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arXiv:1505.05907 · Journal of Applied Physics

Precise determination of micromotion for trapped-ion optical clocks

J. Keller, H. L. Partner, T. Burgermeister, T. E. Mehlstäubler

Published Sep 8, 2015• 124 citations
arXivHTMLPDF

Abstract

As relative systematic frequency uncertainties in trapped-ion spectroscopy are approaching the low 10−18 range, motional frequency shifts account for a considerable fraction of the uncertainty budget. Micromotion, a driven motion fundamentally connected to the principle of the Paul trap, is a particular concern in these systems. In this article, we experimentally investigate at this level three common methods for minimizing and determining the micromotion amplitude. We develop a generalized model for a quantitative application of the photon-correlation technique, which is applicable in the commonly encountered regime where the transition linewidth is comparable to the rf drive frequency. We show that a fractional frequency uncertainty due to the 2nd-order Doppler shift below |Δν/ν|=1×10−20 can be achieved. The quantitative evaluation is verified in an interleaved measurement with the conceptually simpler resolved sideband method. If not performed deep within the Lamb-Dicke regime, a temperature-dependent offset at the level of 10−19 is observed in resolved sideband measurements due to sampling of intrinsic micromotion. By direct comparison with photon-correlation measurements, we show that the simple to implement parametric heating method is sensitive to micromotion at the level of |Δν/ν|=1×10−20 as well.

