Quadrupole Shift Cancellation Using Dynamic Decoupling

Ravid Shaniv, Nitzan Akerman, Tom Manovitz, Yotam Shapira, and Roee Ozeri

Phys. Rev. Lett. 122, 223204 – Published 5 June 2019

Abstract

We present a method that uses radio-frequency pulses to cancel the quadrupole shift in optical clock transitions. Quadrupole shifts are an inherent inhomogeneous broadening mechanism in trapped ion crystals and impose one of the limitations forcing current optical ion clocks to work with a single probe ion. Canceling this shift, at each interrogation cycle of the ion frequency, reduces the complexity in using $N > 1$ ions in clocks, thus allowing for a reduction of the instability in the clock frequency by $\sqrt{N}$ according to the standard quantum limit. Our sequence relies on the tensorial nature of the quadrupole shift, and thus also cancels other tensorial shifts, such as the tensor ac stark shift. We experimentally demonstrate our sequence on three and seven ${}^{88}{Sr}^{+}$ ions trapped in a linear Paul trap, using correlation spectroscopy. We show a reduction of the quadrupole shift difference between ions to the $\approx 10 mHz$ level where other shifts, such as the relativistic second-order Doppler shift, are expected to limit our spectral resolution. In addition, we show that using radio-frequency dynamic decoupling we can also cancel the effect of first-order Zeeman shifts.

Received 11 September 2018

DOI:https://doi.org/10.1103/PhysRevLett.122.223204

Physics Subject Headings (PhySH)

Atomic, optical & lattice clocks Coherent control

Atomic, Molecular & Optical

Authors & Affiliations

Ravid Shaniv, Nitzan Akerman, Tom Manovitz, Yotam Shapira, and Roee Ozeri

Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 7610001, Israel

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Vol. 122, Iss. 22 — 7 June 2019

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Images

Figure 1
Experimental sequences scheme. The sequence pulses are shown in different colors for different rf phases, different pulse times and different rf frequencies. In addition, a geometric representation legend for the angular momentum operators direction for each pulse is displayed. $J_{i}$ represents pulses resonant with the excited large spin manifold, whereas $S_{i}$ represents pulses resonant with the ground state. Red operations mark the opening and closing optical Ramsey $π / 2$ pulses.
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Figure 2
Quadrupole shift cancellation results. (a) Three ions image on an EMCCD camera, along with position-dependent magnetic field (M) and quadrupole shift (Q). (b) Parity oscillations of ions (1,2) and (2,3) in orange and blue circles. Upper and lower plots correspond to Ramsey and quadrupole shift cancellation experimental sequences (see Fig. 1), respectively. The measured relative quadrupole shift is $\sim 3.6 Hz$ . Here the chosen superposition was $\frac{1}{\sqrt{2}} (| 5 S_{\frac{1}{2}}, - \frac{1}{2} ⟩ + | 4 D_{\frac{5}{2}}, - \frac{3}{2} ⟩)$ . Upper and lower plots exhibit different mean parity frequency due to partial cancellation of Zeeman shifts in the DD sequence. (c) Quadrupole cancellation parity fringes with longer integration time. Frequencies were estimated by averaging frequencies from hundreds of minutes-long Ramsey experiments (Grey dots). (I,II) Parity fringes of (1,2) and (2,3) pairs, respectively, for the above superposition after integration of 20.5 hours. (III,IV) Parity fringes of (1,2) and (2,3) pairs, respectively, for $\frac{1}{\sqrt{2}} (| 5 S_{\frac{1}{2}}, - \frac{1}{2} ⟩ + | 4 D_{\frac{5}{2}}, \frac{1}{2} ⟩)$ superposition, after integration of 14 hours. Empty circles are the average of all the data (grey) points and solid lines are maximum likelihood fits to the data, both exhibit similar frequencies agreement.
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Figure 3
Seven ions quadrupole shift and magnetic gradient cancellation experimental results. (a) Seven ion chain imaged on a EMCCD camera. (b) Relative quadrupole shift ( $Δ ω_{Q}$ ) measurement. The shift (red circles) is relative to ion number 4, and it is given by $\frac{1}{2} | | ω_{i, 4} | - | ω_{8 - i, 4} | |$ for the $i$ th ion frequency. (c) Magnetic field gradient shift ( $Δ ω_{M}$ ) measurement. Ion distance is calculated from trap parameters, and the shift (red circles) is calculated as $\frac{1}{2} | | ω_{i, j} | + | ω_{8 - i, 8 - j} | |$ . A linear fit (turquoise solid line) yields $\approx 16.4 Hz / μ m$ . Standard deviations for both (b) and (c) range from 60 to 130 mHz and therefore are too small to be presented. (d) Quadrupole and magnetic field gradient shifts cancellation experimental results, of all 21 correlations. (e),(f) Correlations (1,2) and (1,7) and their fit as examples for inhomogeneous shifts measurement experiment (grey circles and solid lines), respectively, along with their counterparts in the cancellation experiment.
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