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Many-body protected entanglement generation ininteracting spin systems A. M. Rey 1, L. Jiang 2, M. Fleischhauer 3, E. Demler 2 and M.D. Lukin 1,2 1 InstituteforTheoretical Atomic, Molecular and Optical Physics, Cambridge, MA, 02138. 2 PhysicsDepartment, Harvard University, Cambridge, Massachusetts 02138, USA and 3Fachbereich Physik, Technische Universitat Kaiserslautern, D-67663 Kaiserslautern, Germany (Dated:February1,2008) We discuss a method to achieve decoherence resistent entanglement generation in two level spin systems governedbygappedandmulti-degenerateHamiltonians. Insuchsystems,whilethelargenumber ofdegrees offreedominthegroundstatelevelsallowstocreatevariousquantumsuperpositions,theenergygapprevents decoherence. Weapply the protected evolution toachieve decoherence resistent generation of many particle 8 GHZstatesandshowitcansignificantlyincreasethesensitivityinfrequencyspectroscopy. Wediscusshowto 0 engineerthedesiredmany-bodyprotectedmanifoldintwospecificphysicalsystems,trappedionsandneutral 0 atoms in optical lattices, and present simple expressions for the fidelity of GHZ generation under non-ideal 2 conditions. n a J I. INTRODUCTION mentsanddiscuss the significantimprovementin phase sen- 4 sitivitythatitmightprovide.InsectionVweelaborateonthe physicalresourcesrequiredfortheimplementationofthegap Itiswell-knownthatentangledatomicstates(e.g.so-called ] protectedHamiltonianintrappedionsanddiscusstheadvan- r spin squeezed states) potentially allow to significantly im- e tagesanddisadvantagesofitsimplementationwithrespectto prove resolution in Ramsey spectroscopy [1, 2]. Entangled h standardunprotectedHamiltonians. InSec. VIwestudyhow t statesarealsoafundamentalresourceinquantuminformation o to engineer the long range interactions required for the gap andquantumcomputationscience[3,4].However,inpractice . protected evolution in optical lattice systems interacting via t entangledstatesaredifficulttoprepareandmaintainasnoise a short range interactions and discuss the effectiveness of the and decoherence rapidly collapses them into classical statis- m MPM for GHZ generation. In Sec. VII we analyze in such ticalmixtures. Thusoneofthemostimportantchallengesin - systems the effect of non-ideal conditions such as the mag- d modern quantum physics is the design of robust and most- netictrappingconfinementandfinallyconcludeinSec.VIII. n importantly decoherenceresistant methods for entanglement o generation. c We have recently proposed a method [5] that allows for [ noise resistent generation of entangled states. The method II. MULTI-PARTICLEENTANGLEMENTGENERATION 2 uses the energy gap of properly designed gapped-multi- v degenerateHamiltonians. Whilethelargenumberofdegrees A. IdealCase 8 of freedom in the ground state manifold of such systems al- 7 3 lowstocreatevariousquantumsuperpositionsandtoexploit Inthissection we startby reviewinga methodto generate 3 rich dynamicalevolution (suitable for example for precision multi-particleentangledstatesinasystemofN spin1/2atoms . spectroscopy), the energygap preventsdecoherence as local bytimeevolutionunderthesocalledsqueezingHamiltonian. 6 excitationsbecomeenergeticallysuppressed.Asimpleexam- 0 7 pleofamany-bodyspinHamiltonianthatillustratestheidea Hˆz =χJˆz(0)2. (1) 0 ofourschemeisamulti-spinsystemwithisotropicferromag- : netic interactions. These interactionswill naturallyalign the As shown in Ref. [7, 8] the Hˆ Hamiltonian can be imple- v z spins. While all of the spins can be rotated together around mentedintrappedionsbyusingthecollectivevibrationalmo- i X an arbitrary axis without cost of energy, local spin flips are tion of the ions in a linear trap driven by illuminating them r energeticallyforbidden. with a laser field [1, 2, 9]. In Eq.(1) we used Jˆ(0) to denote a α Thispaperpresentsadetailedanalysisofthismethodwhen thecollectivespinoperatorsoftheN atoms:Jˆ(0) = 1 σˆα, appliedtotrappedionsandcoldatomsinopticallattices. We α 2 i i whereα=x,y,zandσˆα isaPaulioperatoractingonPtheith demonstrateitsapplicabilityfordecoherenceresistantgenera- i atomandwehaveidentifiedthetworelevantinternalstatesof tionofN-particleGreenberger-Horne-Zeilinger(GHZ)states the atoms with the effective spin index σ = , . In this [6] and the potentialof the latter to be used for Heisenberg- manuscriptwe willuseunitssuchthat~ = 1 and↑a↓ssumeN limited spectroscopy. The paper is organized as follows: In tobeeven. SecIIwereviewoneofthestandardproceduresusedtogen- An appropriate basis to describe the dynamics of the sys- erate multi-particle GHZ entangled states in ion traps and tem is the one spanned by collective pseudo-spin states de- demonstrate the detrimental effect of phase decoherence in noted as J,M,β [10]. These states satisfy the eigen- suchentanglementgenerationschemes. InSecIIIweexplain z | i theideaofamany-bodyprotectedmanifold(MPM)anddis- value relations Jˆ(0)2 J,M,β z = J(J + 1)J,M,β z and | i | i cusshowandunderwhatconditionsitcansignificativelyre- Jˆ(0) J,M,β = M J,M,β , with J = N/2,...,0 and z z z | i | i ducetheeffectofdecoherence.InSec. IVweshowtheappli- J M J. β is an additionalquantumnumber associ- − ≤ ≤ cability of the gap protectedevolutionto precision measure- ated with the permutationgroupwhich is requiredto forma 2 completesetoflabelsforallthe2N possiblestates. The entanglement generation process starts by preparing 