Fig. 2

Light-mediated coupling. a A localized perturbation of the z-magnetization w (solid black line) will lead to a difference pump rate D (dashed red line) which enhances the original perturbation around x ≈ 0, pumps into the anti-parallel direction providing antiferromagnetic coupling about half a lattice period away (x ≈ ±0.5, the transverse space coordinate is scaled to the lattice period Λ) and pumps into the parallel direction providing ferromagnetic coupling about one lattice period away (x ≈ ±1). The result is an Ising-like antiferromagnetic coupling of super-spins (see text) centered on the lattice sites. (The input pump rate is adjusted for display purposes such that the magnetization and difference pump rate have equal peak amplitude.) b Above threshold, the asymptotic ordered magnetic state is sustained via an optical spin pattern induced, in turn, via diffraction of the transmitted light. c Detailed explanation of mechanism of spin ordering: A spatial modulation in the orientation w changes the refractive index for the σ⁺, σ⁻-components in an opposite direction and hence after the atomic cloud the circular components of the pump acquire an opposite modulation of phase shifts (see Eq. (2)). After propagation in the feedback loop, the resulting amplitude modulations are also opposite. The resulting optical spin structure can then sustain the atomic orientation. Red arrows: Spin-up super-spins, blue arrows: spin-down super-spins, solid arrows: atomic super-spins, dashed arrows: corresponding super-spin of photon helicity