Introduction

Richard Feynman was one of the first to recognize that quantum systems of sufficient complexity cannot be simulated on a conventional computer1. He proposed to use a quantum mechanical system instead. A universal quantum computer would be suitable, but practical implementations are a decade away at best. However, universality is not required to simulate specific quantum models. It is possible to custom-build an analogue quantum simulator (AQS) that allows for preparation of fiducial input states, faithful implementation of the model-specific dynamics and for access to the crucial observables. Simulations on such AQSs could impact a vast variety of research fields2, that is, physics3, chemistry4 and biology5, when studying dynamics that is out of reach for numerical simulation on conventional computers.

Many experimental platforms have been suggested to implement AQSs6,7,8,9. Different experimental systems provide certain advantages in addressing different physics. Results that are not conventionally tractable may be validated by comparing results of different AQSs simulating the same problem10,11. Over the last two decades, many promising proof-of-principle demonstrations have been made using photons6, superconductors7, atoms8 and trapped atomic ions9. Trapped ions in particular have seen steady progress from demonstrations with one or two ions12,13,14,15,16,17,18 to addressing aspects of quantum magnets19 with linear strings of 2–16 ions13,20 and self-ordered two-dimensional crystals containing more than 100 ions21. Ions are well suited to further propel the research since they provide long-range interaction and individual, fast controllability with high precision22.

Two-dimensional trap-arrays may offer advantages over trapping in a common potential, because they are naturally suited to implement tuneable couplings in more than one spatial dimension. Such couplings are, in most cases, at the heart of problems that are currently intractable by conventional numerics10,23. Our approach is based on surface-electrode structures24 originally developed for moving ion qubits through miniaturized and interconnected, linear traps as proposed in refs 25, 26. This approach is pursued successfully as a scalable architecture for quantum computer, see, for example, ref. 27. For AQSs, it is beneficial to have the trapped ion ensembles coupled all-to-all so they evolve as a whole. This is enabled by our array architecture with full control over each ion. Individual control allows us to maintain all advantages of single trapped ions while scaling the array in size and dimension28,29,30.

Optimized surface electrode geometries can be found for any periodic wallpaper group as well as quasi-periodic arrangements, as, for example, Penrose-tilings29. A first step, trapping of ions in two-dimensional arrays of surface traps, has been proposed15 and demonstrated31. Boosting the strength of interaction to a level comparable to current decoherence rates requires inter-ion distances d of a few tens of micrometres. Such distances have been realized in complementary work, where two ions have been trapped in individually controlled sites of a linear surface-electrode trap at d between 30 and 40 μm. The exchange of a single quantum of motion, as well as entangling spin–spin interactions have been demonstrated in this system32,33. The increase in coupling strength was achieved with a reduction of the ion-surface separation to order d and the concomitant increase in motional heating due to electrical noise. Recently, methods for reducing this heating by more than two orders of magnitude with either surface treatments34,35,36 or cold electrode surfaces37,38,39 have been devised.

Here, we demonstrate the precise tuning of all relevant parameters of a two-dimensional array of three ions trapped in individually controlled harmonic wells on the vertices of equilateral triangles with side lengths 80 and 40 μm. In the latter, Coulomb coupling rates32 approach current rates of decoherence. Dynamic control permits to reconfigure Coulomb and laser couplings at will within single experiments. We initialize fiducial quantum states by optical pumping, Doppler and resolved sideband cooling to near the motional ground state. Our results demonstrate important prerequisites for experimental quantum simulations of engineered two-dimensional systems.

Results

Trap arrays and control potentials

Our surface ion trap chip is fabricated in similar manner to that described in ref. 40 and consists of two equilateral triangular trap arrays with side length of ≃40 and ≃80 μm, respectively (Fig. 1a,b), both with a distance of ≃40 μm between the ions and the nearest electrode surface. The shapes of radio-frequency (RF) electrodes of the arrays are optimized by a linear-programming algorithm that yields electrode shapes with low fragmentation, and requires only a single RF-voltage source for operation29,30. To design different and even non-periodic arrays for dedicated trap distances, we can apply the same algorithm to yield globally optimal electrode shapes29. Resulting electrode shapes may look significantly different, but will have comparable complexity, spatial extent and the same number of control electrodes per trap site. Therefore, we expect that different arrays will not require different fabrication techniques (Methods). The two arrays are spaced by ≃5 mm on the chip, and only one of them is operated at a given time. Although we achieve similar results in both arrays, the following discussion is focussed on the 80 μm array.

