nphys287

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A toolbox for lattice-spin models with polarmoleculesA. MICHELI*, G. K. BRENNEN AND P. ZOLLERInstitute for Theoretical Physics, University of Innsbruck, and Institute for Quantum Optics and Quantum Information of the Austrian Academy of Science,6020 Innsbruck, Austria*e-mail: [email protected]

Published online: 30 April 2006; doi:10.1038/nphys287

There is growing interest in states of matter with

topological order. These are characterized by highly stable

ground states robust to perturbations that preserve the

topology, and which support excitations with so-called

anyonic statistics. Topologically ordered states can arise

in two-dimensional lattice-spin models, which were

proposed as the basis for a new class of quantum

computation. Here, we show that the relevant hamiltonians

for such spin lattice models can be systematically

engineered with polar molecules stored in optical lattices,

where the spin is represented by a single-valence

electron of a heteronuclear molecule. The combination

of microwave excitation with dipoledipole interactions

and spinrotation couplings enables building a complete

toolbox for effective two-spin interactions with designable

range, spatial anisotropy and coupling strengths

signicantly larger than relevant decoherence rates. Finally,

we illustrate two models: one with an energy gap providing

for error-resilient qubit encoding, and another leading to

topologically protected quantum memory.

Lattice-spin models are ubiquitous in condensed-matterphysics, where they are used as simplied models to describethe characteristic behaviour of more-complicated interactingphysical systems. Recently there have been exciting theoreticaldiscoveries of models with quasi-local spin interactions withemergent topological order1,2. In contrast to Landau theory wherevarious phases of matter are described by broken symmetries,topological ordered states are distinguished by a homology classand have the property of being robust to arbitrary perturbationsof the underlying hamiltonian. These states do not exhibit long-range order in pairwise operators, but rather they have long-rangeorder in highly non-local strings of operators. A real-world exampleis the fractional quantum Hall eect, which gives rise to stateswith the same symmetry but distinguishable by quantum numbersassociated with the topology of the surface they live on3.

It is of signicant interest to design materials with theseproperties, both to observe and to study exotic phases, and inthe light of possible applications. Cold atomic and moleculargases in optical lattices are prime candidates for this endeavourin view of the complete controllability of these systems in thelaboratory. The idea of realizing bosonic and fermionic Hubbardmodels, and thus also lattice-spin models, with cold atoms inoptical lattices has sparked a remarkable series of experiments,and has triggered numerous theoretical studies to develop coldatoms as a quantum simulator for strongly correlated condensed-matter systems46. However, coaxing a physical system to mimic therequired interactions for relevant lattice-spin models, which mustbe both anisotropic in space and in the spin degrees of freedomand a given range, is highly non-trivial. Here, we show that coldgases of polar molecules, as currently developed in the laboratory7,allow us to construct, in a natural way, a complete toolbox forany permutation-symmetric two-spin-1/2 (qubit) interaction. Theattraction of this idea also rests on the fact that dipolar interactionshave coupling strengths signicantly larger than those of the atomicHubbard models, and than relevant decoherence rates.

Our basic building block is a system of two polar moleculesstrongly trapped at given sites of an optical lattice, where thespin-1/2 (or qubit) is represented by a single electron outside aclosed shell of a heteronuclear molecule in its rotational groundstate. Heteronuclear molecules have large permanent electric dipolemoments. This implies that the rotational motion of molecules

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Si

S1

S2

D2

D1

Sj

E(t )

E(t )

xz

r

y

a b

Figure 1 Example anisotropic spin models that can be simulated with polar molecules trapped in optical lattices. a, Square lattice in 2D with nearest-neighbourorientation-dependent Ising interactions along x and z. Effective interactions between the spins S1 and S2 of the molecules in their rovibrational ground states are generatedwith a microwave eld E(t ) inducing dipoledipole interactions between the molecules with dipole moments D1 and D2, respectively. b, Two staggered triangular lattices withnearest neighbours oriented along orthogonal triads. The interactions depend on the orientation of the links with respect to the electric eld. (Dashed lines are includedfor perspective.)

is coupled strongly through the dipoledipole interactions, whosesignatures are the long-range 1/r3 character and an angulardependence, where the polar molecules attract or repel each otherdepending on the relative orientation of their dipole moments.In addition, microwave excitation of rotational energy levelsallows us to eectively tailor the spatial dependence of dipoledipole interactions. Finally, accounting for the spinrotationsplitting of molecular rotational levels, we can make these dipoledipole interactions spin-dependent. General lattice-spin models arereadily built from these binary interactions.

