Kitaev honeycomb and other exotic spin models with polar molecules

Alexey V. Gorshkov,1 Kaden R. A. Hazzard,2, 3 and Ana Maria Rey2

1Institute for Quantum Information & Matter, California Institute of Technology, Pasadena, CA 91125, USA2JILA, NIST and Department of Physics, University of Colorado, Boulder, CO 80309, USA

3Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA(Dated: January 25, 2013)

We show that ultracold polar molecules pinned in an optical lattice can be used to access avariety of exotic spin models, including the Kitaev honeycomb model. Treating each molecule as arigid rotor, we use DC electric and microwave fields to define superpositions of rotational levels aseffective spin degrees of freedom, while dipole-dipole interactions give rise to interactions betweenthe spins. In particular, we show that, with sufficient microwave control, the interaction betweentwo spins can be written as a sum of five independently controllable Hamiltonian terms proportionalto the five rank-2 spherical harmonics Y2,q(θ, φ), where (θ, φ) are the spherical coordinates of thevector connecting the two molecules. To demonstrate the potential of this approach beyond thesimplest examples studied in [S. R. Manmana et al., arXiv:1210.5518v2], we focus on the realizationof the Kitaev honeycomb model, which can support exotic non-Abelian anyonic excitations. Wealso discuss the possibility of generating spin Hamiltonians with arbitrary spin S, including thoseexhibiting SU(N=2S+1) symmetry.

INTRODUCTION

Recent experimental progress in the control of ultra-cold polar molecules [1–6] has stimulated the study ofmany-body systems featuring strong dipole-dipole in-teractions [7–10]. A particularly promising researcharea [11–24] of implementing lattice Hamiltonians fromdipole-dipole-interacting molecules has emerged follow-ing the pioneering works of Refs. [25, 26]. This re-search area largely owes its promise to the great degree ofcontrollability that external DC, microwave, and opticalfields can provide over rotational levels and dipole-dipoleinteractions [11–15, 18, 26–32]. In Ref. [33], we demon-strated that this controllability can yield a variety of spinHamiltonians, in which spin states are encoded in super-positions of rotational states of the molecules. Whiledipole-dipole interactions are often thought of as beingproportional to 1− 3 cos2 θ, where R = (R, θ, φ) are thespherical coordinates of the vector connecting the twodipoles, we showed that not only the overall amplitudeand sign of the spin Hamiltonian but also its individualterms can depend on R. We harnessed this dependenceto propose the realization of spin models known to harbortopological phases of matter [33].

In contrast to Ref. [33], which focused on the simplestexamples, in the present paper we discuss a range of mod-els that are more interesting but harder to implement ex-perimentally. Specifically, as in Ref. [33], we emphasizethat the interaction Hamiltonian between two moleculescan, in principle, be a sum of five independently con-trollable terms proportional to the five rank-2 sphericalharmonics Y2,q(θ, φ).

We then study a specific class of Hamiltonians with anarbitrary spin S. This class of Hamiltonians can be usedto realize [33] the general SU(2) symmetric S = 1 model,i.e. the bilinear-biquadratic spin model, which has a rich

phase diagram even in one dimension [11, 34, 35]. It canalso give rise to SU(N=2S+1) spin models, which haverecently attracted a great deal of theoretical and exper-imental attention due to their potential to access exoticstrongly-correlated phases in alkaline-earth atoms [36–39]. The strong dipole-dipole interactions of the polar-molecule implementation of SU(N) models can be largerthan the SU(N)-symmetric superexchange interactionsin cold-atom implementations [36].

Then we turn to the primary example of this paper:the implementation of the Kitaev honeycomb model. Incontrast to the proposal of Ref. [28], our implementationof the Kitaev honeycomb model relies on direct – ratherthan perturbative – dipole-dipole interaction and hencegives rise to stronger interactions, which are easier toaccess experimentally. Other proposals for implementingthe Kitaev honeycomb model use solid-state Mott insula-tors [40], trapped ions [41], superconducting quantum cir-cuits [42], weak superexchange interaction between coldatoms in optical lattices [43], and an experimentally chal-lenging system of coupled cavity arrays [44]. We end themanuscript with a discussion of how we use microwavefields to create the dressed states that enable our imple-mentations, followed by an outlook.

