myjournal manuscript No.(will be inserted by the editor)

A trapped-ion local field probe

Gerhard Huber, Frank Ziesel, Ulrich Poschinger, Kilian Singer, Ferdinand Schmidt-Kaler

Institut fur Quanteninformationsverarbeitung, Universitat Ulm, Albert-Einstein-Allee 11, 89069 Ulm, Germany

Received: date / Revised version: date

Abstract We introduce a measurement scheme thatutilizes a single ion as a local field probe. The ion isconfined in a segmented Paul trap and shuttled aroundto reach different probing sites. By the use of a sin-gle atom probe, it becomes possible characterizing fieldswith spatial resolution of a few nm within an extensiveregion of millimeters. We demonstrate the scheme byaccurately investigating the electric fields providing theconfinement for the ion. For this we present all theoret-ical and practical methods necessary to generate thesepotentials. We find sub-percent agreement between mea-sured and calculated electric field values.

1 Introduction

The concept to realize and investigate quantum pro-cesses with single ions in Paul traps was subject to animpressive development over the last years. As for quan-tum information with single to a few ions, there hashardly been an experimental challenge that could with-stand this development, consider for instance high fi-delity quantum gates [1], multi-qubit entanglement [2,3], and error correction [4]. There has been a recent de-velopment in ion trap technology that was primarily ini-tiated by the necessity – and possibility – to scale upthe established techniques to process larger numbers ofions. This idea how to scale up an ion trap computer isto create multiple trap potentials, each keeping only asmall, processable number of ions [6]. Time dependentpotentials can then be used to shuttle the ions betweendifferent sections. To generate such spatially separatedpotentials, the respective trap electrodes are segmentedalong the transport direction and supplied individuallywith time dependent voltages. There have also been pro-posals to incorporate the shuttling process into the quan-tum gate time by letting the ion cross laser beams ormagnetic field arrangements [5]. Another application ofthe shuttling ability is the exact positioning of the ion in

the mode volume of an optical cavity to perform cavityquantum electrodynamic experiments. Recently, multi-trap configurations with locally controlled, individualtrap frequencies were proposed to generate cluster statesin linear ion traps to perform one-way quantum compu-tations [18].

This paper presents the implementation of a methodusing a single trapped ion to investigate local electricfields. The method profits from the outstanding accu-racy of spectroscopic frequency measurements and relieson the ability to shuttle the ion in order to reach differentprobing sites. First, we describe the experimental systemand the way the electric trapping and shuttling poten-tials are generated. After that, the measurement schemeusing the ion as a local field probe is presented. Then,the quantitative experimental results are discussed andcompared to numerical simulations of the trapping fields.

2 Experimental field generation

We confine a single, laser cooled 40Ca+-ion in a linearPaul trap. The radial confinement is generated by a har-monic radio-frequency pseudo potential resulting from a25 MHz drive with an amplitude of about 350 Vpp. Thisresults in radial trap frequencies around 3.5 MHz. Forthe axial confinement, 32 pairs of electrodes are avail-able. Each of these 64 segments can be individually bi-ased to an voltage Vi, |Vi| ≤ 10 V. In the trap region theexperiments were performed in, the segments are 250 µmwide and separated by 30 µm gaps (details can be foundin [12]). The voltage data preprocessed by a computerand transmitted to a home-built electronics device. Thisdevice houses an array of 16 serial input digital-to-analogconverters (dac, 50 MHz maximum clock frequency),each of which supplying four individual voltage lines,so that a data bus conveying 16 bit in parallel sufficesto obtain the desired 64 outputs. The amplitude reso-lution of the converters is 16 bit with a measured noiselevel well below the last significant bit. The serial input

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design of the converters keeps the amount of data beingtransmitted in parallel at a minimum. This makes thedesign extendable to a large number of channels.