DOI
10.1063/1.4930037

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As relative systematic frequency uncertainties in trapped-ion spectroscopy are approaching the low 10−18 range, motional frequency shifts account for a considerable fraction of the uncertainty budget. Micromotion, a driven motion fundamentally connected to the principle of the Paul trap, is a particular concern in these systems. In this article, we experimentally investigate at this level three common methods for minimizing and determining the micromotion amplitude. We develop a generalized model for a quantitative application of the photon-correlation technique, which is applicable in the commonly encountered regime where the transition linewidth is comparable to the rf drive frequency. We show that a fractional frequency uncertainty due to the 2nd-order Doppler shift below |Δν/ν|=1×10−20 can be achieved. The quantitative evaluation is verified in an interleaved measurement with the conceptually simpler resolved sideband method. If not performed deep within the Lamb-Dicke regime, a temperature-dependent offset at the level of 10−19 is observed in resolved sideband measurements due to sampling of intrinsic micromotion. By direct comparison with photon-correlation measurements, we show that the simple to implement parametric heating method is sensitive to micromotion at the level of |Δν/ν|=1×10−20 as well.
In order to improve the short-term stability of trapped-ion optical clocks, we are developing a frequency standard based on 115 In + / 172 Yb + Coulomb crystals. For this purpose, we have developed scalable segmented Paul traps which allow a high level of control for multiple ion ensembles. In this article, we detail on our recent results regarding the reduction of the leading sources of frequency uncertainty introduced by the ion trap: 2nd-order Doppler shifts due to micromotion and the heating of secular motion, as well as the black-body radiation shift due to warming of the trap. We show that the fractional frequency uncertainty due to each of these effects can be reduced to well below 10 -19 .
The uncertainty of the ac Stark shift due to thermal radiation represents a major contribution to the systematic uncertainty budget of state-of-the-art optical atomic clocks. In the case of optical clocks based on trapped ions, the thermal behavior of the rf-driven ion trap must be precisely known. This determination is even more difficult when scalable linear ion traps are used. Such traps enable a more advanced control of multiple ions and have become a platform for new applications in quantum metrology, simulation, and computation. Nevertheless, their complex structure makes it more difficult to precisely determine its temperature in operation and thus the related systematic uncertainty. We present here scalable linear ion traps for optical clocks, which exhibit very low temperature rise under operation. We use a finite-element model refined with experimental measurements to determine the thermal distribution in the ion trap and the temperature at the position of the ions. The trap temperature is investigated at different rf-drive frequencies and amplitudes with an infrared camera and integrated temperature sensors. We show that for typical trapping parameters for In+, Al+, Lu+, Ca+, Sr+, or Yb+ ions, the temperature rise at the position of the ions resulting from rf heating of the trap stays below 700 mK and can be controlled with an uncertainty on the order of a few 100 mK maximum. The corresponding uncertainty of the trap-related blackbody radiation shift is in the low 10−19 and even 10−20 regime for 171Yb+(E3) and 115In+, respectively.
We recently reported a 5×10-17level frequency comparison of two40Ca+optical clocks, one of which has fractional frequency uncertainty of 3.4×10-17, limited by the BBR effect. Here we present progress towards a40Ca+optical clock with uncertainty at the 10-18level. The differential static scalar polarizability Δα0 of the40Ca+ion clock transition is obtained by precisely measuring the “magic” rf drive frequency. Δα0 is measured as -7.2653(44)x10-40J m2/V2, the blackbody radiation shift is then calculated to be 0.3790(2) Hz at 300 K. The contribution of the blackbody shift coefficient to the uncertainty of the optical clock at room temperature is reduced to 5.6×10-19, the excess micromotion induced clock uncertainty is also reduced to the 10-19level by choosing the appropriate trap drive frequency. With a suppression of the BBR field temperature uncertainty to the sub-Kelvin level and improved clock stability, a robust, relatively low-cost40Ca+ion clock with uncertainty at 10-18level can be made.
For experiments with ions confined in a Paul trap, minimization of micromotion is often essential. In order to diagnose and compensate micromotion we have implemented a method that allows for finding the position of the radio-frequency (rf) null reliably and efficiently, in principle, without any variation of direct current (dc) voltages. We apply a trap modulation technique and focus-scanning imaging to extract three-dimensional ion positions for various rf drive powers and analyze the power dependence of the equilibrium position of the trapped ion. In contrast to commonly used methods, the search algorithm directly makes use of a physical effect as opposed to efficient numerical minimization in a high-dimensional parameter space. Using this method we achieve a compensation of the residual electric field that causes excess micromotion in the radial plane of a linear Paul trap down to $0.09\phantom{\rule{4pt}{0ex}}\text{V}\phantom{\rule{0.16em}{0ex}}{\text{m}}^{\ensuremath{-}1}$. Additionally, the precise position determination of a single harmonically trapped ion employed here can also be utilized for the detection of small forces. This is demonstrated by determining light pressure forces with a precision of 135 yN. As the method is based on imaging only, it can be applied to several ions simultaneously and is independent of laser direction and thus well suited to be used with, for example, surface-electrode traps.
We report on the design of a segmented linear Paul trap for optical clock applications using trapped ion Coulomb crystals. For an optical clock with an improved short-term stability and a fractional frequency uncertainty of 10−18, we propose 115In+ ions sympathetically cooled by 172Yb+. We discuss the systematic frequency shifts of such a frequency standard. In particular, we elaborate on high-precision calculations of the electric radiofrequency field of the ion trap using the finite element method. These calculations are used to find a scalable design with minimized excess micromotion of the ions at a level at which the corresponding second-order Doppler shift contributes less than 10−18 to the relative uncertainty of the frequency standard.
The accurate evaluation of the second-order Doppler frequency shift (SODFS) of trapped ions in a linear Paul trap has been studied with experiments and molecular dynamics (MD) simulations. The motion of trapped ions in the trap has three contributions, and we focus on the ion excess micromotion, which is rarely discussed when evaluating the SODFS of ions. Based on the hypothesis that the ion density is uniformly distributed in the radial direction, we propose a model to accurately evaluate the total SODFS for an ion ensemble with temperature less than 100 mK, which consists of three contributions of ion motion. The effectiveness of the model has been verified by MD simulation, and the experimental results of the excess micromotion of trapped ions are consistent with the theory. We believe that our model offers advantages in accurately evaluating the SODFS for the ion trap, especially those of laser-cooled ion microwave clocks based on large ion clouds.