1 the system att = 0 in a fullypolarizedstate alongthe x direction, N/2,N/2 .AsanystatewithJ =N/2isuniquely x characte|rizedbyMi,fordenotingthemweomittheadditional 0.8 β label. Fullypolarizedstatesalongxcanbewrittenasasu- 0.4 perpositionofstateswithdifferentM valuesalongthezdirec- y0.6 ntnlieieoaanndnt:ssi.PmtoApMasritcnhCotesMllsaay|npNsstMee/m2o2,feMdveJoˆpix(lzev0.n)edDs.,euanHrttiofinpwgrhseattvhsteeehret,eovwatothilnteuidmtdiinoeifgnfχeottrhfreeetnhvHtec=apomhmπailpstaeools--l Fidelit0.4 X\(cid:144)JNx-00..220 C0ollaps€Π4€e€€€ G€HΠ2€€€€ZΧt€3€€4€€€Π€€€€ ReΠvival€5€€4€€€Π€€€€ componentsrephasewhithoippositepolarization,andaperfect 0.2 -0.4 revival of the initial state is observed with Jˆ(0) = N/2 x h i − (seeFig.1). Specifically,thetimeevolutionof Jˆ(0) forsys- x h i 25 50 75 100 125 150 175 200 temswithN 1canbeshowntobegivenby: ≫ N N hJˆx(0)i = 2 (−1)ke−N/2(χt−kπ)2 (2) FIG.1: (Coloronline)FidelityofGHZgenerationvs. N withand k=0X,1,2, without gap protection. We assume a noise with step-like spectral ··· densitywithamplitudef andcutofffrequencyωc =χ.Thedashed Rightattimet0 =trev/2thesystembecomesamacroscopic greenand solidredlinesarefor aprotected system, λ = 50, with superpositionoffullypolarizedstatesalongthe xdirection, f = 0.6χ and f = 0.1χ respectively. The dot-dashed blue and ± i.e. aN-particleGHZstateoftheform blackdottedlinesareforanunprotectedsystem, λ = 0, andsame decoherenceparameters.Intheinsetweshow Jˆx(0)(t) /N forN = h i 1 N N N N 50. |ψxGHZi≡ √2(cid:18)e−iφ+| 2 , 2 ix+eiφ−| 2 ,− 2 ix(cid:19), 1 tdt tdt f(t t )[13]. (3) 2 0 1 0 2 1− 2 withandφ realphasesgivenby π/4andπ/4+Nπ/2. RPhaseRdecoherencecauses an exponentialdecay of the re- Recente±xperiments[1,2]have−usedthistypeofschemeto vivalpeakandandN dependentdecayofthefidelity,defined generateGHZ statesintrappedionswith theaimto perform asF(t0) = hψxGHZ|ρˆ(t0)|ψxGHZi(SeeFig. 1). Qualitatively precision measurements of ω , the energy splitting between the effectof phase decoherenceon the evolving state can be 0 and levels. IdeallytheuseofGHZstatesshouldenhance understoodfromtheenergylevelsofHˆ (SeeFig. 2). While z ↑thepha↓sesensitivitytothefundamentalHeisenberglimit[11]. Hˆ commuteswithbothJˆ(0)andJˆ(0)2,Hˆ onlycommutes z z env However, decoherence significantly limited the applicability with Jˆ(0) does not commute with Jˆ(0)2. Therefore, in the z ofthemethod. presence of phase decoherence transitions between different J subspaces are allowed as long as M is conserved. As all the states with the same M value are degenerate, there is | | B. Effectofdecoherence noenergybarriertopreventsuchtransitionsandveryquickly theinitiallypopulatedJ = N/2manifoldisdepletedandthe To understand the detrimental effect of decoherence we fidelity of generating the GHZ state significantly degraded. firstassumethatthedominanttypeofdecoherenceissingle- Moreover,thedegradationscalesexponentialwithincreasing particle dephasing. Such dephasing comes from processes N dueto theexponentialscalabilityof thenumberof acces- that, while preserving the populations in the atomic levels, siblestatestowhichtheinitiallyJ = N/2populationcanbe randomly change the phases leading to a decay of the off- transferredto. diagonal density matrix elements. We model the phase de- Quantitativelythe effectof decoherencecan be calculated coherence by adding to Eq. (1) the following Hamiltonian byusingthe uncoupledspin basisasitdiagonalizesthe total [12] Hamiltonian.Eachstateinthisbasiscanbelabelas n(k) = sk,sk,...,sk ,wheresk = 1for andk =1{,|...2Ni. 1 | 1 2 Ni} i ± ↑↓ Hˆ = h (t)σˆz, (4) If att = 0 the reduceddensity matrixofthe system, ρˆ, is env 2Xi i i givenbyρˆ= k,lρk,l(0)|n(k)ihn(l)|,aftertimet P wsiahnerperohci(ets)seasrewaitshsuzmereodmtoeabneainnddewpiethndaeuntotcstoorcrehlaasttiiocnGfuanucs-- ρk,l(t) = ρk,l(0)e2iχt[(PNi=1ski)2−(PNi=1sli)2]× tion hi(t)hj(τ) = δijf(t τ). Here the bar denotes av- e−iPNi=1R0tdτhi(τ)(ski−sli) (5) − eraging over the different random outcomes. In what follows we will use the propertythat zero mean Gaussian vari- As a consequence, Jˆ(0)(t) = Tr[Jˆ(0)ρˆ(t)] = x x h i ablessatisfy exp[ i tdτh(τ)] = exp[ Γ(t)], with Γ(t) = e Γ(t) Jˆ(0)(t) with Jˆ(0)(t) the expectation − 0 − − h x i|Γ=0 h x i|Γ=0 R 3 FIG.2: SchematicrepresentationoftheenergylevelsoftheχJˆz(0)2 Hamiltonianandtheeffectofthedifferenttypeofnoise. Asstateswith differentJ butequal M aredegenerate,inthepresenceofphasedecoherence(σz noise)theyarepopulatedduringthetimeevolution. σx,y | | noisescouplestateswhichdifferby 1unitsofMandthereforethesmallenergygapbetweenthem,oftheorderofχ,naturallyprotectsthe ± systemfromthesetypeofprocesses. value in the absence of noise (Eq.(2)). The factor e Γ(t) B. Protectionagainstphasedecohernce − comes from the fact that the operator Jˆ(0) = σˆx only x i i probesone-particlecoherence,i.eitonlyconnectsPstateswith exactlyonespinflipped. Assuming that at t = 0 all atoms are polarized in the x direction,i.e. ρ (0)=2 N,onecanshowfromEq. (5)that k,l − thefidelityisdegradedto JJ==NN//22--11 F(to)= 221N e−Γ(4to)PNi=1(ski−sli)2 =(cid:18)1+e2−Γ(to)(cid:19)N Xl,k σσ (6) JJ==NN//22 MM== --33 σσ MM== 33 zz MM== 22 llNN xx,,yy MM== --11 MM== 11 MM==00 cc III. PROTECTEDDYNAMICS FIG.3: SchematicrepresentationoftheenergylevelsoftheHˆprot+ A. Manybodyprotectedmanifold(MPM) χJˆz(0)2 Hamiltonian. Hˆprot liftsthe degeneracy of the different J manifoldsandsuppresses(intheslownoiselimit)couplingsbetween them.Soifat=0thesystemisintheMPMitremainsthere. LetusnowconsiderwhathappensifinadditionofHˆ we z assumethatthereisanisotropicinfiniterangeferromagnetic interactions between the spins so the system Hamiltonian is ThelowenergyspectrumofHˆc isshowninFig. 