Fig. 3: Systematic frequency shifts due to ion motion.
From: Prospects of a thousand-ion Sn2+ Coulomb-crystal clock with sub-10−19 inaccuracy
a Absolute value of the time-dilation shift due to micromotion, averaged over all of the ions, as a function of the number of ions in the trap. As the number of ions increases, some of the ions are located further away from the trap axis, leading to an increase in the average micromotion time-dilation shift. b Absolute value of the net micromotion and secular motion shifts, averaged over all of the ions, as a function of the number of ions in the trap. The net secular motion (micromotion) shift is defined as the sum of the negative time-dilation shift and the positive Stark shift caused by the electric field of the ion trap due to ion motion at the secular motional frequencies (integer multiples of the trap drive frequency). Also shown are estimated uncertainties in how well these shifts can be characterized. The secular motion temperature here and in panel d is assumed to be 120 μK. c Histogram showing the number of ions out of a 1000 ion Coulomb crystal with a net micromotion shift within each bin. Each ion is located at a different position within the trap and thus experiences different micromotion and Stark shifts, but the inhomogeneous broadening associated with these shifts is below 10−18 fractionally and should not impact spectroscopic coherence within the foreseeable future. d Histogram of motional frequency shifts as a function of the frequency of the ion motion, for a 1000 ion Coulomb crystal. All 3000 secular modes have frequencies below 1.5 MHz. For motion at these frequencies, the magnitude of the time-dilation shift is much larger than the Stark shift due to trapping fields and the green bars indicating the net shift are covering up the red bars indicating the time-dilation shift. At the magic trap drive frequency Ωm/(2π) ≈ 225 MHz, the micromotion time-dilation shift is exactly canceled by the corresponding trap Stark shift, but at secular motion sidebands of the trap drive frequency this cancellation is imperfect. At the second harmonic of the trap drive frequency, the Stark shift is much larger than the time-dilation shift and dominates the net shift.