SETUP

Consider a deep optical lattice in the XY plane shownin Fig. 1 and loaded with one polar molecule per site.Each molecule is described by a rigid rotor with rota-tional angular momentum operator N, rotational con-stant B, and dipole moment operator d. In the presenceof an electric field E along z, each molecule is describedby the Hamiltonian

H0 = BN2 − Edz. (1)

arX

iv:1

301.

5636

v1 [

cond

-mat

.qua

nt-g

as]

23

Jan

2013

2

(DC E-field)zZ

X

Y

ΦΦ0

Θ0

R

FIG. 1. (color online). The lattice of molecules is in theXY plane. The direction of the DC electric field is z. Thexyz coordinate system is obtained from the XY Z coordinatesystem by rotating the former around Z by Φ0 and then ro-tating it around y by Θ0. A typical vector R with sphericalcoordinates (R, θ, φ) in the xyz coordinate system has polarcoordinates (R,Φ) in the XY plane.

In general, the molecules can be allowed to hop to giverise to Hubbard-type [25, 45] or t-J-type [46, 47] modelswith highly tunable anisotropic long-range spin-spin in-teractions. In this paper, however, we will assume thatthe lattice is so deep that tunneling is negligible andmolecules are pinned in the motional ground state oneach site. The field-free (i.e. E = 0) eigenstates of H0 arethe simultaneous eigenstates of N2 and Nz with eigen-values N(N+1) and M , respectively. Let us denote with|N,M〉 the eigenstates of H0, which the field-free eigen-states connect to as E is turned on. For each molecule,we define [28] d± = ∓(dx ± idy), which changes the Mof the molecule by ±1, while d0 = dz couples rotationalstates with the same M . There are no selection rules onN for E 6= 0.

Consider two molecules i and j separated by Rij ,which has polar coordinates (Rij ,Θij) in the XY planeand spherical coordinates (Rij , θij , φij) in the xyz coordi-nate system. The dipole-dipole interaction between thesetwo molecules is [48]

Hij =di · dj − 3(di · Rij)(dj · Rij)

R3ij

(2)

= −√

6

R3ij

2∑q=−2

(−1)qC2−q(θij , φij)T

2q (di,dj), (3)

where Ckq (θ, φ) =√

4π2k+1Ykq(θ, φ), Ykp are spherical har-

monics, T 2±2 = d±i d

±j , T 2

±1 =(d0i d±j + d±i d

0j

)/√

2, and

T 20 =

(d−i d

+j + 2d0i d

0j + d+i d

−j

)/√

6, so that T 2q changes

the total M of the two molecules by q. T 2±1 contributes

only at those values of the DC electric field that haveσ± (M → M ± 1) and π (M -conserving) transitions ofmatching frequency.

In each molecule, we couple states |N,M〉 with severalmicrowave fields and choose 2S+ 1 dressed states (linearcombinations of states |N,M〉) to define an effective spinS system. We will focus in this paper on homogeneous

driving, which is easier to achieve as it can be done withmicrowave – as opposed to optical – fields. Extensionsto inhomogeneous driving allow for an even richer classof Hamiltonians [49, 50]. Assuming dipole-dipole interac-tions are too weak to take the molecules out of the 2S+1chosen dressed states, we can project dipole-dipole inter-actions onto these states to obtain a spin-S interactionHamiltonian of the form H = 1

2

∑i 6=j Hij , where

R3ijHij = v(θij , φij) ·H. (4)