3. AsHˆz describedbytheHamiltonianHˆ =Hˆ +Hˆ ,where commuteswithHˆprot,intheabsenceofdecoherencethelatter c prot z doesnotaffectatalltheGHZgenerationdynamics,however Hˆ = λJˆ(0)2 (7) in the presence of decoherencethe latter does significatively prot − reducesthe effect of local environmentalnoise. The protec- Herewehaveassumedthatallatomsareindifferentorbitals tion can be best understood by using the basis of collective but have enough of spatial overlap that every spin interacts states. IntermsofcollectivespinoperatorsHˆ canbewrit- env witheveryotherspin. tenas: TheisotropicHamiltonianHˆ hasa groundstate mani- prot foldspannedbyasetofN +1degeneratestates. Theylieon N 1 thesurfaceoftheBlochspherewithmaximalradiusJ =N/2 Hˆ = 1 − gk(t)Jˆ(k), (8) and are totally symmetric, i.e. invariant with respect to par- env √N z Xk=0 ticle permutations. There is a finite energy gap E = λN g tHhialtbeisrotlsapteascet.heTghriosugnadpsitsattehemkaenyifofoldr fthroemmtahney-rbeostdyofptrhoe- where gk(t) = √1N hj(t)e−i2πNjk and Jˆα(k) = tection against decoherence. Hereunder we will refer to the 21 σˆjei2πNjk. Note that aPllowed transitions must conserve ground state manifold as the many-body protected manifold MPas both the system and noise Hamiltoniancommute with (MPM). Jˆ(0). InthepresenceofalargeenergygapE ,onecandistin- z g 4 guishtwodifferenttypeofprocesses:(i)Decoherenceeffects the MPM. For example, it makes Jˆ(0) to decay N times x,y thattakeplacewithintheMPMduetothecollectivedynam- slower than in the unprotected system: i.e. Jˆ(0)(t) = x,y ics inducedby the k = 0 componentof Hênv, and (ii) tran- e Γ(t)/N Jˆ(0)(t) . h i − x,y Γ=0 sitions across the gap induced by the inhomogeneousterms. h i| Assumingallatomsareinitiallypolarizedinthexdirection, The later couple the MPM with the rest of the system, how- evertheyarenonenergyconservingprocessandconsequently ρM,M˜(0) = 2−N M+NN/2 M˜+NN/2 , using Eqs. (13), the q perUtusrinbgatiaveplyerwtuerabka.tive analysis, and assuming that at t = approximation M+NN(cid:0)/2 ≈ ((cid:1)π(cid:0)2N)1/4e−(cid:1)MN2,validinthelarge N limit and rep(cid:0)lacing t(cid:1)he sums over M and M˜ by integrals 0 the system lies within the MPM, the evolution of the thefidelityatagiventimetcanbeshowntobegivenby: projection of the density matrix on the MPM: ρ MM˜ ≡ N/2,M˜ ρˆN/2,M canbewrittenas z z h | | i 1 ρM,M˜(t)=ρM,M˜(0)eitχ(M2−M˜2)ei(θM−θM˜)e−21(γM−γM˜), F(to)= 1+Γ(to). (14) (9) p Here Theinsensitivityof (t)onN,andtheN timesslowerdecay F rateof Jˆ(0) demonstratetheusefulnessofMPMtogenerate x h i alargenumberofentangledparticles. N t N M τ θ (t) ,M dτHˆ (τ) ,M = g0(τ), M env ≡h 2 |Z | 2 i √N Z 0 0 (10) C. Protectionagainstarbitrarynoise accounts for the dynamics induced by the noise within the MPMand We now discuss the protection against spin flips, which γM(t)= tdτ M eiτωJ,β 2, (11) canbemodeledbytermsproportionalto σix, σiy inthe noise |Z MJ,β | Hamiltonian,Eq.(4). Firstofallnotethatasthe and states J=XN/2,β 0 ↑ ↓ 6 haveafiniteenergysplittingω , lowfrequencynoiseassoci- 0 atedwithsuchtermswillbesuppresseddueit.However,most takes into account the depletion of the J = N/2 levels due to transition matrix elements between N,M and states ofthespinflipsaregenerallyinducedbyimperfectionsinthe |2 iz laser fields and therefore are at frequencies close to ω , i.e. outside the MPM: M = N,M Hˆ J,M,β . ω 0 MJ,β zh2 | env| iz J,β theycorrespondto lowfrequencynoiseintherotatingframe are the respective energy splittings. Because Hênv is a vec- of the laser. In the case involvingGHZ state generation, the tor operator, according to the Wigner-Eckart theorem, Hênv finite energy cost imposed by χJz2 between levels with dif- only couples the states in the MPM with states which have ferent M valuetendstoinhibittheseprocessesasillustrated J =N/2 1andthuswithexcitationenergyλN. in Fig.| 2|. If in addition Hˆ is present this natural pro- − prot Assuming the power spectrum of the noise, f(ω) tection can be enhanced due to the fact that the energy gap dte−iωtf(t),tohaveacut-offfrequencyωc (e.g. f(ω)=≡f suppresses the componentof the noise that cause transitions fRorω ωcand0otherwise),wefindthat betweentheMPMandothermanifolds.Moreprecisely,noise ≤ modeledas hασˆα,whenprojectedintotheMPMreduces γM(t)≈ N2−N4M2fZ ωcdω(cid:18)sin(t(ωω−λλNN)/2)(cid:19)2. to √1Ngα(0)PJˆα0iwithiα=x,y,z. 0 − InFig. 4wequantifytheprotectionprovidedbytheMPM. (12) Intheabsenceofanyprotectionwefindthefiniteenergycost In the limit when the noise is sufficiently slow , i.e. imposed by H helps to protect the system against transver- z ωc ≪ Eg, then γM(t) is bounded for all times, γM(t) < sal noise. Instead of the exponential decay ( e−NΓ(t0) ) (N2−N42M2)(λf2ωNc) ≪ 1 and the atomic populationwithin the of the fidelity observed when dephasing is pres∼ent spin flips groundstatemanifoldisfullypreservedi.e.