ANISOTROPIC SPIN MODELS WITH NOISE-RESILIENT GROUND STATES

Two highly anisotropic models with spin-1/2 particles, whichwe will show how to simulate, are illustrated in Fig. 1a and brespectively. The rst model takes place on a square 2D lattice withnearest-neighbour interactions

H (I)spin =1i=1

1j=1

J(zi,jzi,j+1 +cosxi,jxi+1,j).

Introduced by Duocot et al.8 in the context of Josephson junctionarrays, this model (for = /2) admits a twofold degenerateground subspace that is immune to local noise up to th order, andhence is a good candidate for storing a protected qubit.

The second model occurs on a bipartite lattice constructed withtwo 2D triangular lattices, one shifted and stacked on top of theother. The interactions are indicated by nearest-neighbour linksalong the x, y and z directions in real space:

H (II)spin = Jx-links

xj xk + J

y-links

yj

yk + Jz

z-links

zj zk .

This model has the same spin dependence and nearest-neighbourgraph as the model on a honeycomb lattice introduced by Kitaev9.He has shown that by adjusting the ratio of interaction strengths|J|/|Jz| the system can be tuned from a gapped phase carrying

abelian anyonic excitations to a gapless phase that, in the presenceof a magnetic eld, becomes gapped with non-abelian excitations.In the regime |J|/|Jz| 1 the hamiltonian can be mapped to amodel with four-body operators on a square lattice with groundstates that encode topologically protected quantum memory10. Oneproposal11 describes how to use trapped atoms in spin-dependentoptical lattices to simulate the spin model H (II)spin. There the inducedspin couplings are obtained through spin-dependent collisions insecond-order tunnelling processes. Larger coupling strengths aredesirable. In both spin models (I and II) above, the signs of theinteractions are irrelevant, although we will be able to tune the signsif needed.

SPECTROSCOPY OF POLAR MOLECULES IN OPTICAL LATTICES

Our system comprises heteronuclear molecules with 21/2 groundelectronic states, corresponding, for example, to alkaline-earthmonohalides with a single electron outside a closed shell. Weadopt a model molecule where the rotational excitations aredescribed by the hamiltonian Hm = BN2 + N S, with N beingthe dimensionless orbital angular momentum of the nuclei, andS being the dimensionless electronic spin (assumed to be S = 1/2in the following). Here B denotes the rotational constant and is the spinrotation coupling constant, where a typical B isa few tens of GHz, and is in the hundred MHz regime.The coupled basis of a single molecule i corresponding to theeigenbasis of Him is {|Ni, Si, Ji;MJi }, where Ji = Ni + Si witheigenvalues E(N = 0,S= 1/2,J = 1/2)= 0,E(1,1/2,1/2)= 2B , and E(1,1/2,3/2) = 2B+ /2. Although we ignore hyperneinteractions in the present work, our discussion below is readilyextended to include hyperne eects, which oer extensions to spinsystems S > 1/2.

The hamiltonian describing the internal and external dynamicsof a pair of molecules trapped in wells of an optical latticeis denoted by H = Hin + Hex. The interaction describing theinternal degrees of freedom is Hin = Hdd +2i=1 Him. Here Hddis the dipoledipole interaction given below in equation (1).

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(0, 1/2 ; 0, 1/2)

(0, 1/2 ; 1, 1/2)

(0, 1/2 ; 1, 3/2)

2g1g

1g

1g

1u

1u

2u

1u

0g

0g

0u

0u

0+g

0+g0+u

0+u

1 + (/B)

1 (/B)

1

0

0.50 1.0 1.5 2.0

E/2B

( /4B )1/3 z/rr /r

3/4B

Figure 2 MovrePichler potentials for a pair of molecules as a function of theirseparation r. The potentials E(gi (r )) for the four ground states (dashed lines) andthe potentials E(l(r )) for the rst 24 excited states (solid lines). The symmetries|Y|

of the corresponding excited manifolds are indicated, as are the asymptotic

manifolds (Ni , Ji;Nj , Jj ). The relative coordinate probability densities on a squarelattice are depicted in green on the ground-state potentials. The red arrow indicatesthe coupling of the microwave eld.

