Measurement of electric-field noise from interchangeable samples with a trapped-ion sensor
Kyle S. McKay, Dustin A. Hite, Philip D. Kent, Shlomi Kotler, Dietrich Leibfried, Daniel H. Slichter, Andrew C. Wilson, David P. Pappas
aa r X i v : . [ phy s i c s . a t o m - ph ] F e b Measurement of electric-field noise from interchangeable samples with a trapped-ionsensor
Kyle S. McKay,
1, 2
Dustin A. Hite, Philip D. Kent,
1, 2
Shlomi Kotler, DietrichLeibfried, Daniel H. Slichter, Andrew C. Wilson, and David P. Pappas National Institute of Standards and Technology, 325 Broadway, Boulder, Colorado 80305, USA Department of Physics, University of Colorado, Boulder, Colorado 80309, USA Department of Applied Physics, The Hebrew University of Jerusalem, Jerusalem 9190401, Israel (Dated: February 2, 2021)We demonstrate the use of a single trapped ion as a sensor to probe electric-field noise from inter-changeable test surfaces. As proof of principle, we measure the magnitude and distance dependenceof electric-field noise from two ion-trap-like samples with patterned Au electrodes. This trapped-ionsensor could be combined with other surface characterization tools to help elucidate the mechanismsthat give rise to electric-field noise from ion-trap surfaces. Such noise presents a significant hurdlefor performing large-scale trapped-ion quantum computations.
I. INTRODUCTION
Electric-field noise in ion traps can result in heatingor dephasing of ion motion, causing reduced gate fideli-ties in an ion-trap quantum computer [1]. Measuredheating rates in trapped-ion experiments are typicallyorders of magnitude higher than the heating expectedfrom thermal (Johnson) noise and known technical noisesources. For this reason, the excess heating has beentermed ‘anomalous’ by the ion-trap community. Anoma-lous heating has been attributed to electric-field noisethat emanates from the surfaces of ion-trap electrodes,in large part due to the apparent scaling of the noise as ∼ d − , where d is the distance between the ion and thenearest electrode [1–4]. A d − scaling is expected in thecase of an infinite surface covered with independent fluc-tuating patch potentials, where the radius of the patchesis much smaller than d . Fluctuations of the potentialof an entire electrode, as would be the case for Johnsonnoise, results in noise that scales as d − for electrodeswith dimensions & d [2]. In recent years, further evi-dence for surface origins of the noise has been provided byexperiments that varied the bulk resistivity [5] and tem-perature [6] of trap electrodes and by surface treatments[7–10] that reduced electric-field noise by as much as twoorders of magnitude as compared to electric-field noisefrom untreated electrode surfaces. Despite this improve-ment, the underlying physical mechanisms that result inanomalous heating are not understood.In Ref. [11], a surface-science system was combinedwith a stylus ion trap to enable in situ preparationand characterization of a sample surface combined withelectric-field-noise measurements of the surface. This sys-tem was designed to investigate correlations between theelectric-field noise from surfaces and surface propertiessuch as composition and morphology. However, in thatsystem, electric-field noise from the samples was not de-tectable over the background electric-field noise from theion-trap electrode surfaces. A hypothesis that anoma-lous heating might arise from rf driven surface processeson ion trap electrodes was presented as a possible ex- planation for the inability to detect heating from thesesamples. Here we revisit this experiment with changesto the sample electrode patterning and electrical connec-tions, and present evidence that anomalous heating dueto the sample can be measured by an ion in the stylustrap. We also explore the hypothesis that rf driving maybe correlated with an increase in electric-field noise fromthe sample surface. II. EXPERIMENTAL SETUPA. Stylus Trap