In this work, we restrict ourselves to the investigationof harmonic potentials. This is rectified by the fact thata resting, laser cooled ion only experiences the very min-imum of the potential. Such potentials are generated byapplying control voltages to the trap segments. Thereby,each specific voltage configuration results in an axial trapwith a given trap frequency; the position of the potentialminimum – this is where the ion resides – can be changedto shuttle the ion between different locations on the trapaxis. Doing so, two requirements have to be met by thetime dependent voltages: (i) The amplitudes must rangewithin the limits given by the dac electronics, ±10 Vin our case. (ii) Each segment’s voltage is to change ascontinuous as possible to avoid unfeasibly high frequen-cies. All these requirements are fulfilled by the field sim-ulation and voltage calculation techniques presented insections A and B, respectively. For the experiments pre-sented here, it is advantageous to perform slow, adiabaticpotential changes, and thus avoiding excess oscillationsof the ion. For that reason, the update rate of digital tovoltage converters was chosen with ∼ 3.3 ms, and a par-allel port connection between computer and dac devicewas used. In an alternative operation modus of the trapsupply device, the data transmission and updated rate issignificantly sped up by using a field programmable gatearray. Then we reach 4 µs even when updating all 64channels simultaneously. For the application presentedhere, it is sufficient to use a Doppler cooled probe ion.The probe’s spatial extent can then be estimated fromthe ion temperature and the consequential extent of themotional wave function to be ∼ 60 nm. This value couldbe reduced to ∼ 5 nm by sub-Doppler cooling and atighter confinement of the ion.

3 The local field probe measurement scheme

To test both the numerical methods and the developedelectronic devices for accuracy, we employ a single ionas a probe for the electrostatic potential. We exploit thefact that it is possible to exactly measure the trap fre-quency of the confining axial potential by spectroscopicmeans. Consider, for example, a voltage configuration

{Vi}(zp) to be tested for; this configuration is meant toresult in a trap at position zp with frequency ω(zp).Then we can use time dependent potentials to shuttlethe ion from a loading position to zp [13], whereby the

final voltage configuration must be {Vi}(zp). The spec-troscopically obtained trap frequency at zp can then becompared to the theoretically expected one, ωsim(zp).

The trap frequency is determined by measuring thedifference frequency of the carrier and the first motionalred sideband (rsb) excitation of the ion in a resolved side-band regime. For that, we excite the quadrupole tran-

z0 zP

(i)cooling

(iii) spectroscopypulse

(iv)shuttling back

(v) fluorescencedetection

camera &photomultiplier

a)

b)

c)

(ii)shuttling

V i V i+1 V i+2V i-1

Fig. 1 Illustration of the measurement procedure: (i) coolingand preparatio of the ion at starting position z0 and (ii) trans-port of the ion to the probing position zp. A spectroscopypulse with a certain detuning excites the ion there (iii). Af-ter shuttling the ion back (iv), the quantum state of the ionis read out (v). After many repetitions for different detun-ings an excitation spectrum of the ion at zp is obtained, fromwhich the trap frequency ω(zp) can be deduced.

sition (4S1/2,mJ = +1/2) ↔ (3D5/2,mJ = +5/2) at729 nm. As the carrier resonance frequency does not de-pend on the trap frequency1, it is sufficient to measurethe rsb frequency at different locations zp.

The measurement scheme is illustrated in figure 1. Itconsists of the following steps:

(i) An initial voltage configuration is chosen to trapand cool the ion at the starting position z0. All lasersnecessary for cooling, repumping, state preparation anddetection are aligned to interact with the ion here. Addi-tionally, z0 is the position where fluorescence emitted bythe ion can be detected by a photomultiplier tube and acamera.

(ii) The ion is shuttled to the probing position zp.This is accomplished by applying voltage configurationsresulting in a series of harmonic potentials whose mini-mum positions lead the ion from z0 to zp. For simplicity,

1 Nevertheless, the carrier frequency was monitored to ex-clude drifts in laser frequency or Zeeman splitting.

A trapped-ion local field probe 3

Fig. 2 Resonances of the first red sideband excitation onthe S1/2 → D5/2 transition. The two peaks are measured atdifferent trap positions along the trap axis. The resonancefrequencies are given with respect to the carrier resonancefcar, so that the trap frequency at the respective trap posi-tion can be read off. The full width at half maximum of thepeaks is 8 kHz and determines the measurement accuracy.The different peak heights stem from slightly different fieldintensities experienced by the ion.

these potentials were chosen such that their harmonicfrequencies were almost constant and close to ω(zp).

