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Coherence and Rydberg blockade of atomic ensemble qubits M. Ebert, M. Kwon, T. G. Walker, and M. Saffman ∗ Department of Physics, University of Wisconsin, 1150 University Avenue, Madison, Wisconsin 53706, USA (Dated: July 21, 2015) We demonstrate |Wi state encoding of multi-atom ensemble qubits. Using optically trapped Rb atomstheT coherencetimeis2.6(3)msforN¯ =7.6atomsandscalesapproximatelyinverselywith 2 thenumberofatoms. StrongRydbergblockadebetweentwoensemblequbitsisdemonstratedwith a fidelityof 0.89(1) and a fidelity of ∼1.0 when postselected on control ensemble excitation. These results are a significant step towards deterministic entanglement of atomic ensembles. 5 PACSnumbers: 03.67.-a,42.50.Dv,32.80.Rm 1 0 2 Qubits encoded in hyperfine states of neutral atoms are l are a promising approachfor scalable implementation of u 1 N J quantuminformationprocessing[1]. Whileaqubitcanbe ¯0 = 01...0N , ¯1 = 0102...1j...0N , (1) 8 encoded in a pair of ground states of a single atom, it is | i | i | i √N Xj=1| i also possible to encode a qubit, or even multiple qubits, 1 in an N atom ensemble by using Rydberg blockade to where 0j and 1j aretwogroundstatesofthejth atom ] enforce single excitation of one of the qubit states[2, 3]. in an N| aitom s|amiple[15]. The state ¯1 , which is a sym- h | i Ensemblequbitshaveseveralinterestingfeaturesincom- metricsuperpositionofoneoftheN atomsbeingexcited, p parison to single atom qubits. Using an array of traps it is commonly referred to as a W state in the quantum - | i t is simpler to prepare many ensemble qubits with N 1 information literature. n ≥ a for each ensemble, than it is to prepare an array with Gate protocols for ensemble qubits differ slightly from u exactly one atom in each trap which remains an out- the single atom qubit case [2, 16] as all operations must q standing challenge[4–6]. In addition, a W state en- useblockadetoprohibitmulti-atomexcitation. Gateop- [ | i semble qubit encoding is maximally robust against loss erations are performed via the collective, singly excited 4 of a single atom[7], which can be remedied with er- Rydberg state v ror correction protocols[8], while atom loss is a critical 3 error for single atom qubits. Furthermore an ensem- 1 N 8 r¯ = 0 0 ...r ...0 , 1 2 j N 0 ble encoding facilitates strong coupling between atoms | i √N Xj=1| i 4 and light, an essential ingredient for quantum network- 0 ing protocols[9] and atomic control of photonic interac- where r isthe Rydbergstate ofthe jth atom. Asingle j 1. tions in Rydberg blockaded ensembles[10]. As the atom- qubit r|otaition R(θ,φ) with area θ and phase φ between 0 light coupling strength growswith the number of atoms, ensemblestates ¯0 , ¯1 is implementedasthe threepulse 5 recent experiments[10],[11] and theory proposals[12] are sequence ¯1 Ω| ir¯|,ir¯ ΩN ¯0 , r¯ Ω ¯1 . Note 1 based on ensembles with N > 100. We are focused here | i −→π | i | i ←R−(−θ−,φ→) | i | i −→π | i v: on studying the physics of ensembles for computational that the coupling strength between states ¯1 , r¯ is the | i | i i qubits and therefore work with smaller ensembles with singleatomRabifrequencyΩwhilethecouplingbetween X up to N 10 atoms. ¯0 , r¯ is at the collective Rabi frequency ΩN = √NΩ. r ∼ | i | i a SinceΩN dependsonN,theone-qubitgatepulselengths depend on the number of atoms. A C gate between In this letter we demonstrate and study the coherence Z control and target ensembles c,t is implemented as the andinteractions ofatomic ensemble qubits. We measure the T2 coherencetime ofensemble qubits achievinga ra- three