γM(t) 0inEq. degradesthefidelityas e−Γ(t0)N/4(atleastforthemoder- ≈ ∼ (9). atedN < 12wehavetorestrictoursimulations). Withpro- Consequently,intheslownoiselimittype(ii)processesare tection the fidelity scales even better as the transversalnoise energeticallyforbiddenandonlytype(i)processesareeffec- is restricted to act only within the MPM . Instead of the ex- tiveandthereforethenoiseactsjustasauniformrandommag- ponentialdecayoftheGHZgenerationfidelitywithN, with neticfield: ifatt = 0ρˆ= M,M˜ ρM,M˜(0)|N2,M˜ihN2,M|, protection it decays as ∼ e−AΓ(t0)N0.44 with A a numerical thenaftertimeteachcompoPnentρM,M˜ acquiresanadditional constant, A ∼ 1/3 (forthis resultwe do not have to restrict randomphaseei(θM(t)−θM˜(t)) andonaverage tomoderatedN).Hence,withHˆprotwegainafactoroforder √N. The reason why Jˆ0 and Jˆ0 noise degrade stronger ρM,M˜(t) = ρM,M˜(0)eiχt(M2−M˜)2e−Γ(t)(M−NM˜)2. (13) ∼the fidelity than Jˆz0 noise (wxhich leyads just to a N indepen- dentfidelity)isthattheformerdonotcommutewithHˆ and z The factor of √N in the denominator of θ is fundamen- mixstateswithdifferentM quantumnumber.Additionallyin M tal for the reduction of the effect of decoherence within the figure we show that for noise with long correlation time 5 J = N/2,N/2 or J = N/2, N/2 after applyingthe 1 z z |unitarytransformiation| σx. − i i i 0.9 This generalized RaQmsey sequence can be quantitatively described as a measure of the expectation value of the fol- 0.8 lowingoperator,Oˆ: y Fidelit0.7 Eco+Prot+y-noise hOî=hψ(t0)| [cos(φ)σîz −sin(φ)σîy]|ψ(t0)i (15) 0.6 Prot+y-noise Yi Fit: [email protected] 0.5 yE-con+oiys-enoise with|ψ(t0)i = e−iπ2Jˆz(0)2|N/2,N/2ix. z-noise The phase sensitivity ∆2Oˆ achievable by repeating h i the above scheme during total time T is related to the 0 20 40 60 80 100 N signal variance ∆2Oˆ = Oˆ2 Oˆ 2 and given by: h i h i − h i |δω0| = rt1T (δhO∆ˆ2/Oˆδφi)2 [12]. Because Oˆ2 = 1, we just h i FIG. 4: (Color Online) Fidelity to create GHZ state as a function havetocalculate Oˆ =Tr[ρˆ(t )Oˆ]toevaluateδω . ofNforsystemsinthepresenceofσy noise: withoutMPM(green Experimentallyh,miagnetic fie0ld noise is one of t0he sources dashedline),withoutMPMbutwithspinecho(dot-dashedblueline), ofphasedecoherence. Assumingthatsuchdephasingmainly with MPM (red solid line) and with both MPM and echo (dotted takesplaceduringtheGHZgeneration,asduringtheRamsey black line). Wealso show thedegradation caused bypure dephas- inginanunprotectedsystem(longdashedpurpleline)forcompar- interrogation time the atoms are essentially freely evolving, isonpurposes. Forsimplicityweassumedinfinitecorrelationtime: usingEq.(5)onecanshowthatfortheunprotectedsystem f(ω) = Υδ(ω/χ),Υ = 0.1χ. The echo technique consisted of a perfectsuddenπpulsearoundthexdirectionatχt=π/4.Because the y components of the noise do not commute withJˆz2(0) the dy- hOî=hψxGHZ| [cos(φ)σîz −e−Γ(t0)sin(φ)σîy]|ψxGHZi namicswassolvednumerically. Fortheunprotected systemallthe Yi 2beNrbstyatNes=had10to.Obenctohnesoidtheererdhaannddtwheerheasdtrtioctliiomniotfthtehepadrytnicalmeincusmto- = 1[eiφ(1−e−Γ(t0))+e−iφ(1+e−Γ(t0))]N +h.c (16) 2 2 theMPMintheprotectedschemeallowedustoextendthecalcula- tiontolargerN values. Fortheσy noisethefidelitywithprotection Consequently the maximal phase resolution, achieved at doesnotbecomeN independent,butinsteadscalesase−0.043N0.44 φ =nπ(forintegern),canbeshowntobegivenby opt (seefittedreddots). Neverthelesswedogainafactoroforder√N withrespecttotheunprotectedsystem. Theplotalsoshowsthatthe δω = δω /G, (17) 0 opt 0 sh combinationofMPMwithspinechoprovidesthebestprotection. | | | | with G = ((N −1)e−2Γ(t0)+1) and |δω0|sh = √t1TN the shot noipse resolution. The factor G explains the strong theuseofspinechotechniquescanhelptofurtherreducethe limitationsintroducedbydecoherence.Ifthetimerequiredto effectofdecoherence. generatetheGHZstateissuchthatG 1(i.ewhenΓ(t ) > 0 ∼ ln[√N]), the phase accuracy is reduced to the classical shot IV. APPLICATIONSTOPRECISIONMEASUREMENTS noiseresolution. USINGTRAPPEDIONS However,ifinsteadHˆprot+Hˆz isusedfortheGHZgen- eration, G is replaced by ((N −1)e−2Γ(t0)/N +1). Due Recentexperiments[1,2]havegeneratedGHZstatesmade to the N times slower decpay rate of the atomic coherences, of up to six beryllium ions and used them to perform preci- the same preparationtime that leads to shot noise resolution sion measurements of ω . For the ideal GHZ state prepara- withoutprotection,canleadtoasensitivityatthelevelofthe 0 tion,thespectroscopyshouldleadtoHeisenberg-limitedres- fundamentalHeisenberglimitwithprotection. olution, δω N 1[11]. However,inpractice,evenforsix 0 − | |∝ ions,thephaseaccuracywassignificantlydegradedbydeco- herence. V. IMPLEMENTATIONOFTHEGAPPROTECTED The spectroscopy [1, 2] was realized by first creating HAMILTONIANINTRAPPEDIONS the desire GHZ state by applying to the initial polarized state, J = N/2,N/2 the unitary gate operation U = Wenowproceedtoreviewandcomplementtheimplemen- z N | i eiπ/2Jˆy(0)eiπ/2Jˆz(0)2e−iπ/2Jˆy(0). ThentheGHZ state was letto tation of the protected Hamiltonian, χ(Jˆ(0)2 Jˆz(0)2), pro- − to freely precess in the z direction for time t so each atom posedinRef.[14]. Consideralineartrapwithastringofions accumulated a phase difference φ = (ω ω )t (in a ref- withtworelevantinternallevels. Theionsareassumedtobe 0 − erence frame rotating with the frequency ω, the frequency cooled such that only the in-phase collective center of mass of the applied field). The phase difference was then de- oscillation of all ions is excited. The correspondingoscilla- coded by measuring the collapse probability into the states tion frequency is denoted by ν. The two internal levels of 6 the ions are coupled by a laser field with a slowly varying pictureofHˆ ,assumingthattheundesiredsecondandthird eff RabifrequenciesΩandwithfrequencyω =ω δ,beingδ termscanbecanceledbythetechniquesdescribedabove,and 1 o − thedetunningfromresonance.Assumingthatthefieldcouple treatthesmallnon-idealdeviationsbyperturbationtheory. all ions in the same way, we can describe the system by the Directcoupling HamiltonianHˆ = Hˆ0 +Hîn, where H0 = νaˆ†aˆ+ωoJˆz(0), • aˆ beingthe annihilationoperatorofthe quantizedoscillation Going from Eq.