Figure 1: Surface-electrode ion trap featuring three individual traps.
figure 1

(a) Scanning electron microscope (SEM) image of a cleaved copy of our chip; white scale bar, 20 μm. It provides a cross-sectional view vertically through the trap chip (bottom half of image) and a top view of the horizontal, planar trap electrode surface (top half of image) of the 40 μm array. Buried electrode interconnects as well as the overhangs of electrodes that shield trapped ions from insulating surfaces are exposed in this view. A loading channel, vertically traversing the chip, collimates a neutral atom beam from an oven on the backside of the chip. (b) SEM top-view of the 80 μm array, dark lines indicate gaps between individual electrodes and dashed circles highlight the three trap sites at T0, T1 and T2 that lie 40 μm above the electrode plane (white scale bar, 80 μm); corresponding loading holes appear as dark spots. A vertical plane connecting T0 and T2 is shown as a dotted line and labelled with δ0. The single RF electrode extends beyond the image area and encloses 30 control electrodes grouped into four islands depicted in the central part of the image, enabling the control of individual trap sites. Laser beams (coloured arrows) are parallel to the chip surface and wave vectors kP/D of preparation and detection beams are parallel to the magnetic quantization field B0 (white arrow). (c–e) The pseudopotential φps(x,y,z) of the 80 μm array in different planes is shown, trap sites marked by red dots, motional mode vectors with j={1,2,3} at T0 are represented by white arrows, and saddle points are illustrated by white crosses. In e, the heights of T0, T2 and the ancillary trap are indicated by red and blue lines, respectively.

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Three-dimensional confinement of 25Mg+ ions is provided by a potential φRF oscillating at ΩRF from a single RF electrode driven at ΩRF/(2π)=48.3 MHz with an approximate peak voltage URF=20 V. Setting the origin of the coordinate system at the centre of the array and in the surface plane of the chip, the RF potential features three distinct trap sites at T0≃(−46,0,37) μm, , and . Owing to the electrode symmetry under rotations of ±2π/3 around the z-axis, it is often sufficient to consider T0 only, as all our findings apply to T1 and T2 after an appropriate rotation. Further, the RF potential exhibits another trap site at ≃(0,0,81) μm (above the centre of the array); this ‘ancillary’ trap is used for loading as well as for re-capturing ions that escaped from the other trap sites. We approximate the RF confinement at position r by a pseudopotential , cp. ref. 41, where Q denotes the charge and m the mass of the ion, and ERF(r) is the field amplitude produced by the electrode. Calculations of trapping potentials are based on ref. 42 and utilizing the software package43. Equipotential lines of φps are shown in Fig. 1c–e.

Near T0 we can approximate φps up to second order and diagonalize the local curvature matrix to find normal modes of motion described by their mode vectors u1, u2 and u3, which coincide (for the pure pseudopotential) with x, y and z; we use uj with j={1,2,3} throughout our manuscript to describe the mode vectors of a single ion near T0. We find corresponding potential curvatures of κps,1≃3.0 × 108 V m−2, κps,2≃5.9 × 107 V m−2 and κps,3≃9.2 × 107 V m−2, whereas mode frequencies can be inferred from these curvatures as , with j={1,2,3}: ω1/(2π)≃5.4 MHz, ω2/(2π)≃2.4 MHz and ω3/(2π)≃3.0 MHz. Further, the Mathieu parameters , where κRF,i(r) denotes the curvature of φRF along direction i={x,y,z}, at T0 are: qx≃−0.32, qy≃0.14 and qz≃0.18.

To gain individual control of the trapping potential at each site, it is required to independently tune local potentials near T0, T1 and T2 (Methods), that is, to make use of designed local electric fields and curvatures. To achieve this, we apply sets of control voltages to 30 designated control electrodes (see Fig. 1 for details). In the following, a control voltage set is described by a unit vector , with corresponding dimensionless entries with n={1,…,30}, and result in a dimensionless control potential

where is the potential resulting when applying 1 V to the n-th electrode following a basis function method44,45. We scale by varying a control voltage Uc and yielding a combined trapping potential

Bias voltages applied to the control electrodes are, in turn, fully described by .

