Preparation of decoherence-free cluster states with optical superlattices

Transcription

1 PHYSICAL REVIEW A 79, Preprtion of decoherence-free cluster sttes with opticl superlttices Ling Jing, 1 An Mri Rey,,3 Oriol Romero-Isrt, 4 Jun José Grcí-Ripoll, 5 Ann Snper, 4,6 nd Mikhil D. Lukin 1, 1 Deprtment of Physics, Hrvrd University, Cmridge, Msschusetts 0138, USA Institute for Theoreticl Atomic, Moleculr nd Opticl Physics, Cmridge, Msschusetts 0138, USA 3 JILA nd Deprtment of Physics, University of Colordo, Boulder, Colordo , USA 4 Deprtment de Físic, Universitt Autònom de Brcelon, E Bellterr, Ctloni, Spin 5 Fcultd de CC. Físics, Universidd Complutense de Mdrid, Ciudd Universitri s/n, Mdrid E-8040, Spin 6 Institució Ctln de Recerc i Estudis Avnçts, E Brcelon, Spin Received 19 Novemer 008; pulished 9 Ferury 009 We present protocol to prepre decoherence-free cluster sttes using ultrcold toms loded in two dimensionl superlttice. The superlttice geometry leds to n rry of plquettes, ech of them holding four spin-1/ prticles tht cn e used for encoding single logicl quit in the twofold singlet suspce, insensitive to uniform mgnetic field fluctutions in ny direction. Dynmicl mnipultion of the supperlttice yields distinct inter- nd intrplquette interctions nd permits us to relize one quit nd two quit gtes with high fidelity, leding to the genertion of universl cluster sttes for mesurement sed quntum computtion. Our proposl sed on inter- nd intrplquette interctions lso opens the pth to study polymerized Hmiltonins which support ground sttes descriing ritrry quntum circuits. DOI: /PhysRevA PACS numer s : Lx, Jk I. INTRODUCTION Quntum technology, in prticulr quntum informtion processing nd quntum metrology, requires the precise preprtion of quntum sttes tht outperform given tsk etter thn ny clssicl strtegy. As shown in recent yers, the unprecedented control nd precision provided y ultrcold gses in opticl lttices mkes these systems optiml cndidtes for such technology. Furthermore, since the dynmicl control over the opticl lttice prmeters permits the simultneous coupling etween nerest tom-lttice sites, these systems re lso incresingly used s quntum simultors to mimic distinct complex condensed mtter Hmiltonins 1. The controlled genertion of doule well lttices, i.e., lttices whose unit cells contin two sites, hs opened the possiility to isolte nd ddress individully pirs of toms, nd hence to mnipulte the interctions etween them. Seminl results re the demonstrtion of controlled exchnge interction etween pirs of neutrl toms in n opticl lttice when the toms re forced to e in the sme loction, nd the demonstrtion of superexchnge interctions 3, showing tht the interctions etween toms trpped in two djcent sites of the opticl lttice cn e mde nlogous to the interctions etween tomic spins in mgnetic mterils. While Ref. sets sis to perform in controlled mnner two quit gtes etween neighoring toms in the doule well lttice, Ref. 3 opens direct pth towrds the reliztion of low-temperture quntum mgnets nd vriety of mnyody spin models with ultrcold toms. The tomic interction control chieved with opticl lttices lso hs direct pplictions to quntum computtion. A prticulrly well suited pproch tht exploits the innte mssive prllelism of such systems to perform quntum computtion is the mesurement sed quntum computtion MBQC, where informtion is processed y mens of sequence of mesurements on highly entngled initil stte. It requires the cpility to crete universl cluster stte, tht is, multiprtite quntum stte le to reproduce ny entngled quntum stte in two dimensions, nd to perform locl single quits opertions. Using s quits two internl sttes of toms in two-dimensionl D opticl lttice, it is possile to crete highly entngled quntum stte y mens of controlled collisions 4, which is indeed prerequisite for the genertion of universl cluster stte. Although neutrl toms couple wekly to the environment nd they hve reltively long coherence times compred with the time scle ssocited with the chievle coupling strength, when toms re rought to n entngled stte decoherence will rpidly destroy ny quntum superposition of toms. The lrger the entngled system is, the fster it will decohere. To fight ginst decoherence one should prepre the toms in quntum sttes tht re roust ginst externl perturtions. For periodic rrys of doule wells 5 7, resilient encoding schemes using two internl sttes two Zeemn levels of the toms hve een proposed. In these schemes, ech doule well trps two two-level prticles to encode logicl quit. The logicl spce is spnned y the singlet nd triplet sttes of the two spin-1/ prticles long the quntiztion xis here denoted y z, S = 1 nd T 0 = 1 +. Since these sttes hve zero z component of the totl spin, such encoding is insensitive to fluctutions of the mgnetic field long the quntiztion xis. In prctice, such n encoding scheme is very well suited for roust controlled interctions long one, let us sy horizontl, direction. To crete universl D cluster stte cluster sttes in one dimension re not universl, interctions nd hence entnglement etween neighoring toms long oth the horizontl nd the verticl directions should e performed in such wy tht roustness is preserved. In the ove encoding, interctions long the verticl direction will leve the suspce of zero spin components long the quntiztion xis, ecoming very frgile in front of externl mgnetic field fluctutions noise. In this pper we /009/79 / The Americn Physicl Society

2 JIANG et l. y V y (y) V x (x) 1 FIG. 1. Color online The opticl superlttice consists of periodic rry of plquettes. The D opticl trpping potentil is creted y dding two superlttice potentils V x x nd V y y illustrted in the ottom nd left pnels. The intrplquette coupling is represented y the solid lines nd the interplquette coupling is represented y the dshed lines. x PHYSICAL REVIEW A 79, show how this limittion cn e overcome y using D opticl superlttices. In pssing, let us point out tht superimposing secondry opticl lttices or superlttices on top of the primry ones to further modify the potentil in which the toms re trpped permits us in generl to crete polymerized lttices. By polymerized lttices we men lttices consisting of wekly coupled groups of neighoring tomic sites denoted s plquettes. An exmple of polymerized lttice is squre lttice mde of smller squres. The intrplquette interctions in such lttices might e strong nd my even e designed to include mny -ody terms, while the interplquette interctions might e much weker. Polymerized