(18) to Eq.(19) the off-resonant term H = d mode.TheinteractionHamiltonianHînisgivenby ΩJˆ(0)eitδ+h.c.wasneglected.Thistermcorrespondtodirect + singleatomspinflipswithoutanyvibrationalexcitation. Hîn =ΩJˆ+(0)eiδteiη(aˆ†eiνt+aê−iνt)+h.c. (18) Changingto the interactionpictureof Heff and using the factthatH oscillates a muchhigherfrequencythatUˆ(t) = whereη isthe Lamb-Dickeparameter. Thedetuningδ is as- d eiHefftsothatthelattercanbetreatedasconstantintheinte- sumedtobelargecomparedtothelinewidthoftheresonance gralsusedintheDysonseries,onecanshowthat but sufficiently differentfrom the frequencyof the center os mass oscillation. As a consequence,the dominantprocesses are two-photon transitions leading to a simultaneous excita- Ω2 tionofpairsofions. F(to)=1− δ2 N2sin2(δto)+4Nsin4(δto/2)+... WefirstassumethattheiontrapisintheLamb-Dickelimit, (cid:0) (2(cid:1)1) i.e., that the ions are cooled sufficiently enough, such that The degradation of fidelity is a factor of N larger than the for all relevant excitation numbers n of the trap oscillation degradation caused by direct couplings in the standard real- (n+1)η2 1holds. Inthislimitonecanexpandtheexpo- ization of J(0)2 where (t ) = 1 NΩ2 sin2(δt ). There- nentin Eq.≪(18) to first orderin η. Confiningthe interest to fore it is imzportant forFtheoimplem−entaδt2ion of theoprotected time averageddynamicsovera periodmuchlongerthan any hamiltoniantouseweaklaserpowerortocontrolthesystem of the oscillations present in Hˆ , then the oscillatory terms parametersuchthatδt =2KπwithK aninteger. in 0 maybeneglectedandweareleftwithamoresimpleeffective HamiltonianRef.[15]: Lamb-Dickeapproximation • In Ref.[8] it has been shown that relaxing the Lamb-Dicke 2Ω2 approximation and including higher order terms results in H =χ(Jˆ(0)2 Jˆ(0)2)+ Jˆ(0)+Λ(2n+1)Jˆ(0) (19) eff − z δ z z an effectiveχn which dependson the vibrationalnumberof phonos in the collective mode: χ = χ[1 η2(2n+1)+ n where χ = 2νη2Ω2, Λ = χδ and n the number of phonons η4(5/4n2 +5/4n+1/2)]. As this effect i−s global, the gap δ2 ν2 ν in the vibration−almode. The first term in H is the desire doesnotprotectagainstitanditleadstoadegradationofthe eff protectedHamiltonian. The secondterm acts as an effective fidelitygivenby magnetic field which can be canceled by adding an external magnetic field or by echo techniques. The third term comes thefromtheacStarkshiftoftheatomiclevelsduetothelaser N(N 1)(π/2 χnto)2 −1/2 (t ) = P 1+ − − fields. IncontrasttothestandardschemeusedtocreateJˆ(0)2, F o n(cid:18) 4 (cid:19) z Xn wherethendependenceexactlycancels,hereitdoesnotand π2N(N 1)η4 ifnotcorrectedcancertainlydegradethefidelity. Thedegra- 1 − P (2n+1)2 (22) n ∼ − 32 dationcanbeshowntobegivenby: Xn Othervibrationalmodes • N2Λ2(2n+1)2t2 (to)= Pnexp[ o] (20) WithNionsinthetrap,assumingthatthetransversalpotential F − 8 Xn isstrongenoughtofrozenthetransversaldegreeoffreedom, onlytheNlongitudinalvibrationalmodesarerelevant.Sofar whereP istheinitialpopulationofthestatewithnphonons. n wehaveassumedthatonlythecollectivecenterofmassmo- In order to prevent this effect one has to cool the ions to tion is excited and neglectedother modes. If we include the thegroundstate,P = δ ,whichmightbefeasiblewiththe n n0 effect of other modes into account, the fidelity is decreased. state of the art technology or alternatively one can use spin The main sources of decoherence are :a) off resonant direct echotechniques. Forexampleif at time t /2 the sign of the o couplingstoothermodesandb)reductionofthecouplingto laserdetuningδischanged,thenthedifferentcomponentswill thecenterofmassmode,χ, duetothevibrationofthe other rotateintheoppositedirectionandatt theneteffectdueto o modes. Howeveralltheseeffectsarelocalandthegapener- theextrasecondandthirdtermsinHˆ willbecanceledout. eff geticallysuppressesthem. SofarwehaveusedtheLamb-Dickeandtherotating-wave approximation. Nowweperformamoredetailedanalysisof Spontaneousemission thevalidityoftheseapproximationsandestimatetheeffectof • deviations from the ideal situations in an actual experiment. Additionally a more fundamental source of decoherence To do that we follow Ref. [8] and change to the interaction arisesfromspontaneousemission effects. Intypicaliontrap 7 hermite conjugates and then use the projected equations of motiontosolveforσˆ˙j . ↑↓ eÒ D DDD σˆ˙j = FH +i[Hˆnoise,σˆj ] (29) ↑↓ ↑↓ 1 Hˆ (t) = [hs (t)σˆz +hs (t)σˆx+hs (t)σˆy(30) noise 2 jz j jx j jy j W n g ggg Xj 2 2 2222 g ggg 1111 W n where FH accounts for the the Hamiltonian part of the dy- 1 1 Ò namicsandwith hsjz(t) = Ωi∆1(fˆje† −fˆje)− Ωi∆2(fˆje† −fˆje), hsjx(t) = −Ωi∆2(fˆje† −fˆje)− Ωi∆↓1(fˆje† ↓−fˆje) and↑hsjy(t)↑= −Ω∆2(fˆje†+fˆje)+↓ Ω∆1(fˆ↓je†+fˆje). ↑ ↑ Ò From↓thepr↓eviousexp↑ression↑sonecanestimatethedegra- (cid:0) dation of the fidelity due to dephasing ( similar degradation of the fidelity is caused by x or y type of noise). In the adiabatic limit, i.e. ∆ Ω Ω ,γ ,γ , hs are indepen- ≫ 1 2 1 2 jz FIG.5: Ramantransitiontoathirdlevelwithatomicdecay. Here∆ dent stochastic white noise processes with zero mean and piRshatobhtieofndreerqteuusnoeinnncagineocsfe.t,h∆el=aseωrefi1el−dsνw1it=hfωreeq2u−encνi2es,ωane1d,2Ωf1r,o2martehelaosneer γauspto=cor(rγe1lΩat21∆i+o2nγ2fΩu22n)catinodnchosinzs(etq)husjezn(tτly)th=eγgasδpijcδa(ntn−otτp)rowteitcht against this broad band noise. From Eq.