(iii) Applying a spectroscopy pulse at the probing

position zp. Now, the voltages are exactly {Vi}(zp). Rest-ing at zp, the ion is exposed to a spectroscopy pulse offixed duration (100 µs). The pulse is detuned from the(carrier) resonance by ∆f . This excites the ion with aprobability P into the upper state |D5/2〉.

(iv) Shuttle the ion back to z0, inverting step (ii).

(v) Having arrived back at the starting position, thestate of the ion is read out by illuminating it on the cool-ing transition. Whenever we detect a fluorescence levelabove a certain threshold, the ion is found in the groundstate |S1/2〉, while a low fluorescence level indicates thatthe ion has been excited to the state |D5/2〉.

The excitation probability for a specific detuning,P (∆f), at the remote position zp is obtained by averag-ing over many repetitions of steps (i) to (v). By varying∆f , a spectrum of the quadrupole excitation at the re-mote position is obtained without moving any lasers orimaging optics but the spectroscopy laser used in step(iii). This is a significant advantage, because that wayeven such trap positions can be investigated that cannotbe directly observed by the imaging devices or reachedby all lasers.

The frequency difference between the red sidebandand the carrier transition yields the trap frequency. Fig-ure 2 shows two rsb resonance peaks obtained at differentpositions within the trap volume.

It is noticeable that each iteration cycle (i-v) includ-ing two ion transports results – due to the binary nature

of the projective readout – in exactly one bit informationabout the spectrum. Therefore, thousands of transports,each relying on the calculated potentials, are performedfor the determination of one frequency ω(zp). The trans-port, however, can be performed so fast (∼ 100 µs) thatits contribution to the overall experiment duration is sec-ondary; this duration is still dominated by cooling anddetection times (∼ ms).

4 Implementation and results

To implement the measurement scheme, we first calcu-

lated voltage sets {Vi}(z), where z covers the whole ex-tent of the trap in steps of 5 µm (For arbitrary positions,the voltages can be interpolated). Each set results in acertain, wanted frequency ωsim(z). That means that foreach arbitrary position z in the trap, there can be founda set of voltages resulting in a potential with its min-imum at z and with trap frequency ωsim(z). Then, inorder to shuttle the ion, we simply subsequently applythe voltage configurations for z = z0 ... zp.

The calculated voltages are tested with high axialresolution, i.e. in small steps of zp, in two far distant re-gions of the trap. Doing this, both small local deviationsshould be detectable and the stability over the wholetrap structure can be tested for. To see a variation inω(zp) when increasing zp, it is of advantage that smallvariations of the trap frequency around its means valueoccur. This is a reliable way to ensure that the ion infact probes the remote position.

Figure 3 shows the expected, simulated trap frequen-cies and the measured ones. The data are in excellentagreement with the predicted frequencies. On both endsof the investigated trap structure, the predicted courseof ωsim(z) is confirmed within the spectroscopic accuracyof 0.6%. The mean deviation of all measured data pointsis only 0.73%. Note that the solid line shown in figure 3is based solely on geometric data from a technical draw-ing of the trap; there is no free parameter that was usedto match the simulations with the measurement.

5 Discussion and outlook

The high accordance between the simulated and the realtrap frequencies, as can be observed in the presented ex-periments, indicates the reliability of at least four inde-pendent contributions: First, the numerical field simula-tion is accurate. This simulation is independent from aspecific set of voltages, but relies on a model of the threedimensional trap design.