lttices llow us, for instnce, to engineer vlence ond solids on demnd, to study topologicl spin liquids, nd one might envisge them s potentil quntum circuits. It is lso very ppeling to try to use the plquettes s quits or qudits elementry systems with more thn two internl sttes for quntum informtion processing nd implement quntum logicl gtes, quntum protocols, nd quntum error correction in such systems y employing either intertomic interctions nd/or interctions with externl electric, mgnetic, lser fields. Here we tke dvntge of the two-dimensionl superlttices to present new schemes to prepre universl D cluster sttes using the plquettes s logicl quits. The superlttices crete periodic rry of plquettes, i.e., potentil wells s shown in Fig. 1, ech of them filled with n tom with two internl degrees of freedom spin-1/ prticle. On ech plquette, we encode single logicl quit using the twofold singlet suspce of the four 1/ spins, s shown in Fig. 1. Thus y douling the physicl resources in comprison with the two physicl quit encoding previously mentioned, we otin the desired encoding tht is decoherence free ginst uniform mgnetic field fluctutions in ritrry directions. The encoding scheme using the singlet suspce of four quits hs een previously studied for the quntum dot systems, nd it is lso clled the supercoherent quit Also, it hs een shown tht in such configurtion, tunle Heisenerg superexchnge interctions etween neighoring spins, including the digonl nd off digonl ones re sufficient for universl quntum computtion 11,1. Inthe model we propose here, the genertion of universl cluster stte demnds i the ility to perform one quit gtes to prepre ll logicl quits in the initil stte + =1/ 0 + 1, nd ii the reliztion of controlled-phse gtes, U =dig 1,1,1, 1, etween nerest logicl quits, i.e., etween plquettes, to crete mximlly entngled D cluster stte. Notice tht the superlttice geometry of Fig. 1 does not induce interctions long the digonls sites on ech plquette, nd thus the most chllenging ingredient for universlity is, indeed, the two quit two plquette gte. In this pper, we propose three different pproches to couple the logicl quits tht either preserve the singlet suspce t the end of the gte opertion or keep the stte within the singlet suspce even during the whole completion of the gte. In ll pproches, we tke into ccount relistic ville tools nd discuss the prcticl limittions in opticl superlttices. In our first pproch, we exploit the dditionl virtionl mode of the opticl trp to fcilitte the logicl coupling gte. In our second pproch, we extend the erlier proposls 9,10 y removing the requirement of equl coupling strengths for ll six pirs within the plquette more fesile for D opticl superlttices, while we still otin the ective Hmiltonin within the logicl suspce sufficient for universl gtes. In our lst pproch, we include tunle Ising-type interctions etween neighoring spins ttinle with neutrl toms in opticl lttices 13 nd use the optiml control techniques to find icient nd roust pulse sequences for the logicl coupling gte. We notice tht other proposls which exploit the superlttice structure in two dimensions to crete universl resource which is different from the universl cluster stte for MBQC y connecting Bell entngled pirs y entngling phse gte hve een proposed recently 7. The pper is orgnized s follows: first, in Sec. II we present the generl ides for generting cluster sttes within decoherence free suspce DFS using opticl superlttices. Then, in Sec. III we riefly review the singlet DFS of the plquette nd descrie opertions of single logicl quit using superexchnge couplings. Finlly, in Sec. IV we consider the key chllenge of implementing the logicl controlled-phse gte with the singlet DFS encoding. We propose three interesting pproches for the controlled-phse gte: the geometric phse pproch, the perturtive pproch, nd the optiml control pproch. A detiled comprison mong the three pproches is summrized t the end of this section Tle I efore we present our conclusion in Sec. V. II. DECOHERENCE-FREE CLUSTER STATES One promising pproch to quntum computtion is the MBQC 14,15, which uses universl resources such s the cluster sttes. The cluster sttes cn e iciently prepred y initilizing ll lttice spins in the product stte of

3 PREPARATION OF DECOHERENCE-FREE CLUSTER PHYSICAL REVIEW A 79, TABLE I. Comprison mong three pproches. See Sec. IV D for discussion. Geometric phse pproch Perturtive pproch Optiml control pproch Time scle for CZ gte 1/J J/J 1/J Durtion out of DFS 1/t 0 1/J Systemtic errors t/u J /J 0 Inhomogeneity errors t/t, /t J/J, J /J J/J Sites per CZ gte Simultneous coupling Yes Two steps Yes Mjor interctions Superexchnge Single prticle tunneling Superexchnge Superexchnge Ising interction Superlttice wvelengths,, 4,, 4, 4 /3, Virtionl levels Ground+ excited Ground Ground Physicl process Cler Cler Hrd to interpret Control complexity Medium Low High = nd performing the controlled-phse gte etween ll pirs of neighoring spins. The controlled-phse gte induces n dditionl fctor 1 if oth input quits re in stte 1. Up to some individul quit rottions, the controlled-phse gte cn e chieved y Ising-type interction etween two input quits. The preprtion of cluster stte hs een demonstrted using opticl lttices 4, with logicl quits directly stored in individul spins. Controlled collisions re used to implement the controlled-phse gte etween neighoring spins. However, such tomic quits for the controlled-collision scheme re vulnerle to mgnetic field fluctutions, which limits the prcticl implementtion of the MBQC. The opticl superlttice inducing periodic rry of plquettes s shown in Fig. 1 cn e creted y superimposing two opticl lttice potentils with short nd long wvelengths differing y fctor of 16 long oth the x nd y directions. The ective opticl trpping potentil ecomes crete the decoherence-free cluster sttes, we will lso need interplquette couplings dshed lines in Fig. 1 to implement the controlled-phse gtes. There re eight sites for two neighoring plquettes. If only four middle sites re involved for the controlled-phse gte e.g., for the geometric phse pproch in Sec. IV A or the optiml control pproch in Sec. IV C, it is possile to simultneously pply controlled-phse gtes to couple ll horizontl or verticl neighoring plquettes with no overlp of sites involved for different controlled-phse gtes s shown in Fig.. Menwhile, if ll eight sites from oth plquettes re involved for the controlled-phse gte e.g., V = V x x + V y y, 1 c where V u =V u,s cos u u,s +V u,l cos u u,l for u =x,y. The short-lttice wvelength is, nd the prmeters V u,s, V u,l, u,s, nd u,l re controlled y the intensities nd phses of the lser ems. For integer filling with one prticle per site, ech plquette hs four prticles. The four spin-1/ prticles hve twofold singlet suspce with totl spin zero long ll directions i.e., S tot =0. Thus the singlet suspce is the DFS insensitive to uniform mgnetic field fluctutions. The intrplquette couplings solid lines in Fig. 1 enle opertions of the single logicl quit encoded in the plquette. Note tht y mnipulting intrplquette couplings minimum instnces of topologicl mtter cn e demonstrted in the sme opticl superlttice 17. In order to FIG.. Color online Simultneous coupling etween neighoring plquettes. It is possile to simultneously implement the controlled-phse gtes etween neighoring plquettes long the horizontl direction, if only four sites re involved for ech controlled-phse gte e.g., for the geometric phse pproch or the optiml control pproch. The opticl superlttice with interplquette coupling nd lternting energy offset for odd nd even plquettes, creted y,,4 cn yield the interplquette coupling for the geometric phse pproch. c The spin-dependent opticl superlttice indicted y thin nd thick lines of potentil profiles cn generte Ising interction 13 useful for the optiml control pproch

4 JIANG et l. the perturtive pproch in Sec. IV B, two steps re needed to couple the plquettes long the horizontl direction: first couple ech even plquette with the neighoring odd plquette on the left, nd then couple ech even plquette with the neighoring odd plquette on the right. In order to use the prepred cluster stte for the MBQC, we should lso e le to mesure the individul quits. This cn e chieved y first converting the spin singlet or triplet sttes into different prticle numer configurtions,16, nd then using vrious techniques of coherent opticl control with suwvelength resolution 18 0 to projectively count the prticle numer t specific site without compromising the coherence for the remining sites. In the next two sections, we will consider the rottion of single logicl quit using intrplquette couplings, nd the controlled-phse gte etween two logicl quits using the dditionl interplquette couplings, respectively. III. LOGICAL QUBIT ENCODED IN THE PLAQUETTE In this section, we focus on the opertions within the plquette vi intrplquette coupling. For concreteness, we consider osonic prticles, nd similr results cn e otined for fermionic prticles s well. The following Hurd Hmiltonin governs the dynmics of single plquette, with spin-independent tunnelings nd interctions we will introduce n dditionl virtion level in Sec. IV : H = t ij i i,j, j + H.c. + 1 U i n i n i 1 + i n i, i, where i i is the nnihiltion cretion opertor, n i is the prticle numer opertor for site i=1,...,4 with spin =,, nd n i =n i, +n i,. The tunneling mplitudes t H =t 1 =t 34 nd t V =t 3 =t 41 nd the offset energies i cn e chnged y tuning the superlttice prmeters. The lrge onsite interction U t H,t V ensures tht the system is in the Mott insultor regime with fixed prticle numer for ech site. Prticle tunneling only occurs virtully etween neighoring sites, which leds to the superexchnge coupling H = J H s 1 s + s 3 s 4 J V s s 3 + s 4 s 1, where s i re Puli opertors for the spin t site i. The coupling strengths J H =t H /U nd J V =t V /U cn e chnged independently, y tuning the rriers etween the sites s illustrted in Fig. 3. Controlling the superexchnge couplings is sufficient to perform ritrry rottions of the logicl quit encoded in the plquette. A. Singlet suspce for four spins The spce of four 1/-spin prticles spn suspce of totl spin, three suspces of totl spin 1 nd two suspces of totl spin 0. We use the twofold singlet suspce of the plquette to encode the logicl quit. The singlet suspce is spnned y H = S 1, S 3,4, 3 4 PHYSICAL REVIEW A 79, t H V = S,3 S 4,1, 5 with S i,j 1 i j i j. H or V is the product stte of two singlet pirs long the horizontl or verticl direction, which cn e prepred using the procedure demonstrted in 3. The singlet suspce is decoherence free, ecuse it is insensitive to the uniform mgnetic field fluctutions. In ddition, mesuring single spin will not distinguish the sttes from singlet suspce, nd this is source of protection ginst locl perturtions. Since H V =1/ 0, it is more convenient to use the orthogonl sttes 0 V nd V H. We cn lso write the orthogonl sttes in terms of the singlets nd triplets for verticl pirs, 3 nd 4, 1 11 : 0 = S,3 S 4,1, 1 = 1 T +,3 T 4,1 T 0,3 T 0 4,1 + T,3 T + 4,1, 3 7 where T +,0, =, 1 +,. For such choice of sis, the susystem of two spins, 3 is sufficient to determine the logicl sttes 0 nd 1, ecuse the corresponding reduced density mtrices, 0,3 =Tr 4,1 0 0 = S S,3, 6 8 1,3 =Tr 4,1 1 1 = 1 3 T + T + + T 0 T 0 + T T,3, 9 elong to orthogonl singlet nd triplet suspces, Tr 0,3 1,3 =0. The Puli opertors ssocited with the logicl quit re x , y i 0 1 i 1 0 nd z Within the singlet suspce the logicl opertor z cn e chieved y operting the, 3 spins z 1 1+s s 3, 10 where we use to represent the specil equlity vlid within the singlet suspce. Since s s 3 s 4 s 1 i.e., either oth pirs re singlets or oth re triplets, z cn lso e implemented y operting the 4, 1 spins, z 1 1+s 4 s 1. t V 1 FIG. 3. Color online Intrplquette superexchnge couplings. The coupling strengths re J H =t H /U etween horizontl neighors nd J V =t V /U etween verticl neighors. They cn e chnged independently y tuning the rriers etween the sites

