Oscillating magnetic field effects in high precision metrology
H. C. J. Gan, G. Maslennikov, K. W. Tseng, T. R. Tan, R. Kaewuam, K. J. Arnold, D. Matsukevich, M. D. Barrett
OOscillating magnetic field effects in high precision metrology
H. C. J. Gan, G. Maslennikov, K.-W. Tseng, T. R. Tan, R.Kaewuam, K. J. Arnold, D. Matsukevich, and M. D. Barrett ∗ Center for Quantum Technologies, 3 Science Drive 2, Singapore, 117543 andDepartment of Physics, National University of Singapore, 2 Science Drive 3, Singapore, 117551
We examine a range of effects arising from ac magnetic fields in high precision metrology. Theseresults are directly relevant to high precision measurements, and accuracy assessments for state-of-the-art optical clocks. Strategies to characterize these effects are discussed and a simple techniqueto accurately determine trap-induced ac magnetic fields in a linear Paul trap is demonstrated using Yb + . PACS numbers: 06.30.Ft, 06.20.fb
An ion trap is a widely used tool in atomic physics anda cornerstone system in high precision metrology. Thekey advantages of the ion trap system are the high degreeof control of individual ions and the rigorous assessmentof systematic effects from the environment including thetrapping apparatus itself. In a linear Paul trap, effectsthat have not been given sufficient attention are thosedue to magnetic fields arising from trap-induced rf cur-rents in the electrodes. To our knowledge there are veryfew instances in which these fields have actually beenmeasured or at least an attempt made to quantify theirinfluence on experiments [1–5]. When values were quan-tifiable they were typically a few µ T. Such values wouldhave a significant contribution to many error budgets inhigh precision metrology.The primary effect of ac magnetic fields is to shiftatomic energy levels. However, as observed in [4], theycan also influence the assessment of micromotion, whichcan have further consequences to the validity or accuracyof an experiment. High precision measurements can alsoserve as reference points for other measurements. Hencewe consider it useful to provide a clear description of theeffects these fields have, and provide suggestions as tohow they might be experimentally assessed.The paper is divided into two main sections. In thefirst section the various influences of ac magnetic fieldsare discussed: specifically, the effect on measured Zee-man splittings and shifts of both microwave and opticalclock transitions. For completeness, a brief discussion onthe magnetic blackbody radiation shift is also given. Inthe second section, two methods to measure the ampli-tude of an ac magnetic field are discussed. Both methodsare sensitive to the orientation of the oscillating field rel-ative to an applied static field. The discussion is focussedmainly on ion trap systems, but the effects are relevantto other time-varying fields, such as line noise, which isalso relevant to neutral atom systems. ∗ [email protected] I. AC MAGNETIC FIELD EFFECTS
Throughout the rest of the paper, an applied staticmagnetic field is denoted B and its direction is takenas the quantization axis. The amplitude of an oscillat-ing magnetic field is denoted B and its components or-thogonal to and along the quantization axis are denoted B ⊥ and B z respectively. For any quantity specifying asensitivity to (cid:104) B (cid:105) , the unit µ T − is in reference to theroot-mean-square amplitude of the field. For quantitiesspecifying a sensitivity to B or one of its components,the unit µ T − is in reference to the amplitude of theapplicable field component.The energy shift of | a (cid:105) due to an oscillating magneticfield coupled to | b (cid:105) can be found by direct analogy withan ac stark shift from an oscillating electric field [6]. Withthe magnetic dipole operator M and polarization vector u , the shift is given by δE a = − (cid:104) B (cid:105) (cid:126) (cid:18) |(cid:104) b | u · M | a (cid:105)| ω ba − ω + |(cid:104) a | u · M | b (cid:105)| ω ba + ω (cid:19) (1)where (cid:104)·(cid:105) denotes time averaging and ω ba = ω b − ω a .This expression is simply the magnetic counterpart ofthe expression for an ac stark shift from an oscillatingelectric field [6]. When ω (cid:28) | ω ba | and u = e , thisexpression reduces to the static quadratic Zeeman shiftof | a (cid:105) due to the magnetic coupling to | b (cid:105) . In this case,the effect of the oscillating field can then be accounted forby using B = B + (cid:104) B (cid:105) in the assessment of quadraticZeeman shifts. This appears to be commonly used inthe assessment of magnetic field effects in high accuracyclocks today [2, 7–9]. However it must be stressed thatit only applies when the oscillating field is collinear withthe static field.Coupling between fine-structure levels can be treatedin exactly the same way as for the electric dipole po-larisability and all expressions given in [6] have a mag-netic analogue. However, this is not the case within afine-structure manifold. Oscillating fields collinear witha static magnetic field couple only to neighbouring hy-perfine states with ∆ F = ± , ∆ m = 0. As hyperfinesplittings are often much larger than frequencies of in-terest, the static limit applies and the static quadratic a r X i v : . [ phy s i c s . a t o m - ph ] J u l shift can be used as noted above. Hence, results hereconcern the influence of ac magnetic fields orthogonal tothe applied static field. A. Coupling within a single hyperfine level
The most straightforward case is the coupling betweenneighbouring m states of the same hyperfine level. Fromthe Wigner-Eckart theorem |(cid:104) F, m ± | e ± · M | F, m (cid:105)| = ( g F µ B ) F ∓ m )( F + 1 ± m ) (2)and Eq. 1 gives δE (cid:126) = ± (cid:104) B ± (cid:105) m ± ω z − ω (cid:16) g F µ B (cid:126) (cid:17) , (3)where ω z = g F µ B B / (cid:126) and B ± are the spherical com-ponents of B . Further assuming the field is linearly po-larised, the contributions from each circular componentare equally weighted giving δE (cid:126) = (cid:20) ω z ω z − ω (cid:104) B ⊥ (cid:105) B (cid:21) mω z . (4)This is a modification of the Zeeman shift mω z , with theterm in square parentheses having the interpretation of afractional change in the applied magnetic field. In the dclimit ( ω → B ⊥ . B. Microwave clock transitions
Consider an S / ground-state with a half integer nu-clear spin I . Using the Wigner-Eckart theorem, the shiftin energy (cid:126) δω ± of | I ± / , (cid:105) is (cid:126) δω ± = ± ω ω − ω |(cid:104) S / || M || S / (cid:105)| (cid:126) × (cid:18) (cid:104) B z (cid:105) + 2 I + 1 ∓ I + 1) (cid:104) B ⊥ (cid:105) (cid:19) . (5)As |(cid:104) S / || M || S / (cid:105)| ≈ µ B , the net shift of the clocktransition is δω = | ω | ω − ω µ B (cid:126) (cid:0) (cid:104) B z (cid:105) + (cid:104) B ⊥ (cid:105) (cid:1) . (6) Note that the approximation for the reduced matrix ele-ment uses g J ≈ g I . The clock shift variesby a factor of 2 depending on the orientation of the os-cillating field. In the limit that ω (cid:28) | ω | , the expressionreduces to δω = α z (cid:18) (cid:104) B z (cid:105) + 12 (cid:104) B ⊥ (cid:105) (cid:19) , (7)where α z is the quadratic shift of the clock transition dueto a dc field.This is particularly relevant for the assessment of themagnetic field shifts in the Al + clock at NIST. In theirexperiments, the oscillating field from rf currents inducedin the trapping electrodes was measured by determiningthe shift of the microwave clock transition in either Be + or Mg + as the rf drive power is varied. As discussedin [7], the analysis is based on the Breit-Rabi formula,which is equivalent to assuming the orientation is along z . Consequently the inferred contribution could be twotimes larger. From the numbers given in [2] this would bean error of 1 . × − in their clock assessment. Whilethis doesn’t significantly change the total systematic un-certainties of the clocks reported in [2], future Al + clockswith total uncertainty near 10 − will need to take thisinto account. C. Optical clock transitions
