LETTERdoi:10.1038/nature10981

Engineered two-dimensional Ising interactions in atrapped-ionquantumsimulatorwithhundredsofspinsJoseph W. Britton1, Brian C. Sawyer1, Adam C. Keith2,3, C.-C. Joseph Wang2, James K. Freericks2, Hermann Uys4,Michael J. Biercuk5 & John J. Bollinger1

The presence of long-range quantum spin correlations underlies avariety of physical phenomena in condensed-matter systems,potentially including high-temperature superconductivity1,2.However, many properties of exotic, strongly correlated spinsystems, such as spin liquids, have proved difficult to study, in partbecause calculations involving N-body entanglement becomeintractable for as few as N < 30 particles3. Feynman predicted thata quantum simulator—a special-purpose ‘analogue’ processorbuilt using quantum bits (qubits)—would be inherently suited tosolving such problems4,5. In the context of quantum magnetism, anumber of experiments have demonstrated the feasibility of thisapproach6–14, but simulations allowing controlled, tunable inter-actions between spins localized on two- or three-dimensionallattices of more than a few tens of qubits have yet to be demon-strated, in part because of the technical challenge of realizinglarge-scale qubit arrays. Here we demonstrate a variable-rangeIsing-type spin–spin interaction, Ji,j, on a naturally occurring,two-dimensional triangular crystal lattice of hundreds of spin-halfparticles (beryllium ions stored in a Penning trap). This is a com-putationally relevant scale more than an order of magnitude largerthan previous experiments. We show that a spin-dependent opticaldipole force can produce an antiferromagnetic interactionJ i,j!d{a

i,j , where 0 # a # 3 and di,j is the distance between spinpairs. These power laws correspond physically to infinite-range(a 5 0), Coulomb–like (a 5 1), monopole–dipole (a 5 2) anddipole–dipole (a 5 3) couplings. Experimentally, we demon-strate excellent agreement with a theory for 0.05= a= 1.4. Thisdemonstration, coupled with the high spin count, excellentquantum control and low technical complexity of the Penning trap,brings within reach the simulation of otherwise computationallyintractable problems in quantum magnetism.

A challenge in condensed-matter physics is the fact that manyquantum magnetic interactions cannot currently be modelled in ameaningful way. A canonical example is the spin liquid, an exotic statepostulated1 to arise in a collection of spin-1/2 particles residing on atriangular lattice and coupled to each other by a nearest-neighbourantiferromagnetic Heisenberg interaction. The spin liquid’s groundstate is highly degenerate, owing to spin frustration, and is expectedto have unusual behaviours including phase transitions at zero tem-perature driven by quantum fluctuations15. However, despite recentadvances16,17 a detailed understanding of large-scale frustration insolids remains elusive2,18–20.

Atomic physicists have recently provided a bottom-up approach tothe problem by engineering the relevant spin interactions in quantumsimulators5,21,22. The necessary experimental capabilities—laser cool-ing, deterministic spin localization, precise spin-state quantum con-trol, high-fidelity read-out and engineered spin–spin coupling—werefirst demonstrated in the context of atomic clocks (see, for example,ref. 23). In the domain of quantum magnetism, this tool set permits

control of parameters commonly viewed as immutable in naturalsolids, for example lattice spacing and geometry, and spin–spin inter-action strength and range.

Initial simulations of quantum Ising and Heisenberg interactionswith localized spins were done with neutral atoms in optical lattices6,11,atomic ions in Paul traps9,10,13,14 and photons12. This work involvedsimulations readily calculable on a classical computer: interactionsbetween N < 10 qubits localized in one-dimensional (1D) chains.The move to quantum magnetic interactions on two-dimensional(2D) lattices and between larger, computationally relevant numbersof particles is the crucial next step but at present requires moretechnological development24.

