Abstract
A precisely controlled quantum system may reveal a fundamental understanding of another, less accessible system of interest. A universal quantum computer is currently out of reach, but an analogue quantum simulator that makes relevant observables, interactions and states of a quantum model accessible could permit insight into complex dynamics. Several platforms have been suggested and proofofprinciple experiments have been conducted. Here, we operate twodimensional arrays of three trapped ions in individually controlled harmonic wells forming equilateral triangles with side lengths 40 and 80 μm. In our approach, which is scalable to arbitrary twodimensional lattices, we demonstrate individual control of the electronic and motional degrees of freedom, preparation of a fiducial initial state with ion motion close to the ground state, as well as a tuning of couplings between ions within experimental sequences. Our work paves the way towards a quantum simulator of twodimensional systems designed at will.
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Introduction
Richard Feynman was one of the first to recognize that quantum systems of sufficient complexity cannot be simulated on a conventional computer^{1}. 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 custombuild an analogue quantum simulator (AQS) that allows for preparation of fiducial input states, faithful implementation of the modelspecific dynamics and for access to the crucial observables. Simulations on such AQSs could impact a vast variety of research fields^{2}, that is, physics^{3}, chemistry^{4} and biology^{5}, when studying dynamics that is out of reach for numerical simulation on conventional computers.
Many experimental platforms have been suggested to implement AQSs^{6,7,8,9}. 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 problem^{10,11}. Over the last two decades, many promising proofofprinciple demonstrations have been made using photons^{6}, superconductors^{7}, atoms^{8} and trapped atomic ions^{9}. Trapped ions in particular have seen steady progress from demonstrations with one or two ions^{12,13,14,15,16,17,18} to addressing aspects of quantum magnets^{19} with linear strings of 2–16 ions^{13,20} and selfordered twodimensional crystals containing more than 100 ions^{21}. Ions are well suited to further propel the research since they provide longrange interaction and individual, fast controllability with high precision^{22}.
Twodimensional traparrays 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 numerics^{10,23}. Our approach is based on surfaceelectrode structures^{24} 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 alltoall 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 dimension^{28,29,30}.
Optimized surface electrode geometries can be found for any periodic wallpaper group as well as quasiperiodic arrangements, as, for example, Penrosetilings^{29}. A first step, trapping of ions in twodimensional arrays of surface traps, has been proposed^{15} and demonstrated^{31}. Boosting the strength of interaction to a level comparable to current decoherence rates requires interion 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 surfaceelectrode 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 system^{32,33}. The increase in coupling strength was achieved with a reduction of the ionsurface 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 treatments^{34,35,36} or cold electrode surfaces^{37,38,39} have been devised.
Here, we demonstrate the precise tuning of all relevant parameters of a twodimensional 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 rates^{32} 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 twodimensional 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 radiofrequency (RF) electrodes of the arrays are optimized by a linearprogramming algorithm that yields electrode shapes with low fragmentation, and requires only a single RFvoltage source for operation^{29,30}. To design different and even nonperiodic arrays for dedicated trap distances, we can apply the same algorithm to yield globally optimal electrode shapes^{29}. 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.
Threedimensional confinement of ^{25}Mg^{+} ions is provided by a potential φ_{RF} oscillating at Ω_{RF} from a single RF electrode driven at Ω_{RF}/(2π)=48.3 MHz with an approximate peak voltage U_{RF}=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 zaxis, 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 recapturing 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 E_{RF}(r) is the field amplitude produced by the electrode. Calculations of trapping potentials are based on ref. 42 and utilizing the software package^{43}. Equipotential lines of φ_{ps} are shown in Fig. 1c–e.
