A large scale quantum simulator on a diamondsurface at room temperature

ARTICLESPUBLISHED ONLINE: 20 JANUARY 2013 | DOI: 10.1038/NPHYS2519

A large-scale quantum simulator on a diamondsurface at room temperatureJianming Cai1,2, Alex Retzker1,3, Fedor Jelezko2,4 and Martin B. Plenio1,2*

Strongly correlated quantum many-body systems may exhibit exotic phases, such as spin liquids and supersolids. Althoughtheir numerical simulation becomes intractable for as few as 50 particles, quantum simulators offer a route to overcome thiscomputational barrier. However, proposed realizations either require stringent conditions such as low temperature/ultra-highvacuum, or are extremely hard to scale. Here, we propose a new solid-state architecture for a scalable quantum simulatorthat consists of strongly interacting nuclear spins attached to the diamond surface. Initialization, control and read-out of thisquantum simulator can be accomplished with nitrogen-vacancy centers implanted in diamond. The system can be engineeredto simulate a wide variety of strongly correlated spin models. Owing to the superior coherence time of nuclear spins andnitrogen-vacancy centers in diamond, our proposal offers new opportunities towards large-scale quantum simulation atambient conditions of temperature and pressure.

Many intriguing phenomena in condensed-matter systemsoriginate from the interplay of strong interactions andfrustrations. A representative example is frustrated quan-

tummagnetism, where the spins cannot order to minimize all localinteractions, and the ground state is highly degenerate1,2. Togetherwith long-range interactions between non-nearest neighbours, thefrustrated quantummodels give rise to intriguing quantum phases,for example supersolids3,4. Moreover, they can also stabilize thelong-sought quantum spin liquid5,6, which has connections withhigh-temperature superconductivity7. However, the properties ofthese systems have proved to be very hard to understand from nu-merical calculations, partly owing to the combination of long-rangequantum correlations and the superposition principle of quantummechanics. This principle implies that the required computationalresources grow exponentially with the number of particles, mak-ing numerical approaches inefficient. Richard Feynman’s idea8 ofquantum simulation provides a powerful solution to this problem:one could gain a deep insight into complex states of matter byexperimentally simulating themwith anotherwell-controlled quan-tum system9. Large-scale quantum simulations are expected to be apowerful tool10 for the investigation of fundamental problems incondensed-matter physics.

Quantum simulation has attracted extensive research interestin the past decade. Various architectures for quantum simulationhave been constructed based on different systems, ranging fromultracold neutral atoms11–13, trapped ions14,15, and photonicsystems16 to superconducting circuits in solid-state devices17.The still challenging goal is to realize a large-scale quantumsimulator (with less demanding technical requirements) whichcannot be efficiently solved numerically with classical computers.In this work, we propose a scalable architecture that is ofpractical interest for large-scale quantum simulation. Our quantumsimulator is based on lattices of interacting nuclear spins,which can be fabricated chemically on the diamond surface18,19or by depositing functionalized graphene films20. We proposeto exploit the nitrogen-vacancy (NV) centres21 beneath thediamond surface as an efficient control element for cooling,

1Institut für Theoretische Physik, Albert-Einstein Allee 11, Universität Ulm, 89069 Ulm, Germany, 2Center for Integrated Quantum Science and Technology,Universität Ulm, 89069 Ulm, Germany, 3Racah Institute of Physics, The Hebrew University of Jerusalem, Jerusalem 91904, Givat Ram, Israel, 4Institut fürQuantenoptik, Albert-Einstein Allee 11, Universität Ulm, 89069 Ulm, Germany. *e-mail: [email protected].

spin–spin interaction engineering, and read-out of the nuclear-spin quantum simulator.

We explain how this simulator is constructed, and establishschemes for its initialization, control and read-out. We analyse itsvalidity by detailed numerical studies, thus demonstrating the feasi-bility of our proposal within the current experimental capabilities.