In our Penning trap apparatus, laser-cooled 9Be1 ions naturallyform a stable 2D Coulomb crystal on a triangular lattice with ,300spins (Fig. 1). Each ion is a spin-1/2 system (qubit) over which we exerthigh-fidelity quantum control25. In this paper, we demonstrate the useof a spin-dependent optical dipole force (ODF) to engineer a continuouslytunable Ising-type spin–spin coupling Ji,j!d{a

i,j . This capability, intandem with a modified measurement routine (for example by moresophisticated processing of images such as that in Fig. 1), is a keyadvance towards useful simulations of quantum magnetism.

A Penning trap confines ions in a static quadrupolar electric potential(Methods) and a strong, homogeneous magnetic field B 5 B0z(B0 5 4.46 T). Axial trapping (along z) is due to the electric field. Ionrotation at frequency vr (about z) produces a radial restoring potentialdue to the velocity-dependent Lorentz force (qv|B, where q and v arerespectively the ion’s charge and velocity). Tuning the ratio of the axialto radial confinement allows controlled formation of a planar geometryand, after Doppler laser cooling, the formation of a 2D Coulomb crystalon a triangular lattice26 (Methods). We routinely generate crystals withN ions (100=N= 350), where the valence-electron spin state of eachion serves as a qubit25. Following techniques developed in linear (1D)Paul traps27, spins confined in the same trapping potential are coupledthrough their shared motional degrees of freedom.

Using well-controlled external fields, we engineer spin interactionsof the form

HB~X

iBm?si

HI~1N

Xivj

Ji,jszi sz

j

ð1Þ

where si~(sxi ,s

yi ,sz

i ) is the vector of Pauli matrices for ion i. We labelthe qubit spin states j"æ ; jms 5 11/2æ and j#æ ; jms 5 21/2æ, wherems is the spin’s projection along the quantizing field B0z, such thatsz

i :ij i~ :ij i and szi ;ij i~ ;ij i. The Hamiltonian HB encodes an inter-

action due to an effective magnetic field, Bm (generated by externallyapplied microwaves at 124 GHz), that couples equally to all spins andpermits global rotations (Fig. 1). The interaction HI describes a generalcoupling, Ji,j, between spins i and j a distance di,j apart28,29. For Ji,j . 0

1US National Institute of Standards and Technology, Time and Frequency Division, Boulder, Colorado 80305, USA. 2Department of Physics, Georgetown University, Washington DC 20057, USA.3Department of Physics, North Carolina State University, Raleigh, North Carolina 27695, USA. 4National Laser Centre, Council for Scientific and Industrial Research, Pretoria 0001, South Africa. 5Centre forEngineered Quantum Systems, School of Physics, The University of Sydney, New South Wales 2006, Australia.

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the coupling is antiferromagnetic and for Ji,j , 0 the coupling isferromagnetic.

We implement HI using a spatially uniform, spin-dependent ODFgenerated by a pair of off-resonance laser beams with difference fre-quency mR (Fig. 1 and Supplementary Information). The ODF coupleseach ion’s spin to one or more of the N transverse (along z) motionalmodes of the Coulomb crystal by forcing coherent displacements ofthe ions that in turn modify the ions’ Coulomb potential energythrough the interaction

HODF~{XN

i

Fz(t)ziszi

Here Fz(t) 5 F0cos(mRt) is the ODF; zi~PN

m~1 bi,m

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiB=2Mvm

p(ame{ivmtza{meivmt) is the axial position operator for ion i; bi,m areelements of the N transverse phonon eigenfunctions, bm, at frequencies

vm, normalized asPN

m~1 bi,mj j2~PN

i~1 bi,mj j2~1 (refs 28, 29); M isthe ion mass; and B is Planck’s constant divided by 2p. The modesinclude the centre-of-mass (COM) mode (v1) as well as an array ofmodes of higher spatial frequencies that may be derived from atomisticcalculations (Fig. 2a) and confirmed by experimental measurement30.