The analysis can be easily applied to other hyper-fine structures. As noted earlier, coupling between fine-structure levels can be treated as for an electric dipolepolarizability [6]. In the limit that the detuning is largerelative to the hyperfine splitting of the upper state, theshift can be broken down into scalar, vector and tensorcomponents. The vector term only applies for circularlypolarised field components and even then do not applyfor m = 0 states or cancel when averaged over Zeemanstates with m values of opposite sign. The tensor termhas a similar dependence as for an electric polarisability.In Lu + for example, coupling to the D level gives ashift for each clock state in D of∆ ω F = − (cid:18) |(cid:104) D || M || D (cid:105)| (cid:19) ω fs ω − ω (cid:104) B (cid:105) (cid:126) × (cid:18) − C ,F (cid:0) θ − (cid:1)(cid:19) , (8)where θ is the angle between the ac field direction and thequantization axis, ω fs is the fine-structure splitting and C ,F is a coefficient that depends only on the angularmomentum quantum numbers for the state of interest.Under various averaging schemes [10–12], only the usualscalar term remains, which has the same quadratic de-pendence as for a static field in the limit that ω (cid:28) ω fs .For clock transitions involving levels with a hyperfinestructure, such as Yb + , Hg + , and Lu + , the clock shiftalso has an orientation dependence not cancelled by av-eraging. For Hg + , the clock shift is given by δω c = α z ( D / , , (cid:18) (cid:104) B z (cid:105) + 23 (cid:104) B ⊥ (cid:105) (cid:19) − α z ( S / , , (cid:104) B (cid:105) , (9)where α z ( D / , ,
0) and α z ( S / , ,
0) are the staticquadratic Zeeman shift coefficient for the upper and lowerclock states, respectively. Note that the shift for the lowerstate is proportional to (cid:104) B (cid:105) , which is a consequence of itszero angular momentum. The clock frequency is averagedover three orthogonal field directions, which replaces eachcomponent with one third of the total, giving δω c = (cid:18) α z ( D / , , − α z ( S / , , (cid:19) (cid:104) B (cid:105) . (10)This gives ∼ − . /µ T compared to the staticvalue of ∼ − . /µ T calculated in [10]. The av-eraging therefore restores the assumed dependence on (cid:104) B (cid:105) albeit at a modified shift coefficient. This wouldnot affect the order of magnitude estimate given in [13].A similar consideration applies to Yb + . However, ow-ing to a near cancellation of the quadratic Zeeman coef-ficients for the upper and lower states, the effect is morepronounced. The clock shift after averaging is given by δω c = (cid:18) α z ( F / , , − α z ( S / , , (cid:19) (cid:104) B (cid:105) , (11)From the values of hyperfine splittings given in [9], thecoefficient is 2 .
24 mHz /µ T compared to − .
18 mHz /µ T for the static case. Thus the correction effectively has thewrong sign when simply adding (cid:104) B (cid:105) as suggested in [9].It is unclear how much this would affect clock assessmentsas reports [9, 14–16] do not elaborate on how or if theac fields are assessed. Measured quadratic Zeeman coeffi-cients vary substantially with values differing by as muchas 12 σ of the claimed uncertainties [15, 16], but it is notalways stated what value is being used. The most cur-rent and accurate value of the quadratic shift coefficientis given in [16], but the reported clock shifts are consis-tent with zero contribution from ac fields. Although itmay well be the case that rf currents are significantly re-duced at different operating conditions, the sensitivity toac currents is 30-fold larger for Yb + compared to Al + .Thus it would seem prudent to consider this effect, partic-ularly in light of experiments investigating the variationof fundamental constants [16, 17].For lutetium, calculations can be easily extended toinclude more hyperfine levels. For each level, the shiftcan be written∆ f F = α F (cid:104) B z (cid:105) + α (cid:48) F (cid:104) B ⊥ (cid:105) . (12)Under hyperfine averaging [12], α F averages to zero butnot α (cid:48) F . In table I, α and α (cid:48) are listed for each hy-perfine level of each clock transition and the hyperfine averaged α (cid:48) is also given. The values quoted are deter-mined from measured hyperfine splittings and do not in-clude the much smaller contributions from neighbouringfine-structure levels. For comparison, the coefficients forother ion-based clocks, under the appropriate averagingschemes, are given in table II. Clearly those candidateshaving a hyperfine structure are significantly more sensi-tive in general and the value for the 848-nm transition in Lu + may seem anomalously small in this regard. Thisis owing to a fortuitous hyperfine structure that balancesthe splittings and suppresses the shift. TABLE I. Quadratic Zeeman shift coefficients for ac mag-netic fields for clock transitions in Lu + : α F applies to fieldsaligned along the quantisation axis, α (cid:48) F applies to perpendic-ular fields. All values are expressed in mHz /µ T . F α F (mHz /µ T ) α (cid:48) F (mHz /µ T ) D (cid:104)·(cid:105) F - 0.20 D (cid:104)·(cid:105) F - -3.98 D (cid:104)·(cid:105) F - 6.38 D. Blackbody magnetic fields