The hamiltonian describing the external, or motional, degrees offreedom is Hex = 2i=1 P2i /(2m) + Vi(xi xi), where Pi is themomentum of molecule i with mass m, and the potential generatedby the optical lattice Vi(xxi) describes an external connement ofmolecule i about a local minimum xi with 1D r.m.s. width z0. Weassume isotropic traps that are approximately harmonic near thetrap minimum with a vibrational spacing hosc. Furthermore, weassume that the molecules can be prepared in the motional groundstate of each local potential using dissipative electromagneticpumping12, perhaps beginning with a two-species Mott insulator13.It is convenient to dene the quantization axis z along theaxis connecting the two molecules, x2 x1 = zz with zcorresponding to a multiple of the lattice spacing.

The near-eld dipoledipole interaction between twomolecules separated by r = x1 x2 is

Hdd = d2

r3

(q=1q=1

(1)qD1qD2q 3D10D20 +h.c.). (1)

The dipole operator coupling the ground and rst rotational statesof molecule i is Di =

1q=1 |N = 1, qi iN = 0,0|eq, with the

spherical basis vectors {e0 = z, e1 = (x iy)/

2}, and d is thedimensionful dipole moment.

Although the present situation of dipoledipole coupling ofrotationally excited polar molecules is reminiscent of the dipoledipole interaction between electronically excited atom pairs14, thereare important dierences. First, unlike the atomic case whereelectronically excited states are typically anti-trapped by an opticallattice, here both ground and excited rotational states are trappedby an essentially identical potential up to tensor shifts15,16, whichcan be compensated by applying a static electric eld. Hence,motional decoherence due to spin-dependent dipoledipole forcesis strongly suppressed by the large vibrational energy hosc. Second,spontaneous emission rates are drastically reduced. The decay rateat room temperature from excited rotational states is 103 Hz(ref. 17) versus a comparable rate of MHz for excited electronicstates. There are other, stronger, sources of decoherence, the mostimportant being photon scattering from the optical trapping laser.

For reasonable traps the scattering rate can be of the order of0.2 Hz (ref. 16).

The ground subspace of each molecule is isomorphicto a spin-1/2 particle. Our goal is to obtain an eectivespinspin interaction between two neighbouring molecules.Static spinspin interactions due to spinrotation and dipoledipole couplings do exist but are very small in our model:HvdW(r) = (3d4/2Br6)

[1+ (/4B)2(1+4S1 S2/32Sz1Sz2)].

The rst term is the familiar van der Waals 1/r6 interaction,whereas the spin-dependent piece is strongly suppressed as/4B 103 1. Therefore, we propose dynamical mixing withdipoledipole coupled excited states using a microwave eld.

The molecules are assumed trapped with a separationz r (2d2/)1/3, where the dipoledipole interaction isd2/r3

= /2. In this regime, the rotation of the molecules is

strongly coupled to the spin. The ground states are essentiallyspin-independent, and the excited states are described byHunds case-(c) states in analogy to the dipoledipole coupledexcited electronic states of two atoms with ne structure.Remarkably, as described in the Methods section, the eigenvaluesand eigenstates of Hin can be computed analytically yielding thewell-known MovrePichler potentials18 plotted in Fig. 2.

Possible candidate polar molecules are, for example, CaF,CaCl and MgCl. The rotational constant, spinrotation couplingconstant and dipole moment are: CaF (B/h = 10.304 GHz, /h =39.66 MHz, d

3 = 3.07 D), Ca35Cl (B/h = 4.565 GHz, /h =

42.21 MHz, d

3 = 4.27 D) and 26Mg35Cl (B/h = 7.005 GHz,/h = 63.52 MHz, d3 = 3.38 D). With a typical optical latticespacing of z 300 nm, it would be dicult to trap near r .However, we note that it is possible to resolve many excited stateseven at larger intermolecular spacings. In the Methods section,we describe how the two spin models considered here can beimplemented at the cost of smaller eective interaction strengths.

ENGINEERING SPINSPIN INTERACTIONS

To induce strong dipoledipole coupling, we introduce amicrowave eld E(x,t)eF with a frequency F tuned near resonancewith the N = 0 N = 1 transition. Because the rotationalstates are spaced nonlinearly, this transition is resolvable withoutcoupling to higher rotational states by multiphoton processes. Inthe rotating-wave approximation, the moleculeeld interactionis Hmf = 2i=1(hDi eFei(kFxiF t)/2 + h.c.), where the Rabifrequency is | | = d|E0|/h. For molecules spaced by opticalwavelengths, all the dipoles are excited in phase.