Construction of hardwareWe discuss two main paths for the fabrication of the hardwarefor our quantum simulator. First, large-scale lattices of nuclearspins can be constructed by chemically controlled terminationof diamond surfaces. The fluorine (19F with spin 1/2), oxygen(17O with spin 5/2) and hydrogen/hydroxyl group (1H with spin1/2 and 2H with spin 1) termination of the diamond surfacecan be obtained from the process of chemical vapour deposition,or by functionalizing the diamond surface. As a representativeexample, we will mainly concentrate on the fluorine-terminateddiamond surface, which has a positive electron affinity. The twomost important diamond surfaces are the (111) and (100) surfaces,which constitute the crystal faces of polycrystalline chemical vapourdeposition diamond films and can be selectively grown withappropriately controlled process parameters18. The (111) surfaceof diamond is the natural cleavage plane of diamond, and hasone dangling bond per surface carbon atom which is terminatedby carbon–fluorine bonds on the fluorine-terminated diamondsurface22. The fluorine atoms naturally arrange in a triangularlattice with nearest-neighbour distances of about 2.5 Å (refs 18,19;Fig. 1a). The (100) surface of diamond shows two dangling bondsper surface carbon atom, which will undergo a reconstruction intoa 2×1 geometry with neighbouring surface carbon atoms forming aπ-bonded dimer. The remaining dangling bonds are terminated bycarbon–fluorine bonds, which leads to a rectangular lattice of fluo-rine atoms18,19, Fig. 1b. Functionalized graphene (fluorographene)provides a double-layer triangular lattice of fluorine atoms, Fig. 1c.This can be obtained through the mechanical cleavage of graphitefluoride, or by exposing graphene to atomic fluorine formed bydecomposition of xenon difluoride (XeF2; ref. 20).

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NATURE PHYSICS DOI: 10.1038/NPHYS2519 ARTICLES

2.5 2.5

2.5

2.5

2.5

2.5

2.5

2.5 2.5

2.5

2.5

a b c

Figure 1 | Lattices of fluorine nuclear spin. a, A triangular nuclear spin lattice on the fluorine-terminated diamond (111) surface. b, A rectangular nuclearspin lattice on the fluorine-terminated diamond (100) 2× 1 surface. The distance between two nearest-neighbour fluorine atoms is about 2.5 Å. c, Adouble-layer triangular lattice from fluorographene. Yellow atoms represent carbon, and green atoms represents fluorine.

In addition to the nuclear spins, we shall introduce NV centresby shallow implantation a few nanometres below the surface ofdiamond23,24. This constitutes a fundamental ingredient of ourquantum simulator, as it allows the initialization and read-outof the nuclear spins. Let us remark that in contrast to graphene,fluorographene exhibits a large band gap ofmore than 3.5 eV, whichis larger than the optical gap of NV centres in diamond (∼2.9 eV).This avoids the unwanted fluorescence quenching of NV centresand is thus crucial for using the NV centres for the control ofthe nuclear spin arrays.

Engineering of the interacting HamiltonianThe nuclear spins interact with each other via magnetic dipole–dipole interactions as Vij = g (rij)[si ·sj −3(si · rij)(sj · rij)], where siare the nuclear spin operators, g (rij)= (h2µ0γiγj)/(4πr3ij ), and γiand γj are gyromagnetic ratios of the ith and jth nuclear spin, whichare connected by the vector rij = rij rij . Because nearest-neighbourfluorine nuclear spins are at a distance of 2.5 Å, a coupling strengthof g (a) ' 6.8 kHz is achieved. A strong magnetic field, whichleads to energy level shifts exceeding the nuclear spin couplingstrength, simplifies the effective Hamiltonian due to the rotatingwave approximation to an XXZmodel