Blackbody radiation also provides a shift contributionfrom the thermal magnetic field. For optical transitionsthis is much less significant than the shift from thermalelectric fields but we include it here for completeness.The thermal magnetic field has a mean squared value of (cid:104) B ( t ) (cid:105) = (cid:126) π c (cid:15) (cid:90) ∞ ω dω exp (cid:16) (cid:126) ωk B T (cid:17) − ≈ (2 . µ T) (cid:18) TT (cid:19) , (14) TABLE II. The quadratic ac magnetic field sensitivities andfractional shifts of different optical frequency standards. For Lu + the dependence is on (cid:104) B ⊥ (cid:105) . All others depend on (cid:104) B (cid:105) . Ion λ (nm) ˜ α z (mHz /µ T ) δf/f ( µ T − ) Hg +
282 -13.7 a − . × − Yb + E2 436 33.8 a . × − Yb + E3 467 2.28 a . × − Sr +
674 0.0031 b . × − Ca +
729 0.014 b . × − Al +
267 -0.072 − . × − In +
236 -0.004 − . × − Lu + ( D ) 848 0.20 c . × − Lu + ( D ) 804 -3.98 c − . × − Lu + ( D ) 577 6.38 c . × − Averaged over three orthogonal axes [10]. b Averaged over Zeeman states [11]. c Hyperfine averaging [12]. For these transitions, dependence ison (cid:104) B ⊥ (cid:105) where T = 300 K. For a given transition it is useful tonote that δω ω = µ B (cid:126) π c (cid:15) (cid:90) ∞ ω − ω ω dω exp (cid:16) (cid:126) ωk B T (cid:17) − µ B (cid:126) π c (cid:15) (cid:18) k B T (cid:126) (cid:19) (cid:90) ∞ y − x x dxe x − − β (cid:18) TT (cid:19) f ( y ) , (17)where y = (cid:126) ω / ( k B T ), β = µ B (cid:126) (cid:126) c (cid:15) (cid:18) k B T (cid:126) (cid:19) ≈ . × − , (18)and f ( y ) = 6 π (cid:90) ∞ y − x x dxe x − . (19)The integral is to be interpreted as the principle valueand is plotted in Fig. 1.For S / microwave clock transitions the fractionalshift is δω ω ≈ . × − (cid:18) TT (cid:19) , (20)where we have used the fact that f ( y ) ≈ f (0) = − ω is the transition frequencyof the contributing M1 transition. For an optical clocktransition, the fractional frequency shift is suppressed by a further factor of ω /ω c . Thus shifts from coupling be-tween hyperfine levels is negligible and we need only con-sider coupling to other fine-structure levels. Even in thiscase, fine-structure splittings are typically one to two or-ders of magnitude smaller than the optical transition, andthere is a further suppression due to f ( y ) for the largersplittings. Hence magnetic BBR shifts are not likely tobe significant in any realistic scenario.To illustrate, the Lu + fine-structure splitting be-tween D to D is approximately 19.2 THz giving, y ≈ .
06 at T = 300 K and f ( y ) ≈ . − . × − or − . × − when including thecontribution from D . Note that y itself is a functionof temperature so this shift is not simply quadratic intemperature as indicated by Eq. 17. Shift of the 804-nmclock transition is similarly found to be − . × − which includes coupling to all other D -states. FIG. 1. The figure shows the the function f ( y ) where y = (cid:126) ω /k B T . II. MEASURING AC MAGNETIC FIELDSHIFTS
With the ever-increasing precision of optical clocks andmeasurements carried out in ion-trap systems, it wouldbe ideal to have a technique to precisely measure theamplitude and orientation of various oscillating fields,specifically the trap-induced rf fields. A standard tech-nique has been to vary the rf confinement and extrapolateany measurable difference to zero [1–3, 5, 7]. This is notalways ideal as averaging times can be very long and amore direct approach would be better. In this sectionwe discuss two complementary approaches: one based onan Autler-Townes splitting induced by B ⊥ [19], and theother on a sideband induced by B z [4]. A. Autler-Townes splitting from an ac magneticfield.