To design a specific , we consider the second order Taylor expansion for a point r0 and small displacements Δr:

where is the local gradient and is the traceless and symmetric matrix with indices k and l={x, y, z} that describes the local curvature; square brackets denote vectors/matrices, ∂ partial derivatives and the superscript T the transpose of a vector. We constrain local gradients in their three degrees of freedom (DoF) and local curvatures in their five DoF at T0, T1 and T2, and solve the corresponding system of 24 linear equations to yield . In principle, it would be sufficient to use 24 control electrodes, however, we consider all electrodes and use the extra DoF to minimize the modulus of the voltages we need to apply for a given effect.

In particular, we distinguish two categories of control potentials, denoted by and , respectively: the first category is designed to provide finite gradients and zero curvatures at T0, with zero gradients and curvatures at T1 and T2; for example, provides a gradient along at T0. Control potentials of the second category are designed to provide zero gradients and only curvatures at T0, whereas we require related gradients and curvatures to be zero at T1 and T2. For example, we design , with the following non-zero constrains with corresponding Uc=Utune. Linear combination of multiple control potentials enable us, for example, to locally compensate stray potentials up to second order, to independently control mode frequencies and orientations at each trap site, and, when implementing time-dependent control potentials, to apply directed and phase-controlled mode-frequency modulations or mode excitations.

Optical setup and experimental procedures

We employ eight laser beams at wavelengths near 280 nm, from three distinct laser sources46, with wave vectors parallel to the xy plane (Fig. 1b) for preparation, manipulation and detection of electronic and motional states of 25Mg+ ions. Five distinct σ+-polarized beams (two for Doppler cooling, two for optical pumping and one for state detection) are superimposed, with wave vector kP/D (preparation/detection) aligned with a static homogeneous magnetic quantization field B0≃4.65 mT (Fig. 1b). The beam waists (half width at 1/e2 intensity) are ≃150 μm in the xy plane and ≃30 μm in z direction, to ensure reasonably even illumination of all three trap sites, while avoiding excessive clipping of the beams on the trap chip. The two Doppler-cooling beams are detuned by Δ≃−Γ/2 and −10Γ (for initial Doppler cooling and state preparation by optical pumping) with respect to with a natural line width Γ/(2π)≃42 MHz. The state detection beam is resonant with this cycling transition and discriminates from , the pseudo-spin states and are separated by ω0/(2π)≃1,681.5 MHz. The resulting fluorescence light is collected with high numerical aperture lens onto either a photomultiplier tube or an electron-multiplying charge-coupled device camera. We prepare (and repump to) by two optical-pumping beams that couple and to states in from where the electron decays back into the ground state manifold and population is accumulated in . We can couple to via two-photon stimulated-Raman transitions25,47,48, while we can switch between two different beam configurations labelled BR*+RR with and BR+RR with . The beam waists are ≃30 μm in the xy plane and ≃30 μmm in z direction.

We load ions by isotope-selective photoionization from one of three atomic beams collimated by 4 μm loading holes located beneath each trap site (Fig. 1). We can also transfer ions from one site to any neighbouring site via the ancillary trap by applying suitable potentials to control electrodes and a metallic mesh (with high optical transmission) located ≃7 mm above the surface. Typically, experiments start with 2 ms of Doppler cooling, optionally followed by resolved sideband cooling, and preparation via optical pumping. We use 30 channels of a 36-channel arbitrary waveform generator with 50 MHz update rate49 to provide static (persistent over many experiments) and dynamic (variable within single experiments) control potentials. Each experiment is completed by a pulse for pseudo-spin detection of duration ≃150 μs that yields ≃12 counts on average for an ion in and ≃0.8 counts for an ion in . Specific experimental sequences are repeated 100–250 times.

Initially, we calibrate three (static) control potentials , and to compensate local stray fields50 with a single ion near T0, whereas we observe negligible effects on the local potentials near T1 and T2 (Methods). Rotated versions of these control potentials are used to compensate local stray fields near T1 and T2. Near each site, we achieve residual stray field amplitudes ≤3 V m−1 in the xy plane and ≤900 V m−1 along z, currently limited by our methods for detection of micromotion.

With the stray fields approximately compensated, we characterize the trap near T0 with a single ion (Methods). We find mode frequencies of ω1/(2π)≃5.3 MHz, ω2/(2π)≃2.6 MHz and ω3/(2π)≃4.1 MHz with frequency drifts of about 2π × 0.07 kHz (60 s)−1; mode frequencies and orientations are altered by local stray curvatures on our chip, in particular, u1 and u3 are rotated in the xz plane, while u2 remains predominantly aligned along y. We obtain heating rates for the modes u1 of 0.9(1) quanta ms−1, u2 of 2.2(1) quanta ms−1 and u3 of 4.0(3) quanta ms−1.