The eective hamiltonian acting on the ground states isobtained in second-order perturbation theory as

He(r)=i,f

l(r)

gf |Hmf|l(r)l(r)|Hmf|gihF E(l(r)) |gf gi|, (2)

where {|gi,|gf } are ground states with N1 = N2 = 0 and {|l(r)}are excited eigenstates of Hin with N1 +N2 = 1 and with excitationenergies {E(l(r))}. The reduced interaction in the subspace ofthe spin degrees of freedom is then obtained by tracing over themotional degrees of freedom. For molecules trapped in the groundmotional states of isotropic harmonic wells with r.m.s. widthz0, the wavefunction is separable in centre-of-mass and relativecoordinates with the relative-coordinate wavefunction

rel(r,)= 13/4(2z0)3/2

e(r2+z22rzcos)/8z20 ,

where cos = r z/r. The eective spinspin hamiltonian is thenHspin =He(r)rel.


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0 1 2 3

0.1

0.15

0

0

0.1

04 0 2

(hF 2B )/4

hprobe/J

Sx Sz, Sy

Sz, Sx

Sy

2S 1.8 1.9

1

0

eF = y eF = x

eF = ya b

Figure 3 Design and verication of noise-protected ground states arising from a simulation of H (I)spin. The system comprises nine molecules trapped in a 33 lattice inthe zx plane with lattice spacing b= r /

2 driven with a eld of frequency F and out-of-plane polarization angle . a, Noise resilience of the ground states as a function

of F, quantied by the r.m.s. magnetizations of the two ground states, 2Sz = 2Sx (solid red lines) and 2Sy (dashed green lines) for = 0. The inset shows the protectedregion, when tuning near the 2g resonance E(2g) 1.9 , which realizes H (I)spin. b, Absorption spectroscopy of ground states (probe) for two spin textures obtained by tuningF near the 2g resonance at hF 2B= 1.88 . For = 0 the spectrum is gapped by J/2, which is a signature of a protected qubit (top), whereas for =/2 theexcitations are gapless spin waves (bottom).

The hamiltonian in equation (2) is guaranteed to yield someentangling interaction for appropriate choices of eld parameters,but it is desirable to have a systematic way to design a spinspin interaction. Fortunately, the model presented here possessessucient structure to achieve this essentially analytically. Theeective hamiltonian on molecules 1 and 2 induced by a microwaveeld is

He(r)= h| |8

3,=0

1 A,(r)

2 , (3)

where {}3=0 {1,x ,y ,z} and A is a real symmetric tensor. See

the Methods section for an explicit form of the matrix coecientsas a function of eld polarization and frequency.

Equation (3) describes a generic permutation-symmetrictwo-qubit hamiltonian. The components A0,s describe apseudo-magnetic eld that acts locally on each spin, and thecomponents As,t describe two-qubit coupling. The pseudo-magnetic eld is zero if the microwave eld is linearly polarized,but a real magnetic eld could be used to tune local interactionsand, given a large enough gradient, could break the permutationinvariance of Hspin.

For a given eld polarization, tuning the frequency near anexcited state induces a particular spin pattern on the ground states.These patterns change as the frequency is tuned though multipleresonances at a xed intermolecular separation. In Table 1, it isshown how to simulate the Ising and Heisenberg interactions inthis way. Using several elds that are suciently separated infrequency, the resulting eective interactions are additive, creatinga spin texture on the ground states. The anisotropic spin modelHXYZ = lxxx + lyyy + lzzz can be simulated using threeelds: one polarized along z tuned to 0+u (3/2), one polarized alongy tuned to 0g (3/2), and one polarized along y tuned to 0

+g (1/2).

The strengths lj can be tuned by adjusting the Rabi frequenciesand detunings of the three elds. Using an external magneticeld and six microwave elds with, for example, frequenciesand polarizations corresponding to the last six spin patterns in

Table 1, arbitrary permutation-symmetric two-qubit interactionsare possible.

