A Trapped-Ion Clock Corrected at the Limit Density Scale

Trapped-ion clocks reach fractional systematic uncertainties near 1e-17, at which level small environmental corrections must each be accounted. This paper asks a bounded question: how large is the correction associated with the density Limit, which caps the spin-density a material region can sustain, when applied to a laboratory clock? For an aluminium-ion quantum-logic clock at 1.121 PHz in a laboratory of ordinary density, the computed fractional correction is 9.41e-49, a frequency shift of 1.05e-33 Hz. The clock's ordinary systematic budget totals a fractional 4.89e-17. The Limit correction is therefore some thirty-two orders of magnitude below the systematics that actually govern the clock, and the paper reports that plainly: at laboratory densities the Limit is not a clock correction that any foreseeable instrument could register. The calculation is exact arithmetic on stated quantities; nothing was measured.