The ion trap in this work (see Fig. 1) has a stylusgeometry [12, 13] to enable samples to be positioned closeto the ion while still allowing for appropriate laser accessto and fluorescence collection from the ion. The trapconsists of a pair of symmetric vertically extruded arc-shaped electrodes to provide rf confinement of an ion,along with one central and four surrounding cylindricaldc electrodes (see Fig. 1). The cylindrical posts, used tocompensate for stray electric fields, have a diameter of60 µ m. The rf electrode arcs both have a central angleof 84 ° and inner and outer radii of 72 µ m and 98 µ m,respectively. The rf arcs and the compensation postsare composed of electroplated Au and are extruded toa height of 180 µ m above the underlying electrical layer.This layer consists of 5 µ m-thick electroplated Au on aquartz substrate. Details of the fabrication process canbe found in Ref. [13].In the absence of a sample, the ion height above thestylus electrodes, d t [see Fig. 1(d)], is determined to be ∼ ± µ m by scanning a laser beam with 20 µ m diame-ter at 1/e intensity across the trap features and/or theion using a servo motor with sub-micrometer step size.The trap electrode position is determined by observingthe obscuration of the transmitted beam by the trap elec-trodes as a function of the beam’s position. The ion posi-tion is determined by measuring the position where thislaser beam, when detuned from a resonant atomic tran- FIG. 1. a) Scanning electron microscope image of an example stylus trap (similar to the trap used in this work). The trapconsists of a pair of symmetric arcs that form the rf electrode, along with one central and four surrounding cylindrical dcelectrodes. The 60 µ m-diameter cylindrical posts and the symmetric arcs are extruded to a height of ∼ µ m. b) Top viewlayout of the 5 mm × ◦ and 86 . ◦ relative to the X - and Z -axis, respectively. c) Schematics of themeander and interdigitated capacitor (IDC) samples along with diagrams of the pattern of the fine featured electrodes in thesample regions. The signal electrode is shown in orange and the grounded electrode is shown in green. The width of the gapsand traces within the ∼ × µ m. d) Schematic showing the distancesinvolved in the heating-rate measurement. The sample distance h relative to the stylus electrode top plane is equal to the sumof the ion-to-trap distance, d t , and the ion-to-sample distance, d s . The view shown in the schematic is along the X -axis. Theschematic is not drawn to scale. sition, induces the maximal ac Stark shift on this transi-tion. The pseudopotential produced by the rf electrodesgives rise to normal-mode directions for a single ion thatare parallel ( x ) and perpendicular ( y ) to the gap betweenthe rf electrodes and perpendicular to the trap substrate( z ). The x , y , and z normal-mode directions are approx-imately overlapped with the X , Y , and Z Cartesian axesshown in Fig. 1(b). The trap frequency ω ratios are ( ω x , ω y , ω z ) /ω z = (0 . , . , Z -axis, see Fig. 1(d)],the sample acts as an additional electrode and affectsthe curvatures and the position of the pseudopotentialminimum. As the trap-to-sample distance h is reduced,the distance d t between the pseudopotential minimum(where the ion is trapped) and the stylus trap electrodesis reduced. Additionally, the motional mode frequenciesincrease and the difference in ω x and ω y ratios decreases.The change in ion height ∆d t is measured by monitor-ing the position of the ion imaged with an objective ontoan electron-multiplied charge-coupled device (EMCCD)camera. The effective resolution of the imaging systemis 13 . d t is given by d t = d t − ∆d t , where ∆d t is a function ofthe trap-to-sample distance h [see Fig. 2(b)].The rf confining potential is provided by an rf syn-thesizer and a helical resonator with a loaded qualityfactor of 160 at 64 MHz to step up the drive voltage.With no sample, ∼
75 V amplitude drive on the rf elec-trode results in secular-motion frequencies of 1 . . . x , y , and z modes of a Mg + ion, respectively. A 1 mT magnetic field is ori-ented at 25 ◦ relative to the X -axis and 86 . ◦ with re-spect to the Z -axis [see Fig. 1(b)]. The magnetic field isused to lift the degeneracies of hyperfine levels and to de-fine a quantization axis that provides well-defined beampropagation directions and light polarizations for drivingcycling transitions and optical pumping. Doppler cool-ing is used to cool the x and y secular modes to a meanthermal occupation n ≈