For small, coherent displacements, where residual spin–motionentanglement can be neglected29 (Methods), HODF is equivalent toHI in equation (1): spins i and j are coupled in proportion to their spin

Top-view

detector

V = 0

V = 0

Top-view image

124 GHz

100 μm

Cooling

laser beam

ODF

laser beams

z

y B0

y

x

ωr B0

ωR

–V0θR

ωR + μR

Figure 1 | The Penning trap confines hundreds of spin-1/2 qubits on a 2Dtriangular lattice. Each qubit is the valence-electron spin of a 9Be1 ion. Bottom:a Penning trap confines ions using a combination of static electric and magneticfields. The trap parameters are configured such that laser-cooled ions form atriangular 2D crystal. A general spin–spin interaction, HI, is generated by a spin-dependent excitation of the transverse (along z) motional modes of the ioncrystal. This coupling is implemented using an optical dipole force (ODF)produced by a pair of off-resonance laser beams (left side) with angularseparation hR and difference frequency mR. Microwaves at 124 GHz permitglobal spin rotations HB. Top: a representative top-view resonance fluorescenceimage showing the centre region of an ion crystal captured in the ions’ rest frame;in the laboratory frame, the ions rotate at vr 5 2p3 43.8 kHz (ref. 26).Fluorescence is an indication of the qubit spin state ( |"æ, bright; |#æ, dark); here,the ions are in the state |"æ. The lattice constant is d0 < 20mm.

10–5 10–310–8

10–6

10–4

10–2

100

1 MHz, a = 2.72

J i,j (a

.u.)

Spin–spin separation, di,j (m)

100 Hz, a = 0.01

1 kHz, a = 0.12

10 kHz, a = 0.75

100 kHz, a = 1.73

a = 3/2

–20 –10 0

10log10(Ji,j)1D nearest-neighbour Isingf

a = 0 a = 3

c d e

a

716 kHz 800 kHzωm/2π

μR

+1

–1

b

ω13, ω14 ω11, ω12 ω9, ω10 ω8 ω6, ω7 ω4, ω5 ω2, ω3 ω1

Figure 2 | Spin–spin interactions are mediated by the ion crystal’stransverse motional degrees of freedom. a, For a 2D crystal with N 5 217 ionsand vr 5 2p3 45.6 kHz, we calculate the eigenfunctions, bm, andeigenfrequencies, vm, for the N transverse motional modes (SupplementaryInformation). Plotted here are vm and bm for the 14 highest-frequency modes.Relative mode amplitude is indicated by colour. The COM motion is the highest infrequency (v1 < 2p3 795 kHz); b1 has no spatial variation. The lowest-frequencymode is v217 < 2p3 200 kHz; b217 has spatial variation at the lattice-spacinglength scale, d0 < 20mm. b, Using equation (2), we calculate Ji,j explicitly forN 5 217 spins and plot it as a function of spin–spin separation, di,j. FormR 2 v1 , 2p3 1 kHz, HODF principally excites COM motion in which all ionsequally participate: the spin–spin interaction is spatially uniform. As the detuningis increased, modes of higher spatial frequency participate in the interaction and Ji,j

develops a finite interaction length. We find the scaling of Ji,j with di,j follows thepower law Ji,j / d{a

i,j . For mR 2 v1? 2p3 500 kHz, all transverse modesparticipate and the spin–spin coupling power-law exponent, a, approaches 3. Thesolid lines are power-law fits to the theory points. For comparison with otherexperiments, the nearest-neighbour coupling (d0 5 20mm) is marked by thedashed line. c–e, The power-law nature of Ji,j is qualitatively illustrated for N 5 19(for larger N, diagrams of similar size are illegible). Spins (nodes) are joined by linescoloured in proportion to their coupling strength for various values of a. f, Forcontext, the graph for a 1D nearest-neighbour Ising interaction, a well-knownmodel in quantum field theory, is plotted.