pulse sequence |¯1ic −→Ωπ |r¯ic, |¯1it ←2→Ωπ |r¯it, |r¯ic −→Ωπ tio of coherence time to single qubit π rotation time of ¯1 . The C gate pulses do not depend on the number c Z 2600. We furthermore proceed to demonstrate strong o|fiatoms. The N dependence of the one-qubit gates can ∼ Rydberg blockade between two, spatially separated en- bestronglysuppressedusingadiabaticpulsesequencesso semble qubits. Together with the recent demonstration thathighfidelitygateoperationsarepossiblewithsmall, ofentanglementbetweenaRydbergexcitedensembleand but unknown values of N[17]. a propagating photon[13] these results establish a path The experimental setting is as described in [18]. In towards both local and remote entanglement of arrays brief we prepare a cold sample of 87Rb atoms in a ofensemblequbits,whichwillenableenhancedquantum magneto-optical trap (MOT) and then load a variable repeater architectures[14]. number of atoms into optical dipole traps. The dipole traps shown in Fig. 1 are formed by focusing 1064 nm The computationalbasis states ofthe ensemble qubits lighttowaists(1/e2intensityradii)of3.0µm. Theatoms 2 a) 795 nm optical x b) m=5/2 0.5 1.0 pumping z, Bz nd5/2 j ak 0.8 ccccccoooooonnnnnntttrrroooll 480 nm pe0.4 P|>100..46 5p3/2 σ+ ak-0.3 0.2 8.7 mµ 777778888800000 nnnnmmmm 444448888800000 nnmm 100R GamHzan R782y0d G bnHemzrg e pe0.2 0.00 2 4 ti6me (8µs)10 12 14 d m tttttaaaaarrrrrggggeeeetttt 780 nσm+ 0 0 0 1 tu0.1 3 µ 78 78 pli 8. 5s f=2 |0> m0.0 tttttaaaaarrrrrggggeeeetttt 1/2 f=1 m=0 |1> a 0 Ra1msey g2ap tim3e (ms)4 5 FIG. 1. (color online) Experimental geometry a) and transi- FIG. 2. (color online) Ramsey interference measurement of tionsusedforqubitcontrolb). TheRamanlightisonlyused qubitcoherenceforN¯ =7.6. Thepeak-peakamplitudeofthe for preparation of product states, as discussed in connection oscillation asafunctionofthegaptimegivesT =2.6(3) ms. 2 with Fig. 3. Thecirclesaredatapointswith±σerrorbarsandthedashed and solid lines are fits to the functions v (t),v (t) defined in a b thetext. ThegaptimeisthetimetbetweentheR (π)pulses 1 are cooled to a temperature of 150 µK in 1-1.5 mK inEq. (2). Alldatahavebeencorrectedfor∼1.5%probabil- deep optical potentials. This giv∼es approximately Gaus- ity peratom of theblow away giving an unwantedtransition from |0i → |1i. The inset shows the Ramsey oscillations for sian shaped density distributions with typical standard gap times of 0 (solid line), 0.5 ms (dashed line), and 2.5 ms deviations σ = 0.7 µm perpendicular to the long trap (dashed-dottedline). ⊥ axis and σ = 7 µm parallel to the long axis. The es- z timated density at trap center is n/N = 5 1016 m 3. − × We apply a bias magnetic field along the trap axis of atomsintooneoftheopticaltraps. Thenumberofatoms Bz =0.24 mT and optically pump into 0 5s1/2,f = loaded for each measurement follows a Poisson distribu- | i≡| 2,mf =0 using π polarized 795 nm light resonant with tionwithmeanN¯. Eachmeasurementstartswithoptical i 5s1/2,f =2 5p1/2,f =2 and 780 nm repump light pumping into ¯0 followed by the pulse sequence | i→| i | i resonant with 5s ,f =1 5p ,f =2 . 1/2 3/2 Rydberg exc|itation coupil→ing| ¯0 , r¯ is pierformed by ψ =R1(π)R0(π/2)R1(π)G(t)R1(π)R0(π/2)¯0 . (2) | i | i | i | i off-resonant two-photon transitions via 5p [19] using 3/2 HereR (θ)isapulseofareaθ betweenstates ¯0 , r¯ and counter-propagating 7800 and 480 nm light. With σ+ R (θ) i0s a pulse of area θ between states ¯1|,ir¯|.iThe pstoaltaerizratio=n fnodr5/b2o,tmhjb=eam5/s2wewchoicuhpliestsoelethcteedRywdibtherag fir1stR0(π/2)pulsecreatesanequalsuperpos|itiio|ni|¯0i√+2|r¯i. iBsz1=|0.i357s1m/|2T,fb=ias1,fimeldf.