(6) they will case a degradationoftheGHZfidelityof experimentsthe and levelsare coupledthroughRa- | ↑i | ↓i man transitions to a third excited level e (see Fig.5). As- | i = 1 γ Nt (31) sumingtwophotonresonanceconditions,thatisω ω = sp o e1 e2 F − − ν ν = ω and ∆ = ω ν = ω ν where ∆ is 1 − 2 0 e1 − 1 e2 − 2 Inorderto reducethestrongdegradationsduetothistype thedetuningofthefieldsfromtheonephotonresonanceand of high-frequency decoherence processes one can increase ω andΩ arelaserfrequenciesandRabifrequenciesre- e1,2 1,2 the Raman-detuning[16] at the expense of slower evolution spectively, the Hamiltonian of the system in the appropriate whichinturnwillmakethesystemmoresusceptibletoother rotatingframecanbewrittenas kindoflocalnoise(eg. magneticfieldinhomogeneities). On theotherhandthelattercanbesuppressedbytheMPM. Hˆ = ∆ σî +Hˆ (23) s − ee Is From this analysis we conclude that overhead of imple- Xi menting Jˆ(0)2 Jˆ(0)2 instead of Jˆ(0)2 is mainly the addi- Hˆ = Ω (σî +σî )+Ω (σî +σî ) (24) − z z Is 1 e e 2 e e tional echo technique required to remove the n dependence Xi ↓ ↓ Xi ↑ ↑ ofH . Besidesthat,onaveragethesametypeofnon-ideal eff disturbancesarefoundinbothHamiltonianswith theadvan- whereσî = e e,σî = e andσî = e. The deeecoh|erieinihce| pr↓oecess|e↓siidihue| to sp↑oentan|e↑oiuiish e|mis- tageofJˆ(0)2−Jˆz(0)2 thatthegapprotectsthesystemagainst thoseofthemwhicharelocalincharacter. sion can be described by means of Heisenberg-Langevin equations[17]givenby: VI. IMPLEMENTATIONINOPTICALLATTICES σˆ˙j = =(iΩ1+ˆfj†e)σˆje−(iΩ2−ˆfje)σêj (25) ↑↓ ↓ ↑ ↑ ↓ A. Engineeringlong-rangeinteractions σˆ˙j = (i∆+Γ /2)σˆj (iΩ ˆfj )(σˆj σˆj ) e − e e− 2− e ee− ↑ ↑ ↑ ↑↑ +(iΩ ˆfj )σˆj (26) Up to now we have explored only the generation of an 1− e ↓ ↑↓ MPMviaisotropiclong-rangeinteractions. Inpractice,how- σˆ˙ej = (i∆−Γe/2)σêj +(iΩ1+ˆfj†e)(σêje−σˆj ) (27) ever, it is desirable to have a similar kind of protection gen- ↓ ↓ ↓ ↓↓ −(iΩ2+ˆfj†e)σˆj (28) eratedbysystemswithshortrangeinteractionssuchasthose ↑ ↑↓ providedbycoldatomsinopticallattices. Thesesystemsof- where Γ = γ + γ with γ and γ are decay rates form ferthepossibilitytodynamicallychangetheHamiltonianpa- e 1 2 1 2 e to and respectively and the noise operators fˆ rametersata levelunavailableinmoretraditionalcondensed | i | ↓i | ↑i have zero mean and are δ correlated[17]:hfˆje(t)fˆke†(t′)i = mlatattitceerssyysstteemmss.anWdecnaonwbeshuosewdhtoowroabnusMtlPyMgencaenrabteeNcr-epaatretdicilne ↓ ↓ γ1δ(t−t′)δj,k andhfˆje(t)fˆke†(t′)i=γ2δ(t−t′)δj,k. GHZstates. Inthelargephoton↑detuni↑nglimit∆ Ω1Ω2,γ1,γ2,one Weconsiderultracoldbosonicatomswithtworelevantin- canadiabaticallyeliminatetheoperators≫σˆj andσˆj andtheir ternalstates confinedin a an opticallattice. We will assume e e ↓ ↑ 8 that the lattice is loaded with one atom per site, and again 1 identifythetwopossiblestatesofeachsite,withtheeffective spin index σ = , respectively. For deep periodic potential ↑ ↓ 0.8 and low temperatures, the atoms are confined to the lowest BlochbandandthelowenergyHamiltonianisgivenby[18] 0.4 y0.6 HˆBH = −τhXi,jiσaˆ†σ,iaˆσ,j + 12Xjσ Uσσnˆσ,j(nˆσ,j −1) Fidelit0.4 \(cid:144)JNx 0.200 Π (cid:143)Χ(cid:143)(cid:143)t 2Π 3Π +U nˆ nˆ (32) X-0.2 σ,j σ,j ↑↓ 0.2 -0.4 Here aˆ are bosonic annihilation operators of a particle σ,j 0 oatvesritneejaraenstdnsetiagtehbσo,rsnˆ.σI,njE=q.(aˆ3†σ2,)jaˆthσe,jp,aarnadmtehteersτumisthhie,jtiunis- 0 5 10 15 20 (cid:143)(cid:143)(cid:143)(cid:143)(cid:143) nelingenergybetweenadjacentsites(whichwe assumespin Λ(cid:144)Χ independenti.e. spinindependentlattices)andandUσ,σ′ are thedifferenton-siteinteractionenergieswhichdependonthe FIG.6:(coloronline)FidelitytogenerateaGHZstatevsλ¯/χ¯.Inthe τscaatrteerfiunngclteinogntshobfettwheeelnattthiceeddifefeprtehn.tsWpeecaierse.iBnotetrheUstσedσ′iannda ignrseeetnwaendshsoowlidhJrêx(d0)c(otr)rie.sTpohnedbtloueλ¯d=ot-0d,a5sh,e1d0,,d2o0ttreedspbelcatcikve,ldya.sThehde unit filled lattice in the regime τ U where the system ≪ σσ′ plotsareobtainedbynumericalevolutionofEq.(33)forN =10. is deep in the Mott insulating phase [19, 20]. In this limit, to zero order in τ the ground state is multi-degenerate and correspondsto all possible spin configurationwith oneatom of Hˆ on it, which correspondsto Hˆ = χ Jˆ(0)2 λ¯N per site. A finite τ breaks the spin degeneracy. By includ- I P I e z − N 1 ing virtual particle-hole excitations one can derive an effec- with χe ≡ N4χ¯1, is effective and HI acts as a long ran−ge tive Hamiltonian that describe the spin dynamics within the Hamiltonian. H−ere we used the relation k=0[Jˆz(k)Jˆz(−k)] = oneatompersitesubspace[21]: Jˆz(0)2 + N2 , with the projectionPin6 tothe J = N/2 − N 1 4(N 1) P Hˆ =Hˆ +Hˆ = λ¯ σˆασˆα χ¯ σˆzσˆz. (33) subsp−ace. The−non zero projection of the latter term comes lat H I − <iX,j>,α i j − <Xi,j> i j from the fact that the operatorsJˆz(k)Jˆz(−k) and Jˆz(0)2 are not independentastheysatisfytheconstrain Nk=−01Jˆz(k)Jˆz(−k) = Here the coefficients are λ¯ = τ2/U and χ¯ = τ2(U−1 + N2/4. P U−1 2U−1). Forsimplicitywewil↑l↓nowrestrictthea↑n↑aly- InFig. 