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states, szi and sz

j , and by their mutual participation in each motionalmode m. The coupling coefficient is given by29

Ji,j~F2

0 N2BM

XN

m~1

bi,mbj,m

m2R{v2

m

ð2Þ

These pairwise interaction coefficients can be calculated explicitly by useof the ion motional modes. We find that the range of interaction can bemodified by detuning away from the COM mode as shown in Fig. 2b. Inthe limit mR 2 v1? 2p3 500 kHz, all modes participate equally in theinteraction and Ji,j!d{3

i,j , as discussed in ref. 28. At intermediate detun-ing, we find a power-law scaling of the interaction range, Ji,j!d{a

i,j ,where a can be tuned within the range 0 # a # 3. That is, by adjustingthe single experimental parameter, mR, we can mimic a continuum ofphysical couplings including important special cases: infinite range(a 5 0); monopole–monopole, or Coulomb-like (a 5 1); monopole–dipole (a 5 2); and dipole–dipole (a 5 3). The choice of a 5 0 resultsin the ‘J2

z interaction’, which gives rise to spin squeezing and is used inquantum logic gates27 (Supplementary Information). In addition, tuningmR also gives access to both antiferromagnetic (mR . v1) and ferromag-netic (v2= mR , v1) couplings10,13.

Experimentally, we demonstrate a tunable-range Ising interactionby observing a global spin precession under the application of HI

(Fig. 3). We compare experimental data with the mean-field predictionthat the influence of HI on spin j can be modelled as a magnetic field�Bj~(2=N)

PNi,i=j Ji,jhsz

i i in the z direction due to the remaining N 2 1spins (Supplementary Information; angle brackets denote expectationvalue). For a general qubit superposition state, �Bj gives rise to spinprecession about z in excess of that expected to result from simpleLarmor precession (Fig. 3b). The experiment sequence shown inFig. 3a measures this excess precession, averaged over all spins in thecrystal. At the outset, each spin is prepared in state j"æ and then rotatedabout the x axis by angle h1. The interaction HI is applied during thearms of a spin echo, each of duration tarm; precession proportional toÆsz

i æ coherently adds throughout the interaction duration, 2tarm. Thefinal p/2-pulse maps precession out of the initial plane (y–z) intoexcursions along z (above or below the equatorial plane of the Blochsphere) that are resolved by projective spin measurement along z.

We detect global, state-dependent fluorescence (j"æ, bright; j#æ,dark) as a function of h1 using a photomultiplier tube. This measure-ment permits a systematic study of the mean-field-induced spin pre-cession averaged over all particles

1N

XN

j

�Bj~21

N2

XN

j

XN

i,i=j

Ji,j

!cos (h1):2�J cos (h1)

The probability of detecting state j"æ at the end of the sequence is

P(j:i)~ 12

(1zexp({C?2tarm) sin (h1) sin (2�J cos (h1)?2tarm)) ð3Þ

and a single-parameter fit to experimental data yields �J . Decoherencedue to spontaneous emission is accounted for by C and is determinedby independent measurement of the ODF laser beam intensities, IR

(C / IR; see Supplementary Information).In Fig. 3c, d, we show representative measurements of excess pre-

cession due to HI for different values of spin coupling strength (deter-mined by Ji,j!I2

R) and interaction duration (2tarm). The excess spinprecession varies periodically with h1 (with period p) and greater inter-action strengths result in more precession, manifested in our experi-ment as a larger amplitude modulation of P(j"æ). Our data agree withequation (3) and allow direct extraction of �J for given experimentalconditions. In Fig. 3e, we plot �J , normalized by I2

R (IR is independentlymeasured), as a function of the detuning mR 2 v1 (N 5 206 6 10 ions).Using no free parameters, we find excellent agreement with the value of�J obtained by averaging over all Ji,j, where the Ji,j were calculated byincluding couplings to all N transverse modes (equation (2)).

The mean-field interpretation of our benchmarking measurementapplies only for weak spin–spin correlations. Therefore, in the bench-marking regime we apply a weak interaction (�J?2tarm=

ffiffiffiffiNp

=4; see

0 5 10 15 20 25 30 35 401

10

100

(μR – ω1)/2π (kHz)

J/2πI

R (H

z W

–2 c

m4)

0

2

a

b 1

2 3

c

e

dWeak ODF (AFM) Strong ODF (FM)

0 π 2π 3π 4π

0.25

0.50

0.7

P(�

)

0 π 2π 3π 4π

0.25

0.50

0.75

P(↑

)