=T0he.i oCtohueprliqnugbbitetgwroeuennd¯1st,art¯e wTahiits aisgtahpentimmaeptpeddestcorib|¯0ei√d+2|b¯1iywanithopaerRa1t(oπr)Gp(ut)ls,em, wape | i≡| i | i | i is performedwith7801 and480nmlightwhere7800 and ¯1 r¯ with a R1(π) pulse, and then perform another 7801 have the same propagation vector and polarization |π/i2→pu|lsie between ¯0 , r¯ . Finally, any population left but a frequency difference of 6.8 GHz corresponding to in r¯ is mapped ba|cki t|oi ¯1 with another R (π) pulse. 1 the 87Rb f = 1 f = 2 clock frequency. In the experi- Ato|mis in state 0 are the|nipushed out of the trap us- mentsreportedb↔elowweusedRydberglevels97d5/2 and ing unbalanced r|adiiationpressure from a beam resonant 111d5/2. In both cases strong blockade was observed in with 5s1/2,f = 2 5p3/2,f = 3 while the dipole individual ensembles with no evidence for double excita- trapl|ightis choppeid→on|andoff. For tihe push out step a tion of the logical ¯1 state[18]. While we do not observe biasfield is appliedalongx the narrowaxisof the dipole doubleexcitationo|fi¯1 ,experimentswithtwoensembles traps, and the circularly polarized push out beam prop- | i do show evidence for double excitation of the Rydberg agates along x. This is followed by a measurement of state r¯ , which plays a role in limiting the fidelity with the number of atoms remaining in the dipole trap. The which|wie can prepare the ¯1 state. resultingdataareshowninFig. 2. Theamplitude ofthe | i We proceed to demonstrate the coherence of the en- Ramsey interference at short gap times is limited by the semble states of Eq. (1) using Ramsey interferometry. W state preparationfidelity ofabout50%forthe atom | i The amplitude of the Ramsey signal is used to quantify numberusedinthefigure. ThefidelitiesoftheR (π)and 0 the presence of N atom entanglement in the ensemble, R (π) pulses used to prepare W are estimated to each 1 | i ashasbeenobservedinotherrecentexperiments[20,21]. be at least 90% on the basis of previous experiments[18] Detailsoftheanalysisshowingthat82 6%oftheatoms and the strong inter-ensemble blockade effect we report ± participate in the entangled W state are presented in below. We attribute the limited W state preparation the supplemental material[22|]. iWe load 3 < N¯ < 10 fidelity to Rydberg dephasing, as| wiill be discussed in 3 0.6 a) 5 4 0.4 ) s m 3 ( 2 T2 >t 10.2 P| 1 0 0 0 2 4 6 8 10 12 0.1 b) mean atom number N FIG.3. (coloronline)Dependenceofensemblecoherencetime 0 on N¯ for |Wi states (red circles) and product states (blue 0 π/2 π 3π/2 θ squares). The horizontal error bars represent the bounds for atom numbermeasurements interleaved between Ramsey FIG. 4. (color online) Ensemble to ensemble blockade for measurements. The open symbols are for preselected N = 1 N¯ = 9.9,N¯ = 6.2. a) Probability of preparing |¯1i without c t t states. The dashed lines are a guide to theeye. blockade(redcircles, solidline)andwithblockade(blackcir- cles,dashedline). Thesolidlineisafittoadecayingsinusoid function from [18]. The dashed line is the same fit scaled by 11%. b)Blockadedatapostselectedondetectionof|¯1i . The the following. Periodicfluorescencemeasurementsofthe c dashed-dotted lines in both panels show the expected signal mean atom number (described in the supplemental ma- duetostateleakageduringblow-awayinthecontrolandtar- terialto[18])bounddrifts to6.7<N¯ <9,duringthe 12 get regions. hour measurement of this data set. The principal sources of decoherence in this experi- ment are expected to be magnetic noise, motional de- Thislinewidthisconsistentwiththeobservedshorterco- phasing, and atomic collisions[23]. For