Here

v(θ, φ) =(−2C2

0 ,−√

6Re[C22 ],√

6Im[C22 ],Re[C2

1 ], Im[C21 ])

is a real five-component vector describing the five differ-ent angular dependences (the prefactors are chosen forlater convenience). Each of the five components of His a Hamiltonian acting on the Hilbert space of the twospin-S particles i and j and comes with its own angu-lar dependence. Specific examples for the componentsof H are given below, for example Eq. (7) for the spin-1/2 case. Due to Hermiticity and symmetry under theexchange of the two particles, each component of H has[(2S + 1)4 + (2S + 1)2]/2 independent real coefficients.With an appropriate choice of rotational states, DC elec-tric field strength, and a sufficient number of microwavefields, one might envision achieving full control over all5[(2S+1)4+(2S+1)2]/2 coefficients, which, together withΘ0, Φ0, and a choice of lattice, determine the system. Re-quiring in addition that the total number of particles inany given internal state is conserved gives only (2S+ 1)2

independent coefficients in each angular dependence.Eq. (4) allows one to access a great variety of exotic

spin Hamiltonians. Since we will discuss S = 1/2 exam-ples below in Eq. (7), here we only point out that, in Ref.[33], we showed how Eq. (4) can be used to realize themost general SU(2)-symmetric S = 1 Hamiltonian (theso-called bilinear-biquadratic Hamiltonian) restricted tothe C2

0 angular dependence. in Ref. [33], we also brieflymentioned the possibility of realizing the Kitaev honey-comb model. In the present paper, we discuss the detailsbehind the realization of the Kitaev honeycomb modeland provide additional insights into how to generate spinHamiltonians with an arbitrary S.

SPIN HAMILTONIANS WITH ARBITRARY S

In this section, we show how to obtain a variety ofspin Hamiltonians with an arbitrary S. For simplicity,in each molecule, we choose Q distinct |N,M〉 statesand label them as |a〉, where a = 1, . . . , Q. We breakthis set of Q states into 2S + 1 disjoint sets labeled byp = −S, . . . , S. We couple the states within each set withmicrowave fields to form dressed states in the rotatingframe. We will show in detail below that ∼ n microwave

3

fields are needed to couple n states and to create anydesired linear combination out of them. We then chooseone dressed state from each set to create the single-spinbasis |p〉 =

∑a(p)

√xa|a〉 with rotating frame energies

Ep. Here∑a(p) means that a is summed over the states

belonging to the set p. The coefficients xa are assumed tobe nonnegative real numbers: allowing for superpositionswith complex coefficients does not allow any additionaltunability in this example [51]. The states |p〉 and |q〉will refer to dressed states, while states |a〉 and |b〉 willrefer to bare states. (One could also choose fewer setsand choose more than one dressed state from the sameset.) For simplicity, we further assume that dipole-dipoleinteractions are so weak and the states are chosen in sucha way that the two-molecule state |pq〉 is connected viaHij only to itself and to |qp〉, while all the other pro-cesses are off-resonant and are negligible. Similarly, nowin terms of the bare non-microwave-dressed states, weassume that the states are chosen in such a way that thetwo-molecule state |ab〉 is connected via Hij only to itselfand to |ba〉, while all the other processes are off-resonantand are negligible. Finally, also for simplicity, we assumethat the states are chosen in such a way that only C2

0

contributes (an extension that includes the other fourcomponents of v is straightforward). The assumptionsin the previous three sentences are generically satisfiedif, for example, all the states |a〉 involved have M ≥ 0and no accidental degeneracies occur. We then have

Hij =1− 3 cos2 θij

R3ij

[∑p

Bp|pp〉〈pp|+∑p,q

ApAq|pq〉〈pq|

+∑p<q

Jp,q2

(|qp〉〈pq|+ h.c.)

], (5)

where Ap =∑a(p) xaµa, Bp =

∑a(p)<b(p) xaxbda,b,

Jp,q =∑a(p),b(q) xaxbda,b, where

da,b =

2µ2

a,b if M(a) = M(b)

−µ2a,b if |M(a)−M(b)| = 1

0 otherwise. (6)

Here µa,b = 〈a|dM(a)−M(b)|b〉 and µa = 〈a|d0|a〉. Apcan be regarded as an effective dipole moment of state|p〉, while Bp encompasses the contribution to the inter-action from the transition dipole moments between thestates a(p). On the other hand, Jp,q describes the flip-flop transition involving states |p〉 and |q〉 and is drivenby transition dipole moments coupling a(p) to b(q). Thefactor of 2 and the minus sign in Eq. (6) arise from thefact that two identical dipoles rotating in phase in thex-y plane experience, on average, an interaction equal tonegative one-half of that felt by these dipoles had theybeen pointing along z [29].

