Experimental demonstration of an efficient number diagnostic for long ion chains

Very long, one-dimensional ion chains are the basis for many applications, in particular in quantum information processing, and reliable diagnostics are needed to quantify them. To that purpose, we have experimentally tested Dubin's model for very long ion chains [Phys. Rev. E 55, 4017 (1997)1063-651X10.1103/PhysRevE.55.4017] which describes the equilibrium state of a one-dimensional correlated system. For chains larger than 100 ions, this diagnostic allows us to determine the number of trapped ions with a precision of 8% in a 95% confidence interval, without counting individual ions. This measurement is based on the experimental determination of the ion-ion distance of the innermost particles and of the trapping potential along the ion chain direction. The agreement of the model with experiments allows one to determine the degree of local homogeneity of the ion crystal in the center of the chain.