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Suitability of linear quadrupole ion traps for large Coulomb crystals D. A. Tabor, V. Rajagopal, Y-W. Lin, and B. Odom Department of Physics and Astronomy, Northwestern University 2145 Sheridan Rd., Evanston, IL 60208 (Dated: February 1, 2012) Growing and studying large Coulomb crystals, composed of tens to hundreds of thousands of ions, in linear quadrupole ion traps presents new challenges for trap implementation. We consider severaltrapdesigns,firstcomparingthetotaldrivenmicromotionamplitudeasafunctionoflocation withinthetrappingvolume;totalmicromotionisanimportantpointofcomparisonsinceitcanlimit crystalsizebytransferofradiofrequencydriveenergyintothermalenergy. Wealsocomparetheaxial componentofmicromotion,whichleadstofirst-orderDopplershiftsalongthepreferredspectroscopy 2 axis in precision measurements on large Coulomb crystals. Finally, we compare trapping potential 1 anharmonicity, which can induce nonlinear resonance heating by shifting normal mode frequencies 0 onto resonance as a crystal grows. We apply a non-deforming crystal approximation for simple 2 calculation of these anharmonicity-induced shifts, allowing a straightforward estimation of when n crystalgrowthcanleadtoexcitationofdifferentnonlinearheatingresonances. Intheanharmonicity a pointofcomparison,wefindsignificantdifferencesbetweenthetrapdesigns,withanoriginalrotated- J endcap trap performing better than the conventional in-line endcap trap. 1 3 I. INTRODUCTION unavoidable driven motion at the rf frequency) which ] results in non-resonant transfer of rf energy into ther- h mal energy of the trapped sample (micromotion heat- p Ion trapping by radiofrequency (rf) electric fields was ing). Second, we compare the component of micromo- - first demonstrated in 1954 by Paul in a trap with hy- m tion along the trap axis (the axis of smallest micromo- perbolic electrodes [1]. Attempts to maximize the field- o freetrappingvolumeledtothedevelopmentofthelinear tion),whichinducesfirst-orderDopplershiftsinprecision t spectroscopyonlargeCoulombcrystals. Finally,wecom- a Paul trap by Prestage in 1989 [2]. Although less har- pare trap anharmonicity, which can lead to excitation of . monicthanhyperbolictraps,thefield-freecentralaxisof s nonlinear heating resonances as a crystal grows and its c linear traps often make theirs the preferred geometry for i experiments involving multiple laser-cooled ions, includ- normal mode frequencies are shifted onto a resonance. s To characterize these anharmonicity-induced frequency y ing precision spectroscopy [3, 4], quantum information shifts,weuseasimplenon-deformingcrystalapproxima- h processing [5, 6], and cavity quantum electrodynamics p applications [7]. tion, which allows a straightforward estimation of when [ crystal growth can lead to excitation of different nonlin- When the thermal energy of a trapped ion cloud is ear heating resonances. 3 cooled sufficiently below the repulsive Coulomb interac- v tion energy, the ions settle into a Coulomb crystal with 2 geometry defined by the trapping potential [8, 9]. Crys- 8 II. TRAP DESIGNS tallizationofcertainatomic ionspecieswithclosed-cycle 7 electronic transitions can be accomplished directly by 0 . lasercooling[10,11];crystallizationofotherspecies(e.g., A linear quadrupole ion trap (Fig. 1) is formed from 1 0 molecular ions) can be accomplished sympathetically by four parallel cylindrical electrodes of radius re, held at 1 laser cooling of a co-trapped atomic species [12] or in a separation r0, with an rf voltage signal applied to the 1 situ formation from a pre-cooled reactant [13]. Crystal- electrodes. The resulting electric field confines the ion : lizationcanbepreventedbyvariousheatingmechanisms. motion radially; axial confinement (in the z-direction) is v Twosuchheatingmechanisms,micromotion heating and provided by endcap electrodes (not shown in Fig. 1), to i X nonlinear resonance heating, become more problematic which a static voltage is applied. r as a crystal grows, due to sampling of larger rf fields Near the trap center, the electric potential