As noted in section I A, matching the Zeeman split-ting to the trap drive rf can result in a Larmor precession.When driven on a connected optical or microwave transi-tion, an Autler-Townes splitting arises [19]. The splittingcan be measured accurately and is a direct measure of B ⊥ . This approach is readily applicable when there is anavailable energy level with an appreciable g -factor and amoderate trap drive frequency. Here we demonstrate thistechnique using Yb + confined in a linear Paul trap. Inthis system, the F = 1 ground-state hyperfine level has g F ≈
1, and the Zeeman splitting can be matched to thetrap drive frequency of Ω rf = 2 π × . ∼ .
15 mT.The experiment is carried out in a four rod linear Paultrap with axial end caps as described in [20, 21]. The trapgeometry and relevant level structure are schematicallyshown in Fig 2. The secular trap frequencies for a sin-gle ion are ( ω r , ω r , ω ax ) / π = (0 . , . , . P / level as described in[22]. Microwave transitions between the F = 0 and F = 1levels are driven using a microwave horn located ∼ . F = 1Zeeman splittings near match the trap drive frequency.The combined field of B ≈ . ∼ (0 . , . , .
42) with respect to the coordinate systemshown in Fig 2. The axes are primed to avoid possibleconfusion with notation introduced earlier for the ac-fieldcomponents. The Z (cid:48) -coil current ( i z ) is used to fine tunethe amplitude of the magnetic field. Over the small tun-ing range used, this primarily changes the amplitude of B by approximately − . µ T / mA with only a smallchange of approximately ± . ◦ in the direction of thefield.The experimental sequence is as follows: for each valueof i z , the ion is first Doppler cooled and optically pumpedinto the | , (cid:105) hyperfine ground state. A 100 µ s mi-crowave pulse is then used to drive the atom to the F = 1level. Successful transfer to F = 1 is determined fromfluorescence collected during resonant excitation of the S / , F = 1 to P / , F = 0 transition and the transferprobability is inferred from 100 experiments. The ampli-tude of the microwave drive is chosen to maximize theresonant population transfer for the target m state ofinterest.Typical microwave frequency scans for fixed i z areshown in Fig. 3. When the Zeeman splitting, ω z , between | , − (cid:105) and | , (cid:105) is near to Ω rf , an Autler-Townes split-ting occurs with the two peaks corresponding to the twodressed states [25] arising from the trap-induced mag-netic coupling. For ω z = Ω rf the peaks are symmetric FIG. 2. Schematic of the experimental setup. A small stackof neodymium magnets is used to augment an existing fieldto create a bias field of B ≈ . Z (cid:48) coil current i z is used to fine tune B with < ◦ change in the field direction over the small scan range ofinterest. A microwave horn is used to drive microwave transi-tions between the F = 0 and F = 1 levels, which has a zero-field separation of ω = 12 , , ,
118 Hz [23]. The first-order Zeeman effect gives ≈
14 MHz/mT for the frequencyshift of | , ± (cid:105) with respect to | , (cid:105) . The quadratic shift ofthe | , (cid:105) → | , (cid:105) transition is α z = 31080 Hz / mT [24]. and the splitting is determined by the strength of the cou-pling. As ω z is tuned away from Ω rf , the peaks becomeasymmetric with a larger separation, and the dominantpeak moves towards the energy of the bare eigenstate.Scans over a range of magnetic fields are shown inFig. 4 for microwave frequencies near to the bare res-onances associated with | , (cid:105) and | , − (cid:105) for the up-per and lower plots, respectively. A small splitting seenat the bottom left of the figure is due to a two-photoncoupling between | , ± (cid:105) when the Zeeman splitting be- FIG. 3. Microwave frequency scans at fixed i z ( B ). De-tunings are given relative to the zero field ground-state hy-perfine splitting. A trap-induced magnetic coupling resultsin an Autler-Townes splitting when the Zeeman splitting, ω z , between | , − (cid:105) and | , (cid:105) is near to Ω rf . The twopeaks correspond to the two dressed states arising from thecoupling. Top, middle, and bottom traces correspond to ω z < Ω rf , ω z ≈ Ω rf , and ω z > Ω rf , respectively. Note theslight difference in the horizontal axis in each case. tween these two states is ∼ rf . Over the magnetic fieldrange used, no other splitting near to the | , (cid:105) bare res-onance is observed due to a ∼