5 PREPARATION OF DECOHERENCE-FREE CLUSTER PHYSICAL REVIEW A 79, n V n V 1 1 c S,3 S 1,4 S,3 T 1,4 n C n g n g 1 T,3 S 1,4 T,3 T 1,4 n H 3 4 FIG. 4. Color online The Bloch sphere representtion for the singlet suspce. The stte 0 1 is ssocited with the north south pole. The superexchnge coupling J H J V is ssocited with the rottion round the n H n V xis. Here n H = 3 1,0, nd n V = 0,0,1. The sequentil rottions round the n H nd n V xes cn rotte the Bloch vector from n V to n g = 1,0, 1 i.e., stte Alterntively comined superexchnge coupling with contriutions from oth J H nd J V cn implement the rottion round the xis n C, which rottes the Bloch vector from n V to n g in one step. B. Rotting logicl quit with superexchnge coupling We now consider ritrry rottions in the singlet suspce using superexchnge couplings. First of ll, the superexchnge coupling Hmiltonin commutes with the totl spin opertor of the plquette H,s 1 +s +s 3 +s 4 =0, due to the identity s i s j,s i +s j =0. Consequently, the superexchnge coupling preserves the singlet suspce with zero totl spin. Within the singlet suspce we hve n H s 1 s + s 3 s 4, n V s s 3 + s 4 s with n H = 3 4. The constnt of 1/ cn e neglected, s it only induces n overll phse during the evolution. The rottions out these xes cn e controlled y switching on/off the superexchnge couplings of J H nd J V, which vries expentilly with the height of the corresponding rriers. Since the ngle etween n H nd n V is /3, ritrry rottion of the Bloch sphere cn e chieved within four opertions. This is specil cse of the generl theorem 1, stting tht k+ opertions re sufficient for ritrry rottion given the ngle etween the two rottion xes stisfies k min,,0, 1 nd n V= 0,0,1 s illustrted in Fig. k+1. The product stte of two verticl singlet pirs 0 = V cn e initilized using the procedure demonstrted in 3. Universl rottion enles dynmicl preprtion of ritrry logicl stte encoded in singlet suspce. For exmple, + = cn e prepred y two-step evolution e in V V e in H H V with H = sin nd V = sin , s shown in Fig. 4. Alterntively, 3 we cn tune the reltive strength etween n H nd n V to chieve the totl coupling n C with n C = 1,0, 1, nd prepre in one step e in C / V, s illustrted in Fig. 4. Note tht ll the plquettes cn e simultneously prepred in the stte. In order to crete the decoherence-free Tilted Potentil FIG. 5. Color online Geometric phse pproch to controlledphse gte step 1. The sites from two neighoring plquettes re leled. The intrplquette trpping potentil long the verticl direction is diticlly tilted. This results in single or doule occupncy t the lower site if the verticl pir of prticles is in the singlet or triplet stte. c Prticle numer configurtions re plotted for four possile of spin sttes: S,3 S 1,4, S,3 T 1,4, T,3 S 1,4, nd T,3 T 1,4. S nd T indicte singlet nd triplet. cluster stte, we need the controlled-phse gte etween the logicl quits encoded in neighoring plquettes. IV. CONTROLLED-PHASE GATE We now consider interplquette couplings dshed lines in Fig. 1. In prticulr, we focus on implementing the controlled-phse gte etween two neighoring plquettes, which induces n dditionl 1 phse if oth encoded quits re in the logicl stte 1. In principle, the controlled-phse gte cn e chieved y the Ising-type interction etween the logicl quits, ut unfortuntely such interction is not immeditely ville from the lttice experiments, s the ective Ising term z z requires four-site interction s s 3 s 4 s 1 see Eqs. 10 nd 11. However, since wht we wnt is the specific unitry evolution rther thn the interction, it is ctully more fesile to implement the unitry evolution directly. In the following, we present three different pproches to implement the controlled-phse gte etween two neighoring plquettes. For concreteness, we only consider coupling two neighoring plquettes long the horizontl direction, while ll three pproches cn lso couple neighoring plquettes long the verticl direction. A. Geometric phse pproch The first pproch uses the virtion levels nd the geometric phse to chieve the controlled-phse gte etween neighoring plquettes Fig. 5. The geometric phse is proportionl the surfce re enclosed y the evolution trjectory in the Bloch sphere ssocited with the two energy levels tht re degenerte. For exmple, if hlf of the Bloch sphere is enclosed, the system cquires geometric phse. We first consider the osonic prticles. It tkes three steps to chieve the controlled-phse gte: Step 1. We lower the interplquette rrier nd diticlly tilt the intrplquette potentil long the verticl direc

6 JIANG et l. 1 Δ Bised Potentil c S,3 S 1,4 S,3 S 1,4 S,3 T 1, S,1 S 3,4 1 1 T,1 T 3,4 S 1,4 T,3 T 1,4 tion Fig. 5. Ech lower site will e occupied y one prticle or two prticles if the verticl pir of prticles is in the singlet or triplet stte Fig. 5 c. For exmple, if the spins, 3 re in the singlet stte denoted s S,3, the trnsfer of prticle from site to site 3 is prevented y the symmetry requirement of osonic prticles, resulting in one prticle in site 3 see the upper two pnels in Fig. 5 c. If the spins, 3 re in the triplet suspce denoted s T,3, the prticle from site is diticlly trnsferred to site 3, leving two prticles in site 3 see the lower two pnels in Fig. 5 c. Similr spin-dependent trnsfer lso hppens to other sites, such s 1,4. Step. We quickly pply defined is to the interplquette lttice potentil nd lower the interplquette rrier long the horizontl direction Fig. 6. This induces single prticle resonnt tunneling with rte t etween the virtionl ground stte t site nd the virtionl excited stte t site 1 33, if there is one prticle t site nd zero prticle t site 1 see the highlighted upper right pnel in Fig. 6 c. By witing for time /t, we otin the geometric phse from the resonnt tunneling for S,3 T 1,4. As detiled in Sec. IV, for ll other three cses S,3 S 1,4, T,3 S 1,4, nd T,3 T 1,4 we only otin trivil geometric phse 0 or. Step 3. We chnge the intrplquette potentil to the initil lnced position long the verticl direction hving one prticle per site nd restore ech plquette to the logicl suspce. A recent superlttice experiment uses the resonnt tunneling nd the lockde induced y on-site interction to count the numer of toms 3. This experiment demonstrtes tht the presence or sence of resonnt tunneling cn e highly T,3 1 1 FIG. 6. Color online Geometric phse pproch to controlledphse gte step. The sites from two neighoring plquettes re leled. A defined is of the interplquette potentil is quickly pplied nd the interplquette rrier is lowered to fcilitte the resonnt tunneling long the horizontl direction. c Resonnt tunneling etween the virtionl ground level of the left site nd the virtionl excited level of the right site cn occur for the following two cses: i ech of the left nd right sites hs exctly one prticle, nd the two prticles re in the singlet stte see the left pnel, ii the left site hs one prticle nd the right site hs zero prticles see the highlighted upper right pnel. All other configurtions re off-resonnt, with negligile tunneling. After time /t, geometric phse from the resonnt tunneling is otined for S,3 T 1,4 the highlighted upper right pnel, while only trivil geometric phse 0 or is otined for the other three cses S,3 S 1,4, T,3 S 1,4, nd T,3 T 1,4. PHYSICAL REVIEW A 79, sensitive to the numer of prticles in the lttice sites. The geometric phse pproch cn e regrded s n extension tht uses the resonnt tunneling to coherently imprint geometric phse for specific prticle numer