7, and Raman sideband coolingis used to ground-state cool below n < . z mode is cooled weakly by Doppler laser beams and isnot addressed by the Raman beams. Heating-rate mea-surements were conducted on the y mode using Ramansideband spectroscopy (as described in Sec. II.C). To im-prove the sensitivity to electric-field noise from samples,the electric-field noise from the stylus trap was reducedby using in situ ion bombardment (Ar + , 2 keV, ∼ ) [7]. This results in a heating rate of ∼
39 quantaper second at 4 . d t = 59 µ m, which corre-sponds to an electric-field noise power spectral densityat the ion position of 7 × − V m − Hz − . B. Sample description
The 4 mm × µ m-thick electroplated Au in either a me-ander or an interdigitated capacitor (IDC) geometry [seeFig. 1(c)]. On each sample, one electrode (green) is elec-trically grounded, while the remaining ”signal” electrode(orange) can have electrical potentials applied to it [seeFig. 1(c)]. The samples were fabricated on the samequartz wafer and postprocessed under the same condi-tions; therefore, it is expected that the surface conditionsof the two samples are similar. The trace width in theregion of interest is 6 µ m with 6 µ m gaps. This geom-etry is designed to produce an exponential decrease inthe electric-field strength with distance from the samplewhen potentials are applied to the signal electrode. Thefield resulting from a potential applied to the sample elec-trodes will decay exponentially with distance normal tothe surface with a characteristic distance of ∼ µ m. Thisallows large potentials to be applied to the sample withminimal electric-field changes at the position of the ion. In this way, the dependence of surface electric-field noiseon potentials or currents applied to the sample electrodecan be investigated. The samples are wirebonded to asmall circuit board, connecting the signal and groundedelectrodes of each sample to the center pin and outershield, respectively, of a coaxial cable. Each cable hasa length of ∼
240 mm in vacuum and is connected to avacuum feedthrough. The signal electrodes on the sam-ple can be terminated or biased by connecting appro-priate circuity to the coaxial cable outside the vacuumfeedthrough. The termination is typically a 50 Ω connec-tion to ground unless described otherwise.The sample position is controlled using an
XY ZΘ manipulator that allows for sub-micrometer step adjust-ments. The samples are mounted such that Θ rotationsof the manipulator can select which sample is positionedinto proximity with the trapped ion sensor. The initialsample height h is calibrated by scanning the verticalposition of a focused laser beam and measuring the ob-scuration of the transmitted beam by the trap featuresand sample chip edges. Changes in the height of thesample ∆h are measured using a high-precision digitalcontact sensor with 0 . µ m resolution. The ion-to-sampledistance d s is then given by d s = h − d t = h − ∆h − d t .The meander and IDC samples are tilted (unintention-ally) relative to the stylus trap plane at angles of ∼ . ± . ◦ and ∼ . ± . ◦ , respectively. This angular mis-alignment, combined with lateral positioning misalign-ments, results in a combined uncertainty in h ,meander of7 . µ m and h ,IDC of 20 µ m. C. Electric-Field-Noise Measurement
Previous measurements of electric-field noise from sur-faces have shown strong dependence on distance [2, 3,11, 15, 16], frequency [2, 3, 7, 17–19], and temperature[6, 16, 17, 19]. All of the measurements collected in thiswork were conducted at room temperature and at a sec-ular frequency of 4 . h by adjusting the amplitude of the rfdrive. For the measurement setup presented in Fig. 1(d),the total electric-field noise power spectral density S E,tot measured can be modeled by S E,tot = S E,t + S E,s , (1) S ≡ S E at d = 59 µ m , (2) S E,tot = S ,t ( d t / µ m) − α t + S ,s ( d s / µ m) − α s , (3)˙ n = q S m ~ ω , (4)˙ n tot = ˙ n ,t ( d t / µ m) − α t + ˙ n ,s ( d s / µ m) − α s , (5)where the addition of subscripts tot , t , and s representstotal, trap, and sample, respectively. S represents theelectric-field noise power spectral density at a distanceof 59 µ m (chosen because d t = 59 µ m), ˙ n is the heatingrate in quanta per second, α represents the distance scal-ing exponent, m and q are the mass and charge of theion, ~ is Planck’s constant divided by 2 π , and ω is theangular frequency of the secular motion. In Ref. [11], asimilar stylus trap (similar geometry and surface treat-ment) was used and α t was determined to be 3 .