θ1

a

← → ← →

θ1|xπ|y

π2

1 2 3

τarm

ODF ODF

z

y

x

θ1

2

τarm

|y

Figure 3 | Benchmarking the 2D Ising interaction. a, Spin-precessionbenchmarking sequence for HI. The spins are prepared at the outset in |"æ(a ferromagnetic state). The spin–spin interaction, HI, is present when the ODFlaser beams are on. We choose mR 2 v1 5 n ? 2p/tarm so that for small detuningsfrom the COM mode (v1), the spins are decoupled from the motion by the end ofeach periodtarm. b, Evolution of a single spin before the application of the spin echop-pulse. c, d, Plots showing spin precession proportional to hszi due to HI as afunction of initial ‘tipping angle’ h1. The error bars are statistical (s.d., 200measurements). The plots show typical experimental data (black) and, in c, single-parameter fits to equation (3). For an antiferromagnetic (AFM) coupling,tarm 5 250ms, mR 2 v1 5 2p3 4.0 kHz and IR 5 1.4 W cm22, we obtain�J=I2

R 5 2p3 25 Hz W22 cm4 (yellow fit). Longer drive periods and higher laserintensity, IR, yield a larger precession. For tarm 5 350ms, mR 2 v1 5 2p3 2.9 kHzand IR 5 1.9 W cm22, we obtain�J=I2

R 5 2p3 55 Hz W22 cm4 (red fit). The data inthis plot is typical of the experiments conducted for benchmarking. For a muchstronger interaction (d), equation (3) cannot be used to obtain�J because the mean-field assumption is no longer valid (Supplementary Information). Also, here weused a small negative detuning (v2=mR , v1), which gives a long-rangeferromagnetic (FM) interaction. For these experiments, we setvr 5 2p3 45.6 kHz. e, Benchmarking results for an ion crystal with N 5 206 6 10ions. Each point is generated by measuring�J as in c) and measuring the laser beamintensity, IR, at the ions. The error bars are dominated by uncertainty in IR

(Supplementary Information). The solid line (red) is the prediction of mean-fieldtheory that accounts for couplings to all N transverse modes; there are no freeparameters. The line’s breadth reflects experimental uncertainty in the anglehR 5 4.8 6 0.25u. The mean-field prediction for the average value of the power-lawexponent, a, is drawn in green (right axis, linear scale).

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Supplementary Information). In a quantum simulation, the same inter-action is applied at greater power, producing quantum spin–spin cor-relations. In the present configuration of our apparatus, spontaneousemission due to the ODF laser beams (parameterized by C in equation(3)) is the dominant source of decoherence. With modest laser powersof 4 mW per laser beam and a detuning of mR 2 v1 5 2p3 2 kHz, weobtain �J<2p|0:5 kHz and C=�J<0:06. The expected spin squeezing(J2

z ) due to this interaction is 5 dB (limited by spontaneous emission).The ratio C=�J can be reduced by a factor of 50 by increasing the ODFlaser beams’ angular separation, hR, to 35u (Fig. 1), which is a likelyprerequisite for access to the shortest-range, dipole–dipole, couplingregime (a R 3). At present, geometric constraints impose the limithRv50; we plan upgrades to our apparatus to permit hR 5 35u. Thusfar we operated with ODF magnitudes equal but opposite for j"æ andj#æ; relaxation of this constraint can further reduce C=�J .

Our work establishes the suitability of Penning traps for simulationof quantum magnetism in a regime inaccessible to classical computa-tion. Our approach is based on naturally occurring 2D Coulomb(Wigner) crystals with hundreds of ion qubits, a novel experimentalsystem that does not require demanding trap-engineering efforts.Experimentally, we used an ODF to engineer a tunable-range spin–spin interaction and benchmarked the interaction strength. Excellentagreement was obtained with the predictions of mean-field theory andatomistic calculations that predict a power-law antiferromagnetic spincoupling, Ji,j!d{a

i,j , for 0.05 = a = 1.4.With this work as a foundation, we anticipate a variety of future

investigations. For example, simultaneous application of the non-commuting interactions HB and HI is expected to give rise to quantumphase transitions; HI may be antiferromagnetic (mR . v1) or ferro-magnetic (v2= mR , v1). Geometric modifications to our apparatuswill permit access to larger values of hR and antiferromagnetic dipole–dipole-type couplings (a R 3). Improved image processing softwarewill permit direct measurement of spin–spin correlation functionsusing our existing single-spin-resolving imaging system (Fig. 1).

