  We realize the dynamical 1D spin-orbit-coupling (SOC) of a Bose-Einstein
condensate confined within an optical cavity. The SOC emerges through
spin-correlated momentum impulses delivered to the atoms via Raman transitions.
These are effected by classical pump fields acting in concert with the quantum
dynamical cavity field. Above a critical pump power, the Raman coupling emerges
as the atoms superradiantly populate the cavity mode with photons.
Concomitantly, these photons cause a back-action onto the atoms, forcing them
to order their spin-spatial state. This SOC-inducing superradiant Dicke phase
transition results in a spinor-helix polariton condensate. We observe emergent
SOC through spin-resolved atomic momentum imaging. Dynamical SOC in quantum gas
cavity QED, and the extension to dynamical gauge fields, may enable the
creation of Meissner-like effects, topological superfluids, and exotic quantum
Hall states in coupled light-matter systems.