configurtion corresponding to certin logicl stte. The procedure for the fermionic prticles is lmost the sme s tht for the osonic prticles, except for the following three differences. First, the is of the energy offset needs to e = +U R for fermionic prticles wheres = for osonic prticles, where is the virtionl excittion energy nd U R is the on-site interction etween ground nd excited levels t the right site 1 or 4. Second, the geometric phse is otined from the resonnt tunneling ssocited the suspce T,3 S 1,4 for fermionic prticles wheres it is ssocited with S,3 T 1,4 for osonic prticles. Third, the geometric phse is 0 for the remining cses for fermionic prticles wheres it might e either 0 or for osonic prticles. It is tempting to consider using = for the fermionic prticles, s we might expect tht y exchnging the roles of singlet nd triplets, the fermionic prticles could e mpped to osonic prticles. However, the roles of singlet nd triplets re not exctly symmetric. For exmple, consider the cse with one prticle per site fter step 1. For osonic prticles, the system is in the suspce S,3 S 1,4 tht hs finite projection to S,1 S 3,4 nd T,1 T 3,4 ut not T,1 S 3,4 or S,1 T 3,4, which yields trivil 0 or geometric phse. For fermionic prticles, the system is in the suspce T,3 T 1,4 tht hs finite projection to T,1 S 3,4 nd S,1 T 3,4, s well s S,1 S 3,4 nd T,1 T 3,4, which thus my yields nontrivil geometric phse. The detiled clcultions for oth osonic nd fermionic prticles re presented in Appendix A. B. Perturtive pproch The second pproch uses oth the intr- nd interplquette couplings cting on the eight sites. The intrplquette coupling induces n energy gp etween the logicl sttes i.e., singlet suspce nd other nonlogicl sttes, while the interplquette coupling cts s perturtion tht induces different phse shifts for different logicl sttes. The interplquette coupling cn e iciently chieved using superexchnge interction etween the interplquette neighoring sites. The key chllenge is to otin the intrplquette interction, with finite Heisenerg interction etween the sites long the digonl nd off-digonl directions. We cn overcome the chllenge y using different design of the opticl lttice. 1. Lttice geometry nd energy levels We wnt to otin the Hmiltonin with intrplquette interction: H intr = J s i s i+1 + d s i s i+, 14 i=1,,3,4 i=1, where the exchnge interction J= t /U d= t /U is induced y the tunneling etween the nerest neighors next

7 PREPARATION OF DECOHERENCE-FREE CLUSTER PHYSICAL REVIEW A 79, = 1 3 H + V, 16 = H V, 17 FIG. 7. Color online Density plot of the lttice potentil see Eq. 15, which genertes n rry of plquettes in the x-y plne with periodicity of /k. Using prmeters V l =60E R, V s =9E r, nd V =10E r, with E R nd E r photon recoil energy of the long nd short lttices, respectively, one cn chieve prmeter regime with d/j 0.. Energy levels of single plquette descried y the Hmiltonin given in Eq. 14. In the plot we ssume fermionic toms, i.e., J,d 0 fermions. For osons J,d 0 the order of the energy levels is the reverse, i.e., the S= is the lowest energy stte. nerest neighors with tunneling rte t t. The positive nd negtive signs re for fermions nd osons, respectively. To simplify the nottion, we hve identified s 5 with s 1. Such type of interction cn e creted y lttice potentil of the form see Fig. 7 V x,y,z = V c x,y + V x s x + V y s y, 15 where V s u u = Vs Vl cos ku cos ku re the typicl doule well superlttice formed y the superposition of two independent sinusoidl potentils which differ in periodicity, = /k, /k, y fctor of 3 nd V c x,y = V (k x y )cos k x+y is n dditionl potentil tht llows us to control the digonl couplings within the plquettes. It cn e constructed, for exmple, from folded, retroreflected em with out-of-the-plne polriztion 4. By vrying the depths of the short V s nd V l long lttices it is possile to control the intr- nd interplquette coupling independently, nd in prticulr to mke the ltter negligily smll nd the plquettes independent. As the intensity of the nonseprle prt of the potentil is rmped up, minim t the center of the plquettes develops. If the strength of the ltter is such tht the energies of ound sttes in this minim re lrger thn the energies of the lowest virtionl sttes t the plquette sites it is possile to tune the rtio t /t without populting the centrl site, which is required for the vlidity of Eq. 14. For exmple, using the prmeters V l =60E R =15E r, V s =9E r, nd V =10E r, with E R = k / 8m nd Er= k / m the photon recoil energy of the long nd short lttices, respectively, one cn chieve prmeter regime with t /t 0.5 with n energy gp to the first virtionl stte in the centrl well of order E g /t 10. It is very difficult to increse t /t close to 1 y just controlling the lttice potentil, ecuse the energy gp disppers nd the centrl sites ecome ccessile. Therefore we will focus on the cse d J. The eigensttes ssocited with Eq. 14 cn e clssified ccording with their totl spin S. As shown in Fig. 7, there re two singlet S=0 sttes, with energies E = 4 J d nd E =+4 J d, respectively. There re three S=1 sttes denoted y 1 1,0,1, with energies E 1 q =4J, 4J, nd 4d, respectively. For fermionic osonic toms the highest lowest energy stte is S= stte, with energy E =4 J+d. We wnt to use the singlet sttes within ech plquette s encoded quits nd perform phse gte etween them y coupling nerest-neighor plquettes into superplquette i.e., 4 potentil wells. A superplquette cn e chieve y superimposing lser ems with periodicities 4 nd 4 /3 long one xis 5. Such wvelengths re experimentlly ville for typicl lkli-metl toms or cn e engineered y intersecting pirs of lser ems t pproprited ngles 6. The 4 isoltes pirs of djcent plquettes long one direction nd the extr 4 /3 lttice is needed to lnce the offset creted when the ltter lttice is dded. When pirs of plquettes re wekly coupled into superplquette the Hmiltonin tht connects the plquettes is given y H c = J s s 1 + s 3 s We wnt to use the coupling to implement controlled-phse gte etween the singlet eigensttes in the two plquettes. To chieve tht we require tht the interplquette coupling is wek i.e., J min 4d,8 J d,4 J d nd derive n ective Hmiltonin y diticlly eliminting the ll S 0 sttes. In the following we discuss the implementtion of the controlled-phse gte for the experimentlly relevnt regime d J. The idel cse of d=j is discussed in Appendix B.. Perturtive pproch with d J For the D plquette implementtion the digonl coupling d is lwys smller thn J nd therefore the singlet sttes within the plquette re nondegenerte, E = E E =8 J d Regrdless of this issue, it is still possile to derive n ective Hmiltonin provided tht the interplquette coupling J is less thn the energy difference etween nd ll other sttes see Fig. 7 : J min 4d,8 J d,4 J d. 0 From this considertion we oserve tht close to d=0,j/, nd J, the perturtive pproch sed on nd reks down nd we should sty wy from these points. As detiled in Appendix B, we cn otin the ective Hmiltonin