small atom num- herence time of the W states compared to the product | i bers and low collision rates we fit the Ramsey signal to states which are referenced to the Raman laser beatnote the expression[24] v (t,T )=v /[1+(e2/3 1)( t )2]3/2 which is in turn locked to a stable 6.8 GHz microwave b 2 0 − T2 and in the collision dominated regime we use a Gaus- oscillator. We anticipate that compensated optical traps sian form v (t) = v e (t/T2)2 where v is the amplitude anddynamicaldecoupling methods together with anop- a 0 − 0 at t = 0. Both functional forms give the same T time tical lattice to reduce collisional effects can be used to 2 within our experimental error bars of T =2.6 0.3 ms. greatly extend these coherence times[27]. 2 The π pulse times were 0.24 µs for ¯0 r¯ , 0.±06 µs for To demonstrate ensemble-ensemble blockade we load thegapbetweenpulses,and0.68µs|foir→r¯ | i ¯1 givinga atoms into control (c) and target (t) dipole traps, opti- | i→| i coherencetoR(π)gatetimeratioofapproximately2600. cally pump into ¯0 ¯0 and apply one of two sequences. c t To further clarify the sensitivity to collisional dephas- Preparation of a| isu|pierposition of ¯0 and ¯1 in the icnlugdFinigg.th3e cshasoewosftNhe=m1eaFsoucrkedstTat2esfowrhdiciffhearernetseN¯le,ctiend- tRa1r,gt(eπt)Rqu0b,ti(tθ)is¯0ecff¯0ectt.ed Tbyhisthsehosueqldu|einidceealUlya||¯0leiiacv|¯0eitth=e using anadditional fluorescencemeasurementbefore the qubits in the j|oiint| sitate ¯0 c[cos(θ/2)¯0 t sin(θ/2)¯1 t] Rtoatmhsee1y/sNeq2usecnaclien[1g8o]b.sWerevesdeefothraGtHTZ2 ∼sta1t/eNs¯[2,5in].cTohnetroabst- wsinit2h(θt/h2e),parsobisasbhiloitwynoifn|pFriiegp.a4rian)g. W|¯1i|etisepe−rotphoerteixopneac|ltietdo seenrcveeddo1m/Ni¯nastceadlinbgyfocorl|lWisiionstsastiensciesethxepeccotleldisifoonrdraecteohpeerr- t0i.m52e, dceopnesnisdteennctewwitihthoaurpeeaarklieprrosbtuabdiylitoyf oFfocPk|¯1is,ttat∼e atom is proportional to N¯. For comparison, the T time preparation[18]. 2 wasalsomeasuredforproductstates ψ (0 i1 ) N. Rydberg blockade between two ensembles ⊗ Thesestateswerepreparedusingatw|o-ifr∼equ|eni−cy|Riaman is observed with the sequence U ¯0 ¯0 = b c t lasercoupling 0 and 1 viathe 5p level[26]asshown R (π)R (π)R (θ)R (π)¯0 ¯0 . He|rei |wie have 3/2 1,c 1,t 0,t 0,c c t inFig. 1. Com|pairison|oifthe ¯1 (W state)andproduct usedstate ¯0 ofthe controle|nsie|mibleto block the target | i | i | i state coherence data suggests that for N & 5 the coher- transfer with the final R (π) pulse ideally leaving 1,c encetimeislimitedbycollisions. ForN¯ <5aswellasfor the qubits in the joint state ¯1 ¯0 . The data in Fig. c t | i | i theN =1Fockstatedatatheproductstatesshowlonger 4a) show a ratio of P¯1 ,t(Ub)/P¯1 ,t(Ua) = 0.11(1), coherence time. The coherence of the W states is mea- i.e. a blockade fidelity o|fi0.89. T|hiis implies that the sured by comparison with a phase ref|ereince defined by successprobabilityofthe transitionR (π)¯0 r¯ is 0,c c c the beatnote of the 780 and 780 Rydberg lasers which bounded below by the ¯1 population ratio| fior→th|eitwo 0 1 t | i have a measured beatnote linewidth of 100 Hz FWHM. sequences. We infer that at least one atom is excited to 4 the Rydberg state r c with probability 0.89(1). 