Near T0 we can approximate φ_{ps} up to second order and diagonalize the local curvature matrix to find normal modes of motion described by their mode vectors u_{1}, u_{2} and u_{3}, which coincide (for the pure pseudopotential) with x, y and z; we use u_{j} 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 κ_{ps,1}≃3.0 × 10^{8} V m^{−2}, κ_{ps,2}≃5.9 × 10^{7} V m^{−2} and κ_{ps,3}≃9.2 × 10^{7} V m^{−2}, whereas mode frequencies can be inferred from these curvatures as , with j={1,2,3}: ω_{1}/(2π)≃5.4 MHz, ω_{2}/(2π)≃2.4 MHz and ω_{3}/(2π)≃3.0 MHz. Further, the Mathieu parameters , where κ_{RF,i}(r) denotes the curvature of φ_{RF} along direction i={x,y,z}, at T0 are: q_{x}≃−0.32, q_{y}≃0.14 and q_{z}≃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 nth electrode following a basis function method^{44,45}. We scale by varying a control voltage U_{c} 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 r_{0} 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 nonzero constrains with corresponding U_{c}=U_{tune}. 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 timedependent control potentials, to apply directed and phasecontrolled modefrequency modulations or mode excitations.
Optical setup and experimental procedures
We employ eight laser beams at wavelengths near 280 nm, from three distinct laser sources^{46}, with wave vectors parallel to the xy plane (Fig. 1b) for preparation, manipulation and detection of electronic and motional states of ^{25}Mg^{+} ions. Five distinct σ^{+}polarized beams (two for Doppler cooling, two for optical pumping and one for state detection) are superimposed, with wave vector k_{P/D} (preparation/detection) aligned with a static homogeneous magnetic quantization field B_{0}≃4.65 mT (Fig. 1b). The beam waists (half width at 1/e^{2} 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 Dopplercooling 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 pseudospin states and are separated by ω_{0}/(2π)≃1,681.5 MHz. The resulting fluorescence light is collected with high numerical aperture lens onto either a photomultiplier tube or an electronmultiplying chargecoupled device camera. We prepare (and repump to) by two opticalpumping 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 twophoton stimulatedRaman transitions^{25,47,48}, 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 isotopeselective 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 36channel arbitrary waveform generator with 50 MHz update rate^{49} to provide static (persistent over many experiments) and dynamic (variable within single experiments) control potentials. Each experiment is completed by a pulse for pseudospin 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 fields^{50} 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 ω_{1}/(2π)≃5.3 MHz, ω_{2}/(2π)≃2.6 MHz and ω_{3}/(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, u_{1} and u_{3} are rotated in the xz plane, while u_{2} remains predominantly aligned along y. We obtain heating rates for the modes u_{1} of 0.9(1) quanta ms^{−1}, u_{2} of 2.2(1) quanta ms^{−1} and u_{3} 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 interion Coulomb couplings. We experimentally demonstrate individual modefrequency 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 ω_{2} 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 Δω_{2} due to at T0 (analogously at T2) by
where we neglect a small misalignment of u_{2} 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) × 10^{7} 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) × 10^{7} 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 U_{tune}(t), to adiabatically tune ω_{2} near T0 within single experiments: we prepare our initial state by Doppler cooling, followed by resolved sideband cooling of mode u_{2} to an average occupation number and optical pumping to . In a next step, we apply a first adiabatic ramp from U_{tune,A}=0 V to U_{tune,B} between 0 and 2.3 V (corresponding to a measured frequency difference Δω_{2}/(2π)≃430 kHz) within t_{ramp}=7.5 to 120 μs and, subsequently, couple and to mode u_{2} 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 motionadding sideband can always be driven, and comparing the spinflip probability of the two sidebands allows us to determine the average occupation of the dynamically tuned mode^{48}. 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 t_{ramp}, that is, the motion is unaffected by the dynamic tuning.
We rotate mode orientations near T0 in the xy plane with a controlpotential , 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 electronmultiplying chargecoupled device images of two ions near T0 that align along u_{2} (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 U_{rot}=0 V (left) and U_{rot}=2.45 V (right). Schematics of control electrodes are overlaid to the images and coloured to indicate their bias voltages U_{rot}. Ion positions (in the xy plane) are obtained with an uncertainty of ±0.5 μm, yielding uncertainties for inferred angles ϕ_{2,y} of ±5°. Here, ϕ_{2,y} denotes the angle between local mode u_{2} and y. Figure 3b shows measured ϕ_{2,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 ϕ_{2,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.