8 JIANG et l. PHYSICAL REVIEW A 79, H i,i+1 Within superplquette the term ˆ i ˆ i+1 commutes with it only introduces phse T = J J8 t c in the ective triplet suspce:,,,, nd, +, / nd S = 3J J8 t c for the ective singlet sttes:,, /. Here tc stnds for the durtion of the J controlled-phse gte, i.e., J t c z 1 8 = n 1 /4, with n integer n=1,,... Consequently H i,i+1 cn e used to perform controlled-phse gte within superplquette. We use the stndrd echo technique i.e., pulses t t c / nd t c for ech of the encoded quits to remove the unwnted ˆ jz term from the ective Hmiltonin in Eq. 1. The controlled-phse gte cn e chieved y the unitry evolution U = Xe i H intr +H c t c / Xe i H intr +H c t c /, 4 where X represents the echo pulses for the encoded quits, which cn e chieved vi intrplquette superexchnge couplings. We use the exct digonliztion to clculte the controlled-phse gte fidelity F= f with f = 1 N Tr U c-phsepup, 5 FIG. 8. Color online Prmeters of the ective Hmiltonin for the generl cse d J. At the point d 0.6J, the Ising term in the ective Hmiltonin vnishes, z = 1 8 see text. The controlled-phse gte fidelity F on superplquette s function of d/j for different rtions of J /J. For sme d/j, the smller J /J the higher the fidelity. There re four criticl points t which the fidelity drops considerly; these re d/j 0,0.5,0.6,1. At d/j 0 nd 0.5, one of the singlet sttes ecomes degenerte with one S=1 stte nd consequently the ective Hmiltonin reks down. At the d/j 0.6 the Ising term vnishes nd t d=j the rotting wve pproximtion used in the simplifiction of the ective Hmiltonin ecomes invlid. The shdow regions re the ones where the chievle fidelity is higher thn H i,i+1 = E J z ˆ J jz j=i,i+1 J 1 ˆ i ˆ i+1 + z 1 J 8 8 ˆ iz z ˆ, 1 i+1 where ˆ re ective Puli mtrices cting on the, sttes, nd z = J d 8J d 3J z = J d + 4J + d + J + J J d, 8J d 3J 8 J J d. 3 In Fig. 8 the prmeters z nd z re plotted s function of d/j. where P is the projection opertor to the singlet suspce for ech plquette. In Fig. 8, we plot F s function of d/j for different rtions of J /J, with n=1. The figure shows tht t the points d 0.5J, d 0.6J, d=0, d=j there is n rupt drop of the fidelity, s the rtio J /J is incresed. The drop t these points is expected since t d 0.5J nd d 0 one of the singlet sttes ecomes degenerte with one S=1 nd consequently the ective Hmiltonin reks down. At d 0.6J, z =1/8 see Fig. 8 the Ising term vnishes nd t d=j the two singlets ecome degenerte nd the rotting wve pproximtion ssumed for the derivtion of the ective Hmiltonin in Appendix B is no longer justified. Awy from these points the derived ective Hmiltonin provides good description of the dynmics nd for vlues of J 0.1J one cn get gte fidelity ove In the plot we highlight with gry shdow the d/j prmeter regime where the chievle fidelity is ove However, mong these shdow regions only the regime d/j 0.5 is experimentlly chievle using the lttice geometry descried erly in this section. The smll fluctutions in the fidelity curves re due to the nonenergy preserving terms neglected to otin the ective Hmiltonin see Appendix B, which cn e suppressed when J 8 J d J. The Heisenerg term ˆ i ˆ i+1, on the other hnd, does not commute with H i+1,i+ nd consequently the phse gte cnnot e pplied simultneously to ll plquettes. Insted it hs to e pplied first etween the superplquettes formed y the plquettes i + 1, i + nd susequently etween the superplquettes formed from plquettes i+, i+3 see Fig. 9. Additionlly, in order to crete cluster stte cross ll the plquette rry, it is required to fine tune the prmeters nd time evolution to eliminte the different phse ccumulted y the triplet nd singlet sttes in the encoded spin sis due to the Heisenerg term t t c. Consequently, for multiprtite

9 PREPARATION OF DECOHERENCE-FREE CLUSTER 1 entnglement genertion not ll d/j vlues re llowed ut only the ones which stisfy the following conditions: J T S = m, J t c z 1 8 = n 1 4, FIG. 9. Color online For the d J, the cluster stte genertion hs to e pplied in two steps nd. 6 7 where n nd m re integers. In Fig. 10, we show set of llowed d/j vlues which stisfy the conditions given y Eq. 6, for different n nd m vlues. Here we lso highlight with gry shdow the corresponding d/ J vlues which yield fidelity higher thn 0.98 for J /J 0.1. In Fig. 10 we show two exmples of trces of the phse gte fidelity vs J /J: n,m = 1,1, 3,4 indicted in pnel y squre computed y exct digonliztion of the superplquette FIG. 10. Color online Exmples of controlled-phse gte with the perturtive pproch. The vlues of d/j tht stisfy the conditions stted in Eqs. 6. The shdow regions highlight the regime where the phse gte fidelity cn e lrger thn 0.98 for J /J 0.1. The controlled-phse gte fidelity F s function of J /J cn e clculted y exct numericl digonliztion. The red dshed or lue solid curve is for n,m = 1,1 or 3, 4, which is lso indicted y the red lue squre in pnel. PHYSICAL REVIEW A 79, Hmiltonin. The figure shows tht it is lwys possile to find prmeters which llow for high gte fidelity. However, here we re only including errors due to higher order terms neglected in the derivtion of the ective Hmiltonin. In relistic experiments other externl errors such s lttice inhomogeneities re lwys present, which cn e minimized t the expense of lrger J /J rtio fster evolution. There is consequently trdeoff etween fster time evolution nd smll perturtive corrections. In conclusion, we hve presented scheme to perform controlled-phse gtes in the encoded singlet suspce. This perturtive scheme hs two dvntges: i the plquettes re lwys in the decoherence free suspce; ii it is esy to implement s it only relys on the coherent dynmicl evolution without further mnipultions. Due to the fct tht the dynmics is determined y second order ective Hmiltonin the chievle fidelity with the proposed schemes cn ecome very high ut t the cost of slower time evolution. If the strongly intercting regime is reched y using Feshch resonnce, one cn chieve vlues of J of order of 100 Hz nd therefore cluster