Vij = g (rij)[szi s

zj −∆

(sxi s

xj +s

yi s

yj

)]with ∆ = 1/2. Our calculations verify that such a rotating waveapproximation can already be well satisfied for a magnetic field aslow as 750G (3MHz), see Supplementary Information. We denotethe diamond surface onwhich the nuclear spin lattice is constructedas the X–Y plane, while the vector perpendicular to it defines theZ axis (spatial axes), and the magnetic field direction as the maxis, which gives the quantization of nuclear spins. The spin–spininteraction strength can be tuned by changing the direction ofthe magnetic field as

g (rij)= g (rij)(1−3cos2θij)

where θij is the relative angle between rij and m. Thus, by changingthe direction of the external magnetic fields, we can controlthe spatial anisotropy and the sign of the interaction strength,which determines whether the interaction is ferromagnetic oranti-ferromagnetic and thereby induces spin frustration. The valueof the spin anisotropy∆ can be tuned with gradient magnetic fieldsor radio-frequency fields.

Cooling of the quantum simulatorThe reliable preparation of the quantum simulator in a low-entropystate is a prerequisite for the observation of quantum phases. Herewe explain in detail how the nuclear spin lattice can be initialized toa well defined spin direction using dynamical nuclear polarizationvia optical pumping NV centres in diamond, Fig. 2, which worksat room temperature (similar to the laser cooling of trapped ions).The entire process consists of repetitive cycles. In each cycle, the NVcentre is first prepared in the |+〉 =

√1/2(|ms = 0〉+ |ms =+1〉)

V

N

532 nm

3E1A1

|±1⟩

|0⟩3A

637 nm

Figure 2 | Initialization and read-out of nuclear spins. NV centre lyingbeneath the nuclear spin layer in bulk diamond is exploited to initialize andmeasure nuclear spins on the diamond surface. The spin-1 state of the NVcentre can be optically initialized and read out via spin-dependentfluorescence, and can be coherently controlled with microwave fields.

state by optical pumping to the |ms = 0〉 state using a green laser(532 nm) and a subsequentπ/2microwave pulse. Amicrowave field(with Rabi frequency ΩNV) on resonance with the electronic transi-tion |ms=0〉↔|ms=+1〉will lock the NV electron spin state. If theHartmann–Hahn condition with nuclear spins25 is satisfied, the NVelectron spin polarization will be transferred to the nuclear spins26.Although an individual NV centre suffices, this process can bemadeswifter by using a fewNVs (Supplementary Information). Owing totheir proximity, the interactions between nuclear spins exceed thosebetween the electron spin of the NV centre and the nuclear spins.Therefore, it will be advantageous to effectively decouple nuclearspin interactions, which also facilitates estimates of the cooling effi-ciency. To this end, we apply a radio frequency field with the ampli-tude Ωp whose frequency is detuned from the Larmor frequency ofthe nuclear spins ωL by∆p. The energy conserving terms of nuclearspin interactions cancel each other when Ωp =

√2∆p (ref. 27),

and the anti-rotating terms are suppressed as long as the energymismatchωf= (Ω 2

p+∆2p)

1/2gij is fulfilled. Thus, the nuclear spins

behave as isolated particles, and the Hartmann–Hahn conditionbecomes ΩNV = (ωL − ∆p) + (Ω 2

p + ∆2p)

1/2, see SupplementaryInformation. Such a mechanism for the isolation of the nuclearspins (as we will discuss in the following section) can also reduce theperturbation on the nuclear spin state during the read-out process,whichwill be beneficial for the accuracy of themeasurement.