1. Wewould expect the trap used in this work to exhibit simi-lar characteristics. In this work, we conduct heating-ratemeasurements as a function of distance to extract valuesfor ˙ n ,t , ˙ n ,s , α t , and α s . Heating-rate measurementsare conducted by first cooling the x and y secular modesnear their ground state ( n < . t delay , the ratio of the red and blue Raman sidebandamplitudes r , is used to determine the mean motional oc-cupation n = r/ (1 − r ). The heating rate is determinedas the slope of a linear fit of n vs. t delay . (For a briefdescription of heating-rate measurements see Ref. [20];for a detailed description see Ref. [1]). Prior to conduct-ing heating-rate measurements with proximal samples, anumber of experiments were conducted to confirm thattechnical noise sources that could contribute to ion heat-ing were minimized or excluded. See Appendix A for adetailed discussion of the technical noise sources consid-ered and the experimental checks conducted to charac-terize them. III. MEASUREMENT RESULTS
Heating-rate measurements as a function of distancewere conducted for the meander and IDC samples, wherethe signal electrode of each sample was connected toground via a 50 Ω termination. Figure 2 shows the mea-sured heating rates of the y motional mode as a functionof d t for the meander and IDC sample. As the sampleheight h is reduced from 5 mm to 110 µ m, d t is reducedfrom its initial value of 59 µ m to 39 µ m, as shown in Fig.2(b). The shaded region in Fig. 2(a) shows the expectedheating background of the stylus electrodes, bounded bycurves predicted from previous measurements of the dis-tance scaling of ion heating rates with electrode distance(3 . < α <
4, see Table 1). The measured heating ratesbegin to increase above the predicted range of the trapheating-rate background for d t < µ m, which corre-sponds to values of d s < µ m. A fit to the meandersample data using Eq. 5 results in scaling exponents of α t = 2 . +0 . − . and α s = 3 . +0 . − . and is shown in Fig.3(a) along with weighted residuals shown in Fig. 3(b).Details on the uncertainties in the measured parametersand in the resulting fit can be found in Appendix C.The fitted value of α t = 2 . +0 . − . is consistent with theprevious measurement of α t = 3 . α s = 3 . +0 . − . is consistentwith the scaling for similar two-dimensional (2D) planar electrode surfaces [15, 16] (see Table 1). The fitted heat-ing rate for the samples ( ˙ n ,s ∼ +308 − quanta per sec-ond at d s = 59 µ m for the meander sample) is within therange of reported heating rates in other measurements ofroom-temperature untreated electroplated Au traps [4].The residuals for the double power-law fit suggest goodagreement between the data and the model, but confir-mation of the model requires an independent method forverifying at least one of the scaling exponents α t and α s or the sample-heating-rate scaling parameter ˙ n ,s . Inthis particular experiment, it was not possible to inde-pendently measure any of these parameters. In futureexperiments, d t could be independently controlled withan rf potential applied to the center electrode, allowingfor an independent measurement of α t or to fix the valueof d t independent of sample position, allowing for directmeasurements of α s or ˙ n ,s . Instead, the double power-law fit results were compared to other potential models,such as a single power-law fit (first term only in Eq. 5)or a single power-law fit where the value of α t changeswith d t [2, 21]. The best fit to a single power law yieldsparameters of α t = 5 . +0 . − . and ˙ n ,t = 36 . +6 . − . for themeander sample data is shown in Fig. 3(a). The dis-tribution of the weighted residuals for the single power-law fit shown in Fig. 3(b) display systematic effectsand generally larger magnitude in comparison to thosefor the double-power-law fit, suggesting that the double-power-law model better describes the data. Additionally,the single-power-law exponent α t = 5 .
15 is considerablyhigher than has been seen in other traps (see Table 1).One could alternatively fit the data to a single power-lawmodel where the scaling parameter α t is a function of d t .A fit using this type of model would result in α t = 3 forvalues of d t > µ m, changing to α t = 6 . d t < µ m. This change in α t is the opposite of theproposed behavior in Ref. [21], where α t is expected toget smaller with decreasing values of d t . The change in α t from 3 to 6.5 would also only be expected to occur formuch larger variation in d t , based on Ref. [20]. IV. DISCUSSION
In Ref. [11], an experiment similar to the one describedin this work was conducted, but the behavior was differ-ent from what was observed in this work. In that work,Au samples were positioned at similar distances to thetrap as the samples in this work, but the heating wasconsistent with a single power law with scaling exponentof α t = 3 .
1. The conclusion of that work was that heat-ing was due to electric-field noise from the trap electrodesand that electric-field noise from the samples was not de-tectable. In this work, the heating is consistent withthe sum of power-law dependent terms for the trap andsample (Eq. 5). The reason for these different behav-iors is unclear, but we performed tests to check possiblehypotheses.Known differences between Ref. [11] and this work
FIG. 2. a) Total heating rate ˙ n tot of the y motional mode for the meander and IDC samples vs. ion-to-trap distance d t . Shadedregion represents expected heating rate due to the stylus trap alone, which is bounded by d − scaling and d − scaling. Errorbars are smaller than the markers if not shown. Details on how uncertainties were derived are presented in Appendix C. Thetop axis scale shows measured values of d s for the meander sample data. b) Plot showing the relationship of ion-to-trap distance d t and trap-to-sample distance h . As the sample is positioned closer to the trap, the rf pseudopotential minimum moves closerto the trap. Note: Uncertainty in values of d t, = ± µ m, h ,meander = ± . µ m, and h ,IDC = ± µ m are not shown in eitherplot. Errors in the reported values due to the these uncertainties would result in a common systematic horizontal shift on allpoints, or a common systematic vertical shift as well in panel (b)Trap Type Scaling Exponent Distance Range ReferenceNeedle − . ± . ∼
40 - 200 µ m [3]2D surface electrode − . ± .
12 29 - 83 µ m [15]2D surface electrode − . ± . µ m [16]Stylus ∼ − . µ m [11]Stylus − . ± . µ m this work2D surface electrode − . ± . µ m this workTABLE I. Summary of previous measurements for the distance scaling of ion heating rates with distance to nearest electrode. are the sample electrode geometry and the wiring usedto electrically bias the samples. The samples in Ref.[11] were connected to a vacuum feedthrough usinga 0 .
26 mm-diameter wire approximately 1 m in length.There was only one electrical connection per sample, andno samples had local ground connections; all biasing orgrounding of a sample was via this wire. This wire waswound around a rod for strain relief purposes, resultingin several microhenries of inductance and a few kiloohmsimpedance at the rf drive frequency of 64 MHz. In con-trast, the samples in this work were directly connected toa 50 Ω coaxial cable (see Sec. II.B) and terminated with a50 Ω termination to ground. To test for any dependenceof measured heating rate on the impedance to groundseen by the sample signal electrode at the rf drive fre-quency, we positioned the IDC and meander samples at h ∼ µ m and used a λ / ∼ ∼ FIG. 3. a) Total heating rate ˙ n tot of the y motional mode vs. ion-to-trap distance d t , for the meander sample. The totalheating rate ˙ n tot is fit to the sum of two power laws described in Eq. 5 as well as to a single power law. Error bars are smallerthan the markers if not shown. Details on how uncertainties were derived are presented in Appendix C. Note: The doublepower-law fit line is an approximate visualization of the fit since the relationship between d t and d s is approximated by a fitfunction. For more details, see Appendix B. The top axis scale shows measured values of d s for the meander sample data. b)Residuals for fits shown in part (a). The residuals for the single power-law fit show systematic “u”-shaped deviation from zero,while the residuals from the double power-law fit show a distribution more consistent with statistical noise. The sum of squaresof the residuals (SSR) for the double power-law fit is 0 .
66. For the single power-law fit, the SSR= 9 .
37. Note: Uncertainty invalues of d t, = ± µ m and h ,meander = ± . µ m are not shown in either plot. Errors in the reported values due to the theseuncertainties would result in a common systematic horizontal shift on all points. pendence on applied rf potentials, the signal electrodesof the IDC and meander samples were driven directly viatheir coaxial lines at 200 MHz, a frequency that is not res-onant with any relevant ion transition frequencies. TheIDC and meander samples were positioned at h ∼ µ mand pulses with shaped rising and falling edges to reduceany off-resonant excitations were applied to the samplesduring the delay time between ground-state cooling andsideband thermometry. Drive powers between 0 and 200mW, corresponding to rf voltages on the electrodes withamplitudes of up to ∼
25 V, were applied, but no signif-icant dependence of the measured heating rate on theapplied rf power was observed.
V. CONCLUSIONS
In this work, we measured electric-field noise from in-terchangeable samples with an ion held in a stylus trap.The measured electric-field noise exhibited scaling withthe ion-to-sample and ion-to-trap distances consistentwith independent power-law distance dependencies foreach surface. The power-law exponent, giving the dis-tance scaling, as well as the multiplicative scaling factor,giving the absolute heating rate, were extracted for boththe ion trap and sample surfaces.In the future, our trapped-ion-sensor system could bedeployed in a vacuum chamber as part of a suite of precision surface-science measurement tools, such thattest samples could be studied using multiple surface-characterization tools without breaking vacuum. In thisway, it may be possible to perform high-throughput mea-surements of electric-field noise from test samples and tocorrelate the results with specific surface treatments orwith data from other tools on surface characteristics suchas morphology or chemical composition.To increase the utility of our technique as part of anintegrated surface-characterization system as describedabove, we must reduce the uncertainty in the ion-sampleand ion-trap distances, along with the overall electric-field noise due to the trap. Such improvements offerpromise for understanding the origin of anomalous heat-ing in ion traps, as well as for testing the performance ofmaterials, treatment methods, and designs aimed at re-ducing anomalous heating. This could address one of themajor outstanding challenges for large-scale trapped-ionquantum computing.
VI. ACKNOWLEDGMENTS
The authors acknowledge helpful discussions with S.Glancy and E. Knill and thank M. Kim, A. McFadden,and R. Goldfarb for helpful comments on the manuscript.K.S.M. and P.D.K acknowledge support as associates inthe Professional Research Experience Program (PREP)operated jointly by NIST and the University of ColoradoBoulder under Award No. 70NANB18H006 from the USDepartment of Commerce, NIST. Contributions to thisarticle by workers at NIST, an agency of the US Govern-ment, are not subject to US copyright.
Appendix A: Technical Noise
When interpreting heating-rate results, it is importantto understand sources of technical noise in the experi-ment and to quantify effects that technical noise couldhave on the results. The sources of technical noise con-sidered in our setup include Johnson noise from the elec-trodes and associated low-pass filter elements, noise fromthe digital-to-analog converters (DACs) and amplifiersused to generate dc potentials on the trap electrodes, en-vironmental noise pickup, micromotion dependent noise[22], and stray resonant light. The calculated Johnsonnoise contribution to the electric-field noise at the ion po-sition is more than two orders of magnitude smaller thanthe total measured electric-field noise. The DACs usedin this experiment are heavily filtered to reduce noiseat the secular frequencies and were tested by replacingthe DACs with static voltages generated by resistive di-vision of the potential from a battery. This configura-tion offers lower voltage noise than the DACs. Therewas no change in the measured heating rate when usingthese battery sources instead of the DAC voltage sources.Pickup on trap electrodes was reduced by eliminatingsources of noise at the secular frequencies by measur-ing the radiation of various lab electronics. In-vacuumlow-pass filters on the dc electrodes are located withina few centimeters of the trap to further reduce noise atthe secular frequencies of the ion’s motional modes. Therf electrode is bandpass-filtered by the helical resonator,and the geometry of the rf electrode means that voltagesapplied to the rf electrode do not produce electric fieldsat the pseudopotential null. If the ion position is offsetfrom the rf null, micromotion is induced, which allowscoupling to noise on the rf electrode at frequencies offsetfrom the rf drive (64 MHz ± secular frequencies). Theheating rate due to this source of noise would be pro-portional to the micromotion amplitude. By measuringheating rates as a function of micromotion amplitude andobserving no dependence, this noise source can be shownto be insignificant. Stray resonant light can be absorbedand cause spontaneous emission during the delay time ofthe heating-rate measurement, resulting in recoil heating.Detectors with picowatt sensitivity were used to check forany stray resonant light present in the beam paths [13].The electric field at the ion position from potentialson the signal electrode of the samples was designed to beminimized by the electrode meander or IDC geometry(see Sec. II.B). However, potentials on the wire-bondingpads, or the leads between those pads and the mean-der/IDC regions, could give rise to electric fields at theion position; this effect would allow technical noise on the sample signal electrodes—which have no in-vacuumfiltering—to cause heating of the ion. To understandthe heating rate contribution from technical noise on thesample signal electrode, we performed a measurement ofthe relative electric field strength from the signal elec-trode at the ion position as a function of h . We appliedan oscillating voltage pulse of fixed duration of the form V ( h ) sin ( ω y t ), resonant with the motional secular fre-quency, to the sample electrode and monitored the am-plitude of the resulting coherent excitation of the ion mo-tion. For each value of h , we adjusted the voltage ampli-tude V ( h ) to produce the same total motional excitationof the ion (determined as the excitation level where theresonant fluorescence from the ion during state detectionwas reduced to half of its value without the applied volt-age), indicating the same electric field amplitude at theion. The values of V ( h ) are thus inversely proportionalto the electric field amplitude at the ion produced bya given voltage on the sample signal electrode, for eachvalue of h .We then consider the case of technical noise on thesignal electrode, with voltage spectral density S / V ( ω )that is independent of h . In this case, the resultingelectric-field noise power spectral density at the motionalfrequency S E,tech ( ω y ), and thus the heating rate contri-bution from technical noise ˙ n tech , will depend on h as[ V ( h )] − . Figure 4 plots [ V ( h )] − vs d t for the meanderand IDC samples.Figure 4 shows that [ V ( h )] − scales very differentlyfor the two samples, with neither sample exhibiting ex-pected technical noise heating rates that scale as ∼ d − s .Additionally, an equivalent magnitude of technical noiseon the two samples would result in a technical heatingrate about five times larger for the meander sample thanfor the IDC at d t ∼ µ m. Since the observed heat-ing rates for the two samples are very similar in bothmagnitude and distance dependence [see Fig. 2(a)], weconclude that technical noise on the signal electrodes ofthe samples is not the dominant source of the observedheating rates. Appendix B: Double Power-Law Fit Graphics
The relationship between d t and h shown in Fig. 2(b)was determined empirically from measurements at theplotted points. The double power-law fit shown in Fig.3(a) [residuals in Fig. 3(b)] was conducted based on theseempirically determined points. In order to plot a fit linefor Eq. 5 in Fig. 3(a), it was necessary to interpolatevalues for d s = h − d t for graphical purposes. While itmight be possible to determine an analytical solution todescribe d t vs. h , we instead used an asymptotic expo-nential Lorentzian function to approximate this relation-ship. The equation used to generate the relationship for FIG. 4. Plot of [ V ( h )] − vs. the ion-to-trap distance d t . Theexpected heating rate contribution ˙ n tech from technical noiseon the sample signal electrode is proportional to the plotted[ V ( h )] − . Both the distance scaling and the magnitude of theexpected ˙ n tech are different between the meander and the IDCsamples, likely due to the differences in bond pad and tracegeometry of the two samples [see Fig. 1(c)]. The top x -axislabel represents measured values of d s for the meander sampledata. Uncertainty in values of d t, = ± µ m, h ,meander = ± . µ m, and h ,IDC = ± µ m are not shown. Errors in thereported values due to the these uncertainties would result ina common systematic shift on all points. d t vs. h is h ≈ − . ∗ . d t − . + 0 . −
197 log(1 − d t .
46 )(B1)(values rounded). The fit line shown used this function toapproximate the interpolated values of d s and is thereforeonly a visual guide to the fit. The weighted residualsshown in Fig. 3(b) are an exact representation of the fit. Appendix C: Measurement Uncertainties andStatistics
This section describes how the reported uncertaintieswere determined and their impact on the reported fit pa- rameters. Section II.C describes how a heating rate (lin-ear fit of n vs. t delay ) is measured. The linear heating-rate fit is weighted based on the standard error of themeans of each value of n . The result of the fit is avalue for the heating rate and an uncertainty estimateon that heating rate. The heating-rate measurementsare repeated a number of times, and then a weightedmean and weighted uncertainty for ˙ n (weighted usingthe uncertainties for each linear heating rate fit) is cal-culated. These weighted values are plotted in Fig. 2and Fig. 3. The uncertainties in the values of h , d t ,and d s were discussed in Sec. II. To quantify the impactof the uncertainties, bootstrapping [23, 24] was used togenerate 5000 simulated datasets, where ˙ n , d t , and h were drawn at random from Gaussian distributions withmeans and standard deviations given by the estimatesand uncertainties, respectively, of the calibrated values.The generated datasets were then fit to Eq. 5, and theresulting distributions in fit parameters were used to gen-erate 68% bias corrected confidence intervals for the fitparameters. These asymmetric 68% confidence intervalswere reported in the main text.The meander dataset consisted of measurements con-ducted at ten values of d t and d s . Fitting to Eq. 5results in five degrees of freedom, which is scarce for mul-tiple power-law dependencies. In future measurements ofthe distance dependence, we plan on measuring heatingrates at a larger number of distance values to increase thenumber of degrees of freedom represented by the data.The IDC dataset only contained eight points, leaving justthree degrees of freedom. The limited degrees of freedom,combined with much larger uncertainty in h , resulted ina fit with very large uncertainty values. The distance de-pendence data for the IDC dataset was conducted duringcommissioning of the experiment, which also added addi-tional uncertainty to the results. The complete distancemeasurement was not repeated prior to disassembling theexperiment, and as a result, the confidence level in themeasurements of d s for the IDC data is low. However, forcompleteness, the measured heating rates as a functionof d t were still shown in Fig. 2(a). [1] D. J. Wineland, C. Monroe, W. M. Itano, D. Leibfried,B. E. King, and D. M. Meekhof. J. Res. Natl. Inst. Stand.Technol. , 103(3):259–328, 1998.[2] Q. A. Turchette, D. Kielpinski, B. E. King, D. Leibfried,D. M. Meekhof, C. J. Myatt, M. A. Rowe, C. A. Sack-ett, C. S. Wood, W. M. Itano, C. Monroe, and D. J.Wineland.
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