Control of mode configurations at individual trap sites

The ability to control mode frequencies and orientations at each site with minimal effect on local trapping potentials at neighbouring sites is essential for the static and dynamical tuning of inter-ion Coulomb couplings. We experimentally demonstrate individual mode-frequency control using . To this end we measure local mode frequencies with a single ion near T0 or T2 (Methods). Tuning of about ±2π × 80 kHz of ω2 near T0 is shown in Fig. 2 as blue data points, accompanied by residual changes of about in the corresponding mode frequency near the neighbouring site T2, depicted by red data points. To infer local control curvatures, we describe the expected detuning Δω2 due to at T0 (analogously at T2) by

Figure 2: Individual control of mode frequencies.
figure 2

Frequency change Δω2/(2π) probed with a single ion near T0 (blue dots) as a function of Utune. The intended control curvature 0.937 × 107 m−2 (along the y direction) yields the blue/dashed line (cp. equation (4)), while a fit to the data (blue/solid line) returns a control curvature of 1.164(3) × 107 m−2 (corresponding residuals shown in the bottom graph). The remaining change of the corresponding mode frequency of a single ion near T2 (red squares) is shown in the inset, for the full range of Utune. A fit to these data (red/solid lines) results in a residual control curvature of −0.012(2) × 107 m−2. Ideally, would create no additional curvature at T2 (red/dashed line). We attribute the difference between designed and measured values to residual ion displacements from T0 and T2. Each data point represents the average of 250 experiments with error bars (for some data smaller than symbols) denoting the s.e.m. Residual variations of experimental parameters, for example, changes of stray potentials, can result in day-to-day variations of measurement outcomes that require recalibration to remain within our stated statistical uncertainties.

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where we neglect a small misalignment of u2 from y. The prediction of equation (4) is shown as a blue/dashed line in Fig. 2. The blue/solid line results from a fit with a function of the form of equation (4) to the data yielding a control curvature of 1.164(3) × 107 m−2. The inset magnifies the residual change in frequency near T2. Here, a fit (red/solid line) reveals a curvature of −0.012(2) × 107 m−2. Residual ion displacements of Δz=−2.95(3) μm from T0 and Δz=−2.9(4) μm from T2, respectively, suffice to explain deviations between experimentally determined and designed curvature values and are below our current limit of precision locating the ions in that direction. In future experiments, curvature measurements may be used to further reduce stray fields.

We also implement a dynamic Utune(t), to adiabatically tune ω2 near T0 within single experiments: we prepare our initial state by Doppler cooling, followed by resolved sideband cooling of mode u2 to an average occupation number and optical pumping to . In a next step, we apply a first adiabatic ramp from Utune,A=0 V to Utune,B between 0 and 2.3 V (corresponding to a measured frequency difference Δω2/(2π)≃430 kHz) within tramp=7.5 to 120 μs and, subsequently, couple and to mode u2 with pulses of BR+RR tuned to sideband transitions that either add or subtract a single quantum of motion. If the ion is in the motional ground state, no quantum can be subtracted and the spin state remains unchanged when applying the motion subtracting sideband pulse. The motion-adding sideband can always be driven, and comparing the spin-flip probability of the two sidebands allows us to determine the average occupation of the dynamically tuned mode48. We find that the average occupation numbers are independent of the duration of the ramp and equal to those obtained by remaining in a static potential for tramp, that is, the motion is unaffected by the dynamic tuning.

We rotate mode orientations near T0 in the xy plane with a control-potential , while setting additional constraints to keep gradients and curvatures of the local trapping potential constant at T1 and T2 (Methods). We determine the rotation of mode orientations from electron-multiplying charge-coupled device images of two ions near T0 that align along u2 (axis of weakest confinement). Simultaneously, we trap one or two ions near T1 and T2 to monitor residual changes in ion positions and mode orientations (and frequencies) because of unwanted local gradients and curvatures of . We take 14 images for five different values, while constantly Doppler cooling all ions and exciting fluorescence. Figure 3a shows two images for Urot=0 V (left) and Urot=2.45 V (right). Schematics of control electrodes are overlaid to the images and coloured to indicate their bias voltages Urot. Ion positions (in the xy plane) are obtained with an uncertainty of ±0.5 μm, yielding uncertainties for inferred angles ϕ2,y of ±5°. Here, ϕ2,y denotes the angle between local mode u2 and y. Figure 3b shows measured ϕ2,y for ions near T0 (blue dots) and T1 (red squares) and compares them with our theoretical expectation (solid lines), further described in the Methods. We tune ϕ2,y between 0° and 45° near T0, enabling us to set arbitrary mode orientations in the xy plane, whereas ion positions (mode orientations) near T1 and T2 remain constant within ±0.5 μm (better than ±5°) in the xy plane.