To verify the validity of this idea, we have used a Chebyshevexpansion28 to calculate polarization dynamics with the exactHamiltonian of a 3×3 nuclear spin lattice, assuming a distance of5 nm from the NV centre. It can be seen from Fig. 3a that the cou-pling between nuclear spins is effectively eliminated. We comparethe exact numerical calculations with the results for isolatednuclear spins under the spin temperature approximation in which

NATURE PHYSICS | VOL 9 | MARCH 2013 | www.nature.com/naturephysics 169




ARTICLES NATURE PHYSICS DOI: 10.1038/NPHYS2519

coherences between nuclear spins are neglected29, Fig. 3b; theseshow a good agreement. To achieve an ultra-high polarization givenby the spin temperature approximation, one can introduce mag-netic noise to remove coherence among nuclear spins in betweenpolarization cycles. From the spin temperature approximation, onecan estimate the polarization rate, and the required polarizationtime scales linearly with the total number of nuclear spins Nand the inverse effective temperature. The ultimate polarizationefficiency will be limited by the relaxation time T1 of nuclei, whichcan be as long as a few hours even at room temperature. Thepolarization cycle time sets the required coherence time of NVcentres and nuclear spins to a few hundreds of microseconds, whichis readily achievable with the current experimental techniques indiamond samples. The polarization efficiency can also be improvedby optimizing the polarization cycle time and exploiting several NVcentres. Once the nuclear spins have been initialized, by performingan adiabatic passage, one can prepare the system in the groundstate of the engineered interacting Hamiltonian (similar to othertypes of quantum simulator30); see Supplementary Informationfor an explicit example which demonstrates that adiabatic statepreparation permits the observation of different quantum phases.During the operation of quantum simulation, a green laser can beused to decouple the nuclear spins from theNV electron spin31.

Measurement of the quantum simulatorBefore discussing the static and dynamical properties of theproposed quantum simulator, let us describe the measurementschemes and demonstrate their viability by means of numericalsimulations. The main goal is to measure observables that canprovide information about the nuclear spin state32. It is challengingto measure nuclear spins directly because of their small magneticmoments. However, NV centres implanted beneath the diamondsurface will provide the solution as a measurement interface fornuclear spin states26. Before the measurement, we apply a radio-frequency pulse to map the nuclear spin basis from |↑〉 and |↓〉 to|↑〉 and |↓〉, in which the nuclear spins can be effectively decoupledfrom each other by a continuous driving field, as described inthe process of initialization. The NV centre is prepared in thestate |µ〉 (µ = ±), and is driven to match the Hartmann–Hahncondition between the electron spin of the NV centre and thenuclear spins. After the electron spin interacts with the nuclearspins for time τ , we measure the population of state |ν〉 ofthe NV centre, which is given by Pνµ = τ

2∑i

∑j(g⊥

i g⊥

j )〈sµ

i sνj 〉 tosecond order in τ , where µ,ν = ± and g⊥i , g

⊥

j represent therate of polarization exchange between the NV center and thenuclear spins (Supplementary Information). The above observablesinclude both local contributions of individual nuclear spins (fori = j) and two-point correlations of nuclear pairs (for i 6= j).With a gradient field, we can estimate quantities such as structurefactors, which are very important for the study of quantum phasetransitions and for inferring entanglement properties33, from theobservables ∆

qαβ = τ

2∑i

∑j(g⊥

i g⊥

j )cos(q · (ri − rj))〈sαi sαj + sβi s

β

j 〉

with α,β = x,y,z . By one single magnetic tip with the state-of-arttechnology34, it is possible to generate a gradient field as large as15G (about ten times larger than the coupling between fluorinenuclear spins) over the lattice constant (2.5 Å). We remark thatthe scheme also works by exploiting an ensemble of NV centres(Supplementary Information). The validity of our measurementscheme is numerically tested in the context of witnessing quantumphase transitions as discussed in the following section.

