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