72 kHz quadratic Zeemanshift of | , (cid:105) .Neglecting the contribution from | , (cid:105) , the observedsplitting is given by √ δ + Ω where δ = ω z − Ω rf and Ωis the coupling strength between | , − (cid:105) and | , (cid:105) dueto the trap-induced rf magnetic field. Using Ω as a fitparameter, gives Ω = 2 π × . g F µ B B ⊥ (cid:126) √ . (21)Using g F = 1 and the measured splitting then gives B ⊥ =4 . µ T. FIG. 4. Observed Autler-Townes splittings as a function of i z for microwave frequencies near to the bare resonances asso-ciated with | , (cid:105) and | , − (cid:105) for the upper and lower plots,respectively. Microwave frequency on the vertical axis is rel-ative to the Yb + zero field ground-state hyperfine split-ting. The vertical dashed line corresponds to the value of i z at the minimum splitting as determined from the data:2 π × . The measured splitting is largely independent of thecalibration of B . Moreover it is also insensitive to theexact values of g F and α z . These parameters determinethe location of the splitting but the size of the splitting isalmost entirely determined by Ω. The small dependenceon g F and α z comes from the location and strength ofthe nearby one- and two-photon resonances associatedwith | , (cid:105) . Inclusion of this state in the analysis shiftsthe estimated coupling to Ω = 2 π × . B ⊥ in accordance with Eq. 21. ● ● ● ● ●■ ■ ■ ■ ■ IonPosition ● ■
175 185 195 205 215 2250.0450.0500.055 Z' - coil current ( mA ) P ea ks epa r a t i on ( M H z ) FIG. 5. Measured Autler-Townes splitting as a function of i z with two ions in the trap. In each case one ion is shelved tothe F / dark state throughout the scan. The bright ion wasalways kept at position 1 (blue dots) or position 2 (orangesquares) as shown in Fig. 2. The curves are the fits to thetwo level result as discussed in the text. Displacement of theplots is due to a spatial inhomogeneity in B . However theminimum splitting and hence B ⊥ is fairly constant. Further corrections due to errors in g F , and α z are lessthan the error in determining the minimum value.Strictly speaking the splitting only depends on the e + component of the magnetic field. In principle the e − component could be checked by reversing the field. How-ever an imbalance in the weight of each component wouldimply a significant phase shift between contributing cur-rent sources that would likely be associated with sub-stantial micromotion that could not be compensated bybias fields.As the rf-currents are driven by the trapping fieldsthemselves, a spatial dependence to the ac magnetic fieldcan be expected. To investigate this, a second ion in thelong lived F / level was used to displace the first alongthe trap axis with the separation between ions estimatedto be 8 . µ m. The measured splitting as a function ofmagnetic field for the bright ion at either position alongthe trap axis is shown in Fig. 5. The displacement of thetwo plots indicates that B has a gradient along the trapaxis of about 35 mT/m but B ⊥ remains fairly constant.A more significant variation in B ⊥ can be expected fordisplacements off-axis. This was investigated by applyinga dc bias voltage to one of the rf trap electrodes to movethe ion off-axis. The results are shown in Fig 6, in whichdisplacements were inferred from camera images with anaccuracy of ∼ ∼ . /µ m on the iondisplacement. The linear dependence is expected fromthe four rod geometry of the trap, and suggests a zero inthe ac magnetic field ∼ µ m from the trap center. Thisis not unreasonable given the machining and fabricationtolerances involved in trap construction.In the fortuitous event that the Autler-Townes split-ting is too small to be resolvable, Larmor precession ● ● ● ● ●■ ■ ■ ■ ■◆ ◆ ◆ ◆ ◆▲ ▲ ▲ ▲ ▲▼ ▼ ▼ ▼ ▼ Off - axisdisplacement ● μ m ■ μ m ◆ μ m ▲ μ m ▼ μ m175 185 195 205 215 2250.0450.0500.0550.0600.0650.070 Z' - coil current ( mA ) P ea ks epa r a t i on ( M H z ) FIG. 6. Measured Autler-Townes splitting as a function of i z as the ion is moved off-axis in the radial direction. Distancesare calibrated from camera images with an accuracy of ∼ B ⊥ . would then apply. Two π -pulses on the | , (cid:105) to | , (cid:105) microwave transition separated by a time τ , would seean oscillation of population in F = 1 as a function of τ with a timescale determined by Ω. In general, the result-ing signal might be more complicated depending on howmuch the quadratic Zeeman shift splits the degeneracyof the two Zeeman splittings. For J = 0 to J = 0 tran-sitions this would allow one to quantify line noise if thesplittings could be tuned to near the line noise frequencyand/or its first harmonic as done in [26]. B. Magnetic field induced sidebands