Frustrated quantum magnetism and supersolidsOur system can simulate quantum spin models with thetunable spin–spin interaction g (rij): positive g (rij) correspondsto anti-ferromagnetic (AF) interactions, and negative g (rij)corresponds to ferromagnetic (F) interactions. In the triangular

1.0

0.5

0.0

P (1)↓

P¬↓

1.0

0.9

0.8

0.7

0.6

0.5

0.0 0.2 0.4 0.6

t (ms)

0.8 1.0

0 5 10 15

t/N (ms)

a

b

Figure 3 | Isolation and initialization of nuclear spins. a, The localization(open circle, with radio-frequency driving) and de-localization (opensquare, without radio-frequency driving) of spin excitations initially createdin one specific site, demonstrating the efficiency of decoupling of thenuclear–nuclear interaction in the former case. The driving parameter isωf= 160 kHz. b, The average nuclear spin-down state population P↓ as afunction of the polarization time t: exact numerical simulation (purplesquare) and spin temperature approximation (yellow diamond) with thecycle time τ = 30 µs. Exact numerical simulation (red circle) withτ = 120 µs where magnetic noise is introduced to remove nuclear spincoherence every 20 cycles. The nuclear spin lattice is 3×3 with a distanceof 5 nm to the NV centre. The magnetic field direction ism= (

√1/2,0,

√1/2).

lattice of the fluorine simulator, the nearest-neighbour nuclearspin interactions are denoted as Ja1 , Ja2 , Ja3 , with a1 = (1,0,0)a2 = (1/2,

√3/2,0) and a3 = (−1/2,

√3/2,0). The long-range

interactions and spin frustrationmake it hard to perform numericalsimulations using the Quantum Monte Carlo method owing tothe subtle sign problem35. Here, we exactly diagonalize the systemon a 6× 4 lattice using the Lanczos algorithm under periodicboundary conditions to provide evidence for various phases ofquantum magnetism.

In the situation where Ja1 ≡ J1 = ga is positive (anti-ferromagnetic), and Ja2 = Ja3 ≡ J2 = ga(1− (9/4)cos2θ) is negativefor cosθ ≥ 2/3 (the magnetic field direction is m= (0,cosθ,sinθ)),the triangle which consists of J2 − J2 − J1 is spin frustrated(Fig. 4a). For small values of |J2/J1|, the system consists of 1D(anti-ferromagnetic) chains with weak intra-chain interactions thatinduce ferromagnetic order in the sublattice of every two 1D chainsand are characterized by the (normalized) spin structure factorSq = (1/N 2)〈

∑ij e

iq·(ri−rj )szi szj 〉 with q= (π,0). As |J2/J1| increases,

the competition between anti-ferromagnetic (J1) and ferromagnetic(J2) interactions leads to the ferromagnetic phase above the point|J2/J1| = 1, corresponding to the spin structure factor Sq(0,0)(Fig. 4a, lower left). Note that the largest non-nearest-neighbourinteraction J3 is ferromagnetic and is essential for the emergence of





NATURE PHYSICS DOI: 10.1038/NPHYS2519 ARTICLES

0.25

F¬F 1D AF chain

q = (0, 0)0.20 F¬F

J1

J1

J1 J

1

J2

J2

J2

J2 J

31 J31

J33

J32 J

32 J32

J33J

31J

33

J2

J2

J3

0.15Sq

0.100.050.00

¬1.0 0.0¬0.5

q = (π, 0)

q = (0, 0)

q = (π, 0)

1D AF chain

0.250.20 F¬F0.15Sq0.100.050.00

¬1.0 0.0¬0.5

←

Sq

←

3 )q = (0, 2π/√

1D AF chain F¬NAF

F¬NAF

F¬NAF

F¬AFF¬AF

F¬AF

0.250.200.150.100.050.00

¬8 ¬7 ¬6 ¬5 ¬4 ¬3 ¬2 ¬1 0 1

Sq

←

0.250.200.150.100.050.00

¬8 ¬7 ¬6 ¬5 ¬4 ¬3 ¬2 ¬1 0 1

q = (0, π/√

q = (0, 2π/√

q = (0, π/√

a b

3 )

3 )

3 )

