Spin and density self-ordering in dynamic polarization gradients fields

We study the zero-temperature quantum phase diagram for a two-component Bose-Einstein condensate in an optical cavity. The two atomic spin states are Raman coupled by two transverse orthogonally-polarized, blue detuned plane-wave lasers inducing a repulsive cavity potential. For weak pump the lasers favor a state with homogeneous density and predefined uniform spin direction. When one pump laser is polarized parallel to the cavity mode polarization, the photons coherently scattered into the resonator induce a polarization gradient along the cavity axis, which mediates long-range density-density, spin-density, and spin-spin interactions. We show that the coupled atom-cavity system implements central aspects of the $t$-$J$-$V$-$W$ model with a rich phase diagram. At the mean-field limit we identify at least four qualitatively distinct density- and spin-ordered phases including ferro- and anti-ferromagnetic order along the cavity axis, which can be controlled via the pump strength and detunings. Nondestructive observation of amplitude and phase of the emitted fields bears strong signatures of the realized phase and allows for real-time observation of the underlying dynamics.