An alternative approach for measuring the ac field is toutilize a magnetic field-induced sideband. First observedin [4], this effect can bias micromotion compensation as italso contributes to the sideband signal. Alternatively, ifmicromotion is properly compensated, the residual side-band could then be attributed to the ac magnetic field.In contrast to the previous section, the effect depends on B z .Far from resonance with a Zeeman splitting, B ⊥ effec-tively modifies the static field, whereas B z modulates theenergy levels. This modulation is formally equivalent to aphase modulation of the driving field with a modulationindex given by β m = ( g (cid:48) F m (cid:48) F − g F m F ) µ B B z (cid:126) ω (22)where the prime denotes excited state quantities and wehave neglected any quadratic shifts. Hence a measure-ment of the sideband to carrier ratio should allow B z tobe extracted. For this to be effective, other sources re-sponsible for a signal at the sideband frequency must beeliminated or at the very least measured. In the case ofthe rf sideband in ion-traps, this is predominantly micro-motion, which has two components: excess micromtion(EMM) and intrinsic micromotion (IMM) [27, 28].To disentangle the contribution from micromotion, itmust be assessed and removed as much as possible. Theobvious strategy would be to first use a transition in-sensitive to magnetic fields to quantify the micromotion,and then use an alternative transition with a large β m to assess the magnetic field contribution. Such a sep-aration is not always possible as in the case of Sr + . Inthat case the techniques demonstrated in [4] can be used.Here we consider Lu + [29, 30] to illustrate the generalconsiderations.For Lu + , the S -to- D clock transition at 577 nmis well-suited to micromotion assessment: power require-ments for driving weak sidebands are reasonable, probingtimes of a few tens of ms are possible without signifi-cant decoherence, and the wavelength provides reason-able coupling to the motion. Any of the clock transitionsconnected to an upper m = 0 state, has a magnetic fieldsensitivity on the order of a few Hz /µ T. Moreover, twoof the transitions are field independent at ∼ . ∼
15 mHz /µ T . At a trapdrive frequency of ∼
30 MHz, β m is completely negligi-ble for these transitions and the sideband signal limitedonly by micromotion. However the | , (cid:105) to | , (cid:105) transi-tion, which has the largest available magnetic sensitivity,has only a modest sensitivity of β m ∼ − /µ T. Thisneeds to be compared to the expected levels of micromo-tion compensation and how well the sidebands could beresolved.A detailed account of micromotion limitations is givenin [28]. The minimum resolvable modulation index is lim-ited by available laser power, laser coherence and IMM.Probing along the trap axis of a linear Paul trap effec-tively eliminates IMM as the rf field amplitude alongthis direction is typically very small. Coherence times onthe D clock transitions would only be limited by theupper state lifetime of ∼
200 ms or thermal dephasing,which can be easily characterized. With a laser powerof 0 . µ m, a 25 ms probe resulting ina near 100% transfer to the excited state at the rf side-band would correspond to a modulation index of ∼ − for either transition. So the accuracy at which assess-ment could be carried out would likely be determined byhow well a π -time can be measured. This is not likely tobe as accurate as the determination of an Autler-Townessplitting.In general, the accuracy via this technique is deter-mined by the available β m : the larger the better. Thisimplies a high g -factor and/or low trap drive frequen-cies both of which facilitate the achievement of magneticfields necessary to observe an Autler-Townes splitting.Nevertheless this approach may still be useful for thoseclocks in which the ac magnetic field shift is small. III. DISCUSSION