The implementation of SU(N=2S+1)-symmetric mod-els is an attractive case to consider given that such models

have great potential for generating exotic phases includ-ing chiral spin liquids [52, 53]. The expression in thesquare brackets in Eq. (5) possess SU(N=2S+1) sym-metry if and only if it is proportional to

∑p,q |pq〉〈qp|

up to an additive constant. Necessary and sufficientconditions for this are that ApAq, Jp,q, and Bp + A2

p

do not depend on p and q for all p < q and thatBp + A2

p = ApAq + Jp,q/2 for all p < q. As an exam-ple, for N = 3, it is sufficient to satisfy A1 = A2 = 0and B1 = B2 = B3 + A2

3 = 12J1,2 = 1

2J1,3 = 12J2,3. In

Ref. [33], we showed how to realize this and, in fact, anarbitrary SU(2) symmetric S = 1 interaction. Findingspecific level configurations for higher N is postponeduntil future work.

THE GENERAL S = 1/2 HAMILTONIAN

The most general S = 1/2 Hamiltonian implementablewith dipole-dipole interactions is

R3ijHij = v(θij , φij) · [Vninj + Wx(niS

xj + Sxi nj)

+Wy(niSyj + Syi nj) + Wz(niS

zj + Szi nj)

+JxxSxi S

xj + JyyS

yi S

yj + JzzS

zi S

zj

+Jxy(Sxi Syj + Syi S

xj ) + Jxz(S

xi S

zj + Szi S

xj )

+Jyz(Syi S

zj + Szi S

yj )], (7)

where we have 10 real 5-component vectors in the squarebrackets. By analogy with Refs. [33, 46, 47], we expectthat these vectors might be independently controllable,an issue whose details we leave for future investigation.For this, we need ∼ 50 microwave fields. In Eq. (7), niis the occupation of site i. In the present manuscript,ni = 1 for all sites. However, this more general form ofthe Hamiltonian is necessary when hopping is allowed,such as in Refs. [25, 45–47].

In general, the interaction in Eq. (7) is also accompa-nied by on-site Hamiltonian terms. In particular, if theon-site energy difference between |↑〉 and |↓〉 (the twodressed basis states making up the spin-1/2) is muchlarger than the strength of Hij , only those interactionterms that conserve the total Sz play a role [46, 47]. Inorder to realize models such as the quantum compassmodel [54], the Kitaev honeycomb model [55], and theKitaev quantum double models [56] implemented withtwo-body interactions [57], we need to break Sz conser-vation and realize terms, such as Sxi S

xj . To do this, we

simply tune |↑〉 and |↓〉 to be degenerate.

For ni = 1, the Hamiltonian in Eq. (7) is the specialcase of Eq. (4): its restriction to S = 1/2. Despite therestriction to S = 1/2, the Hamiltonian in Eq. (7) isstill extremely powerful, as illustrated by the followingexamples. First, the commonly used density-density in-teraction with the 1− 3 cos2 θij angular dependence (seee.g. Refs. [7, 9]) comes from the first component of V,

4

while density-density interactions with more exotic an-gular dependences can be obtained by making use of allfive components of V. Second, the first components ofJxy, Jzz, Wz, and V give rise to the spin-spin, density-spin, and density-density interactions of the generalizedt-J model (referred to as the t-J-V -W model) of Refs.[46, 47]. Third, the first three components of Jxy andJzz give rise to an XXZ model featuring a direction-dependent spin-anisotropy [33]. In addition to provid-ing access to symmetry protected topological phases inspin ladders [33], such an XXZ model can also be usedto obtain a variety of exotic antiferromagnets includinga square-lattice Heisenberg model, in which the ratio be-tween coupling strengths on X and Y bonds is tunable[58]. Fourth, the Jxx and Jzz terms can be used to re-alize the quantum compass model [54]. In the presentpaper, we demonstrate the power of Eq. (7) on anotherimportant example: the Kitaev honeycomb model [55].