METHODS SUMMARYIn a frame rotating at frequency vr, the trap potential is

qQ(r, z)~12

Mv21(z2zbr2)

where b~vrv{21 (Vc{vr){1=2. The 9Be1 cyclotron frequency is

Vc 5 B0q/M 5 2p3 7.6 MHz and the frequency of the ions’ harmonic COMmotion along z is v1 5 2p3 795 kHz. Ion rotation is precisely controlled withan external rotating quadrupole potential26. For 100 = N = 350, we setvr < 2p3 45 kHz so that the radial confinement is weak enough that a cloud ofions relaxes into a single 2D plane (b= 1). When the ions’ motional degrees offreedom are Doppler laser cooled30 (TCOM < 1 mK), the ions naturally form a 2DCoulomb crystal on a triangular lattice, which is the geometry that minimizes theenergy of their mutual Coulomb potential energy. The crystal has N transverseeigenmodes, vm, with eigenfunctions bm; the COM mode, v1, is the highest-frequency mode (Fig. 2a).

The spin-dependent ODF is generated by a pair of off-resonance laser beamswith angular separation hR < 4.8u and difference frequency mR (Fig. 1). The resultis a travelling 1D optical lattice of wavelength lR~2p=Dk<3:7 mm whose wave-fronts propagate along z, traversing the ion crystal at frequency mR/2p. Alignmentof the optical lattice is crucial for proper spin–spin coupling (SupplementaryInformation). The lattice’s polarization gradient induces a differential a.c. Starkshift on the qubit states (a spin-dependent force). We choose operating conditionsthat give F"< 2F#, where F:~F0 cos (mRt)z. For reference, if the single-beamintensity at the ions is IR 5 1 W cm22, we obtain F0 < 1.4 3 10223 N.

Small, coherent displacements that produce negligible spin–motion entangle-ment (as required by equation (2)) are obtained for detunings satisfying

B mR{vmj jwF0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiB(2�nmz1)=2Mvm

pThis is a more stringent criterion than that used by others13,29, because it includesan additional factor of

ffiffiffiffiNp

to account for a typical distribution of composite spinstates. Moreover, we also include a correction factor for finite temperature,�nm<kBT=Bvm, where kB is the Boltzmann constant.

Received 16 November 2011; accepted 21 February 2012.

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Supplementary Information is linked to the online version of the paper atwww.nature.com/nature.

Acknowledgements This work was supported by the DARPA OLE programme andNIST. A.C.K. was supported by the NSF under grant number DMR-1004268. B.C.S. issupported by an NRC fellowship funded by NIST. J.K.F. was supported by the McDevittendowment bequest at Georgetown University. M.J.B. and J.J.B. acknowledge partialsupport from the Australian Research Council Center of Excellence for EngineeredQuantum Systems CE110001013. We thank F. Da Silva, R. Jordens, D. Leibfried, A.O’Brien, R. Scalettar and A. M. Rey for discussions.

Author Contributions J.W.B., B.C.S., H.U., M.J.B. and J.J.B designed the experiment.J.W.B. and B.C.S. obtained the data and analysed it with advice from J.J.B. A.C.K.,C.-C.J.W. and J.K.F. developed the formalism and numerics to calculate the spin–spincoupling. J.W.B. wrote the manuscript with assistance from B.C.S., M.J.B. and J.J.B. Allauthors participated in discussions, contributed ideas throughout the project andedited the manuscript.

Author Information This manuscript is a contribution of the US National Institute ofStandards and Technology and is not subject to US copyright. Reprints andpermissions information is available at www.nature.com/reprints. The authors declareno competing financial interests. Readers are welcome to comment on the onlineversion of this article at www.nature.com/nature. Correspondence and requests formaterials should be addressed to J.W.B. (joe.britton@gmail.com).

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