The eective spinspin interaction along a dierent intermo-lecular axis z can be obtained by a frame transformation inthe spherical basis. Writing z = D1(1, 2, 3)(0,1,0)T, whereDj is the spin-j Wigner rotation, the eective hamiltonianalong z in the original coordinate system is obtained by thefollowing replacements to the eld polarization vector andspin operators: (,0,+)T D1(1,2,3)(,0,+)T and D1/2(1, 2, 3)D1/2(1, 2, 3). For example, using az-polarized eld tuned near 0+u (3/2) and a eld polarizedin the x y plane tuned near 1u(3/2) creates a Heisenberginteraction between any two molecules separated by r witharbitrary orientation in space.

APPLICATIONS

We now show how to engineer the spin model I. Consider a systemof trapped molecules in a square lattice with site coordinates inthe z x plane {xi,j} = {ibz + jbx; i, j [1, ]Z}. Illuminatethe system with a microwave eld with linear polarizationeF = cosy + sinx and eld frequency F tuned such that thepeak of the relative coordinate wavefunction at r = b is near-resonant with the 2g potential but far detuned from other excitedstates. Then the dominant interaction between nearest-neighbourmolecules is of Ising type along each axis and we realize H (I)spinwith J = (h| |)21/8(hF 2B /2 d2/r3)rel. For realisticparameters, this coupling can range from 10 to 100 kHz, withthe strength constrained by the trap spacing (J hosc). Therelative strength of the interactions along z and x can be changedby rotating the angle of polarization out-of-plane. Interactionsbetween more-distant neighbours are relatively weak because thefar-o resonant coupling at larger r cannot distinguish the spindependence of excited states.

Duocot et al.8 show that the ideal spin model I (for =/2)has a twofold degenerate ground subspace, which is gapped withweak size dependence for cos = 1. The two ground states, which


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Table 1 Some spin patterns that result from equation (3). The eld polarization isgiven with respect to the intermolecular axis z and the frequency F is chosen tobe near-resonant with the indicated excited-state potential at the internuclearseparation z. The sign of the interaction will depend on whether the frequencyis tuned above or below resonance.

Polarization Resonance Spin pattern

x 2g z z

z 0+u z 0g x x + y y z z

y 0g x x y y + z z

y 0+g x x + y y + z z

(y x)/2 0+g x y y x + z z

cos x+sin z 1g l1( x z + z x )+l2 z z+ l3( x x + y y )

cos y+sin z 1g l1( y z + z y )+l2 z z+ l3( x x + y y )

we denote |0L and |1L, have zero local magnetizations i,jL. Ourimplementation is not ideal because there are residual longer-rangeinteractions along several directions in the plane, as well as o-resonant couplings to excited state potentials yielding unwantedspin patterns. We note, however, that all of the eective spinhamiltonians described in equation (3) obtained using elds withlinear polarization involve sums of products of Pauli operators,and are therefore invariant under time-reversal. For odd, thedegeneracy present in the ground state of H (I)spin is of Kramers typeand an imperfect implementation will not break this degeneracy,although it may decrease the energy gap.

We have numerically computed the eective interaction ona 2 = 3 3 square lattice with spacings b = r /

2, and we

take the localization to the point dipole limit. In Fig. 3a we plotthe = x, y, z-components of the r.m.s. magnetization for theground subspace,

2S

ij

GG

|LG|i,j|GL|2/22,

as a function of the detuning F 2B/h for polarization angle = 0. This allows for computation of logical qubit errors due toquasi-static noise. Near the bare resonance hF 2B = /2, thesystem shows multiple long-range resonances as all the sites couplenear-resonantly at coupling strength 1/b3. The last of these long-range resonance appears at hF 2B 1.36 for the interactionbetween next-nearest-neighbour sites with spacings of

2b. The

2g-resonance lies at hF 2B 1.9 for nearest-neighbour sites,and shows the remarkable feature of no magnetization on anysite in any space-direction within the ground-state manifold(see inset). The resulting immunity of the system to local noisecan be probed by applying a homogeneous B-eld of frequencyprobe polarized in the direction = x, y, z. The correspondingabsorption spectrum for an arbitrary code state |L is

(probe)h[L|S(hprobe Hspin + ih )1S|L]where S =iji,j/2 and is an eective linewidth. This quantityis plotted in Fig. 3b for two dierent spin textures obtained for thesame eld frequency hF 2B= 1.88 but dierent polarizations,

1

4

3

2

1

0

11

2

01

z/z

y/zx/z

E(t )

x

z

y

Figure 4 Implementation of spin model H (II)spin. The spatial conguration of 12 polarmolecules trapped by two parallel triangular lattices (indicated by shaded planes)with separation normal to the plane of z/

3 and in-plane shift z

2/3.