KITAEV HONEYCOMB

In Fig. 1 and in the inset to Fig. 2(a), we show theKitaev honeycomb lattice with Hamiltonian [55]

H = Jxx∑x−link

Sxi Sxj + Jyy

∑y−link

Syi Syj + Jzz

∑z−link

Szi Szj .(8)

We are interested in phase B, which contains the pointJxx = Jyy = Jzz < 0. This phase is gapless, but, in thepresence of a magnetic field,

HB =∑j

(BxSxj +ByS

yj +BzS

zj ), (9)

with all three components Bx, By, and Bz being nonzero,acquires a gap and supports non-Abelian anyonic excita-tions. Non-Abelian anyons have quantum computing ap-plications and can be used, for example, for topologicallyprotected quantum state transfer [59].

We use Eq. (7) for a system of n = 1 molecules persite on a honeycomb optical lattice – which can be im-plemented with three laser beams [60–62] – to implementthe Kitaev honeycomb model. Specifically, we would liketo obtain Sxi S

xj interactions along Φij = π/6, Syi S

yj in-

teractions along Φij = π/2, and Szi Szj interaction along

Φij = 5π/6 [see Fig. 2(a)]. It is worth noting that inter-actions along Φij and Φij + π are the same.

Let us first point out some general facts that hold inde-pendently of the direction and magnitude of the appliedDC electric field and of the dressed states |↑〉 and |↓〉 thatwe choose. The terms in Eq. (7) involving V play no rolesince they are proportional to identity. Ignoring bound-ary effects, the terms in Eq. (7) involving Wx, Wy, andWz give rise to a uniform magnetic field, precisely asrequired by Eq. (9); we will compute this field below.

Having taken care of the V and Wα terms, we omitthem from the Hamiltonian. We would like the remaining

N = 4

(a)

(c)

|↓|↑

M = −4

3

210

−3 −2 −1 0 1 2 3 4

(d)

xxyy

zzyy

xxzzX

YR

Φ

1

2 3 4

5 6 7 8 9

10 11 12 13 14 15 16

17 18 19 20 21 22 23 24 25

∆

∆

FIG. 2. Implementation of the Kitaev honeycomb model. (a)The honeycomb model: Sx

i Sxj , Sy

i Syj , and Sz

i Szj interactions

along Φij = π/6, π/2, and 5π/6, respectively. (b) The rota-tional levels used. The diagram is schematic: the real systemis anharmonic and levels |N,M〉 with the same N are non-degenerate (unless the levels have the same |M |). The dressedstates are |↑〉 (linear combination of states indicated by ovals)and |↓〉 (the rest). (c) Microwave fields that can be used tocreate the dressed state |↑〉. Notice that level |2, 0〉 is usedoff-resonantly with a detuning ∆. (d) Microwave fields thatcan be used to create the dressed state |↓〉. Notice that level|4, 0〉 is used off-resonantly with a detuning ∆.

terms in the Hamiltonian (Jxx,Jyy,Jzz,Jxy,Jxz,Jyz) tosatisfy

Hij(π6 ,

π2 ,

5π6

) = − J

R3ij

Sxi S

xj , S

yi S

yj , S

zi S

zj

(10)

with J > 0 for the three indicated values of Φij . Since,for any Φij , v(Φij) can be written as a linear combina-tion of v(π/6), v(π/2), and v(5π/6), Eq. (10) completely

5

determines the Hamiltonian at all Φij :

Hij(Φij) =

3∑n=1

1− 2 cos (2Φij − 2(n+ 1)π/3)

3

×Hij

((2n−1)π

6

)(11)

= − J

3R3ij

[1− 2 cos (2Φij − 4π/3)]Sxi S

xj

+ [1− 2 cos (2Φij)]Syi S

yj

+ [1− 2 cos (2Φij − 2π/3)]Szi Szj

.