J2 /J 1 J2 /J 1 J2 /J 1 J2 /J 1

Figure 4 |Quantum magnetic phase transitions of the fluorine quantum simulation on a triangular lattice. a, Tuning from 1D anti-ferromagnetic chains toferromagnetic order with the magnetic field in the direction m= (0,cosθ,sinθ). b, The phase transition from ferromagnetic–antiferromagnetic (F–AF)order to ferromagnetic–(alternative) antiferromagnetic (F–NAF) controlled by the magnetic field in the direction m= (cosθ,0,sinθ). In both a and b thegraphs on the lower left are spin structure factors calculated using the Lanczos algorithm for a 6×4 nuclear spin lattice under periodic boundaryconditions and those on the lower right are a comparison of the spin structure factors from exact diagonalization of a 4×4 nuclear spin lattice (curve) andthe estimated values (rescaled for visibility) via NV measurements (circles and squares) by numerical simulations. The interaction time between theelectron spin of the NV centre and the nuclear spins for read-out is τ =60 µs.

the ferromagnetic phase which is absent for the short-range model(Supplementary Information).

For the other situation, Ja2 = Ja3 ≡ J2 = g (a)(1− (3/4)cos2θ) isalways positive (anti-ferromagnetic), and Ja1≡ J1=g (a)(1−3cos

2θ)is negative (ferromagnetic) for cosθ >1/3, where θ is defined by themagnetic field direction as m= (cosθ,0,sinθ). Considering onlythe nearest-neighbour interactions, the system is non-frustrated,and is expected to be ferromagnetic in the a1 direction andanti-ferromagnetic in the a2 direction (F–AF phase), which isidentified by the spin structure factor Sq with q= (0,2π/

√3). As the

value of cosθ approaches 1, the non-nearest-neighbour interactionsJ31 = 1/3

√3, J32 = (1 − (9/4) cos2 θ)/3

√3 become comparable

to J2. The competition between them leads to a magnetic phaseidentified with the spin structure factor Sq with q = (0,π/

√3).

The spins are ferromagnetic in the a1 direction, whereas they areanti-ferromagnetic between next-nearest-neighbour chains in thea2 direction (Fig. 4b, lower left).

We also check the feasibility of using NV centres to identifydifferent magnetic phases. We numerically calculate the dynamicsduring measurement and obtain the estimation of spin structurefactors. Owing to the limit of computational overhead, we consideran example of a 4× 4 nuclear spin lattice with periodic boundaryconditions. By applying a gradient field we can generate therelative phases between different nuclear spins corresponding tothe spin structure factor, which can then be estimated by theobservable Sq ∝ ∆q

xz +∆qyz −∆q

xy via NV centres. We find thatthe estimation is in good agreement with the results from exactdiagonalization, and that it can witness quantum phase transitionsbetween different magnetic orders, Fig. 4a,b (lower right) andSupplementary Information for more details.

The nuclear spin Hamiltonian can be mapped to the hard-coreboson model by the Holstein–Primakoff transformation. Note thatquantum simulation of similar models with polar molecules inoptical lattices has been proposed36,37 and numerically studiedin38,39 for long-range repulsive dipole interactions (Vij). Our systempossesses long-range interactions (both Vij and tij), thus offeringa platform to investigate rich phases of hard-core bosons withpotentially novel features. Indeed, these models with frustrationand long-range interactions pose considerable challenges to theclassical numerical simulations because of the sign problem for2D systems. We simulate such a model with the directed loopalgorithm in the stochastic series expansion representation of the

0.7

0.6⟨n⟩

0.5

0.2

0.1

0.0

0.2

0.1

0.0

0.0 0.5

S

SFSS

¬ 0

1.0 1.5μ μ

0.0 0.5 1.0 1.5

3)q = (π, π /√

S

SS

SFsρ

sρ

Sq

a

b

¬ 0μ μ

Figure 5 |Quantum phases of superfluids and supersolids. a, The averagefilling 〈n〉 as a function of chemical potential µ−µ0 (in units of ga), whereµ0 is the chemical potential corresponding to half-filling. b, The superfluiddensity ρs and the normalized structure factor Sq(π,π/

√3). The system

size is 12× 12 and t/V=0.2. The magnetic field direction ism= cosθ a3+sinθ Z, with cosθ =

√(1/3) and Z= (0,0,1). The temperature

used in our simulation is T/ga=0.1.