6 wecontrastthe dynamicalevolutionof asystem sis↓↓to−oned↑i↓mensionalsystemsandassumeperiodicboundary in the presence and absence of Hˆ assuming at time t = 0 H conditions. allthespinsarepolarizedinthexdirection. IfonlytheIsing termispresent,λ¯ =0,itinduceslocalphasefluctuationsthat Hˆ issphericallysymmetricandintermsofcollectivespin H leads to fast oscillationsin Jˆ(0) = N/2cos2[2χ¯t]. On the operatorsitcanbewrittenas x other hand, as the ratio λ¯/χ¯h incrieases, the isotropic interac- 4λ¯ 4λ¯ 2πk tion inhibits the fast oscillatory dynamicsand instead Jˆ(0) HˆH =−N Jˆ(0)2− N Jˆα(k)Jˆα(−k)cos(cid:18) N (cid:19) exhibitsslowcollapsesandrevivals.Forλ¯ χthedynhamxicis k=1.X..N−1,α (34) exactlyresembles the one inducedby Hˆz ≫and at χet = π/2 theinitialcoherentstateissqueezedintoaGHZ state. AlltheN+1fullysymmetricstateswithJ =N/2aredegen- erateandspanthegroundstateofHˆ . Hˆ isnotspherically H I symmetricbutwecanalsowriteitintermsofcollectiveoper- atorsas B. MPMinlatticesystems HÎ =−4Nχ¯Jˆz(0)2− 4Nχ¯k=1X...N 1Jˆz(k)Jˆz(−k)cos(cid:18)2Nπk(cid:19). HoHwˆHevearl,soHˆpHroivsidneostparsoetefcfeticotniveagaasinHsˆtprpohtasbeecdaeucsoehtehreenecne-. − (35) ergygapbetweentheMPMandtheexcitedstatesofHˆ van- H Iftheconditionχ¯ λ¯ issatisfied, whichcanbeengineered ishes in the thermodynamic limit as E λ¯/N2 . This is inthisatomicsyste≪msbymeansofaFeshbachresonance,the a drawback of the short range Hamiltogni→an for the purpose effectoftheIsingtermcanbestudiedbymeansofperturba- of fully protecting the ground states from long wave length tion theory. Assuming that at t = 0 the initial state is pre- excitations. Note howeverthat one dimensionalsystems are pared within the J = N/2 manifold, a perturbativeanalysis theworstscenarioasforhigherdimensionsthegapvanishes predictsthatfortimestsuchthatχ¯t<λ¯/χ¯,Hˆ confinesthe as N 2/d with d the dimensionality of the system. Never- H − dynamicstothegroundstatemanifoldandtransitionsoutside theless, the many bodyinteractions can still eliminate short- it can be neglected. As a consequence, only the projection wavelengthexcitationssinceinthelargeN limittheyremain 9 separatedbyafiniteenergygap,8λ¯. sysWteemqaugaanitnifsyt tHhêenevffbeyctiuvseinnegsstiomfethdeepMenPdMenttopeprrtoutrebcattitohne F(to)&eγl0at(t) 1+1Γ(to). (38) theory. For this analysis we restrict to the limit λ¯ χ¯ p wheretheIsingtermcanbetreatedasaneffectiveχ Jˆ(≫0)2 In Fig.7 we plot calculated from Eq.(38)as a function of e z F λ¯N Hamiltonian. In this limit a convenient basis to stud−y N. InthelatticethefidelityisdegradedasN growsbecause N 1 the gap decreasesand the generationtime increases with N. the−quantum dynamics is the collective spin basis. Assum- Moreover,anabruptdropofthefidelityoccursatthevalueof ing that at t = 0 the system lies within the J = N/2 N atwhichE =ω . manifold, the evolution of the matrix elements ρ g c MM˜ ≡ N/2,M˜ ρˆN/2,M canbewrittenas z z h | | i ρM,M˜(t)=ρM,M˜(0)eitχe(M2−M˜2)ei(θM−θM˜)e−12(γlMat−γlMa˜t) VII. NOISEANDDECOHERENCEINLATTICESYSTEMS (36) wherethe randomphase, givenbyEq.(10), characterizesthe dynamicsinducedbythenoisewithintheMPMandγM(t)= In the previoussection we used the effective Hamiltonian lat tdτ M eiτωJla,βt 2, takes into account the de- givenbyEq. (33)tostudytheGHZgenerationinlatticesys- J=N/2,β| 0 MJ,β | tems. Hereweperformamoredetailedanalysisofitsvalidity pPleti6on of thRe J = N/2 levels due to transition matrix ele- andestimatetheeffectofdeviationsfromtheidealsituations ments with states outside the symmetric manifold: M = MJ,β in an actual experiment. For this analysis we restrict to the tzihnNg2s,.MUp|Hˆtoentvh|iJs,pMoi,nβtithz.eωeJxla,pβtreasresitohnesraersepesctrtuivcetuernaellrygyidsepnlitti-- tliivmeitχλ¯Jˆ≫(0)2χ¯whλ¯eNretHheamIsiilntgontiearnm. canbetreatedasaneffec- caltotheonesobtainedforlongrangeinteractions. Thedif- e z − N−1 ferenceappearsintheevaluationofγ . IncontrasttoHˆ , lat prot not only the excitation frequenciesωlat are not degenerated J,β butalsotheybecomesmallerasN isincreased. Asa conse- A. Particle-holeexcitations quenceEq. (12)isreplacedbythefollowingequationforthe latticesystem DerivingEq. (33)fromtheBose-HubbardHamiltonianwe N2 4M2 N−1 ωc sin(t(ω ∆Ek)/2) 2 onlyincludedvirtual-particlehole excitation. However,dur- γM(t) − f dω − (37)ing the time evolution real transitions from singly to doubly lat ≈ N(N 1) Z (cid:18) ω ∆Ek (cid:19) − kX=1 0 − occupiedstatescantakeplaceandtheydegradethefidelity. Toaccountfortheseeffects,wewritethemanybodywave withEktheexcitationenergiesofthestatesthatbelongtothe functionas Ψ(t) = C ψ + B φ ,where ψ J = N/2 1 manifold given by ∆Ek = 8λ¯sin2(πk/N), | i n n| ni m m| mi | ni with k = 1−,...,N 1 [22]. FromEq.(37) we can estimate spantheHilbertspacPe withoneatoPmpersite and|φmispan − thesubspacewithoneparticleandoneholeadjacenttoeach thedegradationofthefidelityduetophasedecoherenceas: otherandN 2singlyoccupiedsites. Thelatterarethestates − that directly coupled to ψ throughtunneling. Solving the n | i timedependentSchro¨dingerequationfromtheBoseHubbard 1 Hamiltonia, using the assumption that U U and that σ,σ′ ≈ at time t = 0 no doubly occupied states are populated one 0.8 obtains y 0.6 elit Fid 0.4 iC˙n = ψn Hˆlatt ψk (1 e−iUt)Ck (39) h | | i − Xk 0.2 Here we also assumed that C change at a rate much n 0 smaller than U and treated{the}m as constants during the σ,σ′ 25 50 75 100 125 150 time integration. Eq.