Second, the physical realization of this geometry isvery good, i.e. the manufacturing and assembling pro-cess of the micro trap is so precise that it is not lim-iting the field accuracy. This aspect is coming more anmore into the experimental focus, since the miniaturiza-tion of the trap layouts still proceeds. For example, a

4 Gerhard Huber et al.

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Fig. 3 (a) Harmonic trap frequency as a function of the po-sition along the trap axis. The electrode voltages were calcu-lated and applied such that the trap frequency shows a smalloscillatory variation around a mean value of about 1.4 MHzto see the ion proceed along the axis. The solid line showsthe frequency of the wanted harmonic potential. The datapoints are the spectroscopically measured, real trap frequen-cies. The error bars show the uncertainty due to the finiteresonance linewidth. (b) Relative deviation of the measuredfrom the simulated frequency |ω(z)− ωsim(z)|/ωsim(z).

possible axial misalignment ∆x of the two dc electrodewings with respect to each other would mainly resultin an axial shift of the oscillations depicted in fig. 3aby approximately ∆x/2. Additionally, the amplitude ofthese oscillations decreases with increasing ∆x. Indeed,the residuals (fig. 3b) show a small oscillatory behavior.This oscillation signal is shifted by at most ∼ 15 µm withrespect to the trap frequency oscillation. Such an effectcould stem from a small misalignment of the two traplayers (see ref. [12]) or from a small offset of the theoreti-cal and the real, axial zero point of position. We attributethe observed shift to an offset error, since in this case,the amplitude of the oscillation is not decreased, as itis the case in the measured data2. This shift is system-atic and could be corrected for. Third, the calculation ofthe voltages leading to the wanted potentials is highlyreliable. This is also qualitatively confirmed by the suc-cessful shuttling process itself. From our measurement

2 The misalignment error can also be verified to be smallerthan ∼ 10 µm from microscope pictures of the trap.

we find that the generation of the voltages and theirsupply to the trap electrodes is indeed highly accurate.We also conclude that trapping fields from the appliedvoltages are not significantly perturbed by backgroundfields from stray charges.

As the generation of trapping fields in the multi-segment trap was found to be accurate over the wholeconsidered volume, it is possible to tailor potentials forspecial purposes. For quantum computing tasks, a har-monic potential of constant frequency is mostly required.Simulations show that it is possible to obtain fixed fre-quency potentials with a relative frequency deviation onthe order of 10−4. The fast transport of quantum in-formation in harmonic wells also requires a high degreeof control over the trap voltages [17]. The applicationof optimal control methods can help optimizing tailoredtime dependent potentials. Alternatively, one can feed-back information, gained from the measurement pre-sented here, into the generation process to refine theresults iteratively [10].

However, the scope of possible applications of thedescribed techniques is much wider than quantum com-puting tasks: Ions in time dependent potentials were pro-posed to be used as a testbed for quantum thermody-namic processes [14] or for quantum simulation [15,16].The presented method is just a first proof of principle ofusing single atoms as ultra-sensitive probes. The mea-surement principle is not restricted to the investigationof the electric trapping fields. Only slight modificationsallow for the detection of any electric or magnetic fieldwith a nano-meter scaled quantum probe, as an alter-native to cold atom sensors [22,21]. Then, a single atomfield probe might be used as well for probing magneticmicro-structures with a relative accuracy better than10−3. Also, a study of decoherence and heating effectsand their dependence on the ion-electrode separation orion location is in reach.

Financial support from the DFG within the SFB/TRR-21, the German-Israel Science Foundation and by theEuropean Commission within MICROTRAP, EMALIand SCALA is acknowledged.

A Field calcuations

To create a specific electric potential φ(z) on the trapaxis, it is necessary to find the right set of voltages{Vi} being applied to the trap electrodes labeled by i =1, . . . , N . The potential generated by such a set of volt-ages is the linear superposition of the N individual elec-trodes, whereby the contribution of each electrode i isweighted by the applied voltage Vi. After subdividing theaxial position into a grid of M points zj , j = 1, . . . ,M ,we can write the overall potential at any zj by

φ(zj) = φj =

N∑i=1

Aij · Vi

A trapped-ion local field probe 5

Here we introduced the electrode potential matrix Aij . Itdescribes the influence of the i-th electrode to the overallpotential at xj . Each row i of A can be seen as a position-dependent function, describing the potential generatedby electrode i (in units of Vi), when all other electrodesare set to zero voltage. This quantity A is independentfrom the specific voltage and is solely given by the trapgeometry, i.e. the shape and size of the electrode and itsdistance from zj , for instance.