We will show how to realize this momentarily. First,observe that, for example, along Φij = π/3, which ishalfway between Sxi S

xj and Syi S

yj ,

Hij(π/3) = − J

3R3ij

(2Sxi Sxj + 2Syi S

yj − Szi Szj ). (12)

Because of the 1/R3 dependence, the strength ofsuch next-nearest-neighbor interactions (along Φij =0, π/3, 2π/3) is reduced relative to the nearest-neighborinteractions by 1/33/2 ≈ 1/5, which will be the largestcorrection introduced by long-range interactions. Whilein some cases long-range corrections are weak enough toensure the survival of the desired phases [33], it is an openquestion whether this holds for the present example.

It now remains to find the DC electric field strengthand direction, as well as the dressed states, that yieldEq. (10). We assume that the DC electric field is alongthe Z axis (i.e. Θ0 = 0) and take Φ0 = 0. In that case,the last two components of v vanish, which means thatC2±1 terms do not contribute (v4 = v5 = 0). This is fine

because those terms are the hardest to generate exper-imentally as they are nonvanishing only at specific val-ues of the DC electric field, and it will turn out theyare not required to generate the Kitaev Hamiltonian.At Θ0 = Φ0 = 0, the uniform magnetic field comingfrom Wα terms is easy to calculate. Indeed, v2 and v3do not contribute to the magnetic field since the honey-comb lattice is invariant under 2π/3 rotations and since∑2n=0v2(Φ1 +n2π/3), v3(Φ1 +n2π/3) = 0, 0 for any

Φ1. Finally, summing over all bonds connected to a givensite, the term W · (Si + Sj) with angular dependence v1gives rise to a uniform magnetic field of strength 6.58W.

Since v4 = v5 = 0, we drop the last two components ofv and H to obtain

v(Φ) = (1,− 32 cos(2Φ), 32 sin(2Φ)) (13)

and

H = d0i d0j + 12 (d−i d

+j + d+i d

−j ),

d+i d+j + d−i d

−j , i(d

+i d

+j − d−i d−j ). (14)

The linear independence of v(π/6), v(π/2), and v(5π/6)allows, in principle, for the possibility of obtaining Eq.(10).

We will use the 25 states shown and numbered inFig. 2(b). We write the two dressed states as |σ〉 =∑a(σ) ya|a〉, where ya are complex,

∑a(σ) |ya|2 = 1,

σ =↑, ↓, and∑a(σ) means that one sums a over the 12

(13) states belonging to |↑〉 (|↓〉) in Fig. 2(b). We nowkeep only resonant terms and project H in Eq. (14) ontodressed states |↑〉 and |↓〉.

We work at a DC electric field E = 10B/d. The fol-lowing transitions contribute to H1: |N,M〉 |N ′,M ′〉 →|N,M〉 |N ′,M ′〉, |N,M〉 |N ′,M ′〉 → |N ′,M ′〉 |N,M〉 (for|M −M ′| ≤ 1, as dictated by electric-dipole selectionrules), and |N,M〉 |N ′,−M〉 → |N ′,M〉 |N,−M〉. Thetransition |N,−M−1〉 |N ′,M〉 → |N ′,−M〉 |N,M + 1〉 isthe only one contributing to d+i d

+j in H2 and H3, while

d−i d−j is the Hermitian conjugate of d+i d

+j . Projecting

these terms on |↑〉 and |↓〉, we obtain an expression forH in terms of ya. This allows us to get an expressionfor W and for the left-hand side of Eq. (10). Since amagnetic field that is too large will eventually take usout of the Kitaev B phase, we first verify that we canobtain Eq. (10) with a minimal magnetic field W. Onecan then easily verify that the set of ya given in footnote[63] gives J = 0.0284d2 and 6.58|W| = 7×10−7d2 (whichis negligibly small).

By adjusting the coefficients ya, one can find pointswith larger values of |W|. For example, the coefficients yagiven in footnote [64] achieve J = 0.0264d2 and 6.58W =0.00072, 0.00084,−0.03937.

MICROWAVE DRESSING

So far, we have not discussed in detail what exactmicrowave couplings are required to create the dressedstates |↑〉 and |↓〉 needed to implement the Kitaev hon-eycomb model. In this Section, we provide such a discus-sion. The methods we present here are generally appli-cable to the creation of generic spin models from dipolarinteractions.