genertion times of order s. These genertion times re slow ut longer thn the encoded quits decoherence time due to their insensitivity ginst environmentl decoherence 3. C. Optiml control pproch We now consider the optiml control pproch to fst, high fidelity implementtion of the controlled-phse gte etween the horizontl neighoring plquettes 1,, 3, 4 nd 1,,3,4 s shown in Fig. 5. The key chllenge here is to identify n icient set of opertors tht i enle the unitry evolution of the controlled-phse gte, nd ii re fesile using opticl superlttices s well. We first provide set of opertors sufficient to chieve the controlledphse gte with ritrry precision. After tht we numericlly find the pulse sequences for these opertors to implement the controlled-phse gte. 1. Choice of opertors In principle, tunle Heisenerg superexchnge interctions re sufficient for the controlled-phse gte y coupling ll eight sites 11. However, for the opticl lttice experiments, we would like to chieve the controlled-phse gte y coupling s few sites s possile, preferly using glol rottions for ll spins nd Heisenerg or Ising interctions etween neighoring sites 3,13. According to Eqs. 10 nd 11, we need t lest two sites from ech plquette to determine the z opertor. Since the controlled-phse gte depends on oth z opertors from the plquettes, we should consider t lest four sites to implement the controlled-phse gte. It turns out tht coupling the four middle sites,3,4,1 is sufficient to chieve the controlled-phse gte, which significntly reduces the complexity compred with the erlier proposl tht couples ll eight sites 11. We consider the Hmiltonin 5 H t = k t O k, k=1 8 where k t re the time-dependent control vriles for the set of opertors

10 JIANG et l. PHYSICAL REVIEW A 79, O 1 = s s 3, O = s 1 s 4, 9 30 f = 1 N Tr U c-phse N U T;x = 1 N n=1 n U c-phse U T;x n, 36 O 3 = s,z s 1,z + s 3,z s 4,z, O 4 = s,x + s 3,x + s 1,x + s 4,x, 31 3 O 5 = s,y + s 3,y + s 1,y + s 4,y. 33 To justify tht H t cn implement the controlled-phse gte U c-phse = exp i 4 1 z 1 z, 34 we show tht 1 z 1 z elongs to the Lie lger generted y O k k=1,...,5. We strt with these five opertors s the ville set AS, nd clculte the commuttors mong the AS opertors. We then expnd the AS y dding new commuttors tht re not liner comintions of the AS opertors. We denote the numer of linerly independent AS opertors s the dimension of the AS. We repet the process of clculting the commuttors nd expnding the AS, until its dimension does not increse ny more. We use MATH- EMATICA to iterte the process of expnding the AS until it sturtes t dimension 80 including the identity opertor tht commutes with ll other opertors. Finlly, we verify tht 1 z 1 z s s 3 s 1 s 4 is liner comintion of the AS opertors. Therefore ccording to the locl properties from the Lie lger the set of opertors O k k=1,...,5 is sufficient to implement the controlled-phse gte. The remining tsk is to find the solution for k t.. Smooth pulses We use n lgorithm which cn e interpreted s continuous version of the grdient scent pulse engineering GRAPE 7,8, though it is developed in n independent wy 9. The lgorithm sed on optiml quntum control is summrized in Appendix C. Compring with the GRAPE method it hs the dvntge tht we cn find solutions with specific oundry conditions e.g., pulses strt nd end t zero nd in terms of smooth finite slope functions of time s well. More specificlly, the prticulr form of the coicients k t re chosen to e finite sums of sinusoidl functions, L k t,x k1,...,x kl = x kl sin l t, l=1 T 35 ech of which depends on L prmeters x kl l=1,...,l.as mentioned efore, note tht they fulfill the convenient property tht k 0 = k T =0 nd tht they hve finite slope. Hence we need to optimize K L with K=5 opertors in our cse prmeters x kl which mximize the fidelity F= f, where for prticulr suspce of N sttes n n=1,...,n of dimension d N. In Fig. 11 we show n exmple of pulses otined with L=0, which cn ttin very low infidelity =1 F, less thn 10 7 the vlue cn e further reduced y improving the precision of the numerics. In Fig. 11 we plot the infidelity s function of the reltive devition J/J. We ssume tht the couplings in Eq. 8 re devited from k to 1 J/J k ; tht is, the system evolves under the devited Hmiltonin 1 J/J H. For simplicity, we consider the cse tht J/J is time independent. For exmple, imperfect clirtion of rrier height or rrier thickness my induce such proportionl, time-independent devition in superexchnge couplings. We find tht the infidelity remins constnt vlue pproximtely 10 7 for very smll devitions with J/J 10 4, while the infidelity scles s J/J for lrger devitions with J/J 10 3 which is lso plotted using the liner scle in Fig. 11 c. Such qudrtic dependence to the devition is not uncommon, s the infidelity for single spin rottions lso scles qudrticlly with the devition. The qudrtic dependence cn e regrded s direct consequence of the optimiztion procedure, which finds locl minimum of the function with first order derivtives eing zero. 3. Experimentl implementtion We now riefly discuss the implementtion of opertors O k k=1,...,5 nd O k for the cluster stte preprtion. The opertors of O 1 nd O cn e chieved y superexchnge interction using superlttice techniques 3, while the opertor of O 3 cn e otined from spin-dependent tunneling in opticl lttices 13. Furthermore, we note tht the evolution of the Ising interctions O 3 etween ll horizontl neighoring plquettes cn e performed simultneously, ecuse they ct on different groups of physicl spins s illustrted in Fig.. For the sme reson, the opertors of O 4 nd O 5 cn e performed simultneously for ll spins y driving the entire opticl lttice with pproprite microwve pulses 4. Therefore the simultneous controlled-phse gtes etween ll horizontlly neighoring plquettes cn e chieved. D. Compring three pproches We compre the three pproches see Tle I in the following spects: i two relevnt time scles: the time to implement the controlled-phse gte, nd the durtion for the plquette not eing protected y the DFS which should e short compred to the coherence time outside the DFS 34, ii two types of errors contriuting to the controlled-z CZ gte infidelity: the systemtic errors from the pproximtions used in our nlysis, nd the inhomogeneity errors due to the fct tht the couplings e.g., t,, J, nd J re not exctly the sme for ll plquettes, iii the numer of sites involved for ech controlled-phse gte: if ech gte only couples four