ALPS library40. It can be seen from Fig. 5 that quantum phasetransitions between the solid (S), supersolid (SS) and superfluid(SF) can be observed in our model.

The nuclear spin defects are the main source of imperfections inthe present set-up. The effect of defects and general imperfectionsof nuclear spin lattices are important for the study of, forexample, quantum phases of doped many-body systems anddisorder-induced delocalization. For the examples of quantumphases present in this work, we have performed numericalsimulations including lattice defects. Our results show that they





ARTICLES NATURE PHYSICS DOI: 10.1038/NPHYS2519

m + 1

d+1

d0

d–1m – 1

m

1Ω

1Δ

2Ω

2Δ

Figure 6 | Tuning spin anisotropy with dressed states. Simulation of aneffective spin 1/2 by the dressed states from a higher spin by applying radiofrequency driving fields (with Rabi frequency Ωk and detuning ∆k).

can persist with an experimentally achievable rate of defects(Supplementary Information).

Discussion and outlookIn addition to the static properties of quantum phases such as spinliquids and topological phases41, the present quantum simulatoris capable of studying quantum many-body dynamics, such asquantum quenches and the generation of multi-particle entangle-ment (Supplementary Information). It would also be interestingto exploit the ideas in room temperature quantum informationprocessing with NV centres42. The prerequisite technologies forthe implementation of such a quantum simulator, such as thetechniques for charge state manipulation of NV centres in (surface-functionalized) diamond43, shallow NV implantation23,24, coherentcontrol and readout of NVs44,45, and Hartmann–Hahn spin lockingwith NVs, are currently being developed. The proposal will stimu-late the interest of material scientists in fabricating other candidatesystems, for example, a few layers of 13C grown inside diamond46and graphene heterostructures assembled layer by layer in a desiredsequence47. The relevant tools in this workmay also be beneficial forfurther research in coherent control on surfaces/mono-layer films.

MethodsTuning spin anisotropy with radio-frequency fields. The value of the spinanisotropy ∆ can be tuned with gradient magnetic fields to modulate the hoppingcoupling (sxi s

xj +s

yi s

yj ) while keeping the repulsive interaction szi s

zj unchanged

48. Inthis way the interplay between geometrical frustration and the effects of quantumfluctuations on the realization of a spin liquid phase could be tested49. As analternative scheme with higher nuclear spin species, for example 2Hwith spin 1 and17Owith spin 5/2, one can tune the spin anisotropy∆ by applying continuous fieldscorresponding to the nuclear spin transitions |m−1〉↔ |m〉 and |m〉↔ |m+1〉with Rabi frequencies Ωk and detuning ∆k (Fig. 6). An effective spin 1/2 can beencoded in two of the induced dressed states with the spin anisotropy ∆ flexiblytuned (see Supplementary Information). We remark that spin Hamiltonians forhigher spins may directly lead to rich quantum phases50.

Measurement observables with NV centres. During the measurement, the NVcentre is prepared in the state |µ〉 (µ=±) and driven tomatch theHartmann–Hahncondition between the electron spin of the NV centre and the nuclear spins. Thepopulation of state |ν〉 of the NV centre is given by Pνµ = τ

2∑i,j (g

⊥

i g⊥

j )〈sµ

i sνj 〉 tosecond order in τ , where µ,ν=±. We can extract information about the averagenuclear spinmagnetization by∆M =P+

−−P−+=2τ 2

∑i(g⊥

i )2〈siz 〉, and the transverse

correlation as ∆xy = P+−+P−

+= 2τ 2

∑i,j (g

⊥

i g⊥

j )〈sxi s

xj + syi s

yj 〉. The correlation

function along the other directions, ∆yz and ∆xz , can be obtained by applyingthe Hadamard operation OH = |↑x 〉〈↑ |+|↓x 〉〈↓ | with |↑x 〉 =