1.2 a) 1.0 b) | i ≥ 1.0 0.8 As a further check on the inter-site blockade fidelity, qevueenntcsewUhbearerethpeosctosnetlreocltesdit.eTenhdesobinsesrtvaetde p|¯1oisct-asfetleerctseed- P|>1c000...468 θdP/d|>1c000...246 target population is shown in Figure 4b), along with the 0.2 0.0 expectedblow-awayleakagerateofthecontrolandtarget 0.00.0 0.5 1.0 1.5 2.0 2.5 -1.0 0.0 1.0 2.0 siteswhichismeasuredtobe 0.2%/atom. Fromthedata θ/π log10(F) itcanbeseenthatthepost-selectedresultsareconsistent FIG.5. (coloronline) Probabilityof preparingstate|¯1i asa c with perfect inter-site blockade. function of the target ensemble pulse area θ. a) Probability The observedhighblockadefidelity exceedsthatorigi- for several parameter sets: (111d5/2, R = 8.3 and 8.7 µm) (reddiamonds),(97d ,R=8.3and8.7µm)(greencircles), nallyachievedinexperimentswithsingleatomqubits[28, 5/2 (97d , R = 17 µm) (yellow squares). The data has been 29],andiscertainlysufficientto createentanglementbe- 5/2 normalized to 1 at θ = 0 for clarity, with typical success tweenensemblequbits. Whathassofarlimitedademon- probability 40-60%. b) Comparison of the slope of the data stration of deterministic entanglement is the relatively in panel(a) with thescaling parameter F from Eq. (3). The low probability of up to 62% [18] with which the ensem- color markers are thesame as in panel a). blestate ¯1 canbeprepared. Inordertogaininsightinto | i what is limiting the state preparation fidelity we looked for signatures of Rydberg-Rydberg interactions concur- tors favor atom pairs separated along z[30]. These con- rently with strong blockade. Ideally the probability of siderations imply that lack of perfect blockade leading preparing ¯1 with sequence U , should be independent to long range Rydberg-Rydberg interactions in a single c b | i of the pulse areaθ applied to the targetensemble. How- ensemble only partially explains the observed maximum ever a clear dependence on θ can be seen in Fig. 5a). of P¯1 = 0.62 [18]. Another candidate explanation is We believe this effect is due to long range interactions, very|sitrong interactions at short range in a single en- where the amplitude for Rydberg atom excitation in the semblewhichmix levelstogetherandopenanti-blockade target site is sufficiently blockaded to prevent it from resonance channels[31]. The doubly excited molecular making the transfer to ¯1 with any significant proba- energy structure becomes difficult to calculate with con- t | i bility, yet the target ensemble Rydberg excitation still fidence at short range, with many molecular potentials interacts with the control ensemble strongly enough to near resonant[32]. For our typical Rydberg state 97d 5/2 disrupt the control ensemble state transfer. A similar thischaracteristicseparationis 5µm,andfora6atom ∼ situation of partial blockade together with decoherence samplewithourensemblespatialdistributionsanaverage of multi-atom ground-Rydberg Rabi oscillations was re- of 7 atompairs out of 15 haveR<5 µm. We conjecture ported earlier in [19]. that the strong, short range interactions give an ampli- A two-atom Rydberg interaction effect should scale tudefordoubleexcitation,resultinginRydberg-Rydberg withtheRydbergdoubleexcitationprobability,i.e. P interactionswhichdephasetheground-Rydbergrotations 2 Ω2 /B2,whereBistheensemblemeanblockadeshift[30∝]. needed for state preparation, thereby limiting the prob- N¯ Tocheckthis,weextracttheslopesfromlinearfitstothe ability of preparing the ensemble ¯1 state. A related re- | i P¯1 c(θ) data for small θ and compare to the scaling pa- ductionofthefidelityofRydbergmediatedatom-photon ra|mieter couplingindenseensemblesduetoRydberg-groundstate interactions has also been observed[11]. F =Ω2 (n/n0)12 −2 P . (3) In conclusion, we have demonstrated the coherence of N¯t(cid:20)(R/R )6(cid:21) ∝ double 0 ensemble qubit basis states. The coherence time scales Here n is the Rydberg principalquantumnumber and R approximately inversely with the number of atoms, but is the site - site separation. The larger F is for a given