Figure 3: Individual control of mode orientations.
figure 3

(a) Electron-multiplying charge-coupled device images of pairs of ions near T0 and T1, and a single ion near T2; white scale bars are 50 μm and blow ups of each site are magnified by four. Schematics of control electrodes are coloured according to their bias voltage Urot. Ion pairs align along u2, the lowest-frequency mode. The left image captures ion positions for Urot=0 V and, here, ω2/(2π)=1.9(2) MHz near T0. The right image illustrates the rotation effect for Urot=2.45 V: Mode u2 near T0 with ω2/(2π)=1.8(2) MHz is rotated by ϕ2,y=31(5)°, whereas ion positions near T1 and T2 remain unchanged; white circles indicate initial ion positions (for Urot=0). (b) Mode u2 orientation in the xy plane for T0 (blue dots) and T1 (red squares), described by ϕ2,y, derived from a total of 14 images as a function of Urot; error bars denote our systematic uncertainty. The data are in good agreement with our predictions of the effect of (solid lines).

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A complementary way of characterizing mode orientations and frequencies, now with respect to Δkx and/or Δky is to analyse the probability of finding after applying (carrier) or motional sideband couplings for variable duration. If all modes of a single ion are prepared in their motional ground state, the ratio of Rabi frequencies of carrier and sideband couplings is given by the Lamb-Dicke parameter48, which is for u1 and Δkx:

where ϕ1,x is the angle between u1 and Δkx. The differences of carrier and sideband transition frequencies reveal the mode frequencies, whereas ratios of sideband and carrier Rabi-frequencies determine Lamb-Dicke parameters and allow for finding the orientation of modes.

We use a single ion near T0 to determine the orientations and frequencies of two modes relative to Δkx. We apply another control potential , designed to rotate u1 and u3 in the xz plane near T0, and implement carrier and sideband couplings to both modes with Δkx after resolved sideband cooling and initializing . In Fig. 4, the probability of is shown for different pulse durations of carrier couplings (top) and sideband couplings to mode u1 (middle) and u3 (bottom). Data points for Urot2=−1.62 V are shown as blue rectangles and for −2.43 V as grey rectangles. We fit each data set to a theoretical model (blue and grey lines) to extract the angles51 and distributions of Fock-state populations of each mode (shown as histograms): we find ϕ1,x=24.7(2)° for Urot2=−1.62 V and ϕ1,x=36.1(2)° for Urot2=−2.43 V, whereas average occupation numbers range between ≃0.05 and ≃0.6. Adding measurements along Δky and taking into account that the normal modes have to be mutually orthogonal would allow to fully reconstruct all mode orientations. With resolved sideband cooling on all three modes, we can prepare a well-defined state of all motional DoF.

Figure 4: Measuring mode orientations via laser couplings.
figure 4

Probability of after applying carrier and sideband couplings via Δkx with a single ion near T0. Mode orientations are set with static (rotations in the xz plane) for Urot2=1.62 V (blue dots) and −2.43 V (grey squares). (a) Shows the carrier transitions, , whereas b represents the u1-sideband transitions, , and the u3-sideband transitions, . From combined model fits to all transitions (for each ), we find angles ϕ1,x=24.7(2)° for Urot2=−1.62 V (blue lines) and 36.1(2)° for Urot2=−2.43 V (grey lines) of mode u1 relative to Δkx. Histograms in b display derived Fock-state populations with thermal average occupation numbers between ≃0.05 and ≃0.6. Each data point is the average of 250 experiments and error bars (for some data smaller than symbols) denote the s.e.m. Residual variations of experimental parameters, for example, changes of stray potentials, can result in day-to-day variations of measurement outcomes that require recalibration to remain within our stated statistical uncertainties.