√1/2(|↑〉+|↓〉),

and the phase transformation OI = |↑〉〈↑ |+ i|↓〉〈↓ | on the nuclear spin statebefore the measurement. To measure observables such as the structure factors, wecan use a gradient field with which the nuclear spin at the position ri experiencesa magnetic field bi = ri ·(bx ,by ) and gains a position-dependent phase ϕi = ri ·qafter time tp, where q= (qx ,qy )= γN tp(bx ,by ) after a certain time tp, and γN isthe gyromagnetic ratio of nuclear spins. By performing the same measurementas before, we can obtain ∆q

xy = τ2∑

i

∑j (g⊥

i g⊥

j )cos(q ·(ri− rj ))〈sxi sxj + syi s

yj 〉 and

∆qyz , ∆

qxz in a similar way.

Map to hard-core bosonmodel. The nuclear spin Hamiltonian can be mapped tothe hard-core bosonmodel by theHolstein–Primakoff transformation as

Hb=∑〈i,j〉

[Vijninj− tij

(a†i aj+aia

†j

)]+µ

∑i

ni

with ni= szi +1/2 (ni= 0,1). Here, the chemical potential is µ= (γNB)−∑

j g (rij ),the repulsive interaction is Vij = g (rij ), and the hopping is tij =∆/2g (rij ).The system can therefore simulate the hard-core boson model, whichdemonstrates interesting phases such as long-range off-diagonal ordersuperfluids, and moreover a supersolid phase characterized by both long-rangeoff-diagonal and diagonal order. With the magnetic field along the directionm= cosθ a3+ sinθ Z = (−(1/2)cosθ,

√3/2cosθ,sinθ), the nearest-neighbour

interactions are Va1 =Va2 ≡V1= ga(1−(3/4)cos2θ) and Va3 ≡V2= ga(1−3cos2θ).By changing the value of the magnetic direction angle θ , we can gradually tunethe geometric frustration as quantified by the ratio V2/V1. For cosθ ∈ [0,

√1/3],

the values of V for all interactions have the same sign (including long-rangeinteractions), and one can simulate such a model with Quantum MonteCarlo methods. By comparison with the short-range model, we find that thelong-range interaction significantly enhances the superfluidity (SupplementaryInformation). One can also use a gradient field to selectively tune hoppinginteractions. For example, with a gradient field that decreases along the direction∆b= (

√3/2,1/2), the hopping interaction will be suppressed except in the

direction a3= (−1/2,√3/2).

Received 24 August 2012; accepted 27 November 2012;published online 20 January 2013

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AcknowledgementsWe are grateful for valuable communications with M. Troyer, L. Pollet andB. Capogrosso-Sansone about the properties of supersolids and QMC simulationswith ALPS. We also thank R. Rosenbach and J. Almeida for their help in numericalsimulations. The work was supported by the Alexander von Humboldt Foundation, theEU Integrating Project Q-ESSENCE, the EU STREP PICC and DIAMANT, the BMBFVerbundprojekt QuOReP, DFG (FOR 1482, FOR 1493, SFB/TR 21) and DARPA. J.C.was also supported by a Marie-Curie Intra-European Fellowship (FP7). Computationswere performed on the bwGRiD.

Author contributionsM.B.P. proposed the idea and developed it further together with F.J. and J.C.J.C. carried out the numerical and analytical work with advice from F.J., M.B.P.and A.R. All authors discussed the results. J.C. drafted the manuscript with inputfrom F.J., M.B.P. and A.R.

Additional informationSupplementary information is available in the online version of the paper. Reprints andpermissions information is available online at www.nature.com/reprints. Correspondenceand requests for materials should be addressed toM.B.P.

Competing financial interestsThe authors declare no competing financial interests.




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