In this work the influences of ac magnetic fields havebeen explored and these should be carefully consideredin any precision measurement. For Paul traps, the trap-induced ac fields can be significant and should be consid-ered a mandatory part of any realistic error budget. Forion-based clocks, not only do the ac fields induce a shiftin the clock frequency, they can also influence the properassessment of micromotion as noted in [4]. This could fur-ther influence clock assessments if induced-micromotionis used to calibrate other systematics, for example, theblackbody radiation shift via the static differential polar-izability as done in [31].More generally, magnetic field calibrations are oftencarried out by measuring Zeeman splittings. Althoughthe effect on Zeeman splittings is typically small, it canstill be important in precision measurements. A notableexample is the high accuracy measurement of the D / g J factor in Ca + , which was reported with a fractionalinaccuracy of 2 . × − [32]. This measurement relieson a comparison of Zeeman splittings between the S / and D / states. In principle the ratio of the Zeemansplittings depends only on the ratio of g -factors betweenthe two levels but Eq. 4 modifies that ratio. The trapdrive frequency was not given in the report, but a valueof 20 MHz would give a sensitivity of − . × − / µ T at the static magnetic field used in the experiment.Possible methods to measure the ac field in an ion trapsystem have been discussed. A simple approach using anobserved Autler-Townes splitting demonstrated a < B ⊥ . This method canbe directly applied to Yb + clock experiments for whichac fields could be metrologically significant. If not prop-erly assessed in this system, it would also have signifi-cant repercussions for experiments testing the variationof fundamental constants [16, 17]. The method is alsoapplicable to Hg + clock experiments, in particular themicrowave clock [1], for which the ac magnetic field shiftwas the leading systematic uncertainty.For systems that require much larger fields to observean Autler-Townes splitting, it may be possible to use adifferent species to first characterise the trap. Howeverthis would depend on how stable and reproducible theeffects are, and how they vary spatially. Measurementsshown here indicate the expected strong correlation withmicromotion but this would have to be more extensivelyinvestigated in any given set up. The alternative ap-proach of using a magnetically induced sideband couldalso be used provided it could achieve sufficient accuracy.As the ac currents in an ion trap are driven by the samesource that determines the trapping potential, micromo-tion and ac magnetic fields should be correlated. As thetrap is a predominately reactive load, micromotion andmagnetic fields should be ∼ ◦ out of phase. Therewould also be a spatial correlation but this would likelyhave a rather complex dependence on design and heav-ily dependent on fabrication imperfections. However, itmay still be possible to mitigate these effects by design,particularly as ion traps move to chip-scale fabricationtechnologies [33, 34].The discussion here has been restricted to magneticfields and is a straightforward application of the Wigner-Eckart theorem. As the Wigner-Eckart theorem appliesto any tensor operator, similar considerations should begiven to other fields. In an ion-trap system, the trap-ping field itself will interact with the ion through thequadrupole moment. In this case it will induce couplingsbetween ∆ m = 0 , ± , ± ACKNOWLEDGMENTS
We would like to thank the ion storage group at theNational Institute of Standards and Technology for fruit-ful discussions and bringing our attention to the work in[4]. We acknowledge the support of this work by theNational Research Foundation, Prime Ministers Office,Singapore and the Ministry of Education, Singapore un-der the Research Centres of Excellence programme. Thiswork is also supported by A*STAR SERC 2015 PublicSector Research Funding (PSF) Grant (SERC ProjectNo: 1521200080) and the Ministry of Education, Singa-pore, under the Education Academic Research Fund Tier2 grant (Grant No. MOE2016-T2-1-141). T. R. Tan ac-knowledges support from the Lee Kuan Yew postdoctoralfellowship. [1] DJ Berkeland, JD Miller, JC Berquist, WM Itano, andDJ Wineland. Laser-cooled mercury ion frequency stan-dard.
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