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Discussion

We characterized two trap arrays that confine ions on the vertices of equilateral triangles with side lengths 80 and 40 μm. We developed systematic approaches to individually tune and calibrate control potentials in the vicinity of each trap site of the 80-μm array, by applying bias potentials to 30 control electrodes. With suitably designed control potentials, we demonstrated precise individual control of mode frequencies and orientations. By utilizing a multi-channel arbitrary waveform generator, we also dynamically changed control potentials within single experimental sequences without adverse effects on spin or motional states. Further, we devised a method to fully determine all mode orientations (and frequencies) based on the analysis of carrier and sideband couplings. Measured heating rates are currently comparable to the expected inter-ion Coulomb coupling rate of Ωex/(2π)≃1 kHz for 25Mg+ ions in the 40-μm array at mode frequencies of ≃2π × 2 MHz (ref. 32). This coupling rate sets a fundamental time scale for effective spin–spin couplings33. To observe coherent spin–spin couplings, ambient heating needs to be reduced. Decreases in heating rates of up to two orders of magnitude would leave Ωex considerably higher than competing decoherence rates and allow for coherent implementation of fairly complex spin–spin couplings. Such heating rate reductions have been achieved in other surface traps by treatments of the electrode structure34,35,36 and/or cryogenic cooling of the electrodes37,38,39. The couplings in question have been observed in one dimension in a cryogenic system32,33.

Currently, we can compensate stray fields, set up normal mode frequencies and directions for all three ions and initialize them for a two-dimensional AQS, that is, prepare a fiducial initial quantum state for ions at each trap site. A complete AQS may use the sequence presented in Fig. 5. A dynamic ramp adiabatically transforms the system between two control sets, labelled as A and B, that realize specific mode frequencies and orientations at each site. Set A may serve to globally initialize spin-motional states of ions, potentially with more than one ion at each site, that could be the ground state of a simple initial Hamiltonian. At all sites, mode frequencies and orientations need to be suitable (bottom left of Fig. 5) to enable global resolved sideband cooling, ideally preparing ground states for all motional modes. A first ramp to set B combined with appropriate laser fields may be used to adiabatically or diabatically realize a different Hamiltonian, for example, by turning on complex spin–spin couplings. Mode frequencies and orientations are tuned such that the Coulomb interactions between ions can mediate effective spin–spin couplings, for example, all mode vectors u1 are rotated to point to the centre of the triangle (bottom right of Fig. 5). During the application of such interactions, the ground state of the uncoupled system can evolve into the highly entangled ground state of a complex coupled system. In contrast, diabatic ramping to set B will quench the original ground state and the coupled system will evolve into an excited state that is not an eigenstate. After a final adiabatic or diabatic ramp back to set A, we can use global (or local) laser beams to read out the final spin states at each site.

Figure 5: Generic experimental sequence for tuneable inter-ion couplings.
figure 5

(a) Time-line of an experiment starting with the initialization of an fiducial quantum state of all ions in the array, followed by an adiabatic (or diabatic) ramp of control potentials between sets A and B that reconfigure the normal-mode structure from the setup, see b. Then, appropriate laser fields implement inter-ion couplings required for the AQS. The simulation completes with ramping the control potentials back to set A, where the individual spin states can be detected. (b) Examples for configurations of motional DoF are illustrated by the arrows that show the orientation of u1 at the three trap sites (red dots). Set A may be applied when globally and/or locally preparing and detecting spin-motional states, whereas set B can establish specific inter-ion Coulomb couplings to mediate, for example, effective spin–spin couplings for AQS, cp. refs 33, 55.

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In this way, our arrays may become an arbitrarily configurable and dynamically reprogrammable simulator for complex quantum dynamics. It may enable, for example, the observation of photon-assisted tunnelling, as required for experimental simulations of synthetic gauge fields52,53 or other interesting properties of finite quantum systems, such as thermalization, when including the motional DoF54. Concentrating on spin–spin interactions, the complex entangled ground states of spin frustration can be studied in the versatile testbed provided by arrays of individually trapped and controlled ions30,55. Arrays with a larger number of trap sites could realize a level of complexity impossible to simulate on conventional computers56,57.

Methods

Design of arrays used in the experiments

The design of arrays used in the expeiments is based on the methods described in ref. 29. In particular, we use the Mathematica package for surface atom and ion traps43 to globally optimize the RF electrode shape for maximal curvature with a given amplitude of the RF drive, whereas producing smooth continuous electrode shapes that require a single RF drive to operate the array. We specify the desired trap site positions as well as the ratio and orientation of normal-mode frequencies as a fixed input to the optimization algorithm for the pseudopotential, that is, we define that the high-frequency mode (for all three sites) lies within the xy plane and points towards the virtual centre of the array. Resulting electrode regions held to ground are subdivided into separated control electrodes that provide complete and independent control over the eight DoF at each site.

