependent dispersive coupling to a driven optical cavity. The
collective spin, precessing near its highest-energy state in an applied
magnetic field, can be approximated as a negative-mass harmonic oscillator.
When the Larmor precession and mechanical motion are nearly resonant, cavity
mediated coupling leads to a negative-mass instability, driving exponential
growth of a correlated mode of the hybrid system. We observe this growth
imprinted on modulations of the cavity field and estimate the full covariance
of the resulting two-mode state by observing its transient decay during
subsequent free evolution.