A complementary way of characterizing mode orientations and frequencies, now with respect to Δk_{x} and/or Δk_{y} 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 LambDicke parameter^{48}, which is for u_{1} and Δk_{x}:
where ϕ_{1,x} is the angle between u_{1} and Δk_{x}. The differences of carrier and sideband transition frequencies reveal the mode frequencies, whereas ratios of sideband and carrier Rabifrequencies determine LambDicke 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 Δk_{x}. We apply another control potential , designed to rotate u_{1} and u_{3} in the xz plane near T0, and implement carrier and sideband couplings to both modes with Δk_{x} 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 u_{1} (middle) and u_{3} (bottom). Data points for U_{rot2}=−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 angles^{51} and distributions of Fockstate populations of each mode (shown as histograms): we find ϕ_{1,x}=24.7(2)° for U_{rot2}=−1.62 V and ϕ_{1,x}=36.1(2)° for U_{rot2}=−2.43 V, whereas average occupation numbers range between ≃0.05 and ≃0.6. Adding measurements along Δk_{y} 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 welldefined state of all motional DoF.
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 80μm 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 multichannel 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 interion Coulomb coupling rate of Ω_{ex}/(2π)≃1 kHz for ^{25}Mg^{+} ions in the 40μm array at mode frequencies of ≃2π × 2 MHz (ref. 32). This coupling rate sets a fundamental time scale for effective spin–spin couplings^{33}. 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 Ω_{ex} 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 structure^{34,35,36} and/or cryogenic cooling of the electrodes^{37,38,39}. The couplings in question have been observed in one dimension in a cryogenic system^{32,33}.
Currently, we can compensate stray fields, set up normal mode frequencies and directions for all three ions and initialize them for a twodimensional 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 spinmotional 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 u_{1} 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.
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 photonassisted tunnelling, as required for experimental simulations of synthetic gauge fields^{52,53} or other interesting properties of finite quantum systems, such as thermalization, when including the motional DoF^{54}. 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 ions^{30,55}. Arrays with a larger number of trap sites could realize a level of complexity impossible to simulate on conventional computers^{56,57}.
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 traps^{43} 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 normalmode frequencies as a fixed input to the optimization algorithm for the pseudopotential, that is, we define that the highfrequency 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, N_{sites}, 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 intersite distance. Further, we specify for all arrays that the highfrequency 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 N_{sites}. To illustrate the optimal electrode shapes, we present four examples of triangular arrays with N_{sites}={3,6,18,69} in Fig. 6a–d. To enable the same level of individual control as demonstrated for both of our threesite arrays, we would have to subdivide the optimized ground electrodes into ≥8 × N_{sites} control electrodes. We find that the inner areas converge to fairly regular electrode shapes for larger N_{sites}, 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).
To quantify the geometric strength of individual trap sites independently of m, U_{RF}, Ω_{RF} and h, we consider the dimensionless curvature κ of the pseudopotential that we normalize to the highest possible curvature for a single site^{29}. We show optimized κ for arrays with N_{sites} between 1 and 102, as well as, the value for N_{sites}=∞ in Fig. 6e; a fully controlled array with N_{sites}=102 should be sufficient to study quantum manybody dynamics that are virtually impossible to simulate on a conventional computer. We find that k for N_{sites}=102 is reduced by about a factor of two compared with κ≃0.87 for N_{sites}=3, whereas κ≃0.07 for N_{sites}=∞; see ref. 29 for a detailed discussion of infinite arrays. The decrease in trap curvature can be compensated in experiments by adjusting U_{RF} and Ω_{RF} correspondingly, or by reducing h. Further, we estimate that trapping depths remain on the same order of magnitude for increasing N_{sites} compared with our demonstrated arrays (cp. Fig. 1d). For an infinite array it has been shown that depths of a few mV are achievable^{30}. Note, that in surfaceelectrode traps the trapping potential is less deep along z than in the xy plane, and ionescape 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 plane^{30}. 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 mm^{2} Si substrate of our trap chip is bonded onto a 33 × 33 mm^{2} ceramic pin grid array (CPGA); the electrodes of the trap arrays are wirebonded 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 aluminum1/2% 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 SiO_{2} 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 50nm 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 micromotionsideband transitions probed with Δk_{x} and Δk_{y}; resulting in residual strayfield 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 U_{RF}. The depth of field of our imaging optics aids to detect changes in zposition via blurring of images of single ions trapped at each site, within an uncertainty of about ±5 μm. This corresponds to residual strayfield amplitudes of ≃900 V m^{−1} for typical trapping parameters.