Nearest neighbours are separated by z= r and next-nearest-neighbourcouplings are at

2z. The graph vertices represent spins and the edges

correspond to pairwise spin couplings. The edge colour indicates the nature of thedominant pairwise coupling for that edge (blue= z z , red= y y , green= x x ,black= other). For nearest-neighbour couplings, the edge width indicates therelative strength of the absolute value of the coupling.

and where we set h = 0.1J . For polarization = 0 (see top inset)the protected qubit is realized, whose spectrum is gapped by J/2.For polarization along the x-direction = /2 (see bottom inset)the ground subspace is given by a set of quantum-Ising stripesalong z, whose spectrum is ungapped with a large peak at probe = 0in response to coupling with a B eld polarized along = x.

Spin model II is similarly obtained using this mechanism.Consider a system of four molecules connected by three length-b edges forming an orthogonal triad in space. A bipartite latticecomposed of such triads with equally spaced nearest neighbourscan be built using two planes of stacked triangular lattices. Sucha lattice could be designed using bichromatic trapping lasersin two spatial dimensions, and a suitably modulated lattice inthe third dimension normal to both planes. A realization ofmodel II using a set of three microwave elds all polarizedalong z is shown in Fig. 4. The interaction obtained is closeto ideal with small residual coupling to next-nearest neighboursas in model I. The eld detunings at the nearest-neighbourspacing are: h1 E(1g(1/2)) = 0.05/2, h2 E(0g (1/2))= 0.05/2, h3 E(2g(3/2)) = 0.10/2 and the amplitudes are|1| = 4|2| = |3| = 0.01/h. For /h = 40 MHz this generateseective coupling strengths Jz/h = 100 kHz and J = 0.4Jz .The magnitude of residual nearest-neighbour couplings are lessthan 0.04|Jz| along x- and y-links and less than 0.003|Jz| alongz-links. The size of longer-range couplings Jlr are indicated by edgeline style (dashed: |Jlr| < 0.01|Jz|, dotted: |Jlr| < 103|Jz|). Treatingpairs of spins on z-links as a single eective spin in the low-energysector, the model approximates to Kitaevs 4-local hamiltonian10 on


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a square grid (shown here are one plaquette on the square latticeand a neighbour plaquette on the dual lattice) with an eectivecoupling strength Je =(J/Jz)4|Jz|/16 h167 Hz.

METHODS

DERIVATION OF THE MOLECULAR POTENTIALS

In the subspace of one rotational quantum (N1 +N2 = 1), there are 24eigenstates of Hin that are linear superpositions of two electron spin states andproperly symmetrized rotational states of the two molecules. There are severalsymmetries that reduce Hin to block diagonal form. First, Hdd, conserves thequantum number Y =MN +MS , where MN =MN1 +MN2 andMS =MS1 +MS2 are the total rotational and spin projections along theintermolecular axis. Second, parity, dened as the interchange of the twomolecular cores followed by parity through the centre of each molecule, isconserved. The =1 eigenvalues of parity are conventionally denoted g(u)for gerade (ungerade). Finally, there is a symmetry associated with reection Rof all electronic and rotational coordinates through a plane containing theintermolecular axis. For |Y |> 0 all eigenstates are even under R but for stateswith zero angular momentum projection there are 1 eigenstates of R. The 16distinct eigenvalues correspond to degenerate subspaces labelled |Y | (J) with Jindicating the quantum number in the r asymptotic manifold(N = 0, J = 1/2;N = 1, J).

For Y = 0, the energies are

E(0+ (11/2))= 2B+[3x/21/4(x/2+1/4)2 +1/2],

E(0 (11/2))= 2B+[x/21/4(3x/2+1/4)2 +1/2],

where x d2/r3. The eigenvector components are

K1(0m )= cos(Om /2), K2(0m )= sin(Om /2),

where the angles satisfy tan(O+ )=

2/(1/2+x), andtan(O )=

2/(1/23x). For Y =1 the eigenvectors and doubly

degenerate eigenvalues are obtained by diagonalizing the 33 matrices:

2B13 + 2

(2x 1 11 4x 11 1 2x

).