Although generating large numbers of precisely tunedmicrowaves is experimentally feasible [65], the effort inobtaining and verifying the spectrum of microwaves in-creases with the number of frequencies. It is thus ex-perimentally desirable to use as few microwave fields aspossible. In order to create a dressed state out of n barestates, one needs at least n−1 microwaves (assuming thesame microwave cannot be used to couple more than onetransition). Therefore, let us first address the questionof whether n− 1 microwaves suffice to prepare a dressedstate at any prescribed rotating frame energy featuringany prescribed superposition of the bare states.

Suppose we would like to prepare a dressed state |D〉 =

6∑na=1 ya |a〉 out of the n bare states |a〉 (a = 1, . . . , n)

with prescribed coefficients ya and a prescribed rotating-frame energy E relative to the bare state |m〉. Sup-pose state |a〉 is coupled to state |a+ 1〉 using a mi-crowave with Rabi frequency Ωa for a = 1, . . . , n − 1.The rotating-frame Hamiltonian is then

∆1 Ω1 0 · · · 0 0Ω∗1 ∆2 Ω2 · · · 0 00 Ω∗2 ∆3 · · · 0 0...

......

. . ....

...0 0 0 · · · ∆n−1 Ωn−10 0 0 · · · Ω∗n−1 ∆n

y1y2y3...

yn−1yn

=E

y1y2y3...

yn−1yn

.(15)

Here ∆m = 0 (recall that |m〉 is the reference sate), while∆a+1 − ∆a is the detuning of the a’th microwave fieldfrom the |a〉 → |a+ 1〉 transition. Writing

Ωa = Ωa exp i arg (ya/ya+1) , (16)

where Ωa are real, we obtain n real-number equations

Ω1

∣∣∣∣y2y1∣∣∣∣ = E −∆1,

Ω1

∣∣∣∣y1y2∣∣∣∣+ Ω2

∣∣∣∣y3y2∣∣∣∣ = E −∆2,

...

Ωa−1

∣∣∣∣ya−1ya

∣∣∣∣+ Ωa

∣∣∣∣ya+1

ya

∣∣∣∣ = E −∆a, (17)

...

Ωn−2

∣∣∣∣yn−2yn−1

∣∣∣∣+ Ωn−1

∣∣∣∣ ynyn−1

∣∣∣∣ = E −∆n−1,

Ωn−1

∣∣∣∣yn−1yn

∣∣∣∣ = E −∆n.

One can satisfy the equations using n−1 Rabi frequenciesΩa and any one nonzero detuning ∆c. Allowing otherdetunings to be nonzero, one obtains freedom in how tochoose the magnitudes of Ωa. This can be useful, forexample, in cases when the magnitudes of Ωa need to belarger than a certain value (e.g. larger than the strengthof hyperfine interactions, as in Refs. [33, 47] – see ourOutlook).

The approach we have just described works nicely inthe cases where the only off-diagonal bare-states pro-cesses that contribute to the final spin Hamiltonian havethe form |a〉 |b〉 → |b〉 |a〉. This holds for all the config-urations studied in Refs. [46, 47]. It also holds [66] forthe level configuration used to implement the bilinear-biquadratic model in Ref. [33].

On the other hand, consider a model where |↑〉 , |↓〉 =|0, 0〉 , y−1 |1,−1〉 + y1 |1, 1〉 (as in Ref. [33]) and wherewe want the process |1,−1〉 |0, 0〉 → |0, 0〉 |1, 1〉 to con-tribute. We can couple states |1,−1〉 and |1, 1〉 with twomicrowave fields via state |2, 0〉. It is crucial that these

two microwave fields have the same frequency; other-wise, in the rotating frame, the process |1,−1〉 |0, 0〉 →|0, 0〉 |1, 1〉 would pick up an oscillatory time dependenceand will average out to zero. This means that, in thecases where off-diagonal bare-state processes other than|a〉 |b〉 → |b〉 |a〉 contribute, there are restrictions on whatdetunings can be chosen as nonzero.

