To implement the two-body interactions of equation (3) we turn to the technique of Rydberg dressing25262728, where off-resonant laser light weakly admixes some Rydberg character into the ground state levels , leading to an effective interaction between them. For large laser detunings, the Rydberg dressing acts as a perturbation and the dressed levels predominantly retain their ground state character and remain trapped29. Interactions between two spins i and j arise, as spatially dependent light shifts and of the dressed pair states and , respectively. These pair states are coupled via two photon excitations to doubly excited Rydberg states. Because of multipole–multipole interactions, the energies of the doubly excited Rydberg states vary strongly as a function of the relative position Rij of the atoms, and can exceed typical laser detunings and coupling strengths even at micrometre distances. This strongly affects the light shifts of the dressed pair ground states, thus endowing them with an effective interaction.