Therewith, the potential generation can be logicallydivided into two parts: The matrix A can be calculatedindependently from voltage constraints and independentfrom the desired potential. Second, for each desired po-tential φ, there has to be found a set of voltages v byinverting the matrix equation above. Tackling the firstproblem, one recognizes that modern segmented trap ge-ometries can be realized in such a geometric complexitythat conventional simulation techniques like the finiteelement method (FEM) fail. Instead, we obtained thepotentials by solving the boundary element problem ofthe segmented trap design. Details can be found in [8,9,11,19].

B Calculation of the voltages

In the following, we address the problem how to obtaina set of voltages {Vi} that generates a given potential φwhen applied to the respective electrodes. The problemis formally solved by matrix inversion as A−1φ. Severalcircumstances make this straight forward ansatz unfeasi-ble: First, there is often no exact solution to the problem,because φ is not an exactly realizable potential (notethat in general, M � N is possible). In this case, anapproximate solution has to be found. Additionally, asa specific electrode does not effectively contribute to thepotential at a far distant point, this electrode’s voltageis ill-determined. These cases have to be treated ade-quately by the algorithm. We solved the inversion prob-lem with a singular-value decomposition of the matrix Ato identify its critical, singular values. The real N ×Mmatrix A is decomposed into the product

A = USWT , (1)

of the unitary matrices U (N ×N) and W (M ×M) andthe diagonal N ×M matrix S with non-negative entriessk, k = 1, . . . ,min(M,N). This decomposition is part ofmany standard numerical libraries and can be performedfor any input matrix A. The inverse can then be writtenas

A−1 = WS−1UT . (2)

This step is numerically trivial, because the unitary ma-trices are simply transposed and the entries of S−1 aregiven by 1/sk. At this point, the advantage of the de-composition becomes obvious, since small values of skindicate an (almost) singular, critical value. One way to

overcome these singular values would be to cut off theirdiverging inverse values. Instead, the Tikhonov regular-ization [20] method implies a more steady behavior asit makes the displacement 1/sk → sk/(s

2k + α2). The

latter expression behaves like the original 1/sk for largevalues sk � α, has its maximum at sk = α and tendsto zero for small, critical values sk � α. From this canbe seen that the choice of the optimization parameter αis a compromise between exactness and boundedness ofthe results. For α = 0, the exact solution (if existent)is obtained, whereas large values of α guarantee smallinverse values and thus bounded voltage results. There-fore, we label the regularized quantities with index α.The approximate solution vα = (V1, . . . , VN ) is

vα = WS−1α UTφ, (3)

with S−1α being the regularized matrix with entries sk/(s2k+

α2).Before the problem of finding an optimal α is ad-

dressed, another constraint regarding time dependentvoltages, i.e. series of voltage configurations, has to beaccounted for. While moving the ion by one step, i.e.from a position zp to z′p, the control voltage should varyas less as possible. This is achieved by extending eqn. (3),

v′α = vα +WDαWTv, (4)

where v represents any previous voltage set, providingtrapping at zp. The second term in eqn. (4) contains adiagonal matrix D with entries dk = α2/(s2k + α2). dktends to zero for sk � α, so that uncritical voltages areaffected only little by the second term. For all criticalvoltages indicated by a value sk << α, however, thefirst term in eqn. (4) vanishes due to the regularizationreplacement and what remains is the contribution fromv0, since then, dk ≈ 1. Here, the choice of α determines,how strong the algorithm tries to generate similar volt-ages in a (time) series of voltage sets.

The algorithm described above minimizes ||Av′α −φ||2 + α||v′α − v||2 with respect to the Euclidian norm.That is, the potential φ is reproduced as good as pos-sible under the constraint that solutions similar to theprevious one are preferred. What remains is to find theproper value of α. Hereby, one has to make a tradeoff be-tween the boundedness of the voltages and their continu-ity. Under practical circumstances requiring |Vi| ≤ Vmax

for some maximal voltage Vmax, α can be iteratively in-creased to fulfill this constraint on the one hand, andto obtain as continuous voltage sets as possible, on theother hand.

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