In the case of the configuration for the Kitaev honey-comb model shown in Fig. 2(b), it is easy to check thatthe microwave couplings shown in Fig. 2(c) and Fig. 2(d)for |↑〉 and |↓〉, respectively, work. In particular, in Fig.2(c) all microwaves are assumed to be resonant exceptfor the microwaves coupled to state |2, 0〉, which are alldetuned by the same amount ∆. This detuning providesthe extra tuning parameter one has to add to the Rabifrequencies in order to fully control the coefficients yjdefining state |↑〉. The fact that |2, 0〉 has M = 0 ensuresthat it is nondegenerate. Hence, any contributing bare-state process containing |2, 0〉 in the initial two-moleculestate will also contain it in the final two-molecule state,meaning that a detuning of this state keeps the processresonant. Similar arguments apply to Fig. 2(d), whereall microwaves are assumed to be resonant except for themicrowave coupling to state |4, 0〉. To ensure that |↑〉 and|↓〉 are degenerate, we use |2, 2〉 and |2,−2〉 as the tworeference states and set the two dressed-state energies tobe equal. It is also worth pointing out that an additionalN = 5 state is introduced in both Fig. 2(c) and Fig. 2(d)to act as an intermediate state for coupling two stateswith ∆M = 2. We also note that one has to be carefulusing π transitions to couple states with M 6= 0 since thesame microwave will act on both M and −M .

A solution alternative to the introduction of a detun-ing is an introduction of additional resonant microwavecouplings, resulting in configurations such as three statescoupled with three microwave fields. At the same time, itis worth pointing out that, in some cases, one may wishto introduce additional detunings on purpose with thegoal of eliminating certain interaction processes (writ-ten in terms of bare non-microwave-dressed states as|a, b〉 → |a′, b′〉), which can be regarded as an additionalcontrol knob.

OUTLOOK

The most straightforward implementation of our pro-posal would involve polar molecules with no nuclear spin.This is possible, for example, in the case of a SrYbmolecule, which seems to be within reach given the avail-ability of quantum degenerate Sr and Yb gases [39]. Inthe case where hyperfine structure is present, such as inalkali dimers, we assume that the Rabi frequencies Ωa arelarger than the strength of hyperfine interactions Hhf. Insituations where this poses a significant restriction, theopposite limit, in which Ωj Hhf, can also be consid-

7

ered.

While we have focused on pinned molecules, allowingthe molecules to hop in the lattice gives rise to t–J-type[46, 47] or Hubbard-type [25] models with highly tunablelong-range spin-spin interactions featuring a direction-dependent spin anisotropy. As a particularly simple ex-ample, one can obtain a single-component system witha density-density interaction whose angular dependenceis given by a linear combination of all five C2

q [see theV term in Eq. (7)]. This may allow one to study, forexample, exotic quantum Hall physics [67].

The approaches we presented can be used to tunedipole-dipole interactions to zero [12, 29]. In that case,one has to consider corrections due to dipole-dipole inter-actions to second order (which give van-der-Waals-type1/R6 dependence) and interactions between higher multi-pole moments [68]. These interactions can also be viewedas extra control knobs.

We have focused in the present manuscript on polarmolecules. However, the ideas that we presented shouldbe extendable in a straightforward manner to other sys-tems interacting via dipole-dipole interactions such asmagnetic atoms [69, 70], Rydberg atoms [71, 72], andmagnetic solid-state defects [73, 74].

ACKNOWLEDGMENTS

We thank S. Manmana, E. M. Stoudenmire, J. Preskill,J. Ye, D. Jin, M. Lukin, N. Yao, J. Taylor, S. Stellmer,W. Campbell, M. Foss-Feig, M. Hermele, and V. Gurariefor discussions. This work was supported by NSF, IQIM,NRC, AFOSR, ARO, the ARO-DARPA-OLE program,and the Lee A. DuBridge and Gordon and Betty Moorefoundations. KRAH thanks KITP for hospitality. Thismanuscript is the contribution of NIST and is not subjectto U.S. copyright.

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