is still several ms and 2600 times longer than our char- setofparameters,the strongerthe Rydberg-Rydbergin- acteristic gate time for N 10. Additionally we have ∼ teraction, and thus the larger the slope of dP¯1 c(θ)/dθ. demonstrated inter-ensemble blockade with a fidelity of Indeed, this is the behavior we observe,as show| in in Fig. 0.89 and 1.0 when post-selecting on control ensem- ∼ 5b), for a range of N¯, R, and n. ble excitation. We identified Rydberg-Rydberg interac- This interaction effect hints at the possible mecha- tionsfromweakdoubleexcitations,eitheratlongorshort nism responsible for the observed reduction in the prob- range,asapossiblemechanismlimitingthefidelityofen- ability P¯1 of preparing the collective qubit state in a semblestatepreparation. Future worktowardsensemble single en|seimble. The spatial extent of one ensemble is entanglementandquantumcomputationwillexplorethe 2σ = 14 µm giving a length scale in between the use of a background optical lattice to better localize the z ∼ lowertwodatasetsinFig. 5a). Theintra-ensembleRyd- ensembles while limiting uncontrolled short range inter- berg interactions are significantly stronger than between actions. atoms located in different ensembles at the same sepa- ThisworkwasfundedbyNSFgrantPHY-1104531and ration because the dipole-dipole interaction angular fac- the AFOSR Quantum Memories MURI. 5 [16] D. Jaksch, J. I. Cirac, P. Zoller, S. L. Rolston, R. 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Supplementary Material The matrices onthe upper andlowerdiagonals couple α states with excitation numbers differing by 1, n k ± | i ↔ n 1 with coupling strength α defined by j | ± i MULTIPARTITE W-STATE ENTANGLEMENT VERIFICATION A(n±1,n) j,k =αjk|(n±1)jihnk| (cid:2) (cid:3) = (n 1) (n 1) Aˆn n , (7) j j k k In order to demonstrate multipartite entanglement it | ± ih ± | | ih | is necessary to show that the results obtained in a mea- where surement cannot be reproduced with a separable state. N Thus we require that the N-particle state in question Aˆ= α Sˆ(m) (8) ψN satisfies m x | i mX=1 |ψNi=6 |ψAKi⊗|ψBN−Ki, (4) andαmisthestrengthofthelight-atomcouplingatatom m. In an ideal Rydberg blockaded ensemble states with for any K in the range N/2 K < N. In this sup- n > 1 are not excited and all α are equal. Departures ≤ m plemental material we evaluate the observed signatures fromtheidealcaseareaccountedforbyallowingforatom of W-state entanglement. These signatures include the specific α and double excitations are included by trun- m √N¯-enhancementofthe Rabifrequencybetween ¯0 and cating the basis at n=2 and adding the doubly excited | i r¯ , and the amplitude of the Ramsey oscillations. interaction energies to ∆ . | i 2 A strong blockade shift, δ(n=2) = δ α reduces m dd m ≫ theavailableHilbertspacefortheproblemton= 0,1 , { } Collective Rabi Frequency Enhancement and H becomes: int 0 α α α The interaction of an ensemble with a light field can 1 2 N ··· be written in the basis of individual atom excitations α1 δ1(1) 0 ··· 0  Ht|{ri0idn,it1a}gd(oe1n)sicar⊗libm|i{na0tg,ri1tx}h.(e2T)eihv⊗eolbu..at.|si{ois0n,s1toa}ft(eNtsh)eia.rseTydshteeenmoHtaeimsdiaaltsobnnloiacnk, Hint =α...2 0... δ2(1) ... 0...  (9) where 0 ≤ n ≤ N is the eigenvalue of the excita|tikoin αN 0 0 ··· δN(1) number operator ˆ = N Sˆ(k) +N/2, and the index k labels the degenNeratePeigke=n1staztes, e.g. 11 = 10 0 , The detunings δm(1) are nominally 0, so it makes sense 1 = 01 0 , etc. . Here Sˆ(k) = 1σˆ(k)| isithe|effe·c·t·ivie to treat the δ(1) entries as a perturbation. Under the | 2i | ··· i z 2 z condition of perfect blockade and no detuning, the en- spin operator for atom k along z. In this basis H is int ergy eigenstates of H = A are the dressed states given by: int 1 (¯0 ¯1 ) with total angular momentum J = N/2 √2 | i±| i H =A+∆= and N 1 orthogonal states with total angular mo- int mentum−J = (N/2 1): 1 (¯0 ¯1 ), (¯1) , where ∆0 A(0,1) 0 0 − {√2 | i±| i | ⊥i} AT(0,1) ∆1 A(1,2) ···  |¯1i ≡ Nk=1 α¯αNk|1ki with α¯2N ≡ Nk=1α2k. The eigen-  ... ... ... ... ...  