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)] 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.

Figure 1

(a) Picture of an ion chain of 155 ions composed from five pictures taken with a translated objective and (b) picture of the central part of the chain. Axes are in pixels whose size is $13\mu \mathrm{m}$ and the optical magnification is 11.58. The scale bar stands for $100\mu \mathrm{m}$. The integration time is 1 s.

Figure 2

Red dots: distance between neighboring ions $a\left(z\right)$ vs their position in the chain, in $\mu \mathrm{m}$, in a 155-ion chain. Blue line: best fit of the inverse of the distance by Eq. (1) where $z$ is replaced by $z-z_0$ and $N$ is fixed to 155. The error bar stands for the $\pm 5$ pixels of the typical full width at half maximum of individual ion picture.

Figure 3

Local density $1/a\left(z\right)$ vs the ion position in the chain, in $\mu \mathrm{m}$, measured for chains containing 134 ions (red circles), 95 ions (blue squares), 47 ions (green triangles). The lines are the best fit by Eq. (1) where $z$ is replaced by $z-z_0$ and $N$ is fixed to the counted value.