(39) yieldsthe followinglostof fidelity N duetorealparticleholeexcitations: FIG.7: InthepresenceofphasedecoherencethefidelityoftheGHZ 4χ¯ statepreparationinthelatticeisdegradedasN growsbecauseinthe (to) 1 sin2(Uto/2). (40) F ≈ − U latticethegapdecreasesandthegenerationtimeincreaseswithN. Inthisplotweassumedthelimitχ¯ λ¯ whereEq. (38)holdsand rememberingthatχ t =π/2.Aslongasχ¯/U 1,wecon- usedasystemwithωc =χ¯,λ¯=100≪χ¯andΓ=0.01χ¯.AtN =90, cludethatparticle-heoloeexcitationsdonotsignifi≪cantlyaffect ∆Eg =ωcanditexplainsthedropof forN >90. theGHZgeneration. F 10 B. Magneticconfinement encountered during their experimental realization. On the otherhandlattice-basedGHZstategenerationfacesthescal- InEq. (32)weassumedatranslationallyinvariantsystem. abilityproblemduetothefactthatthegapdecreasesandthe However in most of the experimentsan additional quadratic generationtimeincreaseswithincreasingN. magneticconfinementisusedtocollecttheatoms.Actuallyis VIII. CONCLUSIONS duetothisquadraticpotentialthataunitfilledMottinsulator has been experimentallyrealized. In its absence it would be Wehaveinthispaperevaluatedthepossibilityforarobust difficult to create a unit filled Mott insulator as in an homo- preparationofmulti-particleGHZentangledstatesoftrapped geneoussystemitonlytakesplacewhenthenumberofatoms ionsorcoldatomsinopticallatticebygeneratingadecoher- is exactly equal to the number of lattice sites. A drawback ence free multi-level manifold corresponding to the ground ofthemagneticconfinementisthatitgeneratesalwayssuper- levels of properlydesigned Hamiltonians. The MPM is iso- fluid regions at the edge of the cloud, so only a fraction of latedfromtherestoftheHilbertspacebyanenergygapwhich thetotaltrappedatomslocatedatthetrapcenterhastobese- energeticallysuppressesanylocaldecoherenceprocesses.We lectedasthequantumregister.Assumingweworkonthisunit havepresentedanalyticalestimatesforthefidelityoftheGHZ filledMottInsulatorsubspace,herewequantifytheeffectof preparation. themagneticpotentialintheGHZgenerationintheλ¯/χ¯ 1 In trapped ions we demonstrated that the fidelity can be ≫ limit. significantly better than the one achievable without any gap The magnetic confinement is accounted for by adding a protectionandthereforethatourschemeisin thepositionto termW j2nˆ intheBoseHubbardHamiltonian.W = improvethespectroscopyresolutionincurrentRamseyspec- j,σ σ,j 1/2mω2Pa2 with m the atom mass, ω the frequencyof the troscopyexperiments. T L T external trapping potential and a the lattice spacing. This We also showed that cold atoms in optical lattices inter- L term modifies the global coupling constants λ¯ and χ¯ when actingviashortrangeinteractionscanbeutilizedtoengineer the effective Hamiltonian is derived and make them site de- long range interactionswhich in turn can be used for gener- pUei−n,1de−nt,2Uλ¯i−,→1).λ¯HWi er≡eUτi,2σ/σUe′i=,↑↓Uaσnσd′/χ¯(U→σ2σ′χ¯−i W≡2τ(22(iUe+i−,1↑1↑)2+). anintoinnthg-eidsmeeaaslnyycs-otbenomddistyiiosenntshtaaennsgdcleacmloabneniclltiu.tydWeadesttchhaaeltcMuthlPaetMemdpatirhnoetreeecfstftieroicncttsdiooen-f Ass↓u↓ming th↑a↓t the gradient of the external potential is weak e e e gradeswithincreasingN. comparedtotheonsiteinteractionenergy,asisingeneralthe Thescalabilitycertainlylimitstheuseoflatticesystemsfor caseforcurrentexperiments,theeffectiveHamiltonianinthe massive entanglement generation, however it is not a prob- presenceofthemagnetictrapbecomes lemforrecentquasi-onedimensionalexperiments[23]where an array of 1D tubes with an average of 18 atoms per tube HˆW = Hˆ +HˆT (41) has been realized. In such systems therefore it should be lat lat 1 possible to create few-particlecollectiveentangledstates us- Hˆ1T = − Ti~σi·~σj (42) ingourschemeandtoperformproof-of-principleexperiments <Xi,j> demonstratingtheimprovementofspectroscopicsensitivity. τ2W2(2i+1)2 We emphasize that, even though we have limited the dis- T = (43) i − U3 cussion to ensembles of spin S = 1/2 particles, the MPM ideas can be straightforwardly generalized to systems com- ThecorrectionsonthefidelityoftheGHZstateintroduced posed of higher spin atoms. Besides entanglement genera- byHˆT canbeestimatedbycalculatingtheeffectiveprojection tion,theMPMmighthavealsoimportantapplicationsforthe 1 of it on the MPM: HˆT . As the latter is just proportional implementationofgoodstoragememoriesusingforexample P 1 to the identity matrix I, HˆT = T I, it effects is just a nuclearspinensemblesinsolidstate[24]orphotons[25]. P 1 i i global phase and it does not causePany main degradation of thefidelity. Similarlyanyotherperturbationinducedbylocal fluctuationsinthemagneticfieldorthelasersusedtogenerate IX. ACKNOWLEDGEMENTS thelatticebecomeirrelevantthankstotheMPM. From this analysis we conclude that except from sponta- This work was supported by ITAMP, NSF (Career Pro- neousemission or heating mechanismsthe MPM effectively gram), AFOSR, ONR MURI and the David and Lucille protectslattice systems againstcommonnon-idealsituations PackardFoundation. [1] D.Leibfriedetal. Science304,1476(2004). [5] A.M.Rey,L.Liang,M.Fleischhauer,E.DemlerandM.Lukin, [2] D.Leibfriedetal. Nature438,639(2005). preprint:cond-matt:0703108. [3] M.NielsenandI.ChuangQuantumComputationandQuantum [6] D.M.Greenberger,M.A.Horne,A.Shimony,A.ZeilingerAm. Communication,CambridgeUniv.Press,Cambridge(2000). J.Phys.58,1131(1990). [4] J.Mod.Opt47,127(2000). [7] K.MølmerandA.Sørensen Phys.Rev.Lett.82,1835(1999)