Array scaling for future realisations

To ensure that our approach can be scaled to more than three trapping sites, we compare designs of arrays containing different numbers of sites, Nsites, that are optimized by the algorithm described in ref. 29. Here, we assume a fixed ratio of h/d=1/2, where h denotes the distance of the sites to the nearest electrode surface and d is the inter-site distance. Further, we specify for all arrays that the high-frequency mode is aligned orthogonally to the xy plane at each site, in contrast to our demonstrated arrays (see Fig. 1 for details). This unique mode configuration permits a fair comparison of geometries with increasing Nsites. To illustrate the optimal electrode shapes, we present four examples of triangular arrays with Nsites={3,6,18,69} in Fig. 6a–d. To enable the same level of individual control as demonstrated for both of our three-site arrays, we would have to subdivide the optimized ground electrodes into ≥8 × Nsites control electrodes. We find that the inner areas converge to fairly regular electrode shapes for larger Nsites, whereas electrodes closer to the border are deformed to compensate for edge effects (see Fig. 6d for details). However, the spatial extent and complexity of all electrodes remains comparable to the arrays used in our experiments and, thus, fabrication of these larger arrays can be accomplished by scaling the applied techniques (see below).

Figure 6: Scale-up the number of sites in triangular arrays.
figure 6

(a–d) Optimized surface-electrode shapes for 3, 6, 18 and 69 trapping sites, where we assume h/d=1/2 (as for our 80 μm array); black scale bars have fixed length d. The high-frequency mode is aligned orthogonally to the xy plane in each trapping site, to permit comparability of the different geometries. RF and ground electrodes are coloured in grey and white, respectively, and individual trap sites are marked with red dots. In a functional array, we will have to further subdivide the ground electrodes to enable individual control at each site, as demonstrated in our work with three sites. (e) Optimized dimensionless curvature κ as a function of the number of trapping sites in the triangular arrays, quantifying the strength of the individual traps. For a single site κ=1.0 and for an infinite triangular lattice κ≃0.07. In experimental realizations, we can compensate this decrease in trap curvature by adjusting URF and ΩRF, or by reducing h.

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To quantify the geometric strength of individual trap sites independently of m, URF, ΩRF and h, we consider the dimensionless curvature κ of the pseudopotential that we normalize to the highest possible curvature for a single site29. We show optimized κ for arrays with Nsites between 1 and 102, as well as, the value for Nsites=∞ in Fig. 6e; a fully controlled array with Nsites=102 should be sufficient to study quantum many-body dynamics that are virtually impossible to simulate on a conventional computer. We find that k for Nsites=102 is reduced by about a factor of two compared with κ≃0.87 for Nsites=3, whereas κ≃0.07 for Nsites=∞; see ref. 29 for a detailed discussion of infinite arrays. The decrease in trap curvature can be compensated in experiments by adjusting URF and ΩRF correspondingly, or by reducing h. Further, we estimate that trapping depths remain on the same order of magnitude for increasing Nsites compared with our demonstrated arrays (cp. Fig. 1d). For an infinite array it has been shown that depths of a few mV are achievable30. Note, that in surface-electrode traps the trapping potential is less deep along z than in the xy plane, and ion-escape points (closest and lowest saddle point of the pseudopotential) typically lie above each site. In experiments, we may apply a constant bias potential to the control electrodes, surrounding ground planes, and the mesh (cover plane) to increase the depth along z to a level where trapping is routinely achieved, while reducing the depth in the xy plane30. With such measures in place, we are fairly confident that ions created by photoionization from a hot atomic beam can be loaded and cooled into the local minima of larger arrays.

Architecture of our trap chip

The 10 × 10 mm2 Si substrate of our trap chip is bonded onto a 33 × 33 mm2 ceramic pin grid array (CPGA); the electrodes of the trap arrays are wire-bonded with aluminium wires to the pins of the CPGA, with independent pins for the RF electrodes of the two arrays. The trap chip contains four aluminum-1/2% copper metal layers, that are electrically connected by tungsten vertical interconnects thereby allowing ‘islanded’ control electrodes in the top electrode layer (Fig. 1). The buried electrical leads are isolated by intermediate SiO2 layers, nominally 2 μm thick, while the surface layer is spaced by 10 μm from the buried layers. All electrodes are mutually separated by nominally 1.2–1.4 μm gaps and a 50-nm gold layer is evaporated on the top surfaces in a final fabrication step. The trap chip fabrication is substantially the same as that described in the Supplement to ref. 40. Each control electrode is connected to ground by 820 pF capacitors located on the CPGA to minimize potential changes due to capacitive coupling to the RF electrodes.