Mode frequency and heating rate measurements
To measure mode frequencies, we Dopplercool the ion and pump to . Then, we apply a motional excitation pulse with fixed duration t_{exc}=100 μs to a single control electrode. The pulse produces an electric field oscillating at a frequency ω_{exc} that excites the motion, if ω_{exc} is resonant with a mode frequency, and we can detect mode amplitudes of >100 nm along k_{D} via the Doppler effect. In the experiments, we vary ω_{exc} and obtain resonant excitations at ω_{j} 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 Δω_{1}/Δt=−2π × 0.090(3) kHz (60 s)^{−1}, Δω_{2}/Δt=−2π × 0.064(1) kHz (60 s)^{−1} and Δω_{3}/Δt=−2π × 0.063(5) kHz (60 s)^{−1}.
For the heating rate measurements, we add multiple resolvedsideband cooling pulses after Doppler cooling to our sequence and determine mode temperatures from the sideband ratios for several different delay times^{58}. In our experiments, we either use Δk_{x} to iteratively address u_{1} and u_{3} or Δk_{y} to address only u_{2}. 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 u_{2} frequency during variation of around our initial mode configurations. The mode configurations in the real array deviate from those derived from the φ_{ps} 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 ϕ_{2,y} of u_{2} near T0 as a function of , we consider the final trapping curvature at T0 (analogously for neighbouring sites):
where φ_{ini}(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 φ_{ini}(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 U_{rot} between 0.0 and 3.0 V. We obtain angles ϕ_{2,y}(U_{rot}) of the eigenvector u_{2} and we show resulting values as an interpolated solid line in Fig. 3b.
Similarly, we model the effect of on ω_{2}. We assume that for U_{tune}=0, the corresponding mode vector u_{2} 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 U_{tune} (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 twodimensional quantum simulations. Nat. Commun. 7:11839 doi: 10.1038/ncomms11839 (2016).
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Acknowledgements
This work was supported by DFG (SCHA 972/61). Sandia National Laboratories is a multiprogram laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the US Department of EnergyÕs National Nuclear Security Administration under Contract No. DEAC0494AL85000. All statements of fact, opinion or analysis expressed in this paper are those of the authors and do not necessarily reflect the official positions or views of the Office of the Director of National Intelligence (ODNI) or the Intelligence Advanced Research Projects Activity. We thank J. Denter for technical assistance. Further, we are grateful for helpful comments on the manuscript given by S. Todaro, K. McCormick and Y. Minet.
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M.M., H.K. and U.W. participated in the design of the experiment and built the experimental apparatus. M.M., H.K., M.W., F.H. and U.W. collected data and analysed results. M.M., U.W., D.L. and T.S. wrote the manuscript. R.S. and D.L. participated in the design of the trap arrays and the experiment. M.B., P.M. and D.L.M. participated in the design and fabricated the trap chips. T.S. participated in the design and analysis of the experiment. M.M. and H.K. contributed equally to this work and all authors discussed the results and the text of the manuscript.
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Mielenz, M., Kalis, H., Wittemer, M. et al. Arrays of individually controlled ions suitable for twodimensional quantum simulations. Nat Commun 7, ncomms11839 (2016). https://doi.org/10.1038/ncomms11839
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DOI: https://doi.org/10.1038/ncomms11839
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