vfoarlue1s P(de¯0term¯1ine)tthhiesssppeeeeddaist wαhiPchimthpelysiynsgteamcoelvleocltvievse,  0 AT(N−02,N−1) A∆T N−1 A(N∆−1,N) enha√n2ce|mie±nt|oif √Nα when t±he Ncoupling strengths are  ··· (N 1,N) N  homogeneous. − (5) Oursystemhaslowinhomogeneouscouplingcontribu- tions as evidenced by the α = 0.96√N¯α scaling ob- N ThematrixHint hasdimensions2N 2N,the dimension servedinourpreviouswork[18],forreferenceanaverage × of N 2-level systems. The dimension of the block di- scaling of 0.972 is predicted from experimental param- agonal sub-matrices is given by the binomial coefficient, eters. The observation of √N¯ scaling of the coupling dim(∆n) = Nn ≡ Nn. The sub-matrices, ∆n, contain strengthisaclassicsignatureofRydbergblockadeandN information c(cid:0)on(cid:1)cerning the sub-systems specific energy participating wavefunctions, as the ¯1 state is the only levels state that can evolve with that cou|piling strength. A statewithk-partiteentanglementconsistentwiththeob- ∆n =XkN=n1δk(n)|nkihnk| (6) wseirlvlesdtipllerofseccitllabtleocaktadtehegivsaemneby√, |Nψ¯Nfire=qu|e¯1nkciy⊗, b|¯0uNt−tkhie, 7 a) energy eigenstates of H for N = 5 atoms with our int experimental parameters. 0.2 y t i l bi Coherence Amplitude a b 0.1 o Since the coupling to the orthogonal subspace is neg- r p ligible for our experimental parameters, the amplitude of the Ramsey fringe oscillations provide a threshold for 0. entanglement. A thermal sample of singly excited states 0.9 0.95 1. |t1otrhifo=r tPheNk=kt1heaıφtko|m1k,iw, willhoenrelyφckouispalerbaancdkomtop¯0hasbeyftahce- amount of overlap with the ¯1 state. The p|riojection ¯11 2 will average to 1/N|,itherefore an oscillation th |h | i| with contrast above 1/N cannot be a thermal sample. b) To generate a thresholdfor k-partite entanglement we perform a numerical simulation along the lines of the 0.2 y analysis in [20]. Briefly, the goal is to generate an upper abilit bPo¯0u=nd|ho¯0n|ψai|m2efoarsusrteamteesn|tψoifwP¯1ith=a|hm¯1|aψxii|m2 ausmaofufnkcteinotnano-f b0.1 gled particles. We establish bounds in two ways. First, o we do not assume Rydberg blocakde so multiple excita- r p tions are possible. This is done by creating a random k-partite entangled wavefunction 0. 0. 0.05 0.1 ψ = ψ(k) ...ψ(k) ψ(km) , (10) | i | 1 i⊗ | m−1i⊗| m i where |ψi(k)i = sin(θi/2)|¯0(k)i+cos(θi/2)eiφi|¯1(k)i, θi andφ arerandomlygenerated,andk =N (m 1)k. i m − − FIG. 6. Monte Carlo calculations for N = 5 atoms for We extractthe maximum P¯1 for a givenP¯0 binobtained 10,000 randomized instances of atom positions and velocities numericallytoarriveatthe thresholdsshowninFig. 7a) consistentwithourexperimentalparametersof(a)Projection fork =3particleentanglementwithensembleatomnum- of a symmetric eigenstate without inhomogeneous broaden- bers N = 4 8. Any state above the threshold must ing, |−(0)i= 1 (|¯0i−|¯1i), onto the energy eigenstate of the − √2 have at least k-partite entanglement. The black cross is full inhomogeneous Hamiltonian H |−i = E |−i and (b) int − an experimental data point recorded for a sample with projection of the symmetric eigenstate onto the orthogonal subspace {|(¯1)(0)i}. N¯ =8.8 atoms, verifying the presence of entanglement. ⊥ RydbergblockadelimitstheHilbertspaceton 1ex- ≤ citations,whichsimplifiesthe calculationandenablesan analytical bound for the k-partite entanglement thresh- amplitude will be reduced to the overlap with ¯1N , ¯1N ψN 2 = k/N, this is discussed further in the|nexit old. The state in Eq. (10) includes kets with multiple |h | i| excitations. To remove these we impose the blockade section. condition P =0 and write the state as Theorthogonalsingly-excitedstates (¯1) donotcou- (n>1) ple to the symmetric states ¯0 , ¯1 u|nde⊥riideal condi- tions (δ(1) = 0). This beco{m| eis|cile}ar when the Bloch |ψi=(cid:16)a1|¯0(k)i+b1|¯1(k)i(cid:17)⊗|¯0(N−k)i. (11) k picture is invoked, since the symmetric states have total angular momentum J = N/2 while the (¯1) states Maximizationof P¯1 for a givenP¯0 canbe readilyaccom- | ⊥i plished analytically to give have J = N/2 1 and a rotation on the Bloch sphere − conserves angular momentum. Inhomogeneous broaden- k ing,includingdifferentialACStarkshifts,Dopplershifts, P¯1 = N(1−P¯0). (12) and finite intermediate state lifetimes, are added pertur- Note that this agrees with the limiting case of batively with ∆ and provide a mechanism for coupling into the (¯1) space. This coupling should be negligi- P¯0 = 0 from [20]. Rearranging (12) to give nb6olsethaodnwisdsplsra|ieymduut⊥hclaeietewcdhiatphrraoicnjtececrrteiisaotsniicns√gofNN1eann(hd¯0anacdedm¯1itei)onnat.alolFlnygigwuthirleel (tNkPha¯0t,Pw≤¯1e) m=e1e−Pt¯1Pm(t¯00mahx.a4ex4,th±rae0nsh.d0o2ld,g0if.vo4er6nc±reoa0ut.0iro3n)eoxwftertehmecaeWn -vssahtaloutwee √2 | i−| i 8 1.0 a) with k = 82 6%. Similar arguments for the presence N ± of entanglement based on the amplitude of Ramsey 0.8 oscillations have been used in [21]. (N,k)=(4,3) 0.6 _1 P 0.4 (8,3) 0.2 0.0 0.0 0.2 0.4 P_ 0.6 0.8 1.0 0 1.0 b) (N,k)=(9,9) 0.8 SUMMARY 0.6 _1 P (9,6) 0.4 In summary we have shown evidence for N particle 0.2 W-state entanglementonthe basisofthe followingthree arguments. First, the excellent agreement of the ob- 0.0 served collectively enhanced Rabi frequency with theory 0.0 0.2 0.4 0.6 0.8 1.0 P_ reportedin our previouswork[18]using the same experi- 0 mental setup and procedures as are used here implies an N component wavefunction. Second, the amplitude of FIG. 7. (color online) a) Numerically determined bounds the Ramsey-style oscillations for N¯ = 8.8 is four times for (P¯0,P¯1) using Eq. (10). Rydberg blockade is not as- larger than the 1/N¯ limit expected from a thermal sam- sumed in the calculation so multiple excitations are allowed. States above the (N,k) line imply there are at least k en- ple of singly excited states. Our data shows entangle- tangled atoms in the N atom ensemble. Calculated bounds ment without making the assumption of perfect block- for N = 4−8 are shown, top to bottom. The dashed black ade. Third, with the assumptions of perfect blockade, lineshowstheamplitudesforthethermalsingly-excitedstate entanglement percentage independent of N, and negligi- |1thiwithN =4. Thesolidblacklinerepresentstherangeof blecouplingto (¯1) ,whichisjustifiedbyFig. 1,then experimentalRamseyoscillation datawith thecrossshowing k =82 6%. I{n|oth⊥eir}words82 6%oftheatomsinthe thevalueattgap =0msfromFig. 2inthemaintext. b)An- N ± ± ensemble are participating in the W-state entanglement. alyticalboundsassumingperfectblockadeusingEqs. (11,12). The entanglement thresholds are thestraight lines shown for Thisresultisnotchangedinastatisticallysignificantway N =9andk=9−6,top tobottom. Thedatashown bythe when compared with simulations based on experimental black line and cross exceedsthe k=7 threshold. parameters that include imperfect blockade.