Compensation of stray potentials at each site

For compensation of local stray fields in the xy plane, we vary the strength of individual control potentials and and find corresponding coefficient settings where we obtain a maximal Rabi rate of the detection transition and/or minimal Rabi rates of micromotion-sideband transitions probed with Δkx and Δky; resulting in residual stray-field amplitudes of ≤3 V m−1. For compensation along z, we vary the strength of individual to minimize a change in ion position due to a modulation of URF. The depth of field of our imaging optics aids to detect changes in z-position via blurring of images of single ions trapped at each site, within an uncertainty of about ±5 μm. This corresponds to residual stray-field amplitudes of ≃900 V m−1 for typical trapping parameters.

Mode frequency and heating rate measurements

To measure mode frequencies, we Doppler-cool the ion and pump to . Then, we apply a motional excitation pulse with fixed duration texc=100 μs to a single control electrode. The pulse produces an electric field oscillating at a frequency ωexc that excites the motion, if ωexc is resonant with a mode frequency, and we can detect mode amplitudes of >100 nm along kD via the Doppler effect. In the experiments, we vary ωexc and obtain resonant excitations at ωj with j={1,2,3}. By repeating measurements, we record ≃50 consecutive frequency values for each mode frequency over the course of Δt≃1 h with a single ion near T0. The results are consistent with linear changes in frequencies, with rates Δω1/Δt=−2π × 0.090(3) kHz (60 s)−1, Δω2/Δt=−2π × 0.064(1) kHz (60 s)−1 and Δω3/Δt=−2π × 0.063(5) kHz (60 s)−1.

For the heating rate measurements, we add multiple resolved-sideband cooling pulses after Doppler cooling to our sequence and determine mode temperatures from the sideband ratios for several different delay times58. In our experiments, we either use Δkx to iteratively address u1 and u3 or Δky to address only u2. For this, we prepare similar mode orientations as presented in Fig. 4, find initial mode temperatures after cooling to , and obtain corresponding heating rates.

Potentials for individual control

As a representative example for designing control potentials, we discuss that serves to rotate the normal modes in the xy plane. At position T0, the constraints are:

for k and l={x,y,z}, while local gradients at all three trap sites and local curvatures at T1 and T2 are required to be zero. We add diagonal elements in to reduce changes of the u2 frequency during variation of around our initial mode configurations. The mode configurations in the real array deviate from those derived from the φps due to additional curvatures near each trap site generated by stray potentials on our chip. Ideally, we would design control potentials for mode rotations such that all frequencies stay fixed. This is only possible if we explicitly know the initial mode configuration. In addition, we keep mode vectors tilted away from z to sufficiently Doppler cool all modes during state initialization. Similarly, we design to rotate modes in the xz plane.

Model for varying mode orientations

To model the rotation angle ϕ2,y of u2 near T0 as a function of , we consider the final trapping curvature at T0 (analogously for neighbouring sites):

where φini(r) represents the initial potential, that is, the sum of the pseudopotential, stray potential and additional control potentials (used for stray field compensation). The local curvatures (mode frequencies and vectors) of φini(r) near T0 are estimated from calibration experiments. For simplicity, we reduce equation (7) to two dimensions (in the xy plane) and find corresponding eigenvectors and eigenvalues for Urot between 0.0 and 3.0 V. We obtain angles ϕ2,y(Urot) of the eigenvector u2 and we show resulting values as an interpolated solid line in Fig. 3b.

Similarly, we model the effect of on ω2. We assume that for Utune=0, the corresponding mode vector u2 is aligned parallel to y. This is the case for pure RF confinement (cp. Fig. 1c) and sufficiently small stray curvatures. We design to tune the curvature along y, and the curvature as a function of Utune (along this axis) is described by: . Finally, we insert this into to find equation (4).

Data availability

The data that support the findings of this study are available from the corresponding author upon request.

Additional information

How to cite this article: Mielenz, M. et al. Arrays of individually controlled ions suitable for two-dimensional quantum simulations. Nat. Commun. 7:11839 doi: 10.1038/ncomms11839 (2016).