Macroscopic singlet states for gradient magnetometry

We present a method for measuring magnetic field gradients with macroscopic singlet states realized with ensembles of spin-j particles. While the singlet state is completely insensitive to homogeneous magnetic fields, the variance of its collective spin components is highly sensitive to field gradients. We compute the dynamics of this variance analytically for a chain of spins and also for an ensemble of particles with a given density distribution. We find an upper bound on how precisely the field gradient can be estimated from the measured data. Based on our calculations, differential magnetometry can be carried out with cold atomic ensembles using a multipartite singlet state obtained via spin squeezing. On the other hand, comparing the metrological properties of the experimentally prepared state to that of the ideal singlet can be used as further evidence that a singlet state has indeed been created.