11 PREPARATION OF DECOHERENCE-FREE CLUSTER PHYSICAL REVIEW A 79, α α α α α t/t c ε δj/j F δj/j FIG. 11. Color online Smooth pulses with L=0 yielding to infidelities smller thn Infidelity =1 F vs the devition J/J for the smooth pulses. The infidelity is constnt t 10 7 for J/J 10 4 nd increses qudrticlly with J/J for J/J c Qudrtic dependence of the fidelity with J/J. sites, the controlled-phse gtes etween ll horizontl or verticl neighors cn e chieved simultneously; otherwise two sequentil steps re needed, iv the mjor interctions, such s singlet prticle tunneling 16, Heisenerg superexchnge coupling 3, nd Ising interction 13, v the wvelength components needed to construct the superlttices, vi the relevnt virtionl levels, vii the interprettion of the physicl process, nd viii the control complexity for time-dependent prmeters. The mximum chievle fidelity for the CZ gte is limited y the systemtic nd inhomogeneity errors. For the geometric phse pproch, the systemtic error t/u is due to the off-resonnt tunneling, which is nlyzed in Tles II nd III for osonic nd fermonic prticles, respectively; such off-resonnt tunneling cn e suppressed y using Feshch resonnces to increse U while keeping the sme tunneling rte t. For the perturtive pproch, the controlled-phse TABLE II. Energy difference ssocited with osonic prticle tunneling E 1 for vrious initil numer configurtions n L,n R nd the finl totl spin t the right site j R. The is is set to e =. The resonnce condition E 1 =0 is fulfilled for oth cses: n L,n R, j R = 1,0,1/ nd n L,n R, j R = 1,1,0. n L,n R j R = n R 1 / j R = n R +1 / 1, 0 0 1, 1 0 U R 1, U R 4U R, 1 U L U R +U L, U R +U L 4U R +U L gte fidelity is F 0.98 for J /J 0.1 Fig. 10. For the optiml control pproch, the systemtic error is only limited y the precision of the numerics Figs. 11 nd 11 c. We note tht it is importnt to suppress the inhomogeneity errors, s ll three pproches re sensitive to such imperfections. Replcing the prolic trp with the flt-ottom trp 30 cn e one possile solution to reduce the inhomogeneity errors. It would lso e interesting to consider other pproches tht re insensitive to the inhomogeneity errors. Overll, the geometric phse pproch hs the dvntge of fst opertionl time, short unprotected durtion, nd comptiility of simultneous coupling. The perturtive pproch hs the dvntge of lwys eing protected y the DFS nd fvorle control complexity. The optiml control pproch hs the dvntge of fst opertionl time, vnishing systemtic errors, nd comptiility of simultneous coupling. TABLE III. Energy difference ssocited with fermionic prticle tunneling E 1 for different initil numer configurtions. The is is set to e = +U R. For n L,n R = 1,1, the energy difference is U R for singlet nd triplet sttes, respectively. The resonnce condition E 1 =0 is fulfilled only if n L,n R = 1,. n L,n R E 1 1, 0 1, 1 U R 1, 0, 1, 1 U R U R +U L U R +U L

12 JIANG et l. V. CONCLUSION In conclusion, we hve discussed the preprtion of lrge cluster sttes for neutrl toms in opticl superlttices. Ech logicl quit is encoded in the decoherence-free singlet suspce of four spins from the plquette, so tht it is insensitive to uniform mgnetic field fluctutions long n ritrry direction. Besides ritrry rottions of single logicl quits chieved y superexchnge interction, we provide three different pproches to couple the logicl quits from neighoring plquettes, with their properties summrized in Tle I. These pproches my lso e pplied to other quntum systems, such s quntum dots or Josephson junction rrys. L,> PHYSICAL REVIEW A 79, Δ R,> R,> FIG. 1. Color online The ised potentil for the sites L nd R. The virtionl ground levels re L, nd R,, with energy difference. The virtionl excited level for the right site is R, with excittion energy. ω ACKNOWLEDGMENTS H L = n L + U L n L n L 1, A We would like to thnk Hns Briegel, Igncio Circ, Eugene Demler, Wolfgng Dür, Géz Giedke, Vldimir Gritsev, nd Belén Predes for stimulting discussions. A.M.R. cknowledges support from the NSF CAREER progrms, ITAMP grnt. O.R.I., J.J.G.R., nd A.S. cknowledge finncil support from the Europen Commission Integrted Project SCALA, from the Spnish M.E.C. FIS , AP , Consolider Ingenio010 CSD QOIT, FIS , CAM-UCM/910758, from the Rmony Cjl Progrm, nd from the Ctln Government SGR APPENDIX A: GEOMETRIC PHASE APPROACH 1. Geometric phse pproch with osonic prticles We now justify the clim tht the geometric phse is otined for S,3 T 1,4, while trivil geometric phse 0 or is otined for the other three cses S,3 S 1,4, T,3 S 1,4, nd T,3 T 1,4. This evolution implements the controlled-phse gte up to it-flip of the logicl quit from the left plquette. We strt y generlizing the on-site interction Hmiltonin for site i tht governs oth the ground nd excited virtionl levels denoted s nd, respectively, H i = i n i + i n i + 1 U i n i n i U i n i n i 1 + U i n i n i + i, i, i, i,, + i, i, i, i,, A1, where i is the energy offset, i is the virtionl frequency, U i is the on-site interction strength etween levels nd for site i. The prticle numer opertors re n i = i, i,, n i = i, i,, nd n i =n i +n i. Given lrge virtionl frequency i U i, we my sfely neglect those energy nonconserving terms, i, i, i, i,. For the ised potentil etween the two horizontl sites L nd R s shown in Fig. 1, we consider one virtionl level for the left site nd two levels for the right site: H R = R n R + 1 U R n R n R U R n R n R 1 + U R n R n R + J R n R + n R n R + n R +1, A3 where = L R is the is in the potentil i.e., energy difference etween the ground levels for the two sites, nd J R is the totl spin for the right site see Appendix A for detiled derivtion. Given quntum numers n L,n R,n R, j R, the on-site energies re E L n L = n L + U L n L n L 1, A4 E R n R,n R, j R = R n R + 1 U R n R n R U R n R n R 1 + U R f n R,n R, j R, A5 where f n R,n R, j R =n R n R n R + n R n R + n R +1 + j R j R +1 A6 for n R +n R / j R n R n R /. Note tht n R =0 implies f n R,0,j R =n R / =0. For n R =1, we hve f n R,1,j R = n R +1 / =n R nd f n R,1,j R = n R 1 / =n R Thus the energy difference for the osonic prticle tunneling from the left site to the right site is E 1 n L,n R, j R = E L n L + E R n R,0,n R / E L n L 1 E R n R,1,j R = + U L n L 1 U R f n R,1,j R. A7 In Tle II, we list the energy difference ssocited with osonic prticle tunneling with =. There re two possiilities to fulfill the condition of resonnt tunneling. The first cse is n L,n R, j R = 1,0,1/. This corresponds to the resonnt tunneling etween the sites,1 in the highlighted upper right pnel S,3 T 1,4 of Fig. 6 c, yielding geomet