For Y =2, the eigenvalues are doubly degenerate with energiesE(2 (3/2))= 2B+/2+x.THE EFFECTIVE SPIN INTERACTION

The eective spinspin interaction, equation (3), between polar moleculesdepends both on the frequency F and polarization eF = e1 +0 e0 ++ e1,(e0 z) of the eld. The explicit form for the coupling coecients is:

A1,1 = |0|2[C(0g ,1,2)C(0+u ,1,2)]+(||2 +|+|2)[C(1g,3,3)C(1u,1,1)]+ [+][C(0g ,2,1)C(0+g ,2,1)],

A2,2 = A1,1 2[+][C(0g ,2,1)C(0+g ,2,1)],A3,3 = |0|2[2C(1g,2,2)C(0g ,1,2)C(0+u ,1,2)]

+ (|+|2 +||2)[C(2g)+C(0+g ,2,1)/2+C(0g ,2,1)/2C(1u,1,1)C(1g,3,3)],

A1,2 =[+](C(0g ,2,1)C(0+g ,2,1)),A1,3 =[+0 0]C(1g,2,3),A2,3 =[+0 0]C(1g,2,3),A0,1 =[+0 +0]C(1g,2,3),A0,2 =[+0 +0]C(1g,2,3),

A0,3 = (|+|2 ||2)[C(2g)C(0+g ,2,1)/2C(0g ,2,1)/2].

The component A0,0 weights a scalar energy shift, which we ignore. Thecoecients C(|Y | ) quantify coupling to excited states withdierent symmetries,

C(0m , j,k)=Kj(0m )2s(0m (3/2))+Kk(0m )2s(0m (1/2)),

C(1 , j,k)=4

a=1Kaj (

a1 (3/2))Kak (

a1 (3/2))s(a1 (3/2))

+2

b=1Kbj (

b1 (1/2))Kbk (

b1 (1/2))s(b1 (1/2))

C(2g)= s(2g(3/2)).

Here the energy-dependent terms s(|Y | (J))= h| |/[hF E(|Y | (J))]quantify the amplitude in the excited states. The energies E(|Y | (J))correspond to eigenvalues of Hint given above, and the sets {Kj(0m (J))}2j=1 and{Kaj (a1 (J))}3j=1 are coecients of the eigenvectors for |Y | = 0,1.

A caveat is that we do not have point dipoles but rather wavepackets withspatial distributions parallel and perpendicular to the intermolecular axis z.Components of intermolecular separations orthogonal to z will couple to stateswith dierent symmetry and an exact treatment would require averaging overthe angular distribution with the appropriate frame transformation. However,we argue that in our regime this nite-size eect is negligible. The relativemagnitude can be estimated by the ratio of the marginal relative coordinateprobability distributions perpendicular and parallel to z. Deningp(r)=

d sin2 r2|rel(r,)|2 and p(r)=

d cos2 r2|rel(r,)|2, the

peak of the distributions is at r =z, where for z0/z 1, the relativeamount of unwanted couplings is p(z)/p(z) 4(z0/z)2. Formolecular wavepacket localization 2z0/ltrap = 0.1, the ratio isp(ltrap)/p(ltrap) 103, and hence it is warranted to compute the couplingsas if the entire weight of the wavefunction were parallel to z.

SPIN COUPLINGS AT LARGER INTERMOLECULAR RADII

Using connement with optical lattices it may not be feasible to trap polarmolecules near z = r . At larger intermolecular separations where Hdd there is less mixing between states in dierent spin-rotational manifolds.However, there can still be sucient splitting between states with dierentsymmetries to obtain designer spin models, albeit with smaller couplingcoecents J . Consider a system with /h= 40 MHz and a lattice spacingz = 5.23r = 300 nm. Spin model I is built by tuning to the 2g potential withthe energy 2g|Hint|2g= 2B+/2+d2/r3. It is possible to resolve the 2gpotential because the nearby 0u state is dark. Residual coupling betweennext-nearest neighbours is small as the eld at r =2z is detuned from allresonances by at least 200 kHz. Spin model II is obtainable using a set of twomicrowave elds polarized along z, one tuned near resonance with a 1gpotential and one near a 1u potential. When the detunings and Rabi frequenciesare chosen so that |1g |C(1g,3,3)|1u |C(1u,1,1)rel = 0 then the resultantspin pattern is H (II)spin with J =h|1g |C(1g,3,3)rel/4 andJz = h||1g |C(1g,2,2)rel/4. The ratio |J|/|Jz | can be tuned either bychanging the lattice spacing or by using a third microwave eld polarized alongz and tuned near the 2g potential, in which case J J + h|2g |C(2g)rel/8.It is advantageous to tune elds near the 1 states that asymptote to the(N = 0, J = 1/2;N = 1, J = 1/2) manifold:

1 |Hint|1= 2B2d2/3r3

0+ |Hint|0+ = 2B+4d2/3r3

0 |Hint|0+ = 2B.

The two eld strengths |1g | and |1u | can be chosen to be nearly equal (towithin less than 1%). Shifts on next-nearest neighbours are small as the elddetuning at r =2z is at least 100 kHz.

We can dene a quality factor Q= J/h quantifying the ratio of coherentcoupling to the decoherence rate for the implementations above. A reasonableestimate of J for spin model I is in the range J/h= 0.757.5 kHz. This rangeensures that o-resonant couplings, which give rise to interactions of dierentspin character are |Jores/J| = 0.010.1. Next-nearest-neighbour interactionsare |Jnnn/J| = 0.030.3. For a decoherence rate 0.2 Hz, the quality factorfor spin model I is then Q= 3,75037,500. For spin model II, we ndJ/h=Jz/h= 0.161.6 kHz, providing |Jores/J|= 0.010.1 and


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ARTICLES

|Jnnn/J|= 0.030.3. This gives a quality factor Q= 8338,330. Theseestimates were obtained without optimization.

Received 6 January 2006; accepted 27 March 2006; published 30 April 2006.

References1. Levin, M. A. & Wen, X. G. String-net condensation: A physical mechanism for topological phases.

Phys. Rev. B 71, 045110 (2005).2. Hermele, M., Fisher, M. P. A. & Balents, L. Pyrochlore photons: The U(1) spin liquid in a S= 1/2

three-dimensional frustrated magnet. Phys. Rev. B 69, 064404 (2004).3. Einarsson, T. Fractional statistics on a torus. Phys. Rev. Lett. 64, 19951998 (1990).4. Jaksch, D. & Zoller, P. The cold atom Hubbard toolbox. Ann. Phys. 315, 5279 (2005).5. Buchler, H. P., Hermele, M., Huber, S. D., Fisher, M. P. A. & Zoller, P. Atomic quantum simulator for

lattice gauge theories and ring exchange models. Phys. Rev. Lett. 95, 040402 (2005).6. Santos, L. et al. Atomic quantum gases in kagome lattices. Phys. Rev. Lett. 93, 030601 (2004).7. Special Issue on Ultracold Polar Molecules: Formation and Collisions. Eur. Phys. J. D 31,

149444 (2004).8. Duocot, B., Feigelman, M. V., Ioe, L. B. & Ioselevich, A. S. Protected qubits and Chern-Simons

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44524505 (2002).11. Duan, L. M., Demler, E. & Lukin, M. D. Controlling spin exchange interactions of ultracold atoms in

optical lattices. Phys. Rev. Lett. 91, 090402 (2003).

12. Sage, J. M., Sainis, S., Bergeman, T. & DeMille, D. Optical production of ultracold polar molecules.Phys. Rev. Lett. 94, 203001 (2005).

13. Jaksch, D., Venturi, V., Cirac, J. I., Williams, C. J. & Zoller, P. Creation of a molecular condensate bydynamically melting a Mott insulator. Phys. Rev. Lett. 89, 040402 (2002).

14. Brennen, G. K., Deutsch, I. H. & Williams, C. J. Quantum logic for trapped atoms via molecularhyperne interactions. Phys. Rev. A 65, 022313 (2002).

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AcknowledgementsA.M. thanks W. Ernst, and P.Z. thanks T. Calarco, L. Faoro, M. Lukin, and D. Petrov for helpfuldiscussions. This work was supported by the Austrian Science Foundation, the European Union,OLAQUI, SCALA and the Institute for Quantum Information.Correspondence and requests for materials should be addressed to A.M.

Competing nancial interestsThe authors declare that they have no competing nancial interests.

Reprints and permission information is available online at http://npg.nature.com/reprintsandpermissions/


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