is given by a and of nonharmonic regions of the trapping potential, V cos(Ω t) respectively. Thus, these heating mechanisms can limit φ(x,y,z,t)= rf rf (x2−y2)+ the minimum temperature obtained for a given crystal r2 0 size and also the maximum obtainable crystal size. Our κV 2z2−x2−y2 work is motivated in part by recent difficulties growing ec( ) (1) z2 2 largeCoulombcrystalsinanon-standardtrapgeometry, 0 leadingtospeculationthataxialrffieldsneartheendcaps where V and Ω are the voltage and frequency of the rf rf (fringing fields) might limit crystal size [14]. applied rf drive, V is the static voltage applied to the ec Inthispaper, wecomparethesuitabilityoffour linear endcap electrodes, z is half the distance between the 0 PaultrapdesignsforgrowthandstudyoflargeCoulomb endcapelectrodes,andκisageometricfactor. Whilethis crystals. First, we compare the total micromotion (the descriptionisaccuratenearthetrapcenterandalongthe 2 FIG.1. Endviewofalinearquadrupoletrap. Asinusoidally FIG.2. Trapdesignsanalyzedinthispaper. DesignsAand varyingvoltageisappliedtotherodelectrodes,withonepair B sharethesamegeometry,butdifferinappliedvoltages. In held 180◦ out of phase with the other. the offset view at right, designs C and D are shown with the nearestplateendcappulledawayfromtherestofthetrapto allow the trap interior to be seen. z-axis, significant deviations can occur elsewhere in the trap interior. The radial motion of an ion in the potential of Eq. (1) TABLE I. Operating voltages applied in different trap de- is described by the Mathieu differential equations, and signs. Here V(t)=+V cos(Ω t). rf rf the stability of this motion is expressed using Mathieu parameters which depend only on V , V , Ω , κ, and V1 V2 V3 V4 rf ec rf the charge-to-mass ratio q/m. Solutions to the Mathieu A V(t) −V(t) Vec Vec equations contain regions of parameter space where ion B V(t) −V(t) V +V(t) V −V(t) ec ec motion is stable [15], and trap parameters are chosen to C V(t) −V(t) V N/A ec operate in one such stable region. D V(t) −V(t) V N/A ec The motion of a trapped ion can be described approx- imately as a superposition of micromotion, in which the ion oscillates at the rf frequency, with secular motion, in optimal ratio r =1.14511r for minimizing the leading- whichtheionoscillatesinatime-independentpseudopo- e 0 tential φ˜ (x,y,z) at slower secular frequencies ω , ω , order contribution to anharmonicity in the rf potential rf x y [18]. DesignC usesthincircularplatesofradius2r +r andω . Ananalyticrelationshipexistsbetweenthetime- e 0 z as endcap electrodes, with a 2 mm radius hole left for dependent rf potential and the time-independent pseu- axial optical access. Design D has four small cylindrical dopotential [16]: endcapelectrodesofradius1.75mm,rotated45◦ around q the z-axis; the central axes of these endcap electrodes φ˜ (x,y,z)= |∇φ (x,y,z,t=0)|2 (2) rf 4mΩ2 rf are on the same radius as the central axes of the rf elec- rf trodes. Our group has built a functional trap whose key The endcap electrodes generate a potential φ , and the geometric features are shared with design D, currently ec total potential governing secular motion is being used to trap 138Ba+ ions. All numerical simulations in this paper use values corresponding to 138Ba+ φ =φ˜ +φ . (3) (q = 1.602×10−19 C, m = 2.292×10−25 kg) and our trap rf ec operating rf frequency (Ω =2π×3.00 MHz). rf Wecomparefourdifferentlineartrapimplementations (Fig. 2, with corresponding voltage given in Table I): a conventional in-line endcap design with two different III. COMPUTATION OF TRAPPING voltage configurations (designs A and B), a plate end- POTENTIALS cap design (design C, similar to a design analyzed in Ref. [17]), and an original rotated endcap design (design Our metrics of trap design comparison, presented in D). In all trap designs, z = 8.5 mm, r = 4 mm, and Sec. IV and Sec. VI, are calculated from the trap po- 0 0 r = 4.5 mm, with r and r chosen to agree with the tentials and their gradients. The potentials φ and φ e 0 e ec rf 3 TABLE II. V and V used in comparisons and calculated rf ec value of κ for each trap. Trap Design V (volts) V (volts) κ rf ec A 800 5.0 0.15 B 800 5.0 0.15 C 800 2.7 0.27 D 800 51 0.014 are found using the finite element method to solve the Laplace equation, with the trap geometries discretized by the meshing software Gmsh [19]. FIG. 3. (Color online.) Contour plots of total micromotion To allow evaluation of the potentials at locations not amplitude,labeledinmicrons. DesignsB andDareindistin- on the mesh, the numerical solutions are fitted to an ex- guishable at this scale. pansion in associated Legendre polynomials l(cid:88)max l(cid:88)max for all designs. We discuss the implications of the result- φ (r,θ,φ)= C rl Pm(cos θ) fit lm l ing uncertainty in Sec. VII. Tuning of other parameters, l=0 m=−lmax such as the fitting order in Eq. (4), is found to cause (cid:40) cos(mφ) for m≥0 negligible variance by comparison. × (4) sin(mφ) for m<0. We exclude terms with odd values of l and m from our IV. MICROMOTION fitting due to symmetry. An analytic form of the gradient of Eq. (4) is straight- Althoughradialandaxialmicromotionvanish,respec- forward to find using tively, along the trap axis and at z = 0, a large crystal necessarily extends beyond these regions. Micromotion ∂Plm(cos θ) = heating results from the transfer of micromotion energy ∂θ into secular energy, which leads to an elevated secular lcosθPm(cos θ)−(l+m)Pm(cos θ)) temperature. Although the scaling is non-trivial, simu- l √ −l , (5) lations show that the micromotion heating rate strongly 1−cos2θ increases with micromotion amplitude and with secular which can be derived from a more general identity given temperature [21, 22]. As a crystal grows, additional ions by [20]. Fitting the Laplace solutions to the above ex- are held at locations with increasingly strong rf fields, pansion thus permits efficient evaluation of φ or its causing them to experience increasingly large micromo- fit gradient ∇φ at arbitrary location from the coeffi- tion. The resulting heating limits the minimum obtain- fit cients {C }. In this way, we determine φ˜ and φ able temperature for a crystal of a given size, and even- lm rf ec up to the scaling set by choosing V and V . Our tually prevents further crystal growth. rf ec choices of these voltages, shown in Table II, are made The electric field created by the rf electrodes may be to equalize the single particle secular frequencies ω and written as z ωr across all designs at experimentally reasonable values →− →− E(x,y,z,t)= E (x,y,z)cos(Ω t). (6) ω =ω =2π×418 kHz and ω =2π×18.9 kHz. 0 rf x y z Single particle secular frequencies are calculated by For an ion held in place in a stationary crystal, in the evaluation of φ˜ and φ along a radial or axial trace →− rf ec limit of small micromotion amplitude E may be taken through the origin from r=−r0 to r=+r0 or z=−z0 to 0 2 2 2 asconstantastheionmoves,andthemicromotionofthe z=z0 respectively. The resulting values fit a polynomial 2 ion is harmonic with amplitude whose quadratic coefficient rapidly converges with finer discretizationofthedomainorincreaseofthepolynomial q A(x,y,z)= |∇φ (x,y,z,t=0)| (7) fitting order. From this coefficient C2, we also find the mΩ2 rf (cid:112) rf values of κ= 2qC /m in Table II. 2 TheeffectofinaccuracyintheLaplacesolutionsisesti- With V as set in Table II, we find φ (x,y,z,t = 0) rf rf matedbyobservingconvergenceofcalculatedvaluessuch and calculate the resulting total micromotion amplitude as κ with increasing meshing density, which is found to A(x,y,z). The resulting micromotion contour is plotted cause variation in all reported results of less than a few in Fig 3. All designs are comparable in the amplitude percent unless otherwise noted as we approach the high- of total micromotion for any realistic trapping volume. est mesh densities eventually used – of order 106 nodes Interpretation of this result is discussed in Sec. VII. 4 it is non-trivial to predict the precise response when it TABLEIII.RMSaverageaxialmicromotionamplitudesover is swept through a given nonlinear heating resonance. the cylindrical volume r <500µm, |z|<z =8.5mm. Listed 0 Predicting the heating response in the non-equilibrium values for A and C change by no more than several percent scenario of a growing crystal, as anharmonic frequency when Laplace solutions are calculated over an order of mag- shifts cause it to cross a resonance, is yet more compli- nitude in grid density. Values for B and D change by up to tens of percent. For averaging, the volume is discretized cated. Modeling of the heating rates and whether cross- by a Cartesian grid with h = 20µm, which is verified to be ing a given subharmonic resonance will actually melt a sufficiently dense. crystal or prevent further growth is beyond the scope of this analysis. Trap Design RMS Axial Micromotion Amplitude (µm) Since it is difficult to determine whether crossing a A 1.5 heatingresonancewilllimitcrystalgrowthorcauseother B 0.04 deleterious effects, in the current analysis, we take a C 2.9 conservative approach of attempting to design a trap in D 0.08 which a growing crystal will avoid low-order resonances altogether. We consider the anharmonicity-induced shifts of the COM frequencies for a non-deforming crystal; the non-deforming approximation is exact in the We also compare the size of the zˆ component of mi- zero-temperature limit. In light of the resonant condi- cromotion in each trap, which can limit the precision of tion of Eq. (8), considering the COM frequency shifts spectroscopy. Due to the smallness of these values (sev- provides a simple estimate of when trap anharmonicity eral orders of magnitude smaller than the total ampli- could limit crystal growth. Other modes of higher order tude) numerical convergence comparable to Fig. 3 is not than the COM mode can also be excited by nonlinear observed. Rather than plot contours, we compute the resonance heating, but the COM shifts are easily calcu- root mean square (RMS) average over a large trapping lable and set the scale for anharmonicity-induced shifts volume for each design in Table III, producing a single in higher modes. quantity whose convergence is verified. See Sec. VII for A previous analysis by the Marseille group (Pedregosa discussion. et. al. [17]) specified a single quantity describing the radial trap anharmonicity averaged over a trapped sample. TheMarseilleapproachhastheutilityofprovidinga V. NONLINEAR RESONANCE HEATING single figure of merit by which to compare different trap designs; however, it does not consider axial anharmonic- Nonlinear resonance heating can occur if a resonance ity (potentially important for long crystals) or provide conditionissatisfiedbetweensecularfrequenciesandthe an estimate of what level of anharmonicity is acceptable rfdrivefrequency. ForlinearPaultraps, theconditionis in order to avoid heating resonances. The analysis we present here is similarly simple to calculate, and at the n ω +n ω +n ω =Ω (8) x x y y z z rf cost of not providing a single figure of merit, seeks to includeaxialanharmonicityandtoprovideamechanism where n , n , and n are integers [23]. The correspond- x y z for determining whether a trap is sufficiently harmonic ingconditionforasingleparticleinahyperbolictraphas to safely work with crystals of some size. been derived [24], with the resonances weakening with larger n. This behavior has been confirmed in both hy- perbolic traps [25] and linear traps [23]. For sufficiently coldtrappedsamples,itisunderstoodthatthefrequency VI. COM FREQUENCY of the center of mass (COM) modes, or other normal modes, should be used in Eq. (8) rather than the single- Forazero-temperaturecrystalinaperfectlyharmonic particlefrequencies[26]. ForaninfinitelylonglinearPaul trapping potential, the COM oscillation frequency is the trap, or for a small crystal in a finite trap, where the ax- same as the secular frequency of a single particle. How- ial rf drive vanishes, a non-deforming crystal is only ex- ever, if the temperature of the crystal is nonzero or if pectedtobeheatedonresonanceswithn =0. However, thetrappingpotentialisanharmonic,thecrystaldeforms z a crystal which is large enough to sample axial fringing during oscillation, and the COM frequency and single fields near the endcaps is expected to be excited also by particle frequency are no longer equal. (Space charge resonances with n >0. shifts, which are well-understood in the limit of each ion z Anharmonicity in the trapping potential causes the movingindependentlyinabackgroundpotentialdescrib- mode frequencies to shift as a crystal grows and samples ing the distributed charge of the other ions, occur at less harmonic regions of the trap potential. Eventually, high temperatures but not for a non-deforming crystal.) theanharmonicity-shiftedmodefrequenciesofagrowing Moleculardynamics(MD)simulationscanfindtheCOM crystal will meet the resonance condition of Eq. (8), and frequency under finite temperature conditions, but this some level of heating will occur, potentially halting fur- approach is computationally intensive for a large crys- ther crystal growth. Even for a crystal of definite size, tal. Tounderstandtheeffectsofanharmonicitywithless 5 computationaloverhead,weintroducetheapproximation of a non-deforming crystal in an anharmonic trap. This approach is computationally simple enough to be useful indesigningtrapsforlargeCoulombcrystalexperiments. A single ion in a zero temperature crystal is held in place by the cancellation of two forces: that due to the trap potential F and that due to the Coulomb repul- trap sionofallotherionsinthetrapF . Intheapproxima- ions tion that the crystal does not deform during oscillation, F remains constant while F +F is, to lead- ions ions trap ing order, a restoring force resulting in simple harmonic motion. F canbecalculatedfromφ ,evaluatedonagrid trap trap (x ,y ,z ) where x = ih,y = jh, and z = kh; i, j, k i j k i j k are integers; h=50µm is the step size used in our calcu- FIG. 4. (Color online.) COM axial secular frequency lation. Toillustrateourapproach, wediscusscalculation as a function of ion number. Curves for Designs A and B of the restoring force for small displacements in the zˆ are overlapping. Crystals are grown until their axial extent reaches±z /2;forequallength,designD growsanapprecia- direction, at fixed x and y . By geometrical symmetry 0 i j bly smaller crystal. Ripples result from finite evaluation grid φ is an even function of z, so the numerically calcu- trap size. (See Sec. VII for discussion of both ion number in D as lated values can be fitted to a polynomial series of even well as finite grid effects.) terms: (cid:12) φ (z)(cid:12) =C +C z2+...+C z10 (9) trap (cid:12) 0 2 10 Fig. 4 shows the change in axial COM frequency in xi,yj eachtrapdesignascrystalvolumegrows. Crystalextent The axial force on an ion in a potential of this form is is determined by evaluation of φ on a Cartesian grid trap withspacingh=50µm,withallpointsbelowafixedcutoff Ftrap(z)=−q(2C2z+...+10C10z9). (10) included, and ion number is then determined by the ion density [27], We now consider an ion whose equilibrium location in a m(cid:15) crystal is z =zk. An expansion of Ftrap about zk gives ρ= 0(ω2+2ω2), (14) q2 z r F (z)=F (z )+(z−z )F(cid:48) (z )+... (11) trap trap k k trap k where ω and ω =ω =ω are the single particle secu- z r x y Since z = z is an equilibrium position for this ion, the lar frequencies. We take this density as a constant; the k Coulomb force due to all other ions F must exactly anharmonicity-inducedpositiondependenceisnegligible. ions cancel the first term of this expansion. In the approxi- Implications of anharmonicity-induced frequency shifts mation of a non-deforming crystal, F is a constant, fornonlinearresonanceheatingarediscussedinSec.VII. ions and the total force on the ion is Radial COM shifts are also computed, with fittings made to radial rather than axial traces of φ . Over trap F +F =−q(cid:0)2C +...+90C z8(cid:1)(z−z ) (12) the volume occupied by the largest crystals in Fig. 4, no trap ions 2 10 k k (cid:12) difference among the designs can be determined at our ≡−k (cid:12) (z−z ), (13) z(cid:12) k level of computational accuracy. By varying all calcula- xi,yj,zk tionparameters,weboundradialCOMshiftsatnomore where k | is a local spring constant describing the than tenths of a percent. z xi,yj,zk strength of the axial restoring force on an ion about its equilibrium location (x ,y ,z ). As seen from Eqs. (10) i j k and (12), when z (cid:54)= 0, anharmonic terms of the trap VII. DISCUSSION k potential contribute to the spring constant k . zk Therestoringforceonacrystalisthesumoftherestor- No difference in micromotion heating rate is expected ing forces on the individual ions. For a non-deforming amongthedifferentdesigns,sincetotalmicromotionam- crystal,theresultisoscillatorymotionatafrequencyset plitudesarecomparable. Indeed,inallcasesaxialmicro- by the averaged local spring constants, with the average motion amplitudes are much smaller than total ampli- taken over the ion locations. In our calculation we aver- tudes, indicating that rf-fringing should not limit crystal age instead over the grid locations, for points interior to growth for these designs. The actual limit on crystal thecrystalvolume. Inthisway,wecanestimatetheaxial size due to micromotion heating is difficult to estimate, COM frequency for a crystal, given φ and a specified butitcaninprinciplebefoundfrommoleculardynamics trap crystal geometry. We compute the COM frequencies for simulations. radial motion by the same method, again using a fitting Axial micromotion, quantified by RMS averaging over polynomial of the same order. a volume bounding any realistic crystal, is at least than 6 an order of magnitude smaller in designs B and D (in which the rf electrodes extend to z > z ) than in A and 0 C (in which the rf electrodes end at z =z ). Since axial 0 micromotion results from fringing of the rf fields, these differences are expected, given the different rf boundary conditions at z = ±z . Given a conservative interpreta- 0 tion of the observed convergence of this average, B and D may be said to have comparable axial micromotion, while the situation in A is much worse and in C worse still. The size of COM frequency shifts during crystal growth, found here to differ among the considered designs,isanindicatorfortheeffectofnonlinearresonance heating on trap operation. As the experimenter grows a crystal, adding ions continuously from an ion source, FIG. 5. (Color online.) Aspect ratio as a function of crystal small COM frequency shifts (say a few percent) are not size,withaspectratioestimatedbycomparisonoftheheights expected to present significant challenge. The COM fre- of the radial and axial potentials along xˆ and zˆrespectively. quencies are unlikely to shift onto a nonlinear resonance A is overlapped by B and not visible. during the intermediate stages of crystal growth, and the experimenter can load the largest crystal possible before checking if different voltages increase the maxi- since the regions nearer the zˆ axis are expected to be mum crystal size. For larger size-dependent shifts, trap- more harmonic than those radially further away. ping voltages could, in principle, be dynamically tuned The ripples in Fig. 4 caused by the finite size of the to avoid resonances as the crystal grows, but dynamic evaluation grid do not interfere with the conclusion that tuning would become increasingly challenging for larger theCOMfrequencyshiftinDissmallerthanintheother shifts. ThusthedesignsexhibitinglargeCOMfrequency designs,asthetrendisclearatmuchsparsergrids,where shifts are expected to be more problematic for growing the ripples are even more pronounced. The Laplace so- large crystals. lutions are found to converge sufficiently to determine the COM shifts to within a few percent, and all other As an example of these concerns, we note that for parameters–the associated Legendre and polynomial se- thefrequenciesconsideredinthisstudy,excitationofthe riesfittingordersandtheevaluationgriddensity–aread- nonlinear resonance |n |+|n | = 7, n = 3 is expected x y z justeduntilthe resultingconvergencein COMfrequency when the axial COM frequency increases by 5-10%, a shifts is not limiting. shiftwhichoccursindesignD onlyforacrystalcontain- ing tens of thousands more ions than in other designs. (Resonances of thisorder and higherhavebeen observed experimentally in a linear trap [23].) The expected abil- VIII. CONCLUSION ity of D to load more ions than the other designs before excitingthisresonanceisadirectresultofthesmallerax- We have compared the suitability of four trap designs ialCOMshiftsinD. Ingeneral,sincetherotatedendcap for experiments on large Coulomb crystals. The com- designDhasthesmallestaxialCOMshiftsitisexpected parison is based on the desire to minimize micromotion tobetheleastsusceptibletononlinearresonanceheating. heating and anharmonicity-induced nonlinear resonance The plots in Fig. 4 terminate at crystals of equal ax- heating. ial length. For a perfectly harmonic trapping potential, We have compared total micromotion as a function of the aspect ratio of a crystal will be independent of size position within the trap volume. Micromotion heating, andequaltotheratioofthesecularfrequencies,whichin related to the total micromotion amplitude, is expected thesetrapsweretunedtobeequalintheharmonictrap- to be comparable in all designs. ping region. Accordingly, eventual differences in aspect WequantifyaxialmicromotionbyRMSaveragingover ratio are due to different degrees of anharmonicity sam- a large trapping volume, finding that the resulting limit pledbythetrappedions. InFig.5, weplottheexpected on spectroscopic accuracy in B and D to be much bet- change in aspect ratio during crystal growth in each de- ter than in either A or C, with C expected to be least sign, calculated by a simple comparison of the heights of suitableofallforprecisionspectroscopyonlargecrystals. theradialandaxialpotentialsalongxândzˆrespectively. In order to compare the susceptibility of the differ- The tendency of design D to maintain larger aspect ra- ent trap designs to nonlinear resonance heating, we have tio than other designs explains the ion numbers seen in used a non-deforming crystal approximation to map the Fig. 4; at equal length, the crystals in designs A, B and anharmonicity-induced axial and radial COM frequency C are radially wider than D and so contain more ions. shifts as a function of ion number. Shifts of COM fre- Intuitively, the tendency of D toward radially narrower quencies, as well as other normal mode frequencies for crystalshelpsexplainthesmalleraxialCOMshiftsinD, which COM shifts set the scale, are of concern since a 7 growing crystal in an anharmonic trap will eventually byagivennonlinearheatingresonancebeingcrosseddur- cross a nonlinear heating resonance, leading to instan- ingcrystalgrowth. Tomakedetailedpredictionsofthese taneously elevated temperatures and potentially halting limitationsonultimatecrystalsizeandtemperature,MD furthercrystalgrowth. Wefindthatanovelrotatedend- simulations could be used. It would also be quite inter- cap trap has the smallest axial COM shifts and observe esting to experimentally investigate the upper limits on that the relative stability of aspect ratio with increasing crystalsizesetbyanharmonicity-inducednonlinearheat- crystal size in this trap makes the result intuitive. We ing resonances. suggestthattrapsaimingtogrowlargeCoulombcrystals ought to minimize axial COM shifts, and we provide a method for quantifying these shifts. ACKNOWLEDGMENTS Theanalysispresentedhereissimpletoimplementand The authors gratefully thank Caroline Champenois, should be predictive of trap behavior, so it is especially Eric Hudson, David Kielpinski, Joan Marler, Steven useful for taking a conservative approach to designing a Schowalter, andStephanSchillerforsharingtheirexper- trap for experiments on crystals of a desired size. How- tise in illuminating discussions. This work is sponsored ever, our analysis is not capable of predicting precisely by NSF Grant No. PHY-0847748 and by NSF IGERT when crystal growth will be limited by micromotion or Grant No. 0801685. [1] W. Paul, Rev. Mod. Phys. 62, 531 (1990). functions, first edition ed. (Clarendon Press, 1947). [2] J. D. Prestage, G. J. Dick, and L. Maleki, Journal of [16] H. G. Dehmelt, Adv. At. Mol. Opt. Phys. 3, 53 (1967). Applied Physics 66, 1013 (1989). [17] J. Pedregosa, C. Champenois, M. Houssin, and [3] T.Rosenband,D.B.Hume,P.O.Schmidt,C.W.Chou, M. 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