Pseudospin and spin-spin interactions in ultra-cold alkali

By Raman coupling two selected hyperfine states of alkali atoms in an external magnetic field cold atom experiments simulate magnetic-like spin 1/2 dynamics in a controlled quantum many-body environment. We describe the effective spin (or pseudospin) degrees of freedom in accordance with choosing one hyperfine state as ``spin-up,'' the other as ``spin-down.'' state We derive the corresponding short-ranged spin-spin interactions which are anisotropic and include the interaction of a particle pseudospin with an effective, short-ranged pseudospin independent internal pseudomagnetic field carried by the other particles. In the degenerate internal state approximation we find that a magnetically controlled Feshbach resonance can vary the ratio and relative sign of spin-independent to spin-dependent interactions. In contrast, the relative magnitudes of the spin-dependent interactions, such as anisotropy, are not affected by a Feshbach resonance. When including the indistinguishability of the interacting particles, we find that the effective interaction takes the form of short-range ising spin interactions. We discuss implications for experimental quantum magnetism simulations.