Effect of trapped-ion heating on generalised Ramsey methods for suppressing frequency shifts caused by a probe field in atomic clocks

Based on a simple one-dimensional model, we consider the effect of trapped-ion heating on the efficiency of generalised Ramsey methods for the probe-induced frequency shift suppression in optical frequency standards based on single ultracold ions. It is shown that ion heating reduces the efficiency of all previously proposed one-loop generalised Ramsey methods, leading in all cases to the appearance of a linear dependence of the reference frequency shift on the residual field frequency shift. Two versions of the most heating stable generalised Ramsey